COSTS, LOSS, AND FORECASTING ERROR: AN EVALUATION. OF MODELS FOR BEEF PRICES Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY LLOYD BOUGLAS TEIGEN 1973 LIBRARY Michigan State , University This is to certify that the thesis entitled Costs, Loss, and Forecasting Error: An EValuation of Models for Beef Prices presented by Lloyd Douglas Teig en has been accepted towards fulfillment of the requirements for _—P_h+D-.——degree in Agricultural Economics KM/W~ I Major professor Date December 71 1972 0-7639 “we ‘- amomc av ‘5 I HUAG a sour IEUUII swam mc. LIBRARY an NDERS rosy, mcmsnu ems-mug 6f Lee-«3&3 M! “‘ Q? \ ABSTRACT COSTS, LOSS, AND FORECASTING ERROR: AN EVALUATION OF MODELS FOR BEEF PRICES By Lloyd Douglas Teigen The problem which is addressed in this thesis is stated in the form of one theoretical and one empirical question. The theo- retical question is what is an apprOpriate measure to evaluate the loss which results from incorrect forecasts when the firm using them places complete reliance in the information provided? Related to this is the question of how other measures of forecasting error are related to this measure. The empirical question relates to the relative performance of four means of generating forecasts for beef cattle prices, namely econometric models, trend models, price dif- ference models, and the futures market price. The hypothesis which is tested is that performance is proportional to the information contained in the forecasting method (and hence to the cost of de- veloping the forecast). The theoretical section consists of the derivation of the following theorem: If the output price of a single product firm with a homogeneous production function and predetermined input prices is being forecast, then the loss (in the sense of the dif- ference between actual and maximum realizable profits) due to forecasting error is preportional to (rl+n - (1+n) r pn +’n pl+n), Lloyd Douglas Teigen where r is the realized price, p is the forecast, and n is the elasticity of the firm's short run supply curve. By factorization, this loss function is shown to be related to the quadratic loss function which forms the basis for many widely used measures of forecasting performance. The procedure followed in the empirical analysis consisted of four steps. The first of these was to choose the econometric models to be evaluated, using the criteria that they were published, accessible, and recent enough that their structures could be used for forecasting. Those selected were developed by Hayenga and Hacklander (l), Myers (2), Trierweiler and Hassler (3), Cram (4), and Unger (5). The second step in the process was to estimate trend models (bOth polynomial and trigonometric) for all of the price series used in the econometric models over sample periods identical with those of the corresponding econometric models. The third step was to invert the structural forms of the econometric models and calculate the forecasts fimplied by the econometric and other models for the test sample period of January 1965 to December 1970. Finally, the forecasts were evaluated over this period using a number of alternative performance criteria: Those used were the cost derived average loss measure derived in the theoretical section of the thesis, mean squared error, both of Theil's inequality measures U1 and U2, the number of incorrect predictions of change (turning point errors), absolute moment measures corresponding to the first through fourth absolute sample moments of the forecasting errors, the average relative error of forecast, the correlation of the fore- cast with the realization, the slope of the linear regression Lloyd Douglas Teigen equation of the forecast on the realization (of the form r = a + b p +-u, where r is the realization, p is the prediction, and u is a disturbance term), and finally the bias, or average error of forecast when the sign of the error is accounted for. The theoretical findings of this study were that the cost derived loss function is quadratic in the case where the supply elasticity is one, and contains a quadratic factor when the supply elasticity is an integer greater than one. A number of consistencies and inconsistencies were determined to exist among the rankings of the forecasts derived by the different performance measures: The mean squared error ranked consistently with the cost derived loss measure and with Theil's U2 statistic, and adjacent moment measures exhibited consistent (but not transitively consistent) rankings of the forecasting performance of the models. It was observed that the correlation and the numbercflfturning points missed gave rise to rankings which were not consistent with those of any other measure of performance. The empirical findings of the study fall into two categories -- the stability of the dynamic econometric models used in this analysis and a comparison of all the means of generating forecasts on a one- step-ahead basis. The conclusion regarding the stability of the models was that both the monthly and quarterly dynamic models in- dicated some degree of instability. Comparing the alternative maintained hypotheses' forecasting performance, it was found that the polynomial trend model diverged shortly after the close of the sample period for estimation, the trigonometric trend model performed about as well as the econometric Lloyd Douglas Teigen models during the test period, correcting the trigonometric trend model for serial correlation of disturbances improved its forecasting performance substantially and that the overall best performing fore- casting methods consisted of projecting either the current cash price or the correSponding futures price as the price to prevail in the forecast period. CITATIONS 1. Hayenga, M.L. and Duane Hacklander. "Monthly Supply-Demand Re- lationships for Fed Cattle and Hogs" American Journal of Agricultural Economics, Vol. 52, No. 4 (November 1970), pp. 535-544. 2. Myers, L.H. et. a1. Short-term:?rice Structure of the HogrPork Sector of the United States, Research Bulletin 855, Purdue Univer- sity Agricultural Experiment Station, February 1970. 3. Trierweiler, John E. and James B. Hassler. Orderly Production and Marketinggin the Beef-Pork Sector, Research Bulletin 240, University of Nebraska Agricultural Experiment Station, November 1970. 4. Crom, Richard. A.Dynamic Price-Output Model of the Beef and Pork Sectors, Technical Bulletin 1426, U.S. Department of Agriculture, Economic Research Service, September 1970. 5. Unger, Samuel G. Simultaneous Equations SystemLEstimation: An Application in the Cattle - Beef Sgctor, Unpublished Ph.D. thesis, Department of Agricultural Economics, Michigan State University, 1966. COSTS, LOSS, AND FORECASTING ERROR: AN EVALUATION OF MODELS FOR BEEF PRICES By Lloyd Douglas Teigen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 1973 ,4 .a’l" @Ibp ACKNOWLEDGEMENTS I gratefully acknowledge the guidance, intellectual stimula- tion, and encouragement which Robert Gustafson gave me throughout my stay at Michigan State as my major professor. The freedom he allowed me in the preparation of this thesis made this a challenging, if sometimes frustrating, experience. His suggestions often were what were needed to overcome the obstacles and always directed my efforts towards their eventual solution. The contributions, both joint and separable, to my educa- tional program and professional development made by Lester Mandar- scheid, Marvin Hayenga, James Ramsey, and James Stapleton, as members of my guidance committee, and also by Roy Black, who served on my thesis committee are recognized with sincere appreciation. The support contributed to my program by the Agricultural Economics Department in the form of computational facilities and secretarial assistance, as well as direct financial support through a research assistantship is acknowledged with grateful appreciation. The programming assistance rendered by Daniel Tsai deserves explicit recognition for his ability to quickly implement the procedures involved in this research with a minimum of detailed instructions. The personal contribution and sacrifice on the part of my wife and daughter is most graciously acknowledged, and it is to them that this thesis is dedicated. ii TABLE OF CONTENTS List of Tables List of Figures I. II. III. INTRODUCTION A. Setting B. Objectives C. Literature Review and Research Approach THEORY OF FORECAST EVALUATION . Forecasting as a Decision Problem Forecasting for a Firm . Analysis of the Cost Derived Loss Function . Measures of Forecast Performance dining» DISCUSSION OF THE MODELS FOR BEEF PRICES A. Overview of the Chapter 1. Econometric Models 2. Trend Models Price Difference Models Futures Market Price Econometric Models Hayenga4Hack1ander Monthly Model Myers Monthly Model Trierweiler-Hassler Monthly Model Crom Quarterly Model , Unger Annual Model Trend Models Taylor Series Approximation Fourier Series Approximation Estimates of the Trend Equations a. Corresponding to the Hayenga- Hacklmnder Model b. Corresponding to the Myers Model c. Corresponding to the Trierweiler- Hassler Model d. Corresponding to the Crom Model e. Corresponding to the Unger Model D. The Price Difference Models a F‘ -§Id C E. . O WNnglJ-‘WN O. 0.. O (D E. The Futures Market Price iii 31 31 33 34 36 37 39 43 43 47 48 49 49 53 54 58 59 64 72 73 86 91 94 IV. VI. VII. EMPIRICAL EVALUATION OF THE FORECASTING MODELS A. B. D. Stability of the Econometric Models One-Step-Ahead Performance of the Competing Forecasting Models 1. The Sample Period 2. Divergence of the Polynomial Trend Models 3. Tabulation of the Performance Statistics a. Variables in the HayengaAHacklander Model b. Variables in the Myers Model c. Variables in the Trierweiler-Hassler Model d. Variables in the Crom Model e. Variables in the Unger Model Effect of Lead Period on Forecasts Based on Futures Market Price and Price Difference Models Summary SUMMARY AND CONCLUSIONS A. Summary and Conclusions 1. Theoretical 2 . Emp ir ical B. Implications C. Recommendations for Further Research BIBLIOGRAPHY APPENDIX A. Observations Regarding the Analysis of and the Data for the Econometric Models 1. Analysis of the Econometric Models 2. Data Generation, Manipulation, and Fabrication B. Appendix Tables 1. Data for, and Performance of, the Econometric Models 2. Structural Forms of the Econometric Models Used 3. Reduced Form Equations Corresponding to the Myers Structural Model iv 100 100 107 107 108 110 112 115 115 126 133 138 143 145 145 145 153 160 163 167 174 174 174 180 188 Tab le 2a. 2b. 10. 11. 12. 13. 14. LIST OF TABLES Measures of Forecasting Accuracy Characteristics of Econometric Models under Consideration Summary of Price Variables in Econometric Models under Consideration Estimated Polynomial Trend Equations Corresponding to Hayenga-Hacklander Model Estimated Tr igonometr ic Trend Equations Corresponding to Hayenga-Hacklander Model Estimated Trigonometric Trend Equations Corresponding t o Hayenga -Hacklander Mode 1 Estimated Polynomial Trend Equations Corresponding to Myers Model Estimated Polynomial Trend Equations Corresponding to Myers Model Estimated Polynomial Trend Equations Corresponding to Myers Model Estimated Trigonometric Trend Equations Corresponding to Myers Model Estimated Trigonometric Trend Equations Corresponding to Myers Model Estimated Trigonometric Trend Equations Corresponding to Myers Model Estimated Polynomial Trend Equations Corresponding to Trierweiler-Hauler Model Estimated Polynomial Trend Equations Corresponding to Trierweiler-Hauler Model Estimated Trigonometric Trend Equations Corresponding to Trierweiler-Hassler Model Page 29 4O 41 60 62 63 66 67 68 69 70 71 74 75 76 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. Estimated Trigonometric Trend Equations Corresponding to Trierweiler-Hassler Model 77 Estimated Trigonometric Trend Equations Corresponding to Trierweiler-Hassler Model 78 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model 80 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model 81 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model 82 Estimated Trigonometric Trend Equations Corresponding to Crom.Simu1ation Model 83 Estimated Trigonometric Trend Equations Corresponding to Crom Simulation Medal 84 Estimated Trigonometric Trend Equations Corresponding to Crom Simulation Model 85 Estimated Polynomial Trend Equations Corresponding to Unger Model 87 Estimated Polynomial Trend Equations CorreSponding to Unger Mbdel 88 Estimated Trigonometric Trend Equations Corresponding to Unger Model 89 Estimated Trigonometric Trend Equations Corresponding to Unger Model 90 Forecasts of Choice 900-1100 1b. Steers at Chicago, 1965 to 1970 113 Forecasts of U.S. No. 2-3 Grade, 200-220 1b. Barrows and Gilts at Chicago, 1965 to 1970 114 Forecasts of Deflated Monthly Retail Price of Beef in Urban Areas, 1965 to 1970 116 Forecasts of Deflated Prices of All Grades of Steers Sold Out of First Bands in Chicago, Omaha, and Sioux City (weighted Average), 1965 to 1970 117 Forecasts of Deflated Average Retail Price of Frying Chickens in Urban Areas, 1965 to 1970 118 Forecasts of the Monthly Deflated Retail Price of Pork in Urban Areas, 1965 to 1970 119 vi 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 45. 46. 47. 48. 49. Forecasts of the Eight Market Weighted Average Deflated Price of Barrows and Gilts, 1965 to 1970 Forecasts of Choice 600-700 lb. Steer Carcasses at Chicago, 1965 to 1970 Forecasts of the Price of 900-1100 1b. Choice Steers at Omaha, 1965 to 1970 Forecasts of U.S. Nos. 1-3, 220-240 lb. Barrows and Gilts at Omaha, 1965 to 1970 Forecasts of Quarterly Average Retail Price of Beef, 1965 to 1970 Forecasts of Quarterly Average Retail Pork Price, 1965 to 1970 Forecasts of the Wholesale Price of Choice Grade Beef Carcasses (LCL) at New York, Chicago, Los Angeles, San Francisco, and Seattle, June 1964 to June 1970 Forecasts of the Utility Cow Beef Price at New York City, June 1964 to June 1970 Forecasts of the Weighted Average Price of Choice Steers at Twenty Markets, 1964 to 1970 Forecasts of the Price of Good and Choice 500-800 1b. Feeder Steers at Omaha, 1964 to 1970 Forecasts of the Weighted Average of Wholesale Prices of Individual Pork.Products at Chicago, 1964 to 1970 Forecasts of the Weighted Average Price of Barrows and Gilts at Eight Markets, 1964 to 1970 Forecasts of Annual Average Retail Beef Price, 1964 to 1970 Forecasts of Annual Average Price of Beef Cattle Received by Farmers, 1964 to 1970 Forecasts of Annual Average Price of Other Meat at Retail, 1964 to 1970 Forecasts of Annual Average Price of Other Meat by Farmers, 1964 to 1970 Price Forecasts for Choice 900-1100 lb. Steers at Chicago, 1965 to 1970 vii 120 121 122 123 124 125 127 128 129 130 131 132 134 135 136 137 140 50. 51. 52. 53. 54. Price Forecasts for Choice 900-1100 1b. Steers at Omaha, 1965 to 1970 Forecasts of Weighted Average Deflated Price of all Grades of Slaughter Steers at Three Markets, 1965 to 1970 Joint Frequency of Occurrence of U1 and U2 Number of Coincident Rankings of Forecasting Devices (of 22 Possible) Comparison of Econometric Model Forecasts with Trigonometric Model Forecasts viii 141 142 149 151 158 A1. A2. A6. A7. A8. A9. A10. A11. A12. A13. A14. A15 O A16. APPENDIX TABLES Exogenous Variables used in the Hayenga and Hacklander Model Hayenga and Hacklander Model: Actual and Estimated Endogenous Variables Hayenga and Hacklander Model: Forecasting Performance Exogenous Variables Used in the Myers Models Myers Structural Model: Actual and Estimated Endogenous Variables (1) Myers Structural Model: Actual and Estimated Endogenous Variables (2) Myers Structural Model: Forecasting Performance Myers Stage I Equations: Actual and Estimated Endogenous Variables (1) Myers Stage I Equations: Actual and Estimated Endogenous Variables (2) Myers Stage I Equations: Forecasting Performance Exogenous Variables Used in the Trierweiler and Hassler Model Trierweiler and Hassler Model: Actual and Estimated Endogenous Variables Trierweiler and Hassler Model: Forecasting Performance Crom Simulation Model: Actual and Estimated Endogenous Variables (1) Crom Simulation Model: Actual and Estimated Endogenous Variables (2) Crom Simulation Model: Actual and Estimated Endogenous Variables (3) ix 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 A17. A18. A19. A20. A21. A22. A23. A24. A25. A26. Bl. B2. B3. B4. B5. B6. B7. B8. B9. BIO. C1. Crom Simulation Model: Forecasting Performance (1) Crom Simulation Model: Forecasting Performance (2) Crom Simulation Model: Forecasting Performance (3) Exogenous Variables Used in the Unger Model Analysis 1 -- Unger OLS -Analysis 2 -- Unger ZSLS Analysis 3 -- Unger UNK Analysis 4 -- Unger 3SLS Analysis 5 -- Unger I3SLS Analysis 6 -- Unger LISE Structural Equations for the Hayenga-Hacklander Model Structural Econometric Model From Myers, et. a1. Estimated Coefficients for Myers' Stage One Equations Structural Equations from Trierweiler-Hassler Model Estimated Coefficients of the Unger Model (Ordinary Least Squares) Estimated Coefficients of the Unger Model (Two Stage Least Squares) Estimated Coefficients of the Unger Model (Unbiased K - Class) Estimated Coefficients of the Unger Model (Three Stage Least Squares) Estimated Coefficients of the Unger Model (Iterated Three Stage Least Squares) Estimated Coefficients of the Unger Model (Limited Information Single Equation) Reduced Form Equations Corresponding to Myers' Structura1.Mode1 204 205 206 207 208 209 210 211 212 213 214 215 217 218 220 221 222 223 224 225 226 LIST OF FIGURES Quarterly Model of Beef and Pork Sectors of the Livestock.- Meat Economy: Richard Crom 50 Characterization of Decision Rules Within the Futures Market 95 xi CHAPTER I INTRODUCTION Setting Most departments of agricultural economics conduct both research and extension efforts involving major segments of the agricultural economy of the respective states. The extension efforts include the public provision of price information and out- look (forecasts) in addition to pertinent information regarding industry structure and economic performance in order to assist the industry adapt to its changing environment. The research effort may be conducted by the same individuals as are involved in the extension, but often there is a division of labor. In particular, there is almost always a separation of reSponsibility between those who are conducting methodologically oriented research and those who are doing the practical work involved in forecasting. There is little question that methodological studies can be useful to those involved in forecasting. Perhaps, a more significant question is whether they are, in fact, useful to those who forecast. The question which is asked in this thesis is "How useful are some selected recent published studies of the fed beef cattle sector in providing a basis for generating price forecasts?" The answer provided by this research should assist forecasting personnel to evaluate the basis for their work. In this research, the focus is on the "scientific,"1 or verifiable aspects of forecasting, rather than the "art" of fore- casting, which in part consists of projecting (or obtaining pro- jections of) the exogenous variables of the system, determining which of the variable factors outside the system should have an impact on the system (violations of ceteris paribus), and appro- priately adjusting the original forecasts to reflect these changes. Specifically, the forecasts studied in this research come from explicit models and identified data sources. In the cases where it would be necessary to project concurrent values of the exogenous variables, an assumption of perfect foresight (made possible by the eagpaacbnature of this analysis) will be utilized.2 Objectives The preceding discussion has been intended as a brief view of the philosophical orientation of this research. The specific objectives to be accomplished are: 1. To deve10p a measure of forecasting performance derived from the actual losses of foregone profit or reduced welfare re- sulting from the error of forecast. This measure will then be A forecast is scientific if it generates "verifiable pre- dictions by means of a method which is also verifiable." - Henri Theil (71, p. 10). See also Mincer and Zarnowitz (55, p. 3). 2 It should be noted that this assumption will tend to put the forecasts based upon structural econometric models in some- what more of a favorable light than they would be if these vari- ables had to be forecast. related to the more conventional measures of forecasting perfor- mance to clarify the relationship between the two types of measures. 2. To determine the forecasting effectiveness of several models with endogenous beef prices which differ from each other by degree of complexity, length of run, and by means of statistical estimation. Their performance in explaining beef prices will be measured using the measures deve10ped and examined under objective one. On the basis of these empirical results, potentially fruitful areas of further research on forecasting theory and practice will be suggested. Literature Review and Research Approach Since there are two separate foci of this thesis - one involving the purely theoretical aspects of forecast evaluation and the other specific to forecasting models of the beef industry - there necessarily will be two corresponding sections to this literature review. The central figure in the literature of forecast evalua- tion is Henri Theil. His research evaluating the Dutch and Scandinavian forecasts culminated in at least three books (71, 72, 73) on the subject, as well as numerous articles, of which (74) is among the earliest. The thrust of his work on forecast evalua- tion revolved around the assumption of a quadratic loss (or wel- fare) function. Consequently, his two inequality measures are transformations of the mean squared forecasting error. His work firmed up the concept of the "failure of a forecast" as the performance of a forecast relative to one of no change in the vari- able under consideration. The approach taken in the theoretical section of this thesis was to perform Theil's analysis starting from assumptions several steps earlier in the process. Namely, an industry is considered which is governed by homogeneous production functions (alternatively, by honogenous cost functions) for which output price is being fore- cast. The analysis then determines the form of the loss function which is appropriate to represent the foregone profits which re- sulted from the forecasting errors. Since the derivation of the cost-related loss function consists of purely mathematical analysis, care must be taken to assure that all the implications and inter- pretations present in such a structure will be obtained. Zarnowitz (86) and Stekler (66) both evaluated a number of macroeconomic forecasts in their research. Their performance criteria were mean squared error, average absolute error and Theil's Inequality Coefficients. The modus operandi in each case was to compare a set of forecasts (or forecasting models) with each other and with several extrapolative devices. Generally the conclusion was that the econometric models and other formal forecasts performed better than the naive extrapolation models. With the methodology of these studies as a pattern, I set out to ascertain the performance of econometric models of the beef subsector in predicting beef prices. This objective guided my search of the literature regarding the beef industry.1 A study 1 By restricting the search to published models, many of the models actually used in the industry to derive forecast upon which decisions are made were excluded from consideration. was determined to be relevant if it 1) was based on an econometric model which 2) contained beef prices among its endogenous variables, and 3) was estimated over a sample period sufficiently recent to render it potentially usable for forecasting in the late 1960's. The articles examined were categorized as monthly, quarterly, or annual depending upon the type of data used. In addition, it was noted whether the futures market was given explicit notice. The monthly analyses that were examined included two studies by Hayenga and Hacklander (29, 30), one by Myers fig, 31. (57), two by Trierweiler and Hassler (79, 80), and one by Bullock and Logan (6). All models were attempts to analyze the price structure in the livestock subsector and both the Myers (57) and one of the Hayenga-Hacklander (30) studies included exercises indicating a use of the models for forecasting. The Bullock-Logan analysis made use of a forecasting equation for Slaughter steers in Cali- fornia to guide feedlot marketing decisions. For this thesis, it was decided to limit the analysis to only three of the menthly studies (29, 57, 80).1 The only representations of the livestock subsector of an intermediate length of run were the work of Richard Crom (9, 10) l . The reasons for exc1u31on of the other models are these. The results presented in (79) and (80) are identical and the analysis of (79) would be redundant. Bullock's study (6) had an orientation toward something other than building an econometric model to explain beef prices -- further there was the problem posed in attempting to relate his California price (El Centro) to the mid- western (Corn Belt) prices more typically analyzed. The other Hayenga-Hacklander study (30) was excluded because the authors presented several alternative forms for a number of the relationships, rather than postulating a single model. A case could be made for inclusion based on the fact that it follows closely the format its senior author uses in his own forecasting work, and because it allows for a number of lengths of forecasts to be generated from a single month's data. and by Crom and Maki (11). The first two studies used quarterly data, while the last used semiannual and annual data. It was decided to analyze only (9), because it appeared to be an update of (11) and because (10) does not deal at all with price vari- ables as endogenous factors - only placements on feed, average weights per head, and numbers of animals marketed. A number of annual models of the livestock sector have been develOped. The classical study of the sector is that of Hildreth and Jarrett (36). Since it was completed in 1955 it is dated, but Richard Feltner (19) has done some work to update the model. Two more recent articles which have gained wide professional recognition in this area are by Egbert and Reutlinger (l6) and by Langemeier and Thompson (48). Unger (81) used a model of the live- stock sector to compare alternative simultaneous equations estimators. Uvacek (83) estimated a demand equation for beef, but did not consider supply factors. Walters (84) used beef prices as exogenous variables in the determination of cattle inventories, but had no demand representation to set the prices. The only annual model selected for analysis in this study was that of Unger, for a number of reasons. It is similar in form to and of more recent vintage than the model of Feltner (19). More significantly, since it presents estimates of the same econo- metric model obtained by six different estimation techniques, it is a natural experiment to determine the effect of estimation technique on forecasting performance. The Egbert-Reutlinger and Langemeier-Thompson models were excluded because of nonlinear identities and equations which they contained, preventing the derivation of an explicit reduced form on which to base the fore- casts. The empirical analysis of the models involved a number of steps in the process: The first of these is to verify that the coefficients of the structural model are (or appear to be) free of any obvious errors of transcription or typography. Then, the data series for which the model was estimated is extended through the test period. At this point the model and data are combined with an appropriate computer program to calculate reduced form estimates and evaluate them with the performance measures. .The models with which the econometric models are compared consist of the extrapolation of past time trends as approximated by both Taylor series and Fourier series,1 weighted averages of previous observations of the price variables (where the weights correspond to assumptions that certain order differences of the price variables are randomly distributed about zero), and, in the case of the monthly models, they are compared with forecasts implied by currently prevailing levels of futures market prices. The comparison of these models against the forecasts implied by cattle futures prices is, I think, unique to this analysis. The literature which has developed around the beef futures market in- cludes only one study of the forecasting effectiveness of the For each type of series, there is necessarily a finite number of terms in the approximation. The Taylor series approximation con- sists of a linear combination of the first n powers of the time vari- able. The Fourier series approximation to a general periodic function (with period 1/f, or frequency f) consists of an intercept, the weighted sum of sine and cosine functions of time - each with frequency f, and the weighted sum of similar sine and cosine functions of time - with frequencies k'f, for integer values of k. market - that by Purcell (61). In his study there was no standard against which the future prices were judged, only the inference that the market was an ineffective forecasting device because the range of errors was greater than zero. The remainder of the thesis consists of four chapters. A theoretical chapter develops a measure of forecasting performance which is related to the cost structure of the firm (or supply elasticity in the industry) and relates this measure to other, more conventional, measures of forecasting performance. The next chapter presents a descriptive discussion of the models (econo- metric, trend, futures market, and price difference) which were used in the analysis. The succeeding chapter presents the evalua- tion of these forecasting models: examining the stability of the econometric models, measuring the performance of all the models in generating one-step-ahead forecast, and examining the relative ability of the price difference models and the futures market price in forecasting prices several (as many as eight) months in advance. Finally the conclusions and implications are presented together with recommendations for fruitful extensions of the theory of forecasting and in the empirical analysis of the beef cattle sector of the agricultural economy of the U.S. CHAPTER II SOME THEORETICAL CONSIDERATIONS IN FORECAST EVALUATION This chapter consists of four sections. The first discusses forecasting as a problem of statistical decision theory. The second carries out the decision analysis for a class of firms which has been the focus of considerable economic theorizing - those with homogeneous production functions - and then aggregates to obtain an industry wide measure of loss. The resulting cost derived loss function is analyzed to determine its relationship to some of the more common measures of forecasting performance. Finally, all the measures of forecasting performance to be used in this thesis are presented and briefly discussed. Forecasting as a Decision Problem A person who makes a forecast is making a statement about an event which is yet to be observed. The forecaster generally suffers embarrassment if the forecasted value and the realized value are too divergent, and in addition, the firms relying upon the forecaster will incur losses which reflect the divergence of the g§.gggg and 35 Egg; production decisions. In the abstract, there is a set of values one of which char- acterizes the market during the forecast period. Just which one will characterize the market is not known with certainty. The forecaster 10 must take an action - make a statement as to which of the values he thinks will prevail in the forecast period. Based on the action he takes and the state of nature which prevails a loss will be incurred both by him (embarrassment) and by the firms relying on his fore- casts (less than maximal profits). The behavioral assumption implicit in this analysis is that the forecaster behaves in such a way as to try to minimize these losses. As thus posed, forecasting is a standard problem of statistical decision theory. Following is a review of the vocabulary of statistical decision theory.1 The decision setting is characterized by the Cartesian pro- duct of the state space and the action space. The state Space is a set whose elements describe the plausible states of nature or char- acterize the phenomenon under analysis. The action space consists of those elements which describe the actions which the decision maker can take. In the case where the action space consists of only two elements the analysis is formally equivalent to hypothesis testing. When there are an infinite number of elements in the action space, the analysis parallels estimation theory. There corresponds to each decision setting a loss function Which measures the consequences of the decision. Formally, the loss function is a function, L(r,p), which is defined on the product of the action and state spaces measuring the loss incurred when action p is taken and the state of nature is characterized by r. Occasionally, analyses are made in terms of regret, rather than loss, 1 This glossary draws quite heavily upon Lindgren (50). 11 where regret is defined to be the difference between the loss for a given state/action pair and the minimum loss for that state: R(r,p) = L(r,p) - min L(r,p). Thus, for each state there is an action which result: in zero regret, and all other values of the regret function are nonnegative. In the analysis presented in this discussion the loss functions will in fact be regret functions. When it has been decided that additional information is needed in the decision process, the set whose elements are these bits of information is called the sample space. The probability of occurrence of each point in the sample space depends upon the state of nature prevailing so that the observation of a particular event will provide additional information on the state of nature. Follow- ing through in this approach, a sample observation should lead to a prescribed action which would be appropriate to the information provided. This mapping from the sample space to the action space is called a decision rule - prescribing an action p which corresponds to each conceivable observation in the sample space. Decision rules may be arrived at in a number of ways, not all of which require explicit use of the sample information. For example, the same action might be chosen in all situations, such as always estimating a parameter to be equal to zero (or some other constant), The minimax principle prescribes a decision rule which varies from situation to situation but need not depend upon the observed sample values. It states that, whatever the data, choose that action for which the maximum expected loss over all states of nature is smallest. 1 See, for example (3, p. 129) or (37, p. 7). 12 When an action is determined independently of sample informa- tion, the rule for determining such action is generally called a strategy instead of a decision rule. Strategies may be mixed (deter- mined by a probability distribution on the action space) or pure (determined by a degenerate probability distribution with all its mass concentrated at a single point in the action space). The Bayes decision rule states that given a probability dis- tribution on the states of nature (such distribution may be either an g_priori or an 5 posteriori distribution), select that action which minimizes the expected loss (or regret). Depending on whether the distribution is prior or posterior, this decision rule may or may not be independent of the sample observations. In the analysis in the remainder of this essay the Bayesian Decision rules will be the primary focus of attention. Returning now to the situation facing the forecaster, the state Space for his situation is the set of possible realizations of his forecasts. The action space is the set of plausible forecast values. The loss suffered as a result of forecast inaccuracy is taken to be the marginal value of perfect information - the difference be- tween maximum profit under perfect information and the profit realized from plans based on the forecasts. This assumption abstracts from any personal embarrassment on the part of the forecaster - unless this were assumed to be prOportional to the losses incurred by his client. Since the forecaster's credibility is dependent (inversely) 1 This dependence may be shown in the relation between the prior and posterior distribution. If g(e) is the prior distribution on the state Space and f(zle) is the conditional distribution of the sample observation 2, given the State of nature is 6, then the osterior distribution is h(e‘z) = f(zIe) g (6)/f(z) where f(z) = f f Ma) S (e) de. 13 upon the losses incurred, the behavioral assumption of minimizing expected loss would seem justified. As a case study, a firm with a homogeneous production func- tion will be analyzed to determine an apprOpriate loss function and type of forecast which minimizes the expected loss based on this function. Forecasting for a Firm In this section a loss function which pertains to a perfectly competitive, single product firm with a homogeneous production func- tion will be derived. This analysis is based on the assumption that only the output price is being forecast, and that input prices are predetermined (not subject to forecast nor influenced by the fore- cast). In this study the loss resulting fran imperfect foresight (forecasts) which is incurred by a firm is defined to be the dif- ference between the profit realized assuming complete belief in the forecast and the maximum profit which can be realized under the price which prevails in the forecast period. The latter is obviously larger, and the order of subtraction is taken so that the loss is always positive or zero. A homogeneous production function of degree h has a result- ing total variable cost function which is also homogeneous,1 but of Henderson and Quandt (33) presents a proof based on knowing the optimal amounts of inputs which will produce a single unit of output. An alternative proof which derives these in the process is as follows: Let q = f(x1,x2) I x2f(l,x2/x1) - ng(x1/x2,l) be the homogeneous production function, and let c(q) I plx1 +pr2 - x(f(x1,x2) - q) be 14 degree 1/h. That is, the total variable cost function takes the form c = k qI/h, where q is the output and the constant of pro- portionality, k, depends upon the input prices and the parameters of the production function, particularly as they affect input sub- stitution. One observes a rising marginal cost curve whenever the production homogeneity, h, is less than one (decreasing returns to size), which is a necessary condition for determination of the optimum level of output by the firm in a competitive market. Equating marginal cost to product price, the profit maximiz- ing level of output for the firm with total variable cost function c = k q1/h is q = {higjh/(l-h), where p is the output price upon which the firm bases is decision. This relationship is, in fact, the short run supply curve for the firm. Note that h/(l-h) the Lagrangian costfunction,representing the total variable costs. The partial derivatives of the TVC function with respect to the input levels and the Lagrangian multiplier are: 35L. = - h']. = 3.2— = - h_1 = axl pl ixz f1(x1/x2,1) 0. 3x2 92 kxl f2(1,x2/x1) o, and %§-= q - f(x1,x2) = 0. Taking the ratio of the first two partials, we , 1 1)1 2 ’ 2- 1 h-l , , , obtain - = ( ) , which is valid whenever h # 1 and x2 p2 f1(x1/x2,l) f1 is bounded away from zero. These conditions, together with con- tinuity of the partials assume a solution to the equation, by Brower's * * Fixed Point Theorem. Denote the fixed point (x2/x1) by r and return to the alternative forms of the production function. These 1/h * imply that the input levels employed are: x1 = qllh/(f(l,r )) * and x2 = qllh/(f(l/r ,l))1/h. Substituting this onto the cost equa- tion, we obtain: 1)1 p2 Uh Uh C = ( 4' ) q or c = k q . «(1.1539,h (15(1/r*.1)>"7h 15 is the price elasticity of supply of this commodity from this firm. In this and further discussions, n will be used to denote this elasticity,1 n = h/(l-h). If the firm perfectly foresees the output price which is realized, r, the maximum profit2 realized will be equal to (n/k)n(l-|-n)'-1"n r1+n . Assume now that there is a divergence of the realized price from the forecasted price, specifically let r re- present the realized price and p represent the forecasted (pre- dicted) price. The firm will determine its output by equating its marginal cost to p, rather than r, but its actual profit will be determined by the realized price r, in conjunction with the output which was determined by the forecasted price p. This realized profit is [(l+n)r pn - n p1+n](n/k9n/(l+n)1+n. Since the maximum profit under perfect foresight has already been given, the loss, as defined, is [r1+“ (n/k)n/(1+n)1+n . L(r,p) = - <1+n> r p“ + n 13”“) Notice that this loss function is quadratic if the price elasticity of supply, n, is equal to one (which occurs when the homogeneity of the production function is one-half), viz. (r-p)2/4k. 1 This transformation maps the [0,1) interval for h into the nonnegative reals for n. Some of the alternative Statements of the defining relationship are h - n/(n+l), 1 +'n B l/(l-h), or 1 - h = 1/(1+n). The sense in which "profit" is used herein is as the return which accrues to the fixed factors of production. Accounting for fixed charges does not affect the loss function which is derived. 16 This analysis of the loss accruing to a firm can be extended to measure the cost to an industry which used the forecasts. As currently construed, this is dependent upon the assumption of an infinitely elastic demand.1 If the m firms of the industry have identical short run scale parameters (i.e. homogeneities), with the possibility allowed that the cost coefficients, ki: i = 1...m, may differ from firm to firm due to for example, differing - but known - input prices, possibly reflecting marketing diseconomies, or differing rates of input Substitution, the potential income lost by the industry result- ing from inaccurate forecasts is m AY = Lr1+n - (1+n) r pn +>n p1+nj Z (n/ki)n/(l+n)1+n . i=1 Notice that the basic loss function in this expression differs from that of the individual firm by only a multiplicative constant. If social welfare were represented as the sum (or for that matter, a convex linear combination) of the individuals' utility functions and those functions were reasonably linear over the range of variation considered important, the change in social welfare would take on the same appearance as AY except that the numerator in the sum would also include the product of the individual's weight in the social welfare function and the marginal utility of income together with n. The obvious economic criterion states that the forecasts should be improved only to the extent where the welfare gain equals 1 I am having some difficulty in wrestling with the dynamics of a "consumer surplus" type of framework for a demand curve with a finite price elasticity. 17 the social marginal costs of improving them. Thus, the economically Optimal forecast need not be the statistically best forecast. The relations of statistical measures of accuracy to this welfare criterion will be examined below. Analysis of the Cost Derived Loss Function Now that the forecaster realizes that the loss from his fore- casting errors is proportional to r1+n - (l+n)r pn +-n pl+n, what would he do to take advantage of this in his forecasts? If he followed a Bayesian strategy, he would minimize the expected loss accruing under his forecasts. Taking the derivative of the expected loss expression with respect to the forecast p, he obtains -n (l+n)E(r)pn-1 +-n (1+n)pn which will be zero only in the cases when p is zero or p is equal to E(r),1 of which the former makes no sense and the latter is the same forecast he would have obtained had he used a quadratic loss function. What has the forecaster gained by participating in this exercise? In terms of the forecasts he derives, nothing. But in terms of evaluating alternative forecasts - possibly based on dif- ferent maintained hypotheses - quite a bit, for he now has a measure The second derivative of the expected loss function is n2(l+n)pn-'1 - (n+l)n(n-1)E(r)pn-2. When p I 0 this is zero, in- dicating a point of inflection of the function, and when p = E(r) the second derivative is positive (assuming that E(r) is positive and the elasticity is either positive or less than -l.0) which implies a local minimum of the loss function. Since a quadratic loss function is the special case of this loss function when n I l, and since nothing in the optimization process relied on n in a crucial manner, we see that the Bayes decision (forecast) is the same with a quadratic loss as with the more general cost derived loss function. 18 of the actual loss resulting from the different forecasts to use as a criterion for choice between the different forecasting models. The minimum value of the expected loss in the case of the quadratic loss function is proportional to E(rz) - (Er)2, which is the variance of the posterior distribution of r. For the cost derived loss function this minimum is proportional to E(r1+n) - (Er)1+n, which might be called a quasi-variance.1 A factorization of the cost derived loss function is in- formative. This is only possible in the case where n is a rational number, which implies that the homogeneity must also be a rational number. If we represent n as a/b, implying that the homogeneity is a/(a+b), we can obtain these relatively prime factors of the loss function: +0.. L(r p) = (rllb _ Pl/b)2[r(l+b-2)/b +D2r(a+b-3)/b pm, a r(b-1)/b p(a-1)/b + + (a/b) ((b-l)r(b'2)/b p(a/b) + )r(b-3)/b p(a+l)/b +j +_p(e+b—2)/b (b-2 )] . Two special cases perhaps provide more insight than this most gen- eral case: when n I i is an integer (the production homogeneity would be h a i/(l+i)), and when n I l/i is the reciprocal of an integer (h - l/(1+i)). When the elasticity of supply is an integer 1 Note that this is not generally the same as the l+n central moment of the posterior distribution of r. For it to be that, the moment of the distribution would have to satisfy (Er)n = Eggo {-1)j (Er)j (Ern-j) (:11), in the case where n is an integer. As we already know, this restriction is satisfied for all distribu- tions if n I l. 19 the loss function factors into 2 '- '- - L(r,p) = (r - p) (r1 1+ 2 r1 2 p1+...+ i p1 1), and when the supply elasticity is the reciprocal of an integer, the loss function factors into L(r,p) = (rlli - plli)2(i 131")” + (i-l) r .+ p(i-1)/i). (i-2)/i p1/1 + In the case where the supply elasticity is an integer (greater) than one), for a given magnitude of forecasting error, the loss is greater if the forecast exceeds the realized price. This means that an overestimate would be worse than an underestimate. When the reciprocal of the elasticity is an integer, the converse is true (the loss is greater when the forecast is less than the realized price). In either case, one should note that mere knowledge of the forecasting error alone is not sufficient to evaluate the forecasting loss. The factorization is instructive in that it illustrates in a fairly precise fashion, the relation between the squared error loss function and the cost derived loss function. In the case where n is an integer, the ratio of the cost derived loss function to the mean squared error is a positive polynomial expression in the fore- cast and realized prices wherever both are positive.‘ In this sense, as mean Squared error increases, the cost derived loss will also in- crease, but not in the sense of a partial derivative, since i E k ri-k pk-l k=l will change as the mean squared error increases. The practical implication of this is that when ranking fore- casts on the basis of the effect their errors have on firm profits, 20 a squared error criterion would probably provide a correct ranking of the forecasts, although it is not sufficient to quantitatively estimate the actual effect of these errors on the firms. Mgasures of Forecast Performance This section will analyze the effectiveness of several of the more widely used measures of forecasting accuracy in achieving a ranking of forecasts consistent with the welfare loss associated with the forecasting device. As has been shown above, the welfare loss of each fore- casting error is proportional to r1+n - (1+n) r pn +'n p1+n where n is the Short run supply elasticity and the forecast is p when the realization is r. This suggests that forecasting performance l+n should be measured by l ENBI (r:+n - (1+n) ri p: +'n pi ), where 1 pi and ri are the i-th forecast and realization, respectively, of the price variable under consideration, and N is the number of observations in the evaluation period. The interpretation of this measure would be the average welfare loss, and the ranking of fore- casting methods so derived would be considered to be consistent with social welfare. A number of widely used measures of forecasting performance can be shown to be simply transformations of a quadratic loss criterion or are related thereto. Most obviously so are Theil's inequality coefficients and the mean squared error, but it can be shown that the forecasting bias, forecast correlation (between fore- cast and realization), and the forecast slope (of the descriptive regression of forecast on realization) all are identifiable com- ponents of the mean squared error of forecasting. 21 The Theil inequality measures are defined as follows: 2 2 /2(A ri - A Di) \/2NA ri - A p.) u = \’ g“, _, 7"” w---“ and U2 = - l 1 /(A >2+ "( )2 J< )2 Consistent with our prior notation r represents the realized price and p the forecasted price level. When we define the change used as the basis for the inequality coefficients as the change from the previous realized price, the numerator in each case becomes ‘V/z(ri - pi)2 since the ri_1 terms subtract out. It is well known that U1 lies in the closed interval zero to one (11). The proof is based on the Schwartz Inequality, which is the basis for proof that the correlation coefficient is in the interval [0,1].1 The lower bound is attained when the forecasts are perfect, and the upper bound either when all forecasts indicate no change or when there exists a negative constant of proportionality between the forecast and the realization (indicating perfect negative correlation). What seems to be less well known is the threshold, below which all values of U indicate positive correlation between forecasts and realizations and above which negative correlations are indicated. This is given by U1 IV/l+w2/(l+w), where w2 is the ratio of the sum of squared actual changes to the sum of squared predicted changes. The minimum'value of this expression is /2/2, when w = l, and the expression approaches one as w becomes large. 1 See for example (6, p. 107). 22 1 Another property of this expression is that f(l/w) = f(w), where F.__...-- I A f(x) év’l + wz/(l + w). The implication of this finding is that any forecasting scheme which resulted in U being greater than 1 0.7 would be suspect, while those resulting in a smaller U1 co- efficient can be said to be at least positively correlated with the realized price. The U1 coefficient is related to the mean squared error as follows: ’0“- —-—- ——-.'._-.... I / U1 =./MSE / (JEQI‘QZ/N “IV/“APQZ/N) . AS can be seen from its definition U1 depends upon the numerical size of both the forecasts and the realized prices, as well as the mean squared error. Thus it is possible for two fore- casting models for the same price series to have the same U 1 values and different mean squared errors. 2 2 1 Ui = (AZ'R'BZ'ZC)/(A+B)2, where A2 = ZAri, B = mpg, and C = ZAriApi. If predicted and actual changes are uncorrelated, the C would be close to zero. Define w = A/B and factor B2 out of both numerator and denominator, the U? I l+w2/(l+w)2. The first derivative of U: is 2 (w-l)/(1+w)3 and the second is r(2-w)/(1+w)4 which is positive when w = 1. f(l/w) I (1+1/w2)/(l+-l./w)2 = w2+l (w+l)2 _ w2+1 (TH w "'"'""2" f(w- w (w+l) 2 This rule of thumb is somewhat contingent on the (believable) assumption that the mean squared successive differences of the fore- casts and realizations are approximately of the same magnitude. For purposes of an approximate probability statement, Theil (72, p. 32) has derived an upper bound assuming independent (ri’pi) pairs on the variance of U, which is 1/n Y2(1-Y2)2, where Y is the same function of the population moments as U1, is of sample moments and n is the number of forecasts which are evaluated. When Y2 I 1/2 (when 23 To overcome this latter objection to UI and to arrive at a measure which is more closely related to what he called the failure of a forecast,1 Theil (74) deve10ped the U inequality measure. __ 2 N/éRtr, - Api) This is represented by the formula U = . Its 2 _ / 2 E(Ari) square is the ratio of the mean square forecasting error to the mean square successive difference of actual prices.’ U2 can take on all positive values and is equal to unity if the forecast performs the same as a no change extrapolation. Obviously, smaller values of U2 are preferred to larger ones. Rankings by U2 for a given forecasted series are expected to be consistent with the rankings by mean squared error. This is to say, that the forecast which minimizes mean squared forecasting error will minimize U2 as well. Both of the inequality coefficients were heavily dependent upon the quadratic loss criterion in their development. Theil's welfare analysis of prediction made extensive use of the assumption of a quadratic welfare function (73) mostly because of its tract- ability and at least partly as an approximation to the results obtained by more general functions. As we say above, only the second inequality measure will always rank forecasts of a given variable consistently with mean squared error. p(r,p)=0) this variance would be 1/8n and a 20 confidence bound would be lA/Z . Such probabilistic approach is not incorporated into the rule of thumb stated above. 1 The failure of the forecast is defined as the ratio of the difference between the welfare (profit) accruing under the realiza- tion and that under the forecast to the difference of realized welfare and the welfare which accrues under a no-change forecast. See (11) or (12). 24 The mean squared error, which has formed the bench mark of our previous discussion, is defined as E(r-p)2, or as a simple statistic as MSE = fi'z(ri-pi)2. It represents the average squared error of the forecasts and can be shown equal to the squared bias plus the variance (loosely defined) of the forecast. If the "des- criptive regression" r = b0 +'b1p + U is formed, the mean squared 1 error can be expressed as - - 2 2 2 2 - - 2 2 2 2 2 = .. .. + = - - + _ . MSE (r p) + (1 b1) op GU (r p) + (1 b1) op (l ppr)or This representation breaks the mean squared error into three2 separate components: due to the forecast bias, the forecast Slope, and the forecast correlation. The smaller the forecast bias, the smaller the mean squared error is expected to be. Similarly, as the slope of the descriptive regression or the correlation of that regression approaches unity, the same effect is observed. Thus there are three measures of forecast accuracy3 which can relate to the mean squared error measure, in the sense of a partial derivative, but whose ranking of different forecasting schemes need not coincide with the ranking by the mean squared error criterion. In addition to these, there are two inequality coefficients both of whose motivation was in the quadratic loss criterion (the second of which 1 See (55) or (86). This partition expands upon the information provided by the usual squared bias-variance partition, by setmenting the variance component into subcomponents representing variation along and about a "regression" equation. Five measures, if the variances of forecasts and realiza- tions are counted. 25 ranks different forecasting schemes identically with mean squared error). In addition to the foregoing measures of forecasting per- formance which were derived from or partitions of a quadratic loss function (which is a special case of the cost derived loss function), there are a number of other commonly used measures of forecasting performance. The motivations of these other measures generally lay in considerations other than the explicit cost related loss. Those to be discussed (and later applied) are the class of absolute moment measures, the average relative error (which is a "normalized" variation of the simplest member of the class of absolute moment measures), and the number of turning point errors. The absolute moment measures correspond to assumed loss functions which are the m-th power of the absolute value of the forecasting error (37, 40), i.e. the class is characterized as l/N g \ri - pi\m. This class of measures of forecasting perfor- mance-includes the mean square error, the average.absolute error and the third and fourth absolute moments of the forecasting error (corresponding to values of the power parameter m equal to two, one, three, and four, respectively). In fact, for any positive real number m there is a corresponding measure in this class. The measures in the class focus on the "distance" of the forecast- ing device from the ideal of perfect prediction1 rather than repre- senting the effects of that "distance" on the user of the forecast. 1The m-th root of the typical member of this class is a discrete analog to the norm of the Lp space where p = m. See Royden (64). 26 The measures within this class differ in the relative weights given to errors. The amount of weight ascribed to "outlying points" increases as m increases. For example, an error of two units would be weighted as two, if m were one (the average absolute error); four, if m were two (the mean square error); eight, if m were three (the third absolute moment); and sixteen if m were four (the fourth moment). Small errors (less than one unit) are weighted less as m increases. Again as an example, when the power parameter goes from one to four the weight ascribed to an error of 0.9 units decreases from 0.9 to 0.6561. These measures, like all those considered earlier excepting the inequality measures, are measured in the same units as the variable being forecast. Thus to use these measures in evaluating forecasts of several variables the user would have to refer back to the original data series to determine how small a "small" forecasting error is. To achieve a dimensionless quantity, the absolute error could be transformed to obtain the relative error (by dividing by the realized value of the variable). There would be an entire class of measures thusly formed analogous to the absolute moment class of measures. The only element of this set which was used in this thesis is the average relative error, which was defined as N r 1. up. a.r.e. = l/N 2 \-l--l i=1 ‘1 Unless there were a substantial degree of variation in the variable whose forecasts are being studied, one would expect a con- siderable amount of agreement between rankings by the relative moment measures and the corresponding rankings obtained by the absolute moment measures with the same power parameters. 27 A turning point error occurs if a positive change in the variable is forecast and a negative change is realized, or vice versa, so that the product of the change in the realized variable and the change in the forecast is negative. The motivation is that it is oftentimes more important to foresee changes in the direc- tion of events than to accurately forecast the movement in the same direction. A loss function which would represent this feeling would be one which is zero if the product of the change in the realization and the change in the forecast is positive or zero and equal to a positive constant if this product is negative. The turning point measured used herein is consistent with a loss func- tion of that type and is defined as t.p. = .213 (Ari, Api)’ where1 1: 1 if xy < 0 g(x,y) = . As such it measures the number of in- \0 otherwise correct predictions of change which were made during the test period. This measure is amenable to statistical analysis in the sense that one2 may calculate the probability that this many or fewer errors would occur in N forecasts, assuming error occurrence to be a Bernoulli process. A random prediction of change would be con- strued as having the probability of error equal to one-half. Since this performance measure treats all turning point errors as being "equal", not differentiating between large errors and small ones, one is led to suspect that the rankings of different In the analysis of the quarterly variables in the Crom model Api was defined as pi-ri-1' For the analysis of the annual vari- ables in the Gram model and in all the other models Ap. was de- fined as pi - pi_1. 1 See, for instance (57). 28 forecasting devices by this measure would not be identical with the other sets of rankings. The explicit mathematical equations used in defining these measures of forecasting performance are summarized in Table 1. The assumed values of the supply elasticity used in the calculation of the cost derived average loss were obtained from the NC-54 study of livestock and feed grain supply response (8). The precise values used were 0.04 (the U.S. supply elasticity of pork production), 0.12 (the price elasticity of beef production in the current year), and 0.32 and 0.34 (the elasticities of beef production two and three years subsequent to the price change). It has already been mentioned that the mean square error falls into several categories - in the tabulation it will be considered as the cost derived average loss with supply elasticity equal to one. The succeeding chapter will use these measures as the basis for evaluating the forecasts of beef prices, and to a lesser extent other meat prices. 29 TABLE 1 Measures of Forecasting Accuracy Measure Range Cost Derived Average Loss N C.D.L. a 1/N E (r1+n-(l-i-n)r.Pt.1+ i 1 1 i=1 n p: ), n was assumed to be (0.04, 0.12, 0.32, 0.34). Non-negative Mean Squared Error 2 . M.S.E. =1/N 2 (r1 - pi) Non-negative i=1 Inequality Measures N 2 lag/2 (ri '13,) ‘l [0,1] N N 2 E(rW-r 2+ {Hint-rm) i=2 i=2 N 2 \//2 (r1 - p.) i=1 1 U2 = Non-negative N 2 2 (r -r._ ) \/é.2 i 1 1 Number of Missed Turning Points (Incorrect Forecasts of Change) N Average Absolute Error N a.a.e. - l/N 2 \r1 pi| Non-negative 1-1 Comments n is the elasticity of supply for the commodity whose price is being forecast. If U1 > /2/2, r and p may be negatively correlated. If U2 < 1.0, the forecast is better than a no- change extrapola- tion. g(x,y) = 1 if xy < o 0 otherwise. 10. 11. 30 TABLE 1 (Continued) Average Relative Error N r.-. a.r.e. = 1/N E L—l- 1‘ i=1 ‘1 Third Absolute Moment N 3 3 A.M. = 1/N 2 ‘ri - 131‘ i=1 Fourth Moment N 4 4 M. = l/N 2 (ri - pi) i=1 . Forecast Correlation N 15.31071 - I”) (pi - P) p(r,p) = T N N - 2 - 2 j: (ti-r) 2 (pi-p) i=1 i=1 Forecast Slope N z (r. - §)(p. -13) b = i=1' 1 1 1 N _ 2 2 (pi '-- 1)) i=1 Forecast Bias N Bias = 1/N Z (r. ' P.) i=1 1 1 Non-negative Non-negative Non-negative [-1.0, 1.0] Plus one is optimal. Real Plus one is optimal. Real Zero is optimal. CHAPTER III DISCUSSION OF THE MODELS FOR BEEF PRICES Overview of the Chapter The type of analysis pursued in this section of the thesis might be considered to be meta-research, research on research. In any inquiry there are those propositions, or assertions, which are taken as given and which form the basis on which the conclusions of the inquiry are based. In mathematics, and logic, these are generally termed axioms. In statistical analysis, they have been termed by at least one author, (46, p. 112) the maintained hypothesis. In economic analysis, the maintained hypothesis correSponds to the model which describes the system (or market) under study. In the execution of a research project, the first of the substantive stages consists of the determination of the maintained hypotheses in the research: This includes the structuring of the (economic) model for the system, as well as stating the assumed structure of the stochastic elements of the system (where they occur, their distributions, and any g_priori information at hand regarding the parameters of the distributions, etc.). Once the maintained hypothesis has been established, the test hypotheses (null and alternative) are formed. These may re- late to the relative influence of factors within the maintained 31 32 hypothesis, or whether theoretical expectations are fulfilled, or a number of other things. Once the test hypotheses have been established, the re- searcher must determine the statistics which have to be calculated and their distributions under the maintained hypothesis. From these distributions, given the acceptable levels of error prob- ability (of both types), the critical region for each test is de- termined. For models whose sole purpose is the estimation of the structure of the relationships within a given phenomenon, the above two steps are not formally carried out. However, even in these cases, hypotheses of no effect are implicitly or explicitly tested as a result of the inclusion of "t-ratios" and coefficient standard errors or variances, etc. The sample of empirical data is then drawn and the appro- priate statistics are calculated. Based on the sample value of the statistic relative to its critical value, one or another of the test hypotheses is tentatively accepted. At this time the research results are ready to be reported to the profession. Since the solutions to the research problems are only as adequate as the maintained hypotheses, or initial premises, on which they are based, a comparison of the adequacy of the hypotheses in addressing the problems subsequent to their develOpment should indicate the relative value of the hypotheses in later research. The methodology for testing alternative maintained (as Opposed to test) hypotheses is not well developed. Indeed, the characteriza- tion of the entire set of alternative explanatory hypotheses for a 33 given problem set in a way which would render it amenable to the standard tools of statistical analysis is a problem in itself which could be addressed in a thesis such as this (23). In fact, this thesis has sidestepped this issue and has selected only four approaches to the forecasting of beef prices and only a small number of examples of each, taken to be representative of the set characterized by that particular approach, although no pretense of randomness is alleged concerning the selection of the examples. The approaches to be considered are econometric modelling, trend analysis, variable difference models, and the use of the price of a commodity futures contract to forecast the price of the commodity. Econometric Models By far the most complicated approach to forecasting is that involved in using an econometric model to generate the forecasts. If this involves beginning from scratch, considerable time, effort, and skill is required to deve10p satisfactory1 estimates of the structure under study (in this case, the beef-cattle subsector). Even when the forecasts are to be developed from existing models, the degree of technical skill required to correctly use the models 2 is usually quite high. The information required in using this 1"Satisfactory" generally is interpreted as meaning that the response to the variables in the study corresponds to the prior expectations, that the individual equations closely approximate the "real world" behavior as evidenced by the empirical data, and that the numerical values of the estimated coefficients either compare favorably with prior studies, or differ for reasons which can be explained. Generally, it includes the fact that the results were obtained by methods possessing the most desirable statistical pro- parties. This requirement is lessened somewhat if the model chosen is recursive in form, but even to recognize this requires a degree of skill. 34 type of a forecasting procedure is quite great: all of the exogenous variables of the model need to be known or projected, and, if the model has to be estimated, the values of all the variables - both endogenous and exogenous - must be obtained for the sample period as well as be projected into the forecast period. Further, the data for the forecast period must be comparable to that in the sample period in terms of sc0pe, content, and coverage, which means that data revisions need some sort of explicit recognition, lest the forecasts reflect more the change in the definitions that the actual changes in the segment of the real world which the system purports to explain. There is, however, another aspect of the in- formation issue - more information can be provided. All endogenous variables in the system can be forecast nearly as easily as any one of them. The forecasts can explicitly reflect some of the changes in the state of the system (those aspects of "certeris paribus" which have entered the model as explicit variables) as well as possibly afford some "guesstimates" regarding the impacts of violations of other aSpects of the ceteris paribus conditions. Trend Models Trend models may possibly be viewed in two ways: Either they represent a state of complete ignorance (or disregard) of the structure, or they constitute approximations to the time path of theoutput of a complicated system operating under reasonably stationary con- ditions with a stable structure. Obviously, the second hypothesis is intellectually more acceptable than the first. A dynamic system which is represented by a system of differential or difference 35 equations involving the endogenous variables together with the external forces represented by the exogenous variables has a solu- tion whose time path depends upon the solution to the homogenous differential or difference equation, the time path of the exogenous variables and the initial conditions on the endogenous variables. The exact form of this path depends on the system at hand, but suffice it to say that it is exceedingly difficult to represent by the sum of simple functions. At this point, the analyst may wish to approximate the actual solution by a more tractible, mathe- matical representation. Two of the numerous mathematical approximations are the Taylor series approximation and the Fourier series approximation. Taylor's theorem states that f(t) can be approximated by a finite degree polynomial whose first n-l coefficients are the k-th order derivatives of the function evaluated at zero and divided by k factorial, but whose n-th term (the remainder) is the n-th deriv- ative evaluated at some point x which is between zero and t divided by n factorial and multiplied by tn (12, p. 82). As n becomes large the remainder tends to zero. The Fourier series for a function is an infinite series of sine and cosine functions of increasing frequencies. If the func- tion is periodic [i.e. f(x+p) 8 f(x)] then the frequencies are integer multiples of the basic frequency (l/p). If the function is not periodic, and the Fourier series is used to describe the function over a period of observations, the frequencies are integer multiples of the reciprocal of the sample period. The coefficients of the individual trigonometric functions are simply continuous 36 analogues of the least squares coefficients for an equation specified 00 as Y(t) = §.+' Z (ak cos k(2nt/p) +'b k=l sample being any integer multiple of the interval [0,p]. k sin k(2nt/p)), with the The functions chosen to approximate the more complicated solution to the general system are relatively simple in nature and provide explicit information regarding characteristics of the original function. The first approximation provides estimates of the derivatives of the original function, and the second is related to the original function, in that the estimated coefficients are integrals of the function (loosely defined). There are, of course, other means of approximating functions besides those chosen. In addition to the trigonometric functions, other orthogonal polynomials which may be used include Laguerre polynomials, (85) and Legendre polynomials. The generalization of the power series which accounts for isolated points of singularity is the Laurent series (1) which allows for both positive and negative powers of the independent variable. Price Difference Models Another, perhaps more specific way of expressing past time trends is with the use of weighted average models. If the possi- bility of negative weights is not excluded this set of models in- cludes stationary variate difference models as well. The solution of the difference equation implied by the weights of the lagged variables gives the explicit form of the time dependence assumed in this type of model. 37 In particular, a model which assumes that the n-th dif- ference is distributed with mean zero implies that the form of the dependence is a polynomial of degree n-l in the time variable. The superposition of seasonal variability on top of a basic dif- ference model complicates the form, but alters little of the sub- stance of the analysis. The forms of variate difference models which were given consideration in this analysis were pure dif- ference models of first, second, and third degree and a first difference model which has annual variation superimposed on it. Futures Market Price One final forecasting device, namely the futures price, was considered to be the standard against which any forecasting device should be compared, from the standpoint of simplicity, authority, and communicability. It is simple, in that all that is necessary to obtain its forecast is to see the quotation for the contract maturing in the period being forecast. It has authority, in that the financial consequences of an error in judgement en- courages somber reflection upon the conditions prevailing and likely to prevail in the market, and the fact that the participant remained in the market implies that he has passed the market's test of accuracy.1 The commodity futures market is communicable, in that it is fairly easy for the lay individual to understand the concept 1He may or may not be using a formal model to derive his strategies, but since the market in the long run is expected to re- ward accuracy with profits and inaccuracy with bankruptcy, survival in the market is a minimal test of a forecaster. This perspective is consistent with the view of a futures price as an aggregate of all current price expectations. 38 (though not always the mechanism) of a contract for future delivery of a commodity and its associated price, and the widespread avail- ability of most contract prices. The price corresponding to the Chicago Mercantile Exchange contracts for live slaughter cattle was used to forecast prices of slaughter cattle. The specific slaughter prices it was used to forecast are the price for choice 900-1100 lb. steers at Chicago and Omaha, and the deflated weighted average price of all steers sold out of first hands at these two markets and at Sioux City.1 For those months in which no contracts mature, the price forecast was taken to be the average of the cash price and the near term futures price. For forecasts more than one month into the future, the prices of the contracts maturing around the forecast month were averaged to generate the forecasted price for months in which no contracts matures. The next four sections of this chapter discuss the forecast- ing models summarized above with greater detail. In addition to this, the section dealing with the trend models presents the estimated trigonometric and polynomial trend equations. For the period of analysis, this contract called for the de- livery of 40000 lbs. of choice grade live steers, with the steers in the weight range 1050-1150 lbs. estimated to dress to 61 percent and those in the range 1151-1250 lbs. estimated to yield 62 percent. Delivery was to take place in Chicago at par, or at Omaha, Nebraska, or Kansas City, Missouri, with discounts of $0.75 and $1.00, respectively per hundred weight. Effective with the August 1971 contracts, par delivery was to occur at Omaha, with delivery allowed at Guymon, Oklahoma, at a $1.00 per cwt. discount, and at Chicago and Peoria, Illinois, allowed at a premium of $0.50 per cwt. Kansas City was eliminated as a delivery point. For this and other pertinent information regarding the cattle contracts, see (7). 39 The Econometric Models The analysis included econometric models based on monthly, quarterly, and annual data. Three of the models were dynamic, in the sense of containing lagged dependent variables, and the two others were not. All of the models chosen considered one or more prices of beef cattle to be endogenous to their structure and pur- ported to explain more than simply that (those) prices. As an indication of the sizes of the models considered, the number of endogenous variables ranged from five to thirty and the number of separate series of exogenous variables ranged from five to thirteen (although the total number of actual exogenous variables was gen- erally much larger due to the includion of trend and dummy variables in the analyses). A brief description of the models considered is presented in Tables 2a and 2b. Table 2a indicates the relative sizes of the models, in terms of dimensions and density of the coefficient matrices, as well as the means chosen to estimate the model. Table 2b gives a rundown of the price variables which were included in each of the models. 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The procedure used in the generation of forecasts from the econometric models consisted of 1) verifying (to the extend possible) the coefficients of the structural equations, 2) inverting the structural equations to obtain the reduced form implied by that system (this is to say, each of the endogenous variables became a function of only predetermined (exogenous and lagged endogenous) variables, rather than being jointly dependent on other contempor- aneous endogenous variables), 3) re-creating and updating the data series used in original analysis to cover the test period (no mean task in view of revisions, deletions, and other editing which is done by the agencies which collect, prepare and disseminate statistical information), and finally 4) using the updated data in the reduced form equations to calculate the forecasted (or esthmated) values of the endogenous variables (since most of the models made use of contemporaneous variables in explaining the system, using the actual values of these factors in the forecasting procedure should be expected to bias the analysis slightly in favor of these types of models over other types of forecasting procedures). The monthly econometric models were the results of work at Michigan State University, Purdue University and at the University of Nebraska. One was presented as a journal article, and the other two were research bulletins published by the respective Agricultural Experiment Stations. 43 Hayenga-Hacklander Monthly Model The Hayenga and Hacklander article (29) appeared in the American Journal;of Agricultural Economics in November 1970. The purpose of the model was to explain the short run variations in supply and demand for both fed cattle and hogs and the variations in the storage holdings of frozen and cured pork. While only nine series of exogenous variables were used in this model, after recognizing the dummy variables which shift both slope and intercept in the model there gets to be thirty-five exogenous variables in the model as estimated. Schematically, the model is of the form A‘Y(t) = B Y(t-l) +-C X(t) +-U(t), where X(t) and Y(t) are the vectors of exogenous and endogenous variables, reSpectively, which are observed at time t, A, B, and C, are matrices of scalar co- efficients, and U(t) is a random disturbance vector which is gen- erated at time t. The lagged endogenous variables in the model are the result of including both levels and changes of the price and storage variables in the model. Myers Monthly Model The second monthly model considered in this research originated in L.H. Myers' thesis at Purdue (56) and was taken from the research bulletin (57) which was co-authored by Myers, Joseph Havlicek, Jr., and P.L. Henderson. It consists of eight behavioral equations and two identities and involves twenty-four exogenous variables, including a time trend, population, and dummy variables to alter_the intercepts in six of the relations. The sample period over which this model was estimated is January 1949 to December 1966. 44 Although it is claimed that two stage least squares was the estima- tion method applied to this model, I will show that what actually occurred was more akin to an instrumental variables approach than it is to the true two stage procedure. Although the system of behavioral equations was itself entirely linear, both the identities in the system were nonlinear. They related the per capita quantities demanded to the total quantities produced: in one case, simple division of an endogenous variable by an exogenous variable was the only process involved, but the other case involved the application of a multiplicative con- stant1 (dressing yield) to the quantity produced before the division by population. Thus the system can be represented as A(p) Y = B X +'U, where A(p) is a matrix whose elements are functions of population (i.e. p), rather than the constants usually assumed. Before the forecasts could be obtained from this system, via the reduced form, it was necessary to determine the inverse matrix A-1(p), so that we could represent the endogenous variables as func- tions of only the exogenous variables (including population). This 2 was done by the process of matrix inversions by partitions: First l . I have some doubt as to whether it was indeed a constant, or rather another variable to be reckoned with. The only mention of this constant, other than as the abstract representation in the text, is in footnote 25 on page 17 which states "average dressing yield for cattle over the period of study = .556". The process of inversion by partitions is as follows: If a non- singular matrix A can be partitioned into (A )) A = ( 11 :12—) , where All is nonsingular, (A21 22 ) its inverse is a matrix B which partitions conformably and whose corresponding matrices are given by 45 A(p) (as well as B and U) was rearranged so that the endogenous variables divided by population correSpondcd to the last two columns and the nonlinear identities were the last two equations. In this way, except for the 2 x 2 matrix elements which were containing the functions of P in the southeast corner, A(P) contained all con- stants. Because of this, the inverse matrix is straightforwardly obtained, with the only particular difficulty involved in the pro- cess associated with the determination of the Z submatrix (see footnote 2, p. 44). The determination of the 2 matrix is essentially a numerical process, with not much gained from purely algebraic pre- sentation, so at this point I will retreat into the arithmetic of the particular problem: A22 8 E-l.3 P-l -.5g6 P'l) the estimated structural coefficients, , and based on A -1 a (10.369339325 -1.480056935 ) x 10.3 “A21 11 A12 (-2.485153142 4.139016516 ) ' From this we observe that the "2" matrix is z = ( _ A A -1 )-1 _ 1000 (4.139016516 - 556 p'l, 1.480056935 _1) A22 21 11 A12 ’ f(p-1) (2.435153142 10.369339325-1000p ) where f(p‘l) = 39.24069853829 - 9904.3691807 p'l +-556000 p‘2 is the determinant of the matrix. With some simplification (and fewer significant digits), 8 ‘p (-556 0 ) +_ P2 (4.139 1.480) g(P) ( 0 -1000) g(P) (2.485 10.369) 2 9.904 P + 0.0392 P . The roots g(P) - 0, and hence points of Z where g(P) = 556 - singularity of A(p), are P 8 84.277 and P - 168.123. -1 -1 B22 ‘ z ' (A22 ' A21A11 A12) -1 21 ’2 (A21(A11 ) _ -1 B12 ‘ '2 (A11 A12) - -1 -1 -1 B11 A11 +'(A11 A12) 2 (A21A11 ) ' cu ll 46 2 If we denote the 2 matrix as Z = B1 P/g(P) + B2 P /g(P), the inverse of the matrix A(P) is A-1(P) = C +C1 P/g(P) + 0 C2 P2/g(P), where CO, C1 and C2 are matrices of constants de- fined as follows: -1 (A 0) c0 = < 11 ) ( 0 0) C _ E(A111A12) 31(A21A111) '(A11-1A12)B1 ; l ( M1( 21 A11 1) B1 ) -l (( ) 82(A1) -(A ) B ) and C2=( A111A12 21:11 11 A12 2) As a result of this derivation we see that the reduced form for this system of equations is Y = (Co B +'C B P/g(P) + 1 C2 B P2/g(P)) X 41A"1 (P) U, where Y, X, and U are vectors of endogenous, exogenous, and random disturbance variables and the B matrix (without subscript) refers to the matrix of the co- efficients of the exogenous variables in the structural system. Obviously, this is not a linear system of equations involving the exogenous variables, and is definitely not the system of equations which were claimed to be used in the two stage estimation process. The equations in the stage one system actually used by Myers, et. al. were not all of the same form, viz. the per capita beef and pork consumption and the retail price of broilers all used per capita income, broiler consumption and pork storage holdings, while the other stage one equations used total values of these variables (the structural system used total pork storage holdings and per capita income and per capita chicken consumption). As a result, the 47 estimation process utilized by Myers can, at best, be described as an instrumental variables approach to the problem, and even this would probably stretch the literal definition of the process. In this thesis I examined the forecasts which were gen- erated by both his stage one equations and the forecasts which were generated by the structural model (as determined by the reduced form equations). In either format there were no lagged endogenous variables to affect the analysis. Trierweiler-Hassler Monthly Model The third monthly econometric model which was used in this analysis was developed by John Trierweiler and James Hassler at Nebraska (79, 80). Their model included retail demand equations for both beef and pork which were estimated from quarterly data, assuming that the per capita supplies were predetermined, and a number of other equations which related carcass and slaughter animal prices to the respective retail prices for beef and pork., these latter equations being estimated using monthly data. The structure of the model included no lagged variables in the analysis and so could be represented in the form A Y(t) = B X(t) + U(t). The model consisted of six equations explaining five endogenous variables (one endogenous variable was explained using two alternative func- tional forms) using five separate series of exogenous variables, which, together with a time trend and dummy variables for quarters, gave the X(t) vector a dimension of 10. 48 Crom Quarterly Model The quarterly model which was analyzed in this thesis was that which Richard Crom of the U.S. Department of Agriculture pub- lished as Technical Bulletin No. 1426 (9). His is a recursive model of the beef and port subsector of the economy which he estimated by means of ordinary least squares over the period 1955 to 1966 (although for one set of equations the period 1957 to 1968 was used).There were twenty-five separate quarterly endogenous data series in this model, but more than that number of equations were estimated, since for some relations separate equations were estimated for each quarter of the year (not always containing the same explanatory variables). There was also an inventroy subsystem' which contained five variables for which only annual observations are made. There were eleven series of exogenous variables used in the model, along with a time trend, dummy variables, and numerous restrictions to be mentioned below. As the system is structured, it is an exceedingly complicated dynamic model. The quarterly relations contain endogenous variables lagged as many as five quarters, and the annual equations contain variables lagged as many as three years. There are two identities which state that one endogenous variable is the product of two other endogenous variables. Finally, when the system as estimated by Crom did not sufficiently approximate the actual market behavior in its original form, he began imposing additional restrictions on the response of the endogenous variables in various circumstances (the model as published contained 147 such restrictions, of which 128 49 were given explicit notice in the publication). The easiest way to express the flow of dependence within the model is in a figure such as he used to describe the model which I have reproduced here as Figure 1. Figure 1 has been reproduced from his bulletin with some modification to make it correspond with the published form of his model. Unger Annual Model The annual model which was analyzed in this thesis was con- structed by Samuel Unger at Michigan State University as part of his dissertation research (81). He constructed an interdependent model describing the cattle and beef sector and then estimates it with a number of alternative estimators.1 Schematically, Unger's model can be represented as A‘Y(t) = B Y(t-l) +'C X(t) + U(t). The Y vector of endogenous variables is ten dimensional and the X vector of exogenous factors (includ- ing the intercept) has dimension of eight. No binary variables were included in his analysis. Trend Models Models of trend extrapolation are generally thought of as something akin to an intellectual "copout", an admission of ignorance These were ordinary least squares, two stage least squares, Nagar's unbiased K-class, three stage least squares, iterated three stage least squares, and limited information single equation es- timators. He used three methods as the intermediary step in the three stage process: two stage least squares (which is the commonly used intermediate step), unbiased K-class and limited information es- timates. In my analysis of the effect of estimation method on fore- cast accuracy, I chose not to evaluate the latter two options in the three stage process. 50 "WI! I QUARTERLY MODEL OF BEEF AND PORK SECTORS OF THE LIVESTOCK-rMEAT ECONOMY Richard Cram you «on .- I non-v10 {:11 an mm: A: .9 I. I 0 fl '0. I“ M” “WK IISIAICI SEIVICI U.S. .I'AITIIIT 0' ASIICIIIWII 51 as to the forces affecting the market under study. Hildreth (3S) commented to the effect that every relation which includes time as an explanatory variable is an unanswered challenge to econometricians. With that as a remark concerning the inclusion of a single trend variable as one of the exogenous variables within a simultaneous equations model, there is little question as to the potential response to a model which contains nothing but time (or functions of time) in an attempt to smooth, "explain", or forecast a series for a particular variable. An alternative explanation for the existence of trend models is the relatively high cost of modelling, estimating, and interpret- ing the complicated interdependent systems of relationships which characterize most economic systems. Elementary economics tells us we should only act up to the point where the marginal benefit is no less than the marginal cost, in whatever the activity at hand, be it growing wheat or, as it were, applying economics. Implicit in any trend analysis is the assumption of a stationary system. Although this assumption restricts the analysis somewhat, it is not as confining as one might suspect. It only means that those things which have been constant remain so, and those which have been changing, contiue to do so in the same way. As a particular example, consider the following equation which might typically be found in a dynamic economic system. m n Y(t) - 2 ak'Y(t-k) +' z Bj X(t-j). Mathematically, it would be k=l j-O described as an myth order nonhomogeneous linear difference equa- tion with constant coefficients. In economic terms, it would be described as a general distributed lag model. For concreteness, 52 consider Y(t) and the ak to be scalars and the Bj and the X(t) to be vectors. (This relation may actually be a part of a larger economic system, but that possible complication will be ignored for the present purposes). The solution to this equation is given by the sum of the general solution to the homogeneous part and a particular solution to the nonhomogeneous equation, subject to the initial conditions and the assumed time path of the exogenous variables (2, p. 192). The general solution to the homogeneous equa- tion is Yh(t) = kglck Rkt, where Rk is the k-th root of the algebraic equation Rm - ale-1 - asz-2 -...- am = 0. The m roots are assumed to be distinct. Note that these ak's are the same ak's as appeared in the difference equation. The ck's are con- stants which are arbitrary until determined by the initial con- ditions on the y's. The determination of the particular solution is basically a matter of diligence and experience once the time path of the forcing functions 2 BjX(t-j) has been determined or assumed. Needless to say, thisj:grt of solution process is quite arduous and becomes more so if the equation is part of a system of other similar equations. But whatever the specifics of the solution for the model, it can be said that there is a definite time path which is followed by the dependent variable which can be represented as Y(t) - f(t). If all we are concerned about is simply projecting the time path of the variable in question within a fairly stable environment, it may be much easier (in a costs and returns sense) to use some method to approximate the time path f(t), than to attempt a direct analytical solution to the entire structural system. The question 53 then distills to one of determining an apprOpriate approximation method. Two commonly used ones are the Taylor series approthation and the Fourier series approximation. The effectiveness of these two methods in generating forecasts outside the estimation period will be investigated in this thesis, both in comparison with each other and as alternative maintained hypotheses to compare the econometric models with. Taylor Series Approximation The basic idea behind a Taylor series expansion of a function is the theorem (12, p. 82) which states: If a function f is con- tinuous on the closed interval [a,b], then f(x) = f(a) + n-l (k) (n) 2‘, L—ki-m (X - a)K + LEI—L2)- (X - a)“, where z is some kel number in the open interval (a,x) and f(k)(a) is the k-th derivative of f evaluated at a. This implies that an n-th degree polynomial can be used to approximate an n-th order differentiable function of this type for variation within the interval of convergence. Since we have posited a time dependence in the variation of the price series (of either an unspecified form or a too-complicated form) which we want to approximate by means of a Taylor series, the statistical model which we assume is of the form: P(t) B a +-b1t +b2t2 +3..+'bntn'+ u(t). There is a direct correspondence between the bi coefficients and the i-th derivatives 1 In this context, the interval of convergence is taken to be those values of the independent variable for which the difference between the actual functional value and the value of the approximation is less than a pre-specified amount. In a more general context, it refers to the ranges of X's for which the power series converges, and hence, to those for which the limit of the remainder is zero as the degree of the polynomial increases without bound. 54 of the time series at the origin. The a coefficient (intercept) represents the sum of the value of the function at the origin and the average value of the remainder. The disturbance u(t) repre- sents the deviation of the remainder at any time t from the average remainder for the observation period and is assumed to possess all the usual statistical properties. One should note that the fre- quently used linear trends are no more than polynomials of degree one (the simplest case). One rather undesirable feature of a polynomial approximation is the fact that as the independent variable (t) becomes large, the approximation approaches either plus or minus infinity, depend- ing on the sign of the highest order coefficient (bn)' This fact leads one to suspect that this means would not work.very well as a way of generating intermediate run projections of bounded series. The only question of substantial interest in evaluating the fore- casts generated by a polynomial approximation is the period of time between the sample period and the point at which the divergence of the series is apparent. Fourier Series Approximation To arrive at an approximation which does not diverge outside the sample period, one may use a series of bounded functions to approximate the actual function. Elementary functions which do possess the property of boundedness over the entire range of defini- tion are the trigonometric functions (sin x and cos x). The theory of Fourier series developed around the question of when does an 55 infinite series of trigonometric functions converge, and what co- efficients are necessary for that series to converge to a partic- l ular function. If f(t) is a function which is integrable over the domain [0,p], the Fourier series corresponding to f(t) is defined as a .9. 2 + =u’uma (an cos (2n n t/p) +bn sin (2n nt/p)), where the co- 1 efficients an and bn are defined by the following integrals: a = if: f(t) cos (2n11 t/p) dt, for n 0,1,2,3,... and b = g'fg f(t) sin (2n n t/p) dt, for n = l,2,3,4,... . This expansion can be used to interpolate any function which is integrable over an interval, or if the function under study is periodic, i.e. f(x+p) = f(x), it can be used to extrapolate future values of the function. A point which is important enough that it bears repeating, is that the Fourier coefficients are no different than the co- efficients which would be derived from ordinary least squares estimation of the trigonometric series from a sample period which is an integer multiple of the interval [0, 2n] when the sums of squares and cross-products are replaced by the values of the corres- ponding integrals: If one were to estimate the function Y(t) - a+b cos(t) +0 1 l 2 2 a sample period in which the time variable corresponds to the range sin (t) +-b cos (2t) +'C sin (2t) + U(t) over From another vantage point, it can be said that Fourier series theory provides the interpretation of the coefficients of the trig- onometric series used to approximate a particular function. 56 O to 2n, the least squares normal equations would be: (1' 1 -Z(:o:(t.) [sin(t) {cos(Qt.) tsin(?t) n it y 12 cor. (t) 3013?“) {sin(t) cos(t) i2cos(2t) cos(t) fisin(2t) cos(t) b1 2 y cos (1.) 1: sin (t) [cos (t)sin(t) 231212 (t) {cos(i‘t) sin(t) {sin (2t) 131-u(t) c1 : 12 y sin (t) i: cos(21.) Eco-(t) cos(2t) B:in(t) cos(2t) Zcosa (2t) Zsin(2t) cos(2t b? t y cos (91.) Kr sin(ZL) )Lco::(t) 31:1(20) min”) sin(2L) £cos(2t) sin(2t) tsinz (2t) / c2 11 y sin (21.) If the time interval between observations is small enough that the discrete sums on both sides of the equation can be replaced by the corresponding integrals1 over the sample interval [0, 2n], the normal equations become: . 2T! '1 f 2 \ 7a \ t dt . f n o o o o \ / \ I0 Y ( ) \ . 1 A 2 | I o n o o o g b1 \ n y (t) cos (t) dt , 1 3 .' 0 '~ 2 o o n o o 3 A c1 f for y (t) sin (t) dt ; a . ' '- , b j K jg" y (t) cos (2t) dt I I o A 1 \ O o o o n,/ \\ From this we observe that these estimated least squares coefficients 2 / \\ 2n / 92/ I6 y (t) sin (2t) dt/ are precisely the coefficients which define the Fourier series for the variable (function) y(t). The mathematical fact which allows this process to simplify in this manner is that the trig- onometric functions and their harmonics are orthogonal (12, p. 221). This means that for all distinct integers 'm and n the following integrals are all zero: f(z)” COS (mt) C08(nt) dt 3 I3" C08 (mt) 81n(nt) dt = $3" sin(mt) sin(nt) dt = 0 One would recall that the process of integration obtains the limit of such discrete sums as the interval between successive observa- tions goes to zero. 57 (when m = n, the first and last integrals are equal to n and the middle integral is still zero). The existence of the Fourier series is simply a matter of definition (i.e. it is defined whenever f(t) is integrable over the interval [0, p]), but the convergence of the series, in particular its convergence to f(t), is a matter which has to be established in each case. The conditions (in addition to in- tegrability) under which the Fouries series converges to the original function are that there are only a finite number of maxima and minima over a finite interval and that there exist at most a finite number of (finite) discontinuities (at which the series con- verges to the average of the right- and left-hand limits) (9, p. 3-14). Generally, these criteria are fulfilled for most empirical time series. Once it has been decided to use a Fourier series to approximate the time series of prices for beef cattle and hogs, there is the question as to the period to use as the basis for estimating these Fourier series for each price series. For the prices of beef cattle, carcasses and retail cuts, a cycle with a period of ten years was assumed. This corresponds with the generally assumed cycle in cattle production and is the same period assumed by another researcher in describing long term price cycles (21, 22). The basic hog and pork price cycle was assumed to be four years long, with an "overtone" of length eight years. The four year cycle was derived from a cobweb-type model of adjustment in an article by Arnold Larson (49). The eight year cycle was added to try to capture some of the longer term swings in the price behavior. 58 For all except the annual relations an attempt was made to include the sine and cosine terms with period one year in order to quantify the pattern of price variation within the year. Since the purpose of the trend analysis is to compare it as an explanatory (or predictive) hypothesis for the beef prices with the econometric models of price structure, the following guide- lines were developed for the use of these approximation methods in this thesis: The same sample period would be used to estimate the polynomial and trigonometric trend equations as was in the research which gave rise to the econometric models which are being studied. Since each added variable (power of t, or cos (nt)) when included in the equation to be estimated increases the explanatory power of that equation, at the cost of reduced degrees of freedom for error, it was decided to structure the equations so as to retain approx- imately the same number of error degrees of freedom as were present in the original econometric models. For the dependent variables in the Gram Model, the restrictions he imposed on the estimates of the particular endogenous variables were considered to reduce the error degrees of freedom by one for each such restriction. Estimates of the Trend Equations In this section of the discussion, I will present and briefly comment upon the polynomial and trigonometric trend approx- imating equations. The models to which these equations correspond are, in order, Hayenga and Hacklander, Myers, Trierweiler and Hassler, Cram and then Unger. One word of caution should be uttered regarding the inter- preation of the estimated coefficient standard errors, the standard 59 error of estimate, and the t-ratios: The positive serial correla- tion of the disturbances indicated by the small values of the Durbin- Watson statistic causes the usual variance calculations (those pre- sented in these tables) to be biased toward zero, so that the t-ratios will also be biased, but away from zero (24). Correspondintho the HayengaéHacklander Model. -- In the Hayenga and Hacklander model the two price variables are the price of choice 900-1100 lb steers at Chicago and the price of Nos. 2-3 200-220 lb. barrows and gilts also at Chicago. The model as specified contained an average of sixteen coefficients per equation, with the two demand equations which were in proce dependent form, having sixteen and seventeen parameters in them. The polynomial trend equations were estimated as polynomials of fifteenth degree in teim as the independent variable. The estimated coefficients of these equations are presented in Table 3, in exponential form, to- gether with their corresponding t-ratios. To illustrate the con- version of the exponential form of the coefficients to the form more frequently encountered, I will present two examples: The co- efficient of T0 (i.e. the intercept) in the hog price equation is presented as .14959233E+02; this is to be read as .14959233 times ten raised to the power +02, or 14.959233. The coefficient of T4 in the same equation is -.67956878E-02 which is the same as -0.0067956878. For these and all other of the estimated equations the coefficient of determination, standard error of estimate and Durbin-Watson statistic are presented as measures descriptive of the residuals from the estimated equation (and not explicitly for purposes of hypothesis testing). 6() Table 3 Estimated Polynomial Trend Equations Corresponding to Hayenga - Hacklander Model Sample Period: April 1963 to June 1968, by Months Price of Nos. 2-3, 200-220 lb. Barrows and Gilts at Chicago Price of Choice 900-1100 lb. Slaughter Steers at Chicago ' 14 Exponent ($/th) ($/th) . of Time Coefficient t-ratio Coefficient t-ratio 0 .14959233E+02 24.2819 .228048028+02 61.8326 1 -.36234268E+00 -1.3050 -.783392988+00 -4.7129 2 .237310005+00 3.1773 .697388543-01 1.5597 3 .788901843-02 0.4062 .305004843-01 2.6233 4 -.67956878£-02 -3.2157 -.35859166E-02 -2.8344 5 .163825078-03 0.3669 -.23282093B-03 -0.8709 6 .727524805-04 1.6756 .561039673-04 2.1584 7 -.52298938£-05 -2.2437 -.234240328-05 -1.6786 8 -.17316589E-06 -0.4328 -.14059199E-O6 -0.5870 9 .394263288-07 1.2407 .200066258-06 1.0516 10 -.23588428£-08 -1.6509 -.10433577E-08 -l.2198 ll .775598278-10 1.9139 .315191198-10 1.2992 12 -.1559757OE-11 -2.1007 c.5963517OE-12 -1.3416 13 .191765968-13 2.2410 .699279023-14 1.3656 -.l3290387E-15 -2.3499 -.46699124E-16 -1.3792 15 .398965778-18 2.4360 .135928762-18 1.3864 R2 0.9224 0.8656 S.E.E. 1.2548 (47 degrees of freedom)- 0.7512 (47 degrees of freedom) 0.0. 1.1724 0.9305 Tin: origin: T 1 1 in January 1964. 61 The estimated trigonometric trend equations for the Hayenga- Hacklander model are presented in Tables 4 and 5. The hog price equation if in Table 4 and the cattle price equation directly corresponding to this model is in the first column of Table 5. The equation in the second column of that table is a substitute estimate with the same dependent variable but a longer sample period. In this and all other tables of the trigonometric approximating equations, the coefficients are presented, as well as the amplitude1 for each cycle considered. As is obvious from perusal of the coefficients, the cattle price equation which was estimated using the same sample period as the HayengaAHacklander model is unrealistic in terms of both the estimated amplitudes of the cycles (the unit of measurement is dollars per cwt.) and the intercept which is to be interpreted as the mean of the dependent variable. Of the following two explanations for this phenomenon, I suspect the second has more bearing on the issue than the first: First, as will be demonstrated below, the model which Hayenga and Hacklander estimated is dynamically unstable which may mean either that the market is unstable which might cause estimates of this type, or that the model itself is deficient in some way, in which case, this would have little bearing on the trigonometric 1 If y = a cos x +'b sin x, then y is also equal to d cos (x-t) where the amplitude d is given by d = (a2 +b2)1/2 and the phase angle t is given by t a arctan (b/a). The phase angle would only be useful in determining the relative displacements from the time origin of the cosine terms of different frequencies, hence was not presented explictly. The difference between the high and low points on the cycle is twice the amplitude. (52 Table 4 Estimated Trigonometric Trend Equations Corresponding to Hayenga - Hacklander Model Price of Nos. 2-3, 200-220 lb. Barrows and Gilts at Chicago ($/th) Sample Period: April 1963 to June 1968, by Months Period Cosine Sine Amplitude Intercept 20.75164 (0.4352) 8 years -2.26952 0.94000 2.45648 (0.2210) (0.7300) 4 years -3.64688 -0.71660 3.71662 (0.4462) (0.2604) 2 years 1.73192 0.34334 1.76563 (0.2337) (0.2137) 16 months -0.56286 -0.08602 0.56940 ' (0.2166) (0.2179) 12 months -0.76331 -0.65761 1.00751 (0.2189) (0.2186) 9.6 months -0.56073 -0.27352 0.62389 (0.2166) (0.2161) 8 months 0.23860 0.71527 0.75401 (0.2147) (0.2147) 6.85 months -0.19733 -0.03386 0.20022 (0.2122) (0.2118) 22 0.9361 5,3,3, 1.1509 (46 degrees of freedom) 0.x. 1.2268 The numbers in parentheses are the coefficient standard errors. Time origin: T s 1 in January 1964. €513 Tehle 5 Estimated Trigonometric fiend neuations Corresponding to Beyengs - Hacklander Model Price of Choice 900-1100 lb. Steers at Chicago 9 by Months ($/th) . Sample Period: April 1963 to June 1968 ¢§Sanp1e Period: January 1957 to December 1969 Period Cosine Sine Amplitude Period Cosine Sine Amplitude Intercept -4290.58135 Intercept 26.05737 (4150.3002) (0.0815) 10 years 3137.548 7319.509 7963.63078 10 years -0 09091 -1.79577 1.79808 (2751.12) (7126.33) (0.1171) (0.1135) 5 years 4300.821 ~4508.904 6231.15311 5 years -0.29768 0.11612 0.31953 (4405.52) (3972.55) (0.1140) (0.1166) 40 months-3824.264 -1442.540 4087.28755 40 months 0.70364 1.70686 1.84621 (3402.35) (1782.57) (0.1161) (0.1145) 30 months 78.306 2191.865 2193.26352 30 months -0.52553 0.29232 0.60136 (399.19) (1985.40) (0.1151) (0.1154) 2 years 840.868 ~380.470 922.93895 2 years -0.00432 0.18099 0.18104 (786.01) (271.51) (0.1151) (0.1154) 20 months -199.896 -193.018 277.87431 20 months--0.64688 -0.02142 0.64724 (146.97) (193.21) (0.1157) (0.1148) 17+months -18.908 42.704 46.70317 17+months -0.00162 0.48699 0.48699 (22.81) (32.07) (0.1149) (0.1156) 1 year -0.954 0.336 1.01145 15 months 0.01909 -0.04381 0.04779 (0.604) (0.477) (0.1153) (0.1150) 2 I 0.8719 13+months -0.09482 -0.25325 0.27042 (0.1152) (0.1150) a.r.r. 0.77.15 (46 degrees of freedom ) 1 year 0.13277 0.04747 0.14100 0.“. 1.0224 (0.1149) (0.1153) The noebers in parentheses are the coefficient standard errors. * Time origin: G Time origin: 2 - 1 in January 1964. T c 1 in January 1962. 11-months 0.89822 (0.1149) 10 months 0.45481 (0.1121) :3 0.0594 s.s.t. 0.9702 0.9. 0.6129 0.16141 0.91261 (0.1144) 0.16251 0.48297 (0.1130) (107 degrees of freedom) 64 estimates. The second hypothesis regarding the unrealistic coef- ficients is that the basic period of the assumed function is longer than the sample period.1 I have not uncovered a reference which deals with the convergence (or failure to converge) of Fourier- type series where the coefficients are integrals identical with those of the Fourier series except that the interval over which they are integrated is not an integer multiple of the period of the function.2 Because the likelihood of the non-convergence of such a series appears substantial, I feel that this latter is the more credible of the explanatory hypotheses. An observation on the relative degree of fit of the trend approximations compared with the structural equations of the model is that the R2 of the trend models was only between .005 and .047 less than the proportion of explained variation corresponding to the demand equations for the particular commodities (which had the best fits of the equations of the model). Corresponding to the Myers Model. -- In the Myers model, the endogenous price variables are the weighted average price of all grades of steers sold out of first hands in Chicago, Omaha and Sioux City, the eight market weighted average price of barrows and In some unpublished previous work, I observed a trigonometric model for a different price series which generated coefficients similarly unstable and the sample period was shorter than the period of the model. However, I also obtained results which appeared reasonable, notwithstanding a sample period shorter than the period of the cycle. This situation covers the case at hand, as well as the case where there are more than n complete cycles, but less than n+1 cycles in the sample period. 65 gilts, the average retail price of choice carcass cuts of beef in U.S. urban areas, the average retail price of retail pork cuts and sausage in U.S. urban areas, and the average retail price of frying chicken in retail stores in U.S. urban areas. All the prices were deflated to eliminate the influence of general trends of prices: The farm level prices were divided by the wholesale price index with the base years 1957-59 = 100, and the retail prices were divided by the consumer price index with the same base period. Since the average number of parameters per equation in the Myers model was fourteen, the polynomial trend equation was estimated by a polynomial of degree thirteen and the trigonometric trend equa- tions consisted of seven pairs of sine and cosine functions. Because of an obvious downtrend in the retail price of frying chickens, a linear time trend was included in this equation. One should note that the independent variables in the polynomial equations are time divided by 100 raised to the appropriate power, rather than time alone raised to the power.1 The estimated polynomial trend equations corresponding to the prices in the Myers model are given in Tables 6, 7 and 8 and the trigonometric trend equations are presented in Tables 9, 10 and 11. The retail and farm level beef prices are in Tables 6 and 9, the retail and farm level hog prices are presented in Tables 7 and 10, and the retail price of frying chicken are presented in Tables 8 and 11. 1 This transformation was necessary to allow observation of the coefficients of the higher order terms. For example, if time alone were the argument, the coefficinet of T13 in the equation for the retail beef price would be 3.6 times 10'23. 665 Table 6 Estimated Polynomial Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966, by Months Price of Beef at Retail Three Market Price of Beef Cattle Deflated by C.P.I. (1957-59 base) Deflated by w.P.I. (1957f59 base) Exponent of (¢/1b.) ($/th) . (Time/100) Coefficient t-ratio Coefficient t-ratio 0 74.26443657 117.5862 22.74260519 59.1572 1 -43.45149882 -6.7256 -24.47224749 -6.2228 2 ~18.21462359 -0.4557 53.60428051 2.2032 3 1464.65842393 6.3418 1042.88517568 7.4183 4 3040.44951987 5.2153 1080.21420574 3.0440 5 ~12566.12527251 -4.5118 -10589.9709l675 -6.2465 6 -44299.36641979 -8.1105 -27794.84918022 -8.3600 7 -10140.94931245 -1.2392 4489.15401316 0.9012 8 141063.85958862 4.9781 109011.24372482 6.3199 9 275958.14494324 6.4660 191653.18920517 7.3773 10 249990.13731003 7.1619 166043.17642593 7.8148 11 124347.19972420 7.5794 80421.51545906 8.0530 12 32855.90013885 7.8587 20866.15546513 8.1992 13 3617.26791179 8.0556 2266.89920938 8.2936 Rz 0.8927 0.8608 S.E.E. 2.6968 (202 degrees of freedom) 1.6416 (202 degrees oi freedom) D.W. 0.4520 0.3686 T e 1 in January 1964 6'7 Table 7 Estimated Polynomial Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966, by Months Price of Pork Cuts and Sausage ' Price of Barrows and Gilts at Retail at Eight Markets _ Deflated by C.P.I. (1957-59 base) Deflated by W.P.I. (1957-S9 base) Exponent of (¢/1b.) ($/th) (Time/100) Coefficient t-ratio Coefficient t-ratio 0 52.07844992 63.7992 14.41254284 27.6190 1 '56.46173995 -6.7618 -22.18714805 -4.1564 2 186.39684981 3.6082 186.87682634 5.6587 3 2605.81573415 8.7297 1442.91201967 7.5615 4 1812.07796925 2.4049 -206.84803542 -0.4294 5 -23856.72619772 -6.6274 -15218.793l9358 -6.6134 6 -52119.69294930 -7.3830 -25606.21305084 '5.6740 7 19166.76462984 1.8121 22572.10520124 3.3383 8 196863.26084137 5.3752 119265.12522888 5.0939 9 303631.69908905 5.5045 165498.88009644 4.6933 10 236289.15169525 5.2376 120272.51917839 4.1703 11 103476.46707535 4.8800 49626.08676147 3.6610 12 24345.78279686 4.5055 11013.57065773 3.1883 13 2402.54230148 4.1397 1022.60519701 2.7562 R2 0.6793 0.6816 5.5.3. 3~4855 (202 degrees of freedom) 2.2232 (202 degrees of freedom) D.W. 0.4902 0.3392 Time origin: T 2 1 in January 1964. 663 Table 8 Estimated Polynomial Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966. by Months Averagf Retail Price of Pr n Chicken a De ted by C.P.I. (195 -5 base) (c/lb.) Exponent of (Time/100) Coefficient t-rstio 0 36.48191563 62.4337 1 -15.68856975 -2.6247 2 -36.94366262 -0.9990 3 288.34719814 1.3494 4 1409.80886060 2.6138 5 -209.13836120 -0.0812 6 -9102.76802707 -l.8013 7 -15273.51571703 -2.0173 8 -2610.82223l83 -0.0996 9 18760.06170797 0.4751 10 24397.19566965 0.7555 11 14034.21023107 0.9246 12 4011.28916478 1.0370 13 463.54278534 1.1158 22 0.9674 3.3.3. 2.4951 (202 degrees of freedom) D.W. 0.6482 Time origin: T a l in January 1964 69 Table 9 Estimated Trigonometric Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966, by Months Price of Beef at Retail Three Market Price of Beef Cattle Deflated by C.P.I.(l957-59 base) (cllb.) ' regigh‘ted clbyai‘h‘ep' I ‘ (19915117069 "‘53.:1‘24985) Period Cosine .Sine Amplitude Intercept 78.47660 Intercept 26.18744 (0.3956) (0.2332) 10 years -2.40950 -7.18848 7.58155 10 years -1.04977 -3.35579 3.51615 (0.5719) (0.5472) (0.3371) (0.3225) 5 years -1.13019 -0.29168 1.16722 5 years -0.59730 0.15935 0.61819 (0.5507) (0.5692) (0.3246) (0.3355) 40 months -1.94470 0.68567 2.06204 40 months -1.11222 0.33443 1.15313 (0.5646) (0.5546) (0.3328) (0.3269) 30 months -0.46907 -O.296€4 0.55511 30 months -0.49525 0,27258 0.56531 (0.5565) (0.5586) (0.3281) (0.3293) 2 years -0.49967 0.75917 0.90885 2 years -0.05599 0.29644 0.30168 (0.5502) (0.5577) (0.3243) (0.3287) 20 months -0.10673 0.25491 0.27635 20 months -0.19097 0.15343 0.24497 (0.5512) (0.5480) (0.3249) (0.3230) 1 year 0.25419 -0.87240 0.90867 1 year 0.07840 -0.77883 0.78276 (0.5413) (0.5424) (0.3191) (0.3197) 2 2 R 0.5361 R . 0.4356 S.E.E. 5.6221 (201 degrees of 5.3.8. 3.3140 (201 degrees of freedom) freedom ) D.w. 0.0884 .V. 0.0736 Harbors in pa: Tine origin: T a 1 in January 1964 entheses are coefficient standard errors. 70 Table 10 Estimated Trigonometric Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966, by Months Eight Market Price of Barrows and Gilts Deflated by 17.2.1. (1957-59 base) ($/th) Retail Price of Pork Cuts and Sausage Deflated by C.P.I. (1957-59 base) (¢/1b.) Peri od Cosine Sine Amplitude Period Cosine Sine Amp 1i tude Intercept 19.00390 Intercept 59.93911 (0.2031) (0.3184) 8 years -l.05848 0.15006 1.06907 8 years 4.10833 0.46703 2.15943 (0.2952) (0.2786) (0.4627) (0.4366) 4 years -3.08353 -0.37184 3.10587 4years -4.25151 -1.55618 4.53082 (0.2871) (0.2856) (0.4500) (0.4476) 2 years -0.00903 -0.05201 0.05279 2years -o.02127 -0.50154 0.50199 (0.2850) (0.2855) (0.4466) (0.4475) 15 months 0.03229 -o.13973 0.14341 16 months 0.19957 -0.43405 0.47773 (0.2852) (0.2850) (0 4471) (0.4467) 1 I7--ar -1.36962 -0.51349 1.46271 lyear 4.44032 -1.69000 2.22050 (0.2849) (0.2852) (0.4465) (0.4470) 9-6 months-0.23007 -0.12025 0.25960 9.6 months 0.10279 -0.30452 0.32141 (0.2851) (0.2849) (0.4469) (0.4465) a mo nths -0.04048 -0.00833 0.04143 8months -0.25130 -0.09783 0.26967 (0.2841) (0.2845; (0.4453) (0.4459) 2 R 0.4465 112 0.4403 c ‘ ~3~r~.. 2.9453 (201 degrees of s.£.8. 4.6163 (201 degrees of I) ~. ““40” freedom) ‘ - 0.1552 0.:'. 0.2413 5:37-le T ime 01-131“; T 1 l in January 1964. rs in parentheses are coefficient standard errors. 7]. Table 11 Estimated Trigonometric Trend Equations Corresponding to Myers Model Sample Period: 1949 to 1966, by Months Average Retail Price of Frying Chicken Deflated by C.P.I. (1957-59 base) (c/lb) Period Cosine Sine Amplitude Intercept 35.57174 (0.5112) Time ~O.20625 (0.0341) 5 years -0.59659 0.72347 0.93773 (0.3570) (0.3640) 30 months -0.35819 -1.45694 1.50032 (0.3602) (0.3612) 20 months -0.15553 -0.21669 0.26673 (0.3618) (0.3590) 15 months 0.08716 0.39823 0.40765 (0.3611) (0.3589) 1 year -0.81505 -0.09434 0.82050 (0.3598) (0.3605) 10 months 0.09399 -0.58984 0.59728 (0.3595) (0.3601) 8.57 months 0.04694 0.16674 0.17322 (0.3593) (0.3579) 2 R 0.9287 5.2.2. 3.7072 (200 degrees of freedom) D.w. 0.2779 Numbers in parentheses are coefficient standard errors. Time origin: T = 1 in January 1964. 72 To indicate the relative degree of fit, during the sample period, the structural equations generally had the highest values of R2, with the polynomial trend next, following by the stage one equa- tions that Myers used, and finally by the trigonometric trend equa- tions. The exceptions to the rule were hog and pork prices, where the stage one equations fit better than the polynomial trend equations. Correspondinggto the Trierweiler-Hassler Model. -- The irrierweiler-Hassler model was structured to explain the variation in tzhe prices of choice 900-1100 lb. slaughter steers at Omaha, choice (500-700 1b. (steer) beef carcasses at Chicago, and the retain price c>f beef, together with the price of nos. 1-3 220-240 1b. barrows eind gilts at Omaha and the retail price of pork (excluding lard). The retail price series were both quarterly variables, and the re- unaining series were reported monthly. The structural equations eaxplaining the quarterly variables required the estimation of six, 61nd in the other case seven, parameters. In three of the monthly Irelationships, three parameters were estimated, and in the fourth, ifour. The polynomial trend equations were estimated as cubic equa- tzions for the monthly series and of degree six for the quarterly 53eries. The trigonometric trend equations contained two pairs of tZrigonometric functions (three in the case of the quarterly relations) in addition to the intercept term. It might be noted that the tzrigonometric relations for the price of slaughter steers and for t>eef carcasses does not contain an annual component to the cycle. 7Dhe component corresponding to a five year sub-cycle was chosen (Iver the annual component based on the insignificance of the annual Component of the cycle. 73 The polynomial trend equations are presented in Tables 12 and 13, with the trigonometric equations in the next three numbered tables. Tables 12 and 14 contain the estimates for the quarterly retail beef and pork prices. Tables 13 and 15 contain the equations estimated for the price of beef carcasses and the price of slaughter steers. The equations for the price of Omaha barrows and gilts are in Tables 13 and 16. The structural equations fit the sample data much better than either of the trend models for the monthly price sereis. For the quarterly price series the polynomial trend equations fit marginally better than the structural equations. In almost all cases the polynomial trend model fit the sample data better than did the trigonometric trend. Corresponding to the Gram Model. -- The endogenous price of choice grade steers at twenty markets, the weighted average price of choice grade carcasses at New York, Chicago, Los Angeles, San Francisco, and Seattle (less than carlot basis), the price of good and choice 500-800 lb. feeder steers at Omaha, the price of utility cow beef at New York, the eight market weighted average price of barrows and gilts, and the weighted average of wholesale prices of individual pork products at Chicago. All of these variables were observed quarterly. The typical quarterly relationship in the Crom model con- tained slightly more than seven parameters1 and had ten restrictions 1 This figure is arrived at by considering the relationships where separate estimates were obtained in each quarter as being equivalent to a single equation in which both slopes and intercepts were allowed to vary over subsets of the sample. This underestimates '74 Table 12 Estimated Polynomial Trend Equations Corresponding to Trierweiler - Hassler Model Sample Period: 1957 to 1966. by Quarters Retail Price of Beef Retail Price of Pork (Excluding Lard) (c/lb.) (c/lb ) Exponent of (Time / 10) Coefficient t-ratio Coefficient t-ratio 0 79.54647007 103.9495 55.67102063 56.6509 1 -0.86992951 -0.5688 -2.19417321 -1.1172 2 3.65874505 1.4507 18.80487956 5.8062 3 3.77505053 1.5638 19.18893340 6.1898 4 -0.14452996 -0.1041 -5.54647561 -3.1113 5 -1.42378125 -1.0888 -9.64289214 -5.7424 6 -0.42940667 -1.5132 -2.20881266 -6.0612 22 0.7664 0.8139 8.3.8. 2.1442 (33 degrees of freedom) 2.7536 (33 degrees of freedom) 0.0. 1.0036 1 5760 I Time origin: T a 1 in the first quarter of 1964. 75 Table 13 Estimated Polynomial Trend Equations Corresponding to Trierweiler - Bassler Model Price of Choice 600-700 # Steer a Carcasses at Chicago ($7CWC) Price of Choice 900-1100 I Slaughter - Steers at Omaha (S/th) Price of Nos. 1-3 220-240 # Barrows and a Gilts at Omaha ($/th) Sample Period: 1957 to 1967, by Months 39.03241635 - 0.10184343 21+ 0.00231009 12 + 0.00004303 23 (111.7277) (-9.0866) (7.7945) (9.0396) R2 . 0.4706 5.2.2. 4 2.3764 0.w. - 0.2813 3 24.07463973 - 0.04853323 T + 0.00127596 T2 + 0.00002404 T (91.1741) (~5.7291) (5.6961) (6.6800) R2 - 0.2757 3.8.2. ' 1.7961 D.W. I 0.2262 2 3 18.12063357 + 0.09144270 T + 0 00052467 T - 0.00001232 T (44.4561) (6.9926) ' (1.5173) (-2.2183) R2 1 0.3417 5.3.3. a 2.7726 D.W. r 0.1934 128 error degrees of freedom in each equation. The numbers in parentheses are the t-ratios for the coefficients. Time origin: T s 1 in January 1964. '76 Table 14 Estimated Trigonometric Trend Fquntion: Corresponding to Trierweiler - Hassle: Model Sample Period: 1957 to 1966, by Quarters Retail Beef Price Retail Pork Price (Excluding Lard) (c/lb) (cllb) Period Cosine Sine Amplitude Period Cosine Sine Amplitude Intercept 80.18500 Intercept 59.76732 (0.6596) (0.6207) 10 years 0.43187 50.96347 1.05583 8 years -1.19720‘ 3.00501 3.23472 (0.9329) (0.9329) (0.9121) (0.8354) 5 years -0.12763 1.51432 1.51969 4 years -4.00467 -2.85365 4.91739 (0.9329) (0.9329) (0.8743) (0.8543) 1 year 0.40000 0.41000 0.57280 1 year 0.39406 -l.58241 1.63073 (0.9329) (0.9329) (0.8461) (0.8461) 2 2 R 0.1156 R 0.6500 S.E.E. 4.1719 (33 degrees of 3.8.8. 3.7764 (33 “8”” °‘ freedom) freedom) 0.”. 0.3692 D.W. . 0.6728 Numbers in parentheses are coefficient standard errors. Time origin: T a 1 in the first quarter of 1964. '77 Table 15 Estimated Trigonometric Trend Equations Corresponding to Trierweiler - Bassler Model Sample Period: Choice 600-700 # Steer Carcass Price, Chicago ($/th) Period Cosine .Sine Intercept 41.34331 (0.2111) 10 years -2.86167 -1.44457 (0.3010) (0.2958) 5 years 0.39638 0.27475 (0.3046) (0.2912) R2 0.4662 SOEOE. D.W. 0.2799 Amplitude 3.20561 0.48229 2.3958 (127 degrees of freedom) 1957 to 1967, by Months Choice 900-1100 # Steer Price, Omaha ($/th) Period Cosine Sine Amplitude Intercept 25.18264 (0.1628) 10 years- -l.37406 -0.55949 1.48360 (0.2321) (0.2280) 5 years 0.20477 0.08375 0.22124 (0.2348) (0.2245) R2 0.2402 3 E E 1 8469 (127 degrees of ' ° ° ' . freedom) D.W. 0.1857 The numbers in parentheses are the coefficient standard errors. Time origin: T w l in January 1964. 78 Table 16 Estimated Trigonometric Trend Equations Corresponding to Trierweiler - Hassler Model Sample Period: 1957 to 1962 by Months Price of Nos. 1-3, 220-240 1b. Barrows and Gilts, Omaha ($/th) Period Cosine Sine Amplitude Intercept 18.24373 (0.2333) 4 years -2.71092 -0.94155 2.86977 (0.3282) (0.3310) 1 year -0.83676 -0.63672 1.05147 (0.3277) (0.3279) R2 0.3981 8.8.8. 2.6617 (127 degreei Of freedom) D.W. 0.1801 The numbers in parentheses are the coefficient standard errors. Time origin: T a l in January 1964. 79 imposed upon it by the so-called "operating rules" which were arrived at after estimation for purposes of improving the performance of the model during his validation period. The subset of price variables of the model had a similar number of parameters per equation (7.5), but averaged only a bit less than five restrictions per variable. When the trend models were specified, the result was that the poly- nomial approximations were Specified with error degrees of freedom equivalent to the typical quarterly relation in the model (i.e. a polynomial of degree sixteen was specified), and the trigonometric trend equations were specified more nearly like the price equations of the model (five pairs of sine and cosine functions were included, along with an intercept). The estimated polynomial trend equations corresponding to the Crom model are presented in Tables 17, 18 and 19. The estimated trigonometric trend equations are presented in Tables 20, 21 and 22. Tables 17 and 20 contain the estimated equations describing the wholesale prices of fed (choice grade) and nonfed (utility grade) beef carcasses. The equations describing the trend relationships involving the slaughter and feeder steer prices are in Tables 18, 19 and 21. The prices at the wholesale and farm levels for hogs and pork are given in Tables 19 and 22. The degree of explanation during the sample period is not significantly different between the structural model and the poly- nomial trend model, with the trigonometric trend model explaining about twenty percent less of the variation of prices than the other models. the number of constraints on the data in that not always are the same variables assumed to affect the dependent variable in different quarters of the year. 8C) Table 17 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model Sample Period: 1955 to 1966. by Quarters Five Market Wholesale Price Price of Utility Grade Beef of Choice Beef Carcasses at New York ($/th) ($/th) Exponent of (Time/10) Coefficient t-ratio Coefficient t-ratio 0 39.23259507 41.4545 29.01807929 40.1653 1 2.18096777 0.3755 -7.81065612 -1.7615 2 25.02384718 1.2137 5.24063487 0.3330 3 -13.55352224 -0.1907 34.90446897 0.6432 4 -3.17692040 ~0.0287 53.21789108 0.6291 5 30.91858055 0.1032 -42.72115294 -0.1868 6 -157.61555137 -0.4979 -183.60300151 -0.7597 7 -148.37667993 -0.3350 -75.13912296 -0.2222 8 256.24698404 0.3927 190.69845625 0.3828 9 367.80439530 1.2469 222.23513558 0-9870 10 25.69369359 0.0710 31.57240013 0.1143 11 -224.09226772 -0.5549 -93.62086531 -0.3037 12 -l88.31574659 -0.8609 -80.64571694 -0.4830 13 -75.18982412 -1.0799 -32.09927221 -0.6039 14 -16.73949883 -1.2554 -7.11812879 -0.6993 15 -2.00410556 -1.4028 -0.85035188 -0.7797 16 -0.10102973' -1.5292 -0.04284127 -0.8494 R? 0.8444 0.9407 S.E.E. 1.6/26 (27 degrees of freedom) 1,2768 (27 degrees of freedom) 0.0. 2.3747 2.3008 Time origin: T 1 1 in the third quarter of 1904. £31 Table 18 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model Sample Period: 1955 to 1966. by Quarters Average Price of Good and Choice 500-800 1b. Feeder Steers, Omaha ($/th) Twenty Market Average Price of Choice Grade Steers ($/th) Exponent of (Time/10) Coefficient t-ratio Coefficient t-ratio 0 21.98648980 35.8453 21.36493299 50.1566 1 2.50169507 0.6645 -10.47419729 -4.0063 2 22.76597259 1.7037 24.72366389 2.6642 3 -15.43054031 -0.3349 65.52407141 2.0480 4 ~32.78097522 ~0.4564 ~37.55666659 -0.7530 5 25.72220891 0.1325 -187.81927765 -1.3929 6 -26.22542929 -0.1278 -68.82075661 -O.4830 7 -55.01168272 -0.19l6 222.68872327 1.1170 8 98.95872124 0.2340 256.50154717 0.8733 9 153.20783325 0.8014 20.32569029 0.1531 10 3.11879280 0.0133 -148.85353096 -0.9138 11 -llO.88038660 -0.4237 -135.577l3332 -0.7460 12 -92.35234346 -0.6514 -61.49479l49 ‘ -0.6246 13 -37.01697881 -0.8203 -l6.67640772 -0.5321 14 -8.28156454 -0.9583 -2.74474704 -0 4574 15 ~0.99585043 -1 0755 -0 25404256 -0.3951 16 -0.05038631 -1 1767 -0.01017687 -0.3422 R2 0.8645 0.9661 S.E.E. 1 0840 (27 degrees of freedom) 0,7523 (27 degrees of freedom) 0.9 2.3057 2.2254 Tin: origin: T a 1 in the third quarter of 1964. Average Price of Wholesale Pork £32 Table 19 Estimated Polynomial Trend Equations Corresponding to Crom Simulation Model Sample Period: Products at Chicago 1955 to 1966. by Quarters Average Price of Barrows and Gilts at E1 ht Markets (s/cm) ( "WU Exponent of (Time/10) Co-ffirient t-ratio Coefficient t-ratio 0 39.83063421 30 9920 15.09957744 22.3712 1 3.27876506 0.4157 2.37664597 0.5737 2 8.26170252 0.2951 13.73501485 0.9341 3 -26.86090324 -0.2785 -4.29564832 -0.0847 4 90.97325320 0.6046 15 97348436 0.2021 5 399.45422717 0.9819 165.80027862 0.7760 6 148.01379465 0.3443 117.56508201 0.5207 7 -722.99661210 -1.2020 -3o3.63425100 -0.9612 8 -9’12 . 11377454 4.0292 -484.51039466 4 .0410 9 -105.52988935 -0.2635 -114.89885991 -0.5462 10 567.86015293 1.1554 271.28665130 1.0510 11 553.39386533 1.0092 302.35815712 1.0499 12 262.27395861 0.8830 153.76009927 0.9857 13 73.39463052 0.7762 45.49518929 0.9162 14 12.34519609 0.6818 8.05157139 0.8467 15 1.15803459 0.5969 0.79387804 0.7792 16 .04667000 0 5202 0 03367125 0.7146 82 0.8885 0 9133 5.5.2. 2.2713 (27 degrees of freedom) 1.1929 (27 degrees of freedom) 0.0. 2.3816 2 3503 Time origin: ‘ = 1 in the thiid quarter of 1904. £33 Table 20 Estimated Trigonometric Trend Equations Corresponding to Crom Simulation Model Wholesale Price of Choice Beef Ca1casses Price of Utility Grade Beef at New York ($/th) ($/th) Period Cosine Sine Amplitude Period Cosine Sine Amplitude Intercept 42.73956 Intercept 32.05093 (0.3874) (0.3771) 10 years -2.38184 -l.22612 2.67891 10 years -3.30143 -2.08255 3.90339 (0.5547) (0.5407) (0.5398) (0.5262) 5 years 1.09833 -0.79358 1.35503 5 years 1.56404 -1.25367 2.00447 (0.5542) (0.5401) (0.5395) (0.5257) 40 months -1.44511 0.68764 1.60038 40 months -1.41521 1.14720 1.82178 (0.5393) (3.5533) (0.5249) (0.5386) 30 months -0.91171 0.53414 1.05666 30 months -0.29548 -1.07882 1.11855 (0.5458) (0.5448) (0.5313) (0.5302) 1 year 0.19074 0.33078 0.38183 1 year 1.14502 0.51959 1.25739 (0.5396) (0.5396) (0.5252) (0.5252) 2 2 R 0.5669 R 0.7319 S.E.E. 2.5242 (33 degrees of 3.8.8. 2.4569 (33 degrees of freedom) freedom) 0.2. 1.0919 0.0. 0.4870 Tne numbers in parentheses are the coefficient standard errors. Tire 011910: I a 1 in the third quarter of 1964. Table 21 Estimated Trigonometric Trend Equations Corresponding to Crom Simulation Model Sample Period: Twenty Market Price of Choice Grade Steers Amplitude 1.94544 0.78961 1.29411 0.66977 0.20020 (33 degrees of ($/th) Period Cosine Sine Intercept 24.83119 (0.2620) 10 years -1.73893 -0.87228 (0.3751) (0.3656) 5 yerr. 0.t7304 -0.50182 (0.3140) (0.3652) 40 months -1.13675 0.61848 (0.3047) (0.3742) 30 months -0.59791 0.30182 (0.3691) (0.3684) 1 year 0.13390 0.14418 (0.3649) (0.3649) :2 0.5895 S.E.E. 1.7070 D.w. 0.9330 freedom) 1955 to 1966, by Quarters Price of Good 6 Choice 500-800 # Feeder Steers (Omaha) ‘ ($/th) Period Cosine Sine Amplitude Intercept 24.32559 (0.2810) 10 years -2.23228 -2.04231 3.02558 (0.4024) (0.3921) 5 years 1.43239 -1.04648 1.77394 (0.4010) (0.3917) 40 months -1.57493 0.60872 1.68847 (0.3911) (0.4013) 30 months -0.45737 -0 52570 0.69681 (0.3959) (0.3951) 1 year 0.58290 0.06307 0.58630 (0.3914) (0.3914) R2 0.7551 S.E.E. 1.8308 (33 degrees of freedom) 0.0. 0.3845 The numbers in parentheses are the coefficient standard errors. ' I Tiae origin: T a 1 in the third geirtur of 1904. €35 Table 22 Estimated Trigonometric Trend Equations Corresponding to Crom Simulation Model Sample Period: 1955 to 1966, by Quarters Wholesale Price of Pork Products, Chicago Eight Market(ngfog and Gilt Price WC (Slewt) Period Cosine Sine Amplitude Period Cosine Sine Amplitude Intercept 43.10998 Intercept 17.20029 (0.5275) (0.3101) 8 years 0.21447 3.05609 3.06361 8 years 0.26459 1.67059 1.69141 (0.7314) (0.7558) (0.4300) (0.4443) 4 years -4.23365 1.33860 4.61565 4 years -2.49406 1.45319 2.88654 (0.7260) (0.7415) (0.4268) (0.4359) 2 years 0.87433 -0.26964 0.91496 2 years 0.35922 -0.10965 0.37559 (0.7175) (0.7196) (0.4218) (0.4231) 16 months -0.12402 0.41533 0.43346 16 months -0.05436 0.26532 0.27084 (0.7066) (0.7272) (0.4154) (0.4275) 1 year 0.57948 1.64793 1.74685 1 year 0.47921 0.82169 0.95122 (0.7142) (0.7142) (0.4199) (0.4199) 2 2 R 0.7034 R 0.7158 6.8.1:. 3.3229 (33 degrees of 8.8.8. 1.9536 (33 degrees of freedom) freedom) D.0 0.9821 D.w. 0.7933 The nuvbcrs in parentheses are the coefficient standard errors. Tim: ori_in: T a 1 in the third quarter of 1964. 86 Corresponding to the Unger Model. -- The annual model de- veloped by Unger contained four endogenous price variables: The average retail price of beef and the average price of "other meat" at retail were analyzed as well as the farm level prices of beef and "other meat". The retail prices were deflated by the consumer price index with base year 1957-59 = 100, and the farm level prices were deflated by the index of prices paid by farmers which includes interest, taxes, and wages, with base year 1957-59 = 100. The "other meat" commodity consists of pork, veal, lamb and mutton, chicken, and turkey. The price aggregates were formed by weighting the retail prices of the individual commodities by the total annual consumption of those commodities, and the farm level prices by the total annual production of the respective commodities. For a more complete description of the process and the exact variables in- volved, the reader is directed to the original work. The typical equation in the model as structured by Unger contained 4.5 parameters to be estimated. This led us to form the polynomial trend as a fourth degree polynomial in time and to include two pairs of sine and cosine functions in the trigonometric trend equations (a linear trend variable was added to the model prior to the final analysis to account for long term price movements). The estimated polynomial trend equations are presented in Tables 23 and 24 and the estimated trigonometric trend equations are in Tables 25 and 26. The relations describing the retail and farm price behavior for beef are in Tables 23 and 25. The relations in- volving the farm and retail behavior of the prices of other meats are presented in Tables 24 and 26. Exponent of Time 7O 10 S.E.E. 13.1)}. Time origin: 637 Table 23 Estimated Polynomial Trend Equations Corresponding to Unger Model Sample Period: 1936-41 and 1949-62, by Years Retail Price of Beef Farm Level Price of Beef Cattle Deflated by Consumer Price Index Deflated by Prices Paid by Farmers Index (1957-59 base) (1957-59 base) (¢/1b) ($/th) Coefficient t-ratio Coefficient t-ratio 81.23041861 14.3939 19.57937025 7.2911 4.96036164 1.5826 0.68988976 0.4626 1.04586688 2.1235 0.12154435 0.5186 0.06463987 2.3431 0.00359832 0.2741 0.00117387 2.3167 -0.00001244 -0.0516 0.6755 0.4223 7.1696 (16 degrees of .reedom) 3.4115 (16 degrees of freedom) 1.0062 0.7748 T a 1 in 1964. Exponent of Time Time origin: 883 Table 24 Estimated Polynomial Trend Equations Corresponding to Unger Model Sample Period: 1936-41 and 1949-62, by Years Retail Price of Other Meat Price of Other Meat Animals at Farm Deflated by Consumer Price Index Deflated by Prices Paid by Farmers Index (1957-59 base) (1957-59 base) (c/lb) ($/th) Coefficient t-ratio Coefficient t-ratio 53.29343890 24.8702 15.82203122 11.9844 1.56279570 1.3131 0.64905361 0.8852 0.73372322 3.9233 0.31178420 2.7060 0.05449259 5.2021 0.02205252 3.4170 0.00111397 5.7898 0.00043926 3.7056 0.8685 0.8127 2.7224 (16 degrees of freedom) 1.6773 (16 degrees of freedom) 1.9575 2.2695 T a 1 in 1964. £39 Table 25 Estimated Trigonometric Trend Equations Corresponding to Unger Model Sample Period: 1936-41 and 1949-62. by Years Annual Average Retail Price of Beef Deflated by Consumer P:ice Index Annual Average Price of Beef at Pa Deflated by Prices Paid by Farmers In ex (1957-59 base) (cllb) (1957-59 base) (S/th) Period Cosine Sine Amplitude Period Cosine Sine Amplitude Intercept 81.91370 Intercept 20.26538 (3.7547) (1.2790) Time 0.70656 Time 0.13386 (0.2624) (0.0894) 10 years -0.03213 -5.42237 5.42246 10 years -1.98654 -3.19449 3.76180 (3.3536) (3.1270) (1.1577) (1.0652) 5 years 0.16478 -1.21075 1.22191 5 years -1.10591 -0.07925 1.10875 (3.1i07} (3.1188) (1.0290) (1.0624) 2 2 R 0.4588 R 0.5063 freedom) freedom) D.w. 0.7233 D.W. 0.7752 Numbers in parentheses are coefficient standard errors. Tire origin: T = 1 in 1964. Annual Average Retail Price of Other Meat Deflated by C. P. 1. (1957-59 base) (Cllb) Cosine Period Intercept Time 8 years 4 years SOEOE. D.W. -4 (2 -1 (2 .05316 .1290) .21480 .0249) Tab1e26 9() Estimated Trigonometric Trend Equations Corresponding to Unger Model Sample Period: 58.86935 (2.4671) -O.13326 (0.1685) -0. (l. -2. (2. 0.2801 Sine 00795 9922) 88606 1281) Amplitude 4.05317 3.13131 6.5782 (15 degrees of 0.6841 freedom) 1936-41 and 1949-62, by Years Annual Average Price of Other Meat at Farm Deflated b Period Intercept Time 8 years 4 years 8.5.5. 0.1% .P.F.I. 1957-59 base .'C t {agine ine Ampaigidéa) 17.61304 (0.9594) -0.23978 (0.0655) -1.73966 (0.8279) 0.04381 (0.7747) 1.74021 -l.31563 (0.7874) -0.23978 (0.8275) 1.48853 0.5916 2.5580 (15 degrees of freedom) 0.6140 Numbers in parentheses are coefficient standard errors. Time origin: T a 1 in 1964. 91 In terms of the relative degree of explanation during the sample period, the structural equations, excepting those estimated by the limited information single equation method, had the highest degree of explanatory power, followed by the polynomial trend model and then by the trigonometric trend model. The evaluation of the trend model, as well as econometric and other, forecasts for each set of price variables over a period which includes more nonsample than sample points is presented at a later point in this chapter. Price Difference Models The previous two means of generating forecasts are relatively eXpensive in terms of the resources required to generate the fore- casts. Technical skill is required to formulate and analyze the econometric models used in the first case to generate the forecasts, the required data for both the trend and the econometric models must be collected, and for either case computational skill and/or computing facilities are required to effect the estimation and forecasting pro- cess. This section of the chapter and the next discuss a variety of forecasts (which some might characterize as naive) for which the resource cost is much less than the previous two methods. The fore- casting schemes discussed in this section involve price projections which use only the most recent observations from the data series, making certain assumptions regarding the distribution of the dif- ferences of the price series. The next section discusses the fore- casts which are implies by the prices which prevail in the futures market for the commodity in question, namely beef cattle. 92 The cash price difference models are actually only another, perhaps more specific, way of representing the time trend in the data series. In particular, if a time series Y(t) can be repre~ sented by a polynomial of degree n in the time variable, then the n+l-st difference as well as all higher order ones, Of the series will be zero except for whatever stochastic elements remain in the trend. As a particular example, consider a quadratic trend equation y = a + b t +'c t2: The first difference is y(t) - y(t-l) 8 2c t- (c-b), the second difference is y(t) - 2 y(t-l) +~y(t-2) = 2c, and the third difference is y(t) - 3 y(t-l) +13 y(t-2) - y(t-3) = 0. To facilitate the discussion Of the difference models the notion Of a shift Operator will be introduced. The Operator to be used in this discussion is a backward shift Operator, call it B, which Operates in the following fashion: B Y(t) = Y(t-l), BkY(t) = Y(t-k), B-kY(t) = Y(t+k), and a3 Y(t) = a Y(t-l), where a is any multiplicative constant. In general terms, all of the algebraic Operations that can be performed with scalar constants can be per- formed on the shift Operator. The process Of Obtaining the k-th difference of a variable Y(t) can be represented quite compactly using the shift Operator, as (1 - B)k Y(t). Seasonal variation can also be represented within the context Of a difference model as the product of the basic model and a factor involving the operator raised to the power equal to the period of the seasonal variation: For example, a second difference model involving monthly data and seasonal variation within the year would be represented as (1 - B)2(1 - 812)Y(t) 8 U(t). General weighted average models can be represented as a polynomial in B multiplied by Y(t) which 93 2 equals the disturbance: (a0 + 813 + 823 +...+*aan)Y(t) = U(t). The condition under which the variance Of the limiting value Of Y is finite, given homoskedastic disturbance terms, is that the largest root of the algebraic polynomial equation xn-l 1 be touched upon later in a discussion of stability of dynamic systems. (aoxn +1a +x..+-an) = 0 be less than unityl, a fact that will In this thesis, the difference models which were considered to be plausible alternative maintained hypotheses for beef prices were three pure difference models, Of first, second and third order, and a hybrid model which represented the product of a first order process and an annual seasonal variation component. When these were compared with the formal models, i.e. the trend approximations and the econometric models, they were used only on a one-step-ahead basis. But in comparing them with the futures market as a forecast- ing device, the difference models took the algebraic forms: (1 - B)kY(t) = U(t), in the first difference case; (1 - Bk)2Y(t) = U(t), in the second difference case; (1 - Bk)3Y(t) = U(t), in the third difference case; and (1 - Bk)(l - Blz)Y(t) = U(t), in the hybrid difference model.2 For these comparisons with the futures market, the lead time involved in the forecast (k) was permitted to vary and took on the values 1, 2, 3, 6, and 8 months. NO higher The pure difference models are not stable in this sense, as can be noted. See (5). The explicit forecasting equations are as follows: Y(t) = Y(t-k) + U(t) in the first difference model, Y(t) = 2 Y(t-k) - Y(t-Zk) + U(t) in the second difference model, Y(t) = 3 Y(t-k) - 3 Y(t-Zk) +-Y(t-3k) + U(t) in the third difference model, Y(t) = Y(t-k) + Y(t-l2) - Y(t-k-lZ) +'U(t) in the hybrid difference model. 94 values Of k were chosen because the futures market did not gen- erate a complete time series for comparison at lead times greater than eight months.1 The difference equation models were applied tO all Of the beef price variables in the various econometric models in the one step ahead analysis. The analysis of the effect Of lead times on forecasts involved only the beef price series which a) corresponded reasonably with the commodity traded in the futures contract, namely slaughter cattle, and b) were reported on a monthly basis, also corresponding to the time period for which the futures prices were reported. Futures Market Prices The futures market is an institution which enables cash market participants to reduce the risk Of their enterprises by sub- stituting a certain price in the futures market for an uncertain price in the cash market. As is indicated in Figure 2, producer hedgers would sell futures contracts if the futures price plus their personal risk premium exceeds the expected cash price. Consumer hedgers would buy futures contracts if the futures price minus the risk premium is less than the expected cash price. The buying and selling activities of long and short speculators, reapectively, create excess demands and supplies Of futures contracts which force the futures price back into the range of the expected cash price plus or minus the risk premium. 1 . . The minimum number Of cattle futures contracts outstanding during the sample period (1965-1970) was four (each maturing every other month) and the maximum number was nine. The smaller number was the relevant constraint. 95 Futures Price minus Expected Cash Price Short Speculators Sell at These and Larger Differences Risk Premium Of Consumers Consumer Hedgers Buy at These and Lower Differences Zero -b Producer Hedgers Sell at These and Larger Differences - Risk Premium of Producers Long Speculators Buy at These and Lower Differences Figure 2: Characterization of Decision Rules Within the Futures Market 96 As this figure indicates, the equilibrium difference between the expected cash and futures price is determined only to be within a range around zero specified by the risk premium allowed by con- sumer and producer hedgers.1 The actual equilibrium is established by the actions Of the hedgers, but the equilibrium is stabilized by the action Of the speculators in the manner described above. With the interpretation Of the futures price as an aggregate expectation Of the cash price (plus or minus the risk premium) which will prevail at a future date (not necessarily the maturity period), we have an easily communicable, simple and, presumably, authoritative source of information concerning prices in the future. These char- acteristics have already been alluded to, but should be mentioned again. The idea of using the futures market as a source Of infor- mation on prices is one which is easily communicable, in that most lay individuals can grasp the concept Of a contract for future delivery of a specific commodity and the price which is associated with the delivery.2 Moreover the price quotations for the contracts most directly affecting market participants in an industry, commodity group, or locality are generally quite available. The process of taking advantage of this information or using this as an explicit forecasting device is simple in that all that is required is to 1 The actual level at which equilibrium is established is dependent on the relative market power of the producer and consumer hedgers. In the absence of knowledge to the contrary, one is tempted to assume equal market power which would give rise to an equilibrium difference at zero, i.e. the futures market would be expected to equal the expected cash price. The actual terms of the contracts and the mechanism of actual participation in the particular futures market would probably be less well communicated. 97 locate the quotation for the commodity Of interest. The price quota- tion which is available is authoritative, in that it represents a kind Of average Of the best judgments1 Of the market participants regarding the price to prevail in or near the maturity month. That the futures market does not reflect unforeseen events, such as drought, crop failure (domestic and foreign), or institutional changes (e.g. the banning Of DES from feeder cattle) is not a fault unique tO it as a forecasting device, indeed these events are the reason that virtually all social and economic models are stochastically structured. The product price whose forecasts are being evaluated in this thesis is that Of beef cattle, and there is a live cattle futures contract which is traded at the Chicago Mercantile Exchange,2 for which data were available at Michigan State University.3 The three monthly slaughter steer prices which were analyzed in the models These may be pure judgments or may be based upon the framework of a formal structure (model, if you please) whereby the effects of present and potential circumstances Of the market are traced through tO their impact on the market price. Since being right is more profit- able than being wrong, the ability Of a forecaster tO survive in the market would be a minimal criterion for accuracy for an individual. Since the futures price represents the aggregate performance Of all such individuals it would be expected to be a "stern standard" by which other individual forecasting devices can be measured. 2 The contract calls for the delivery of 40,000 lbs. of choice live steers, weighing between 1050 and 1150 lbs. and yielding 61 per- cent carcasses, or between 1151 and 1250 and yielding a dressing per- centage Of 62 percent. Par delivery for the contract during the period under study (1965 to 1970) took place in Chicago, with delivery allowed at Omaha, Neb., at a discount of $0.50 per cwt., and at Kansas City, MO., at a discount of $1.00 per hundredweight. The basic data were compiled on a weekly average basis by Keith Holaday Lacy in conjunction with his Master's thesis research (47). This author compiled these into monthly averages. 98 considered in this thesis are the price of choice 900 - 1100 lb. steers at both Chicago and at Omaha (both delivery points under the contract) and weighted average price of all grades of steers sold out of first hands in Chicago, Omaha, and Sioux City, deflated by the index of wholesale prices with base years 1957-59. The weighted average price posed several interesting problems in its analysis. The first question is how it relates to the other two monthly price series. The second is what is the effect of the price deflation on the predictability of the series. Finally, since the series is deflated, there is the question of what the appropriate means of deflating the futures price is to make its forecasts in units com- parable to those of the realized series. (In response to the last question, the procedure employed was to deflate the futures price by the price level in the month the forecast is being made, effec- tively assuming that the effects of anticipated subsequent price level changes are negligible.) As was indicated above, the futures market was evaluated by all of the chosen measures against all of the competing (maintained) hypotheses for the one-step-ahead (one month lead) forecasts, and on the basis of certain selected characteristics of forecast performance with the price difference models for lead periods of one, two, three, six and eight months. This chapter has discussed both generally and in some detail the four characteristic maintained hypotheses to be used to generate forecasts of beef prices. These maintained hypotheses were that the price movements for beef and to a lesser extent, other meat could best be predicted by 1) an econometric model, 2) an approximation 99 to the time trend, 3) the current cash market, adjusted to reflect various assumptions concerning the form of the difference model, and 4) the current price of the futures market contract. In the next chapter, these hypotheses will be compared with each other by examining the forecasts generated by each of the models over the 1965 - 1970 time period. CHAPTER IV EMPIRICAL EVALUATION OF THE FORECASTING MODELS The empirical results which are presented in this chapter consist of a discussion of the stability characteristics of the econometric models used in the subsequent analysis, an evaluation of the one-step-ahead forecasts generated by the competing maintained hypotheses for the price variables in each of the econometric models, and finally, an evaluation of the forecasts for differing lead periods of the monthly slaughter steer prices obtained using the futures market price and the case price difference models. Stability of the Econometric Models This study of the stability of the models under consideration focuses on the three models which are dynamic, in the sense that lagged endogenous variables are included in their structure. These models were developed by Hayenga and Hacklander, Crom, and Unger.1 Before the particular empirical results are presented, I will briefly describe the process of determining the dynamic stability (or in- stability) of a system of difference equations describing, as in this In his thesis, Myers reached the conclusion that his model was statically stable in the Walrasian sense, since the supply equations were more inelastic than the demand equations (both the cattle and hog supply equations had negative price elasticities. The trierweiler- Hassler model assumed an infinitely inelastic supply of beef and pork on a per capita basis, and hence is stable in the sense of Walras, as well as in the sense of Marshall. 100 101 case, a market. (One notes that any econometric model which contains lagged endogenous variables is, in purely mathematical terms, a dif- ference equation system.) Any purely linear system of difference equations of any order can be represented as a larger dimension linear system of first order, so the system to be discussed is Y(t) = A Y(t-l) +'B X(t), where Y(t) is an n dimension vector of endogenous variables observed at time t, X(t) is an m dimension vector of exogenous and stochastic variables influencing the system at time t, and A and B are matrices of coefficients with dimensions n X n and n X m, respectively. The general solution to this system of difference equations depends on the initial values of the endogenous variables and on the time path of the exogenous variables in the following manner: Y(t) = At'Y(0) + tgl AkB X(t-k). This solution will be stable, in the sense that Edgnded values of the exogenous variables result in bounded values of the endogenous variables, if (and only if) the eigenvalues (characteristic roots)1 of the A matrix are all less than one in absolute value. The rate of convergence of di- vergence of the solution depends on the relative magnitude of the largest eigenvalue. To determine if the underlying market is stable or unstable one would need to know the pOpulation values of the The eigenvalues of a square matrix A are those values of r such that the determinantal equation det(A - r I) = O is satisfied. There will be the same number of eigenvalues as the dimension of the matrix. The eigenvectors of a matrix A are those vectors X, each corresponding to one of the eigenvalues, such that A x = r x. The eigenvectors corresponding to distinct roots are orthogonal, and the vectors can be normalized to length one. If X is the matrix whose columns are the normalized eigenvectors of A and S is the diagonal matrix whose diagonal elements are the eigenvalues of A, then the k following relations hold: A X = X C and A = X CkX'. 102 parameters of the system, and inferences based on the estimated parameters would be subject to sampling variation because of the stochastic nature of the estimated parameters (59, 75). However, if the market under study is apparently stable and the estimated structure is unstable,1 then this would be an important bit of information in the evaluation of the model. To determine the eigenvalues of the dynamic models, the determinant of the (A - r I) matrix was directly evaluated for various values of r, because no computer program was available for the calculation of the eigenvalues of a general nonsymmetric matrix. No attempt was made to estimate asymptotic standard errors for these eigenvalues, although an algorithm has been developed (59) which could be applied if the estimated covariance matrix for the structural coefficients were known. For most cases, the conclusions were clear enough without the necessity of formal statistical inference (which would hold only asymptotically in this case, anyway). The Hayenga-Hacklander MOdel, as has already been mentioned, is of the form A Y(t) = B Y(t-l) +-C X(t) +-U(t). The A matrix is of full rank (five) and the B matrix is of rank three (but dimension five). As a consequence there is an eigenvalue of order two at zero, and three other eigenvalues. By the process of numerical approximation and interpolation, these other roots are 0.5240, 0.6719 and 2.6034. By virtue of the root at 2.6034, the system is dynamically unstable without any expectation of oscillation in the path to 1 Allowing, of course, for the sampling variation to which that inference is subject. 103 divergence.1 While this root is subject to a certain amount of sampling variability, the coefficient of variation (the ratio of standard deviation to mean) which would be necessary for there to be any significant probability of stability of the structure is much larger than this writer believes prevails for that variable. The Crom simulation model involved twenty-five quarterly endogenous variables lagged as many as five periods in the analysis, and five annual variables which were lagged as many as three periods. The quarterly relations contained two nonlinear identities, of the form log Y1 = log Y2 + log Y3, which forced me2 not to consider the stability of the overall model, much as I would have wanted to. The annual relations were entirely linear in all the variables and this subset of the model was subjected to the eigenvalue test of stability, based on the supposition that the overall model cannot be any more stable than any subsystem within the model. The annual inventory relations involved five endogenous vari- ables, with the rank of the augmented matrix of coefficients of the lagged endogenous variables being three. This implied that these equations had an eigenvalue of order two at zero. The operating rules Crom imposed upon the system subsequent to estimation give rise to three different sets of coefficients for this subsystem. The When previously calculated values of the endogenous variables were used as the lagged endogenous variables, rather than the values which actually were observed in the previous period, the path to di- vergence looked just as this theory projected. As a matter of fact, this divergence motivated the investigation of the eigenvalues of the system as the probable explanation of the phenomenon. 2 I am not aware of any general theorems which form the basis for determining the stability of nonlinear systems of difference equations. 104 eigenvalues determined for this subsystem are as follows: For those years prior to June of 1961 when the annual average price of feeder steers was less than or equal to $27.00, the nonzero eigenvalues were estimated to be -l.5053, 1.2472, and -O.4579. For those years prior to June 1961 when the average price of feeder steers exceeded $27.00, the only real-valued characteristic root other than zero which was determined in a direct search between 100 and minus 100 was 1.2733. For all values of the feeder price in the years sub- sequent to June 1961, the eigenvalues were calculated to be -l.5170, 1.2305, and -O.457l. The large negative eigenvalue estimated in cases one and three above implies that the model if left to operate in an un- modified recursive fashion would tend to generate endogenous vari- ables which oscillate with increasing amplitude as the subsystem diverged. In the second case there would be an unwavering trend towards infinity. Figure Two of Crom's bulletin illustrates the oscillating divergence of two of the annual inventory variables of this sub- system, in the original run before he imposed any restrictions on the operation of the estimated structure as a simulator of the market. There is also an indication that the quarterly variable representing the number of sows farrowing may have been tending on a path towards infinity with some oscillation imposed over the trend. Both of these observations are consistent with an unstable model. What this would lead us to conclude is that the 147 Operating rules which Crom developed for this particular model were designed to, in effect, stabilize a model with an inherently unstable estimated structure (the instability of which Crom was apparently unaware). 105 The third dynamic model which was analyzed in this thesis was constructed by Unger, and estimated by six different estimation methods. The model contained ten endogenous variables with the matrix of coefficients of lagged endogenous variables being of rank three. For that reason each of the estimated structures has an eigenvalue of order seven at zero. The three nonzero estimated eigenvalues corresponding to each estimate of the system structure are: For the ordinary least squares estimates, 0.95175, 0.425 and 0.000135; for the two stage least squares estimates, 0.96655, 0.505, and -0.0379; for Nagar's unbiased K-class estimates, 0.97469, 0.514, and -0.100; for the three stage least squares estimates, 0.98753, 0.550, and -0.400; for the iterated three stage least squares estimates, 0.98840, 0.550, -0.507; and for the limited information single equation estimates, the roots were calculated to be 1.16635, 0.089, and -36.276. The interpretation of these estimated eigenvalues is that all except the limited information estimates imply stability in the system, although there is some probability (though likely less than 1/2) that an unstable structure could have generated estimates which are characterized by eigenvalues such as those calculated. The limited information estimates of the structure yielded eigenvalues which imply that, if left to Operate by itself, this estimate of the model would generate endogenous variables which would oscillate to infinity.1 It was apparent that the LISE estimate differed considerably from the other methods of estimation, even prior to the stability analysis. The estimated coefficients of the other methods tended to cluster around common values with the LISE estimates sometimes not 106 By way of a summary of the results determined in the stability analysis of the dynamic models considered in this tehsis, it was determined that both the model developed by Crom and the one de- ve10ped by Hayenga and Hacklander are dynamically unstable, and the one deve10ped by Unger is nearly unstable notwithstanding fair degrees of fit during their sample periods. This would seem to indicate that much more work is necessary to describe accurately the dynamics of the beef cattle market. The concept of stability which was applied to the dynamic econometric models in this section is that which is used in the stability theory of linear systems of difference equations having constant coefficients. All of the inferences drawn from this analysis are obviously contingent upon the premises of this theory being fulfilled. In particular, there are two situations in which incorrect inferences of overall market instability may be drawn: The market forces prevailing may actually be nonlinear, or the model as analyzed may be a part of a larger linear system in which some of the exogenous variables in the current model may become endogenous to the larger model in such a way that a stabilizing feedback loop may be formed in the process.1 One notes that in either of these circumstances, the model as originally presented would not represent the "true" state of reality. even agreeing in sign with the others. The "k" values (of the k-class interpretation of LISE) seemed to be consistently larger than their asymptotic value at unity. (It may be noted that this divergence of LISE estimates from other types of simultaneous equations estimates is not an uncommon experience.) 1 . . It is also poss1ble that the larger model may become more un- stable, even to the extent of converting a stable subsystem into a component of an unstable overall system (53). 107 To see if the performance of the models which have been shown to be unstable is relatively any different from that of the other models, the next sub-section will describe the predictive perfor- mance of each of the econometric models compared to the performance of the trend and difference models, and where applicable, to the futures market. One-Step-Ahead Performance of Competing Forecastinngodels The alternative maintained hypotheses upon which the price forecasts to be evaluated are based on a) an econometric model, b) four alternative forms of a price difference model,1 c) the polynomial trend model and d) two forms of the trigonometric trend model, one as estimated, and the other formed by correcting the original estimate by adding the previous period's error. The Sample Period Each of the performance measures used in the evaluation pro- cess can be construed as a random variable which has a sampling dis- tribution characterized by the maintained hypothesis to which it is applied. As such, it would be conceivable that an "Optimal sample size" could be developed corresponding to each of the measures and The price difference models were not applied to any of the non- beef prices, e.g. hogs, pork, chicken, etc. One reason these forecasts were corrected in this manner is the extent of the serial correlation apparent in the residuals. Rather than solving explicitly for the estimate of each serial correlation coefficients separately, this correction procedure is equivalent to assuming it to be near one. GMost of the Durbin- Watson statistics were less than 1.0 which is equivalent to serial correlation coefficients greater than 0.50). This correction pro- cedure was also applied to the sets of polynomial estimates which did not immediately diverge. 108 each of the maintained hypotheses under study. The determination of an "optimal sample size" for all measures simultaneously for all of the maintained hypotheses may also be conceivable once the set of maintained hypotheses is characterized. However, these considerations did not enter formally into the decision regarding the sample size or sample period used for this analysis. The main concerns were data availability and data comparability. The actual test period chosen was generally the period January 1965 to December 1970, with exceptions noted in the appendix. This period is such that it contained some observations from both the sample period and from outside the sample. It was constrained at the beginning by the nonexistence of the Live Cattle Futures Market before November 1964 and at the end by the nonexistence of the 1971 edition of the Livestock and Meat Statistics, the data from which formed the backbone of this analysis. This six year period is slightly more than half the length of the generally accepted cattle cycle and presented the Opportunity of evaluating the forecasts in periods of both rising and falling prices. The whole question of data comparability is dealt with at considerable length in the appendix. Divergence of the Polynomial Trend Models Of the maintained hypotheses the polynomial trend model was the easiest to evaluate, namely because for all except two cases there was considrable divergence from the realized values within the period of evaluation. The actual definition of divergence which I used in making this judgment is "a model is judged to have diverged 109 if there is generated an error of at least $10.00 per hundredweight (106 per lb.) which is sustained until the end of the evaluation period". For the Hayenga4Hacklander model, divergence of the cattle price polynomial model occurred two months after the sample period and there was divergence of the hog price series six months prior to the end of the sample (perhaps due to the truncation of the co- efficients). The trigonometric trend approximation corresponding to the sample used by Hayenga and Hacklander for the steer price diverged one month after the sample period, as was expected from the coefficients of that equation. The polynomial approximations to the trend in each of the price variables in the Myers model, except the retail price of chicken, diverged from the actual series two months after the close of the sample period. The chicken price in this model diverged after three months. The Trierweiler-Hassler model was one of two cases where the polynomial trend model was reasonable successful (in the sense of begin less unsuccessful). The polynomial trend approximation to the momthly butcher hog price did not meet the definition of divergence for the entire test period (thirty-six months post sample), although the error at the end of the test period was $6.00 per hundredweight. The slaughter steer price approximation did not diverge until thirty- one months after the sample, and the approximation to the steer carcass price series diverged twenty months after the sample period. The approximations to the quarterly price series in this model did fit more in line with the pattern of the other results, with the 110 retail beef price diverging after four quarters and the retail pork price approximation diverging one quarter after the sample period. One hypothesis regarding the longer time required by the monthly approximations to diverge is that the approximating functions are of low degree (they are only cubic equations) and good sample period fits required coefficients such that the time derivatives were close to zero in absolute value. The trend approximations corresponding to the price variables in the Gram simulation model, with exception of the price of utility cow (nonfed) beef, were all divergent on the first quarterly observa- tion outside the sample period. The divergence of the nonfed beef price occurred three quarters after the close of the sample period. In the Unger model, the price of beef at retail diverged one year's observation after the sample period and the farm price diverged seven years (i.e. on the last year's Observation of the test period) after the sample. The price series for other meat at retail diverged three years post-sample and the farm level price of other meat diverged six years after the close of the sample. Because of the length of time required for the divergence of the farm level prices in this model, I consider this to be the other case where the polynomial trend approximation was less unsuccessful. Tabulation of Performance Statistics The general set of performance measures for the polynomial trend models are presented along with those for the other maintained hypotheses only in the cases of monthly variables in the Trierweiler- Hassler Model and for the Unger model. For the other models the 111 performance criteria corresponding to this hypothesis were not calculated, and hence are not presented. The difference between the forecasts generated by models indicated by (l) and those denoted by a (2) is that the former con- stitute the forecasts as actually generated by the estimated equation and the latter forecasts were generated by adding the previous period's error of the basic model to the forecast generated by that particular model in this period. If Y(T) is the actual dependent variable at time T, and the first forecast is Y(1,T) at time T, then the second forecast is generated as Y(2,T) = Y(1,T) +-Y(T-1) - Y(1,T-1). The interpretation of the measures of forecast performance should be fairly straightforward, with all of the measures discussed in Chapter Two above. Because of a possible difference between my usage of the terms and other uses of them, I will repeat the defini- tions of the Bias measure and the Turning Points measure. The bias is the average difference, accounting for the sign of the difference between the actual price and the forecasted price, so that a model which consistently underestimates the actual price will have a positive bias and one which overestimates the realized price will have a negative bias. The measure of turning point errors might more descriptively be labelled number of incorrect predictions of the direction of change. Since a turning point is said to occur at time t for a series Y when the product (Yt+l - Yt)(Yt - Yt-l) is negative, the number of turning points in the test period as a whole is equal to the number of "Turning Point Errors" reported for the First Difference model. The larger of the two numbers presented 112 in the entries along this line is the number of forecasts which went into the analysis (and not the number of changes - which is one less than this number). Probability statements can be made regarding the likelihood that this many or fewer incorrect predictions of change would be generated by a purely random process, such as coin tossing. The values of the cost derived average loss were calculated based on five alternative assumptions regarding the price of elasticity of supply. The cost derived average loss which is estimated when the supply elasticity is one is precisely the mean square error of the forecasts. Variables in the Hayenga-Hacklander Model. -- The statistics describing the performance of the alternative maintained hypotheses in forecasting the price variables in the Hayenga and Hacklander model are found in Tables 27 and 28. The beef price forecasts are described in Table 27 and the hog price forecasts in Table 28. Note that the econometric model performs least well of the means compared, and that the best devices appear to be the first and second difference models and the futures market price. The futures market and the second difference model appear to generate nearly the same perfor- mance criteria. The corrected trigonometric trend model is better than a no-change model in the case of the hog price (U2 equal to 0.8266), but slightly worse in the case of the beef price forecasts. 1.1.3 0500.0 0~00.0 a~00.0 nuoo.n 0n00.~ 00.~ 0000.0 «5 00 us 00am.“ onus.0 N~n0.~ ~0N0.0 0-0.0 nn00.0 0000.0 an» acne! vaouh can 00N«.0 ~a~0.0 0~n¢.0 -00.~u nn00.o~ a~.o “was.” «n we on anon." 0045.0 0003.0 00-.0 «n0~.0 0~00.0 0000.0 adv dove: vodka can uneconomuuh nuoaocouuua 0n-.0 ~u00.00 u¢~0.0 oo~0.00 nnn0.0 000N.n 0000." 0000.0 «unn.0 0000.0 00N0.0 0000.0 na~0.0 0050.0 «N00.0 0000.0 0000.0 0~n~.0 -n0.~ agon.0 coon.nu Gama." ns00.n 0N5¢.00n 5000.0 onoa.n 0000.0 0nn~.a 0000.0 n0~c.00 Na.~ ea.n . 0~.n on.“ -.N na.uu 0000.0 n~00.0 ~000.0 0000.0 nuan.0 00-.n Os we «a 00 «0 ON 00 no «N am we nu an me an Nu uo 5N «~00.~ ooum.a sumo.“ «aso.~ n~00.~ 50~0.¢ 0000.0 s0n~.0 0050.0 0000.0 n~00.a ~s~0.0 ~0-.0 n~0¢.~ cane.“ a~0~.0 000n.0 o~00.u~ ~0~0.0 non0.0 0000.0 -~0.0 na~0.0 0000.0 anao.o cano.o aano.o neao.o xuao.o noon.o 0~00.0 ~n00.0 5000.0 a~00.0 -00.0 0000.0 0000.0 «000.0 n~00.0 0000.0 «000.0 a-0.0 done: . ~Ovoz “can: move: dove: yuan-x ouaoueuuan ouaeuouuun ouaeuouuuo ounOuOuuaa uevnaaxoqu census» vane»: vaunk vacuum uuuum season-u anon on «can .ouauaau a. .uoosm .Aa coda-ooa .uaoau no .u.-0.uo~ nu Mum—<9 Auouuu ounuesdv e400 .0000 souuauouuou uncle: nause- uaulax Ounuoen< chunk Auneou0hv noun» sauna—em encased nouua 09:06:04 ensuesi Chou...” ufluom ”can“ «a a: nousouex huaueaveon a.a.m.zV 00.— I I an.0 I a «n.0 I a -.0 I a 00.0 I I ..ou essence unoo coaQBuOuueh no chance: 114 Table 28 Forecasts of U.S. NO. 2-3 Grade, 200 - 220 1b. Barrows and Gilts at Chicago, 1965 to 1970 Hayenga- Trigonomct- Trigonomet- Measure of Performance Hacklander ric Trend ric Trend Model 1:0ch (1) Model (2) Cost Derived Loss n a 0.04 0.0751 0.0026 0.0015 n a 0.12 0.3042 0.0108 0.0063 n a 0.32 1.7079 0.0633 0.0371 n = 0.34 1.9541 0.0727 0.0426 n a 1.00 (M.S.B.) 65.4714 2.5220 1.4752 Inequality Measures 01 0.8492 0.4958 0.5199 02 5.5396 1.0872 0.8266 Turning Point Errors 33 of 72 19 of 72 20 of 71 Average Absolute Error 6.4805 1.3078 0.9283 Average Relative Error 29.11 5.61 4.14 Third Absolute Moment 859.1794 6.1359 3.0132 Fourth Moment 13348.6795 17.6494 7.2176 Correlation 0.5588 0.8991 0.9317 Slope 0.2011 1.0416 0.9325 Bias (Average Error) -1.8292 0.6232 -0.0209 115 Variables in thegMyers Model. -- The performance of the alternative hypotheses in describing the prices studied in Myers' model are presented in Tables 29 throuth 33. The retail and farm level beef prices are presented in Tables 29 and 30, respectively. The forecasts of the retail price of chicken are described in Table 31, and those of the price of pork at retail and hogs at farm level are described in Tables 32 and 33, respectively. His stage I model ("reduced form equations") performed uniformly better than the in- verted structural form model. The no-change, or first difference model appeared to be the model to beat, for these variables, and actually was by the corrected trigonometric trend model in the pork and hog price comparisons. The corrected trigonometric approximation model seemed to be the general second choice after the no-change models. After being deflated by the wholesale price index in the month in which the forecast is being made, the futures market fore- casted the deflated slaughter steer price to approximately the same degree of accuracy as the current deflated slaughter price of the slaughter price corrected by the change from the preceding month. Variables;ig_£he Trierweileréflassler Model. -- The descrip- tion of the forecasting performance of the hypotheses as regards the dependent variables in the Trierweiler and Hassler model is given in Tables 34 through 38. The monthly carcass price forecasts are in Table 34, the monthly slaughter price forecasts are in the next table and the monthly slaughter hog price forecasts are in Table 36. The forecasts of the quarterly retail beef and pork prices are des- cribed in Tables 37 and 38. In the description of the slaughter steer price forecasts, the difference between the equations described 1165 mmmm.a med . muao. owoo. 3000. Ame Hence 88.... 30 IuOEocowuha :mao.=l ommo.| H0m0.I 0000.0020 00:5.Nmm ms.» sass.m .Na no o: mmsm.m «000. mazm.m= :mS. 0Hmm. H000. Amflo. AHV amooz cache oak luoeocomuna 0n.~ 000~.~ on 00 an n0~n.u 00mm. sham.“ vane. sung. 0m00. 0000. Bren: ~0uo.- acne. 00m0. 0-0.-~ n~00.0N -.~ anun.~ 00 00 an 0Nno.~ memo. mama.s 5000. 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The futures market was very similar to the cash market for predicting the slaughter steer price, except for the bias expression which reflected 41.7 cents of the 50 cent per hundred- weight delivery discount at the Omaha market. The corrected trigono- metric model performed consistently well for all of the prices in this subset, reflecting at least as much the influence of the no- change (first difference) model as it does that of the trigonometric model. Variables in the Cerel. -- The performance of the fore- casting models for Crom's price series is summarized in Tables 39 through 44. The carcass price forecasts corresponding to fed and nonfed beef are in Tables 39 and 40. Forecasts of the slaughter and feeder steer prices are presented in Tables 41 and 42, and the whole- sale and slaughter prices of hogs are presented in Tables 43 and 44. In terms of the general performance Crom's simulation model was a more accurate forecaster of prices than the uncorrected trend 1In the sample period the relation involvigg time had an R2 of .951 while the one with the wage rate had an R of .760. This evidence shows that sample fit need not correlate with forecasting ability (or effectiveness). 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An observation which is unique in the analysis is that the Hybrid Difference model proved to be the best predictive hypothesis in the case of the price series for utility grade beef, while for all the other prices series considered, this model did quite poorly.1 Other than this apparent "fluke" in the data, the first difference model appeared to be the most effective here, as elsewhere, of the maintained hypotheses considered, for purposes of generating forecasts. Variables in the Unger Model, -- The performance of the alternative hypotheses generating forecasts of the price variables created in the Unger thesis is tabulated in Tables 45 through 48, with the retail and farm level beef prices discussed in the first two and the retail and farm level prices of other meat in the last two of this set of tables. This set is interesting, because it allows the comparison of forecasts which differ solely on the basis of means of estimation, having the same structural model, the same data series for both the endogenous and exogenous variables and identical test and sample periods. In this regard, the estimation method whose coefficients generated forecasts with the smallest degree of error, measured both by bias and mean squared error were the two systems methods (three stage least squares and iterated This may have resulted possibly from some sort of "standard Operating procedure" guiding the purchase prices paid by major pro- cessors of nonfed beef (e.g. for hamburger, T.V. dinners, and canned meat products) or it may just have been an abberation of the particular data series and sample period. 134 33.? 9.3.3. 22.2. 33...... can... 82... 435.3: $8.9: Sonia: 9.3.9.: 2.8.3.. 2.1.4.? 19586.»: 83. 53... 2.3. :3. 23. RS. 8R... 83. as. «who. 98. 38. .88. . .81.. 33.- 2.8. 3:. 33. 28. 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However, ranking the methods on the basis of the correlation between the forecast and the realized price for each of the four variables did not yield a consensus, because the ranking on this measure for the beef price variables is nearly perfectly correlated in a negative sense with the ranking for the other meat price variables. In comparison with the trend models, the best estimates gen- erated by the econometric model appear better than the polynomial trend model and worse than the trigonometric trend model. Correcting the trend models to reflect the error of the previous year, had the effect of reducing the mean square error at the expense of a lower correlation of forecast with realization. Among the forecasting methods applied to the variables used in Unger's thesis, there did not appear to be any method which proved to be significantly better than an extrapolation calling for no change in the dependent variable. One will recall that this observation has characterized the forecasts generated by all the different sets of models, and as such, constitutes a real challenge to price analysis and other researchers concerned with explaining this subsector of the agricultural economy. Effect of Lead Period on Forecasts Based on Futures Market Price and Price Difference Models we have already seen the performance of the different means of generating price forecasts for one month into the future. In considering forecasts with different lead periods between the fore- cast and the realization the comparisons which will be presented 139 involve only monthly data, and are generated by the futures market price and the cash market price difference models. Although there are four separate price difference models which were considered, the comparison is effectively between the first difference cash price model and the futures market forecast, because of a uniform domination of the performance of the other difference models by the first difference model (i.e. which states that the current price is expected to prevail in the forecast period). The different lead periods which are considered in this analysis are one, two, three, six and eight months from the forecast date. Tables 49, 50 and 51 present this analysis for the price of choice 900-1100 lb. slaughter steers in Chicago, and Omaha, and the weighted average price of all grades of steers sold in Chicago, Omaha and Sioux City, deflated by the wholesale price index with 1957-59 as base respectively. Some of the observations Which seem apparent from these tables are: First, forecasts get worse the longer the period be- tween the forecast and the realization. Second, for all lead periods the nondeflated cash prices are more easily forecast (as indicated by the correlation) than are the deflated prices, but are subject to more variation (as indicated by the mean square error). Third, for short term forecasts (i.e. for less than three months into the future) the futures price and the current cash price seem to be virtually equivalent means of forecasting the nondeflated prices. For forecasts more distant than this, it appears that the current cash price is somewhat a better device than the futures price. Note that the futures market price is virtually uncorrelated with 140 Table 49 Price Forecasts For Choice 900 - 1100 lb. Steers at Chicago 1965 - 1970 Performance of First Second Third Hybrid Futures FOYBCBSCS With Lead Diffzrenc: Diff::ence D ffercnce Difference .Earket Period EQual to K Nodel Fodel model Hodcl Model X I 1 fican Scurre eror 0.5960 0.7019 1.6358 1.4923 0.72E7 3133(Lt'rt;3 1 -er 0.0535 -0.0169 0.0147 -0.0672 0.1730 Averéi; LE:01--: 2-:or 0.5971 0.6063 0.9007 0.8925 0.6830 Corrci;:i:n 0.9461 0.9444 0.8322 0.8734 0.9293 Turning Point Errors 21 of 72 15 of 71 22 of 70 20 of 60 17 of 70 K I 2 Kean Square Error 1.6914 3.1199 8.3553 4.0728 1.9176 Bias (LVCYCQQ Error) 0.1158 -0.0712 '0.0796 I0.1188 0.3379 Avertfe Absolute Error 1.0265 1.3480 2.2255 1.1(42 1.1211 Correlation 0.2469 0.7895 0.3436 0.7003 0.8028 Turning Point Errors 31 of 71 32 of 69 29 of 67 30 of 59 24 of 69 K I 3 u666 Square Error 2.7341 6.3732 19.4623 6.1004 3.2648 Bias (Average 2:262) 0.2020 -0.1663 -o.osoo -0.1678 0.5025 Averag: ibsolute Error 1.2500 1.8925 3.3503 1.5650 1.4341 Corrclitioa 0.7513 0.6177 0.4583 0.5923 3.6624 Turning Point Errors 40 of 70 36 of 67 33 of 64 32 of 58 30 of 69 K I 6 Mean Square Error 4.3350- 11.4760 38.7134 8.0754 6.8758 Bias (Average Error) 0.4501 -O.2969 ~0.1329 -0.3020 0.7219 Average Absolute Error 1.6327 2.4943 4.4060 2.2504 2.1638 Correlation 0.6065 0.4279 0.2461 0.5413 0.2601 Turning Point Errors 34 of 67 25 of 61 26 of 55 28 of 55 32 of 67 K I 8 {can Square Error 4.7284 11.7937 37.6170 9.0802 8.0653 Bins (Average Error) 0.5868 -0 2474 0.6982 -0.3881 0.7524 Average Absolute Error 1.7618 2.7288 4.8292 2.3836 2.3053 Correlation 0.5800 0.4240 0.3054 0.5321 0.0959 Turning Point Error; 34 of 65 33 of 57 27 of 49 27 of 53 35 of 65 Pric: Performance of Forecasts with Lead Period Equal to K 00 I3 1 K:f1 8' ‘ £:TO' . <'. ‘‘‘‘‘ -‘Cr) 0;.. T2- 13:: K I 2 3.xn C"'3rt .;ror 2 .5 (. ~ erer) .4: -2. c LL;.1;:2 Crror CC: - L.1:-;5.: .ll TurnLrg ?.Lnu Trrorc K - 3 {2:1 Stu..- Prror bias (: 3 :rror) Irrr: .e ercr C..-1.‘-1...;....: Tun.£:.31w X I 6 nzcn Square Error Dies (are are Error) Averace Abcnlute Error Ccrrclztica Turning Point Errors K I 8. Kean Squarc [rrar Din: (Aver:a: Lzrcr) Average 55::lurr Error Correlaticn Turning Point firrors forecasts ‘141 Table 50 for Choice 900 - 1100 lb. Steers at Omaha 1965 - 1970 First Second Third Hybrid Difference Difference Difference Difference £3531 £0031 chel Hodel 0.7473 0.8259 1.7432 1.6446 0.0322 -0.0173 0.0177 -0.0622 0.5231 0.675’ 0.9826 0.9492 0.9-;3 0.93'7 0.2770 0.8523 25 of 72 15 of 71 20 of 70 25 of 60 2.1915 4.1807 11.2001 4.5523 0.112 -0.0555 -0.0769 -O.1168 1.1644 1.6187 2.7115 1.6042 0.791“ 0.7320 0.5948 0.6533 41 of 71 33 of 69 37 of 67 34 of 59 3.5494 8.4450 25.0589 6.5781 0.1950 -0.2034 .0.1;64 -0.1836 1.4206 2.2437 3.9342 1.9053 0.633 0.5163 0.4136 0.5450 44 of 70 39 of 67 34 of 64 32 of 59 5.2995 14.9822 51.4689 7.9138 0.4596 -0. 333 -0.0713 -0.3236 1.8258 2.9664 5.4640 2.2029 0.4824 “0.2544 0.0557 0.5059 30 of 67 28 of 61 25 of 55 31 of 55 5.1487 12.5148 38.4921 8.3854 0.5843 -0.2298 0.6873 -0.4164 1.7705 2.6975 4.5722 2.2225 0.5073 0.3539 0.289 ' 0.4992 36 of 65 29 of 57 27 of 49 22 of 53 Futures Market Model 0.7558 -0.3652 0.6735 0.9326 25 of 70 1.6924 o0.3766 1.0624 0.8198 30 of 69 2.6993 -J.2120 1.3001 0.6852 36 of 69 5.9349 0.0065 2.0253 0.2735 32 of 67 7.1843 0.0227 2.1959 0.0911 33 of 65 1112 Table 51 Forecasts of weighted Average Deflated Price of All Grades of Slaughter Steers at Three Markets 1965 - 1970 First Second Third Hybrid Futures Difference Difference Difference Difference Market Model Model Model Model Mbdel x - 1 Mean Square Error 0.5555 0.6252 1.3218 1.2977 0.6188 Bias (Average Error) 0.0035 -0.0016 0.0026 -0.0612 -0.3833 Average Absolute Error 0.5838 0.5885 0.8389 0.8490 0.6293 Correlation 0.8647 0.8990 0.8162 0.7585 g 0.8730 Turning Point Errors 24 of 71 16 of 70 17 of 69 19 of 59 21 of 70 r - 2 Mean Square Error 1.6176 3.0691 8.2805 3.6523 1.3240 Bias (Average Error) 0.0194 -0.0735 -0.0768 -0.1350 -0.4443 Average Absolute Error 0.9944 1.3746 2.2729 1.4284 0.9401 Correlation 0.5991 0.6104 0.5272 0.4754 0.6515 Turntnz Point Errors 41 of 70 34 of 68 35 of 66 32 of 58 30 of 69 x - 3 Mean Square Error 2.5968 6.0913 18.0127 5.3915 2.0793 Bias (Average Error) 0.0509 -0.2081 -0.0040 -0.2068 -0.3492 Averase Absolute Error 1.1967 1.8898 3.2897 1.6780 1.1378 Corrrlation 0.3535 0.3552 0.2983 0.3347 0.3599 Turning Point Error: 44 of 69 38 of 66 32 of 63 33 of 57 34 of 69 x - 6 Mean Square Error 3.7007 10.4644 35.6286 6.8115 4.5601 Bias (Average Error) 0.0897 -0.2424 0.0200 -0.2443 -0.3105 Average Absolute Error 1.5128 2.5097 4.5836 2.0251 1.7805 Correlation 0.1000 0.0966 -0.0594 0.2650 -0.3961 Turning Point Errors 35 of 66 29 of 60 26 of 54 ' 31 of 54 38 of 67 r - 8 ‘Mean SQusre Error 3.6479 9.9929 31.8110 7.3094 5.3916 Bias (Average Error) 0.0960 -0.1881 0.7214 -0.3382 o0.4135 Average Absolute Error 1.4949 2.3865 4.2720 2.0775 1.9070 Correlation 0.1321 0.1423 0.2039 0.1971 -0.5802 Turnins Point Errors 29 of 64 27 of 56 21 of 48 25 of 52 34 of 65 143 the cash price eight months away and only slightly correlated with the price six months away. An analysis of the average errors of the first difference and the futures market models reveals that the contract prices average about twenty cents per hundredweight below the Chicago cash price and about four cents below the Omaha price adjusted for the delivery allowance (fifty cents per hundredweight) at nearly any point in time. The difference in forecasting accuracy between the cash and futures prices suggests that there may exist an opportunity for speculation on distant contracts to improve the forecasting performance of the market. It remains to be seen whether a strategy of Operating in the distant futures market as if the current cash price is the price at maturity would be profitable enough to attract sufficient speculators to potentially affect its predictive perfor- man ce 6 Summary As a short summary of the empirical findings of this chapter, it has been observed that the econometric models did not perform as well as some of the other forecasting devices which were con- sidered, indeed there were cases where the trigonometric trend models performed much better than the structural models. The most general observation on the forecasting performance of the alternative main- tained hypotheses is that the low cost (in the sense of private costs) methods, particularly the no-change (first difference) model and the futures market price, proved to be devices at least as good as the more sophisticated models for purposes of price forecasting. If 144 these models are at all representative of the set of all econometric models of beef prices this might be taken as an indicator of dis- equilibrium in the price research market, since one would expect that the marginal benefits attributable to a model should tend to approach the marginal cost of the model. With negative benefits and positive costs, the situation does not appear as if it should be permitted to remain in its current state. By viewing the rankings of the different devices based on the different measures of performance, the inconsistencies and similarities of the different measures with each other become apparent. For example, the rankings of the cost derived loss func- tion, mean square error and Theil's U2 inequality measure are identical, while the separate rankings based on the correlation, bias and mean squared error are not necessarily the same. In the chapter which follows a brief summary of the research will be presented together with the major conclusions of the study and recommendations for further research. CHAPTER V SUMMARY AND CONCLUSIONS This thesis has two separate orientations within its organiza- tion. Following an introductory chapter, the second chapter of the thesis presented a theoretical analysis of the forecasting process, as it affects the measures which are used to evaluate forecasting performance. The fourth chapter applies the measures which are de- ve10ped and discussed in the second chapter to evaluate the alternative forecasting devices for prices affecting the beef industry whidh were described in Chapter III. The main body of this summary and conclusions chapter will likewise be organized around this theoretical - empirical dichotomy. At the end of this discussion, I will recommend further work which can build upon this analysis and would improve upon the state of the empirical arts in beef price models. Theoretical The theoretical chapter begins by examining the process of forecasting in general, and price forecasting in particular, from the viewpoint of decision theory. It is observed that the forecaster is making a statement (taking an action) about the price to prevail (in the face of uncertain states of nature) and must face the con- sequences of his error (incur a loss when the action is not appro- priate to the state of nature). Since the forecasting device 145‘ 146 determines the action taken, and the realized prices are what they are, the process of forecast evaluation is one which must focus on the appropriate function to represent the losses incurred when the forecast is p and the realization is r. In this thesis it was assumed that the appropriate measure of loss from forecasting error is the difference between the maximum possible profits accruable to a firm under the realized price and the actual profits realized by the firm when it determines its output assuming the forecast would be realized. Assuming the firm which is using the forecast can be repre- sented by a homogeneous production function, that input prices are predetermined and that only the output price is being forecast, I was able to derive the function which measures the loss (in the sense referred to above) incurred by the firm as a result of incorrect forecasts. This loss function is given by the expression (r1+n - (1+n) r pn‘+ n p1+n) (n/K)n (l “inf-1-u where r and p are the realized and forecasted prices, n is the elasticity of the firm's short run supply curve (related to the production homogeneity (h) by n a h/(l-h)), and K is the constant of proportionality of the total variable cost function, c - K qllh, depending upon the input prices and the production parameters. It was shown that if the supply elasticity were equal to one, i.e. the production homogeneity were one-half, this loss function would be quadratic. Further, it was shown that the estimator, namely the posterior expected price, which minimized the expected squared error also minimized the cost derived loss function. By factoring the cost derived loss function, a particular type of 147 "squared error" term was one of the factors of the expression. All of these results led one to suSpect that there should be a high degree of correspondence between the rankings of the forecasts by the cost derived loss function and the mean squared error criterion. In the empirical chapter, an examination of the tabulated performance measures showed than there were no inconsistencies between the two rankings. The one area where the "robustness" of the least squares type of criteria is not as evident, without substantial assumptions, is in the determination of the relative gains from using one estimator over another when the costs of the two are unequal. In this case, one would have to know the supply elasticity and hence calculate the cost derived loss, to balance the additional costs of one estimator relative to another with the additional gain in performance provided by the first estimator over the second. Thus, there is the necessity of obtaining technological information about an industry, e.g. pro- duction homogeneities or supply elasticities, before one can recommend an apprOpriate forecasting device for prices in that in- dustry. With the relation of mean squared error to the cost derived loss function thus established, the analysis in the theoretical chapter sought to relate other common measures of forecasting error to the mean squared error criterion. It was noted that the mean squared error could be partitioned into expressions which involve the forecasting bias, the slope of the.re1ation of forecasts to realizations and the degree of fit of that relationship. This partition indicates that, certeris paribus, anything which reduces 148 the bias, makes the slope closer to plus one, or increases the correlation between the forecast and the realization will decrease the mean squared error. Since the ceteris paribus condition is not expected to hold in all cases it is expected that these measures of performance would rank the forecasting devices somewhat dif- ferently than the mean squared error criterion. In the empirical chapter, evidence substantiating this expectation is found. See also Table 53 of this chapter for a summary of that evidence. The inequality coefficients, U1 and U2, developed by Theil were motivated by the idea of a quadratic loss function and are functions of the mean squared error as was shown in the theoretical chapter. Because U1 depends on the forecasted changes as well as the forecasting errors and can therefore take values independent of the mean square error, Theil concluded that U2 was a preferable measure of forecasting performance. Besides, the U2 statistic provides both a relative and an absolute measure of performance, since the denominator is the mean squared error of the successive differences. Consequently, values of U2 greater than one indicate that the forecast under study was no better than the last period's price as a forecaster. A similar threshold for interpretation of U1 was derived in the theoretical chapter which indicates that values of U1 less than the square root of two divided by two are associated with fore- casts which are positively correlated with the realized prices. That this threshold is related to, but not identical with, the threshold of U2 at one is demonstrated in Table 52 which tabulates the joint frequency of occurrence of U1 and U as calculated in 2 149 Table 52 Joint Frequency of Occurrence of U1 and U2 112 < 1.0 1.0 s 02 < 1.3 1.3 < 02 Total 01 < 0.5 1 2' o 3 0.56 01 < 0.6 5 11 8 24 0.6‘ 01 < 0.7 4 5 9 18 0.75 01< 0.8 2 4 34 40 0.84 ”1‘ 0.9” 0 0 23 23 0.9< u 3 11 31 45 1 Total 15 32 105 153 150 the evaluations of the empirical chapter.1 Apparently, the dif- ference between the two threshold points is that the previous period's observation is generally quite highly correlated with the current observation, which would mean that a much smaller value of U1 would be necessary to be equiValent to the U2 threshold. Of the other measures of forecasting performance, the moment measures are most closely related to each other. These include the mean squared error, the average absolute error, the third absolute moment, and the fourth moment measures. Of the same form, but not directly comparable to the others, is the average relative error. The moment measures differ from each other in terms of the weight that they each give to outlying points. Consequently, one would believe that adjacent sample moment measures would rank the fore- casting hypothesis similarly. The evidence obtained in the empirical chapter supports this belief except possibly in the case of the com- parison of mean square error with the average absolute error. Table 53 shows the number of times that each of the in- dicated pairs of performance measures identically ranked each element of the set of forecasting devices. Also shown in this table is the number of times the same elements received the same ranks plus the number of times that there was only one error in the ranking, i.e. when one pair of ranks was inverted. Since the probability of obtaining sixteen or more successes in twenty-two Bernoulli trials with equally likely outcomes is approximately .0274, using the Central Limit Theorem, this was taken to be the cutting point for The chi square statistic corresponding to the null hypothesis of independence is approximately 41.6. The .9995 point for the chi square distribution with 10 degrees of freedom is 31.4. 151 Table 53 Number of Coincident Rankings of Forecasting Devices (of 22 Possible) U2 Inequality Measure 16 Average Absolute Error 12 13 Average Relative Error 11 8 19 Third Absolute Moment 18 13 11 10 Fourth Komcnt 13 10 8 8 l7 Correlation 5 7 6 6 5 5 Turning Point Errors 1 1 1 1 1 1 1 MSE 02 AAE ARE 3AM 4M p Number of Coincident Rankings Allowing the Inversion of One Pair of Ranks . 02 Inequality Measure 22 Average Absolute Error 19 18 Average Relative Error 18 14 22 Third Absolute Moment 20 19 17 17 Fourth Moment 19 16 14 15 21 Correlation ll 12 14 12 ll 9 Turning Point Errors 4 3 4 4 4 4 3 ass 02 AAE ARE 3AM 4M 9 152 assessing agreement among the rankings in the top portion of the table. With this criterion for agreement, it is apparent that ranking by mean squared error is substantially the same as ranking by the U inequality measure and by the third absolute moment 2 and that ranking by the third absolute moment is substantially the same as ranking by the fourth moment (although the rankings relation- ship is not transitive). Further, we observe that the average absolute error resulted in rankings which matched the rankings of the average relative error. Allowing for a single error, as was done in the lower portion of the table, does not affect the general conclusions of the analysis, although the critical region is changed. Note that the rankings based on correlation of the forecast with the realization are not significantly related in this sense with the rankings which resulted from the error'moments.1 Further note that there is no relation of the turning point rankings with the rankings using the other measures, which is a rather surprising result. With this analysis of the different performance criteria completed, three measures were used to assess the performance of the alternative models for forecasting beef prices: mean squared error, the U2 inequality measure, and the correlation of the forecast with the realization. ‘1 I believe that defining the significance of the relationship as I did is a stronger measure of relationship than some sort of aggregate rank correlation measure. In my sense, there has to be a perfect rank correlation of plus one before a success is determined to have occurred in the Bernoulli trial. I have not been able to prove or disprove my belief, however. 153 Empirical In the empirical analysis of the forecasting models three subsets of work were done: First, the stability characteristics of the econometric models were studied. Then the alternative models for the forecasts on a one-step-ahead basis were examined. Finally, the effectiveness of the low-cost forecasting models, i.e., the futures market and the price difference models, in correctly fore- casting the prices up to eight months ahead was studied. The results of these analyses follow. The stability analysis showed that the two static models, namely of Myers and Trierweiler and Hassler, were both stable in the sense of Walras. This means that as long as prices adjust to changes in excess demand or supply, the model is stable. Since the implicitly assumed supply function of the Trierweiler-Hassler is independent of price, this model is also stable in the sense of Marshall, that is, equilibrium can be attained by quantity adjust- ments to situations of excess supply and excess demand. Because the supply functions in the Myers model are negatively sloped with respect to price, this model is not stable in the sense of'Marshall. The stability theory for the dynamic models is based in the stability theory of linear difference equations, namely the model is stable if the absolute value (or modulus) of the largest characteristic root of the system is less than one. If this dominant root is negative and less than -1.0, the endogenous vari- ables in the system will oscillate with increasing amplitude, and if positive and greater than 1.0, the endogenous variables will follow a direct path toward either of the infinities. 154 The results of the stability analysis of the dynamic models showed that all the dynamic models were either unstable or exhibited tendencies towards instability. The monthly Hayenga-Hacklander model had its dominant root at 2.6, which meant that recursively operating this model would have resulted in direct divergence of the calculated endogenous variables. The Crom quarterly model included two nonlinear identities which prevented the analysis of the stability of the overall model. Operating with the assumption that the overall model cannot be any more stable than any subset of it, I studied the stability char- acteristics of the annual inventory subsystem. The dominant root of this system was found to be approximately -1.5, with another root at 1.25. These roots would cause the solutions to the system to oscillate with increasing amplitude over time. The annual model developed by Unger was determined to be stable, but the calculated eigenvalues were close to the critical value of one. In cases such as these, estimates of the variances of these roots would have been helpful in making probability state- ments regarding the likelihood of an unstable structure given these estimated eigenvalues (type II error). The estimate of this model obtained by the limited information single equation estimator was in fact unstable with roots at -36.3 and 1.17. The other five estimates of the model gave rise to estimated dominant roots of between 0.951 and 0.988. Because the limited information estimate of the model gave coefficients which were virtually unrelated to the estimates obtained by the other methods, less faith is placed in the "clear indication" of instability of the model based on this 155 estimate than in the marginal indications of stability of the model provided by the estimates of the model which were obtained by the other means. The concluding observations regarding the stability analysis which were performed in this thesis should include at least two points. First, it appears that the beef cattle subsector of the economy is either at the brink of instability or the models which have been studied in this thesis in some way misrepresent behavior in it, especially as regards the stability characteristics in the industry.1 Second, it would appear that at least in the case of Crom model, there was a degree of recognition that the model by itself was not representing the recorded history, without knowing that a major part of the problem was the instability of estimated structure of the model. The 147 operating rules which he sub- sequently imposed upon the model constitute, at least partly, an attempt to impose stability on a structure of estimated coefficients which is, itself, unstable. Perhaps a better allocation of resources would have been to determine the cause of the model instability, than to attempt to "coerce" stability out of an unwilling structure. In the assessment of the different maintained hypotheses as bases for price forecasting, the underlying judgment was that the returns from the model in terms of its performance should be roughly proportional to the costs2 (at least in the marginal sense) if there 1 The basis for this judgement is this writer's belief in the inherent stability of the beef industry. 2 The information (either in amount or in value) obtained from a forecasting device should be related to information put into that device (again, either in total quantity or in terms of cost). 156 is an approximation to equilibrium in the market for price forecasts. In this sense, econometric models were viewed as high cost, but potentially high valued forecasting devices. Trend models, both polynomial and trigonometric, were seen as lower cost substitutes for and approximations to the solutions of the econometric models. These were compared with models which in this continuum are lowest cost, namely price difference models and futures market prices. The expectation of model performance was roughly in that order, that the econometric models would perform better over the test period than the trend models which were expected to perform better than the simple models. It was expected that the futures market would be a better forecaster than the price difference models, and pre- sumably than the other devices as well, because the participants in the futures market could use these methods as the basis for their expectation of price in or near the maturity month for the contract. The results of the evaluations of the alternative forecasting models were surprising, in that not only were the expectations not fulfilled, there was a negative rank correlation between the expectations and the conclusions on the order of -0.3. In particular the results showed that the polynomial trends diverged to a degree that evaluation of the descriptive statistics involving these fore- casts was not even possible in all but two cases. In terms of the operational definition of divergence (a sustained error of 10 cents per pound maintained after occurrance) even one of these two cases diverged shortly after the close of the sample period. The conclusions regarding the comparison of the econometric models with the uncorrected trigonometric trend equations is that 157 the econometric models are not superior to this trend model. As shown in Table 54, the trigonometric equations produced better price forecasts than the econometric models in slightly more than half of the cases, but not enough to be significantly better than the models.1 Correcting the trigonometric trend model to reflect the forecasting error of the previous period resulted in improved forecasting performance in all cases but one (and this is the last comparison presented in the empirical chapter, Table 48). The comparison of the trigonometric trend models (corrected and uncorrected) with the first difference model was operationalized by examining the U2 inequality coefficient. Values of this statistic less than one indicate that the forecasting device being considered is better than the no-change extrapolation. For the uncorrected trend model, none of the U2 statistics were less than one, while the corrected trigonometric trend model had 12 of the 22 U statistics in that range. The insignificance of that difference 2 left us with the conclusion that the corrected trend equation yielded forecasting performance approximately equal to the no-change or first difference model. The performance of the difference models, other than the first difference model, was dominated: by the performance of the It might be instructive to examine at a later date the perfor- mance of these trend models vis a'vis all variables of the econometric models. For example, in the Cram model, the forecasts of the quantity variables were much better than the price forecasts. Whether this was because of the model's superior performance, or due to better behaved data series, could be determined by such a study. 2 The verb "dominate" is used in this context to describe the situation in which all the measures of performance show the "dominating' procedure to be superior to the other procedure, for example, having 158 Table 54 Comparison of Econometric Model Forecasts with Trigonometric Model Forecasts Model Number of Price Variables For Which the Econometric Model Uncorrected Trig Model Undecided Forecasts Better Forecasts Better According to According to According to . Mean Square Correlation Mean Square Correlation Mean Square Correlation Error Error. Error Hayenga Hacklander 0 1 2 1 0 0 Myers 1 O 3 2 1* 3* Trierweiler Hassler 4 2 1 3 0 0 Cram 4 3 2 3 O O Unger 0 2 4 2 0 0 Total 9 8 12 11 1 3 * The trig model was better than the structural model, but worse than the Stage I (reduced form) model. 159 first difference model on all the measures considered. The only exception to this is the performance of the hybrid difference model (first difference corrected by last year's forecast error) in fore- casting the quarterly proce of utility cow beef at New'York, from Crom's model. The explanation of this "deviance" suggested here, is the possibility that certain standard operating procedures on the part of meat processors may induce this "fluke" into the price series. On the basis of the meager sample of three series of pre- dictions, the conclusion regarding the futures market as a one month ahead forecaster of prices is that it is not significantly better than the current cash price in forecasting the nondeflated price of slaughter steers in the two terminals (Omaha and Chicago). The de- flated futures market price; likewise, did not perform significantly better than the current deflated three market average cash price. The relative performance of the low cost forecasting models did not change noticeably as the lead period between the forecast and realization increased. Specifically, the no-change model con- tinued to dominate the other types of difference formulations. The futures price did not perform significantly different than the cash price (first difference formulation) over any of the lead periods, although at intervals of two and three months the futures market price had a slightly smaller mean square error than the current cash price as forecaster. The deflated futures price was negatively and smaller values of the cost derived loss measures, smaller mean square error, smaller average relative error, larger correlation, etc. 160 significantly correlated with the three market deflated average price, but otherwise the pattern of performance for this variable paralleled that for the Omaha price. Of the two cash markets con- sidered, the futures market price seemed to more closely forecast the price in the Omaha market. Perhaps the most interesting of all the observations regard- ing the effect of length of lead period on the forecasts is that the futures market price in the early months of the contract, i.e. eight or more months prior to maturity, is virtually uncorrelated with the cash price in the month of maturity of the cantract, and much less so than the prevailing cash price that same month. If this is more than just a sampling fluctuation, there is an opportunity for some sort of cash-futures arbitrage activity that in the short run might be personally quite profitable and in the long run, should possibly improve the ability of the futures market to predict prices near the maturity months. Implications The major finding in this research is that the value of the sophisticated econometric models studied in this analysis is less than that of the simpler models for generating price forecasts on a one-time-period-ahead basis, in spite of the reduced amount of information required for these simpler models. In particular, the econometric models performed less well than either the cash market price or the futures price in forecasting the cash price in the next time period (month, quarter, year), while the futures price and the current cash price each forecasted the cash price in the next 161 month about equally well. In light of this result, certain actions on the part of the producers, distributers, and consumers in the market for price information seem appropriate. Price researchers should realize that, with their greater information content, econometric forecasts have the potential to provide much better information about future price levels than the current price levels, and certainly, than the econometric models studied in this thesis. To realize this potential, continued or increased efforts in the development of econometric models for the beef industry should be supported. While realizing that the current poor forecasting performance may have resulted from an orientation of the current models toward purposes other than forecasting, or simply reflected an underinveshment in beef price forecasting models, to improve the perfonmance of future models an explicit attention to the forecasting performance of models deve10ped should be a part of the evaluation of these developmental efforts. Those charged with keeping the forecasts current and dis- seminating the price information (e.g. extension outlook specialists) should maintain a record of the performance of the different methods they used to derive their forecasts. In this way they would be able to tell when a particular model, or the price and income elasticities and other information derived from it, are out of date and need to be revised or updated. Some of this same information can be used to extend the life of the models by adjusting them to correct for the serial correlation of disturbances which typically affects many 162 forecasting models.1 (When the trend models in this study were corrected in this manner, the degree of improvement was substantial.) The implications for farmers and other consumers of forecast information are threefold: First, they should maintain some degree of skepticism regarding the information they receive on prices from all sources (basically reflecting the fact that the corresponding forecasting errors have nonzero variances). Second, they should give explicit consideration to the cash and futures market prices for forecasting information in addition to other sources. Finally and most importantly, with the current state of the art as indicated by the models studied herein, they cannot afford to pay very much for private forecast information without substantially documented information regarding the past performance of the model and the service. (The successful commercial forecasting models are so as a result of the resources which are committed toward developing, revising, and updating the basic model and the expertise and judg- ment used in conjunction with the model's use.) Thus, the user is faced with the decision as to how to aggregate the individual fore- casts to obtain one which would perform better than any single fore- cast. One relatively simple scheme would weight the individual forecasts inversely by the respective standard deviations of the forecasting errors (or root mean squared errors). A minimum variance unbiased aggregate would use weights based upon all the Some of the forecasting devices currently being used consist of looking at the current price and then correcting that in accordance with the changes foreseen by some model, more or less reversing the process of correcting for serial correlation as indicated above. .163 elements of the covariance matrix of the forecasts, rather than just the diagonal elements of that matrix. For those who service the users of this price forecast in- formation (e.g. the managers of a "felplan"-type operation or con- sulting group), an appropriate format for the information which they provide might include a statement of each of the individual price forecasts, including the current prices in both the cash and futures markets, together with one or more aggregates of the fore- casts. It would also be instructive to include with the actual forecasts an indication of how the respective methods performed in the past. Recommendations for Further Research This set of recommendations will also be subdivided along the theoretical-empirical dichotomy which has been used in the organization of this thesis. There are a host of minor points and suggestions sprinkled throughout the body of the thesis, but the recommendations suggested in this section achieved this position be- cause of the fairly direct relation that they have to the intellectual development outlined in this thesis. Teigen, Lloyd D., "How should projections by various sources be consolidated into a single value for purposes of computerized consulting?" unpublished manuscript, July 24, 1970. The major drawback to an aggregation process of this form is that the elements of the covariance matrix are almost never known 5 priori. To empirically determine these elements, one would have to undertake an evaluative process similar to that pursued in this thesis to develop the data from which the variances and covariances are calculated. 164 In the theoretical vein, there are two main suggestions for research, namely to extend the concept of the cost derived loss function, and to establish the statistical properties of the Bayesian estimators which use the current and extended loss functions derived by representing the loss as the profit foregone from the inaccuracy of the forecasts. The current cost derived loss function has been derived assuming a one product firm whose output price is being forecast, and the aggregate interpretation of this function implicitly assumes an infinitely elastic demand function. The obvious extensions of this analysis include the evaluation of the loss resulting from the imperfect forecasting of input prices, the losses which accrue when both input and output prices are imperfectly forecast, and to the case of a multiproduct firm. In the aggregate analysis, there is the problem of representing the welfare loss, in a producer or consumer surplus sense, resulting from forecasting error when the demand elasticity is finite. Each variation of the basic assumptions would be expected to result in a different form of the loss function representing the costs of forecasting error. For each of these loss functions there is a corresponding Bayesian estimator or forecasting rule. In the case of the current cost derived loss function it is the same as the estimator derived by the Bayesian analysis of the quadratic loss function, namely the posterior expectation of the dependent variable under study. The properties of the estimators depend in part upon the prior distribu- tion of the dependent variable, as well as the loss functions assumed, so consequently there are many opportunities for econometric research involved in Bayesian estimation and analysis. 165 Virtually all the empirical results obtained in this thesis are contingent upon the assumption that the data used in this analysis are comparable with the data used in the original re- searchers' estimation process. To eliminate data problems as a factor contributing to the probability of (in this case, type I) error in the analysis, if this experiment were to be replicated, I would recommend the chosen models be re-estimated over their original sample period with the data defined in the same manner as that used in the test period. The re-estimated models would then be the subjects of evaluation, rather than the estimates as pub- lished in the source of the original models. In the empirical analysis, the two surprising points arising in this research were the apparently poor performance of the econometric models and the unexpectedly good performance of the low cost models. These points form the basis for the recommended empirical research. It appears that there is yet to be developed and published a stable dynamic model of the beef industry which can be used for short term forecasting of prices. The real challenge of this recommended direction of research is to obtain such a forecasting model which will consistently out perform the naive extrapolations of a no-change model, or better yet, from a financial standpoint, out perform the commodity futures market. The unexpectedly good performance of the models which con- tained no more than present and previous cash prices and the corrected trend models as forecasting devices suggests that perhaps a better still forecast might be obtained by an "optimal" weighting of such 166 observations and past errors. The time series modelling techniques deve10ped by Box and Jenkins consist of algorithms to efficiently estimate the parameters in such weighted average types of models, including provision for the incorporation of autoregressive dis- turbances of various forms. These estimates may result in an even better sort of "naive" standard against which the econometric models should be compared. In view of the disappointing performance of the econometric models considered in this comparison, one must fervently hope that the future models of this and other industries, which involve reasonably large commitments of resources for their develOpment and analysis, would perform more in accord with the costs-returns expectations which formed the basis for the expectations within this research. B IBLIOGRAPHY 10. 11. 12. BIBLIOGRAPHY_ Ahlfors, Lars, Complex Analysis; New York: McCraw Hill Book Company , 1 966 . Allen, R.G.D., Mathematical Economics; New York: St. Martins Press, 1963. Anderson, T.W., Entroduction to Multivariate Statistical Analysis; New'York: John Wiley and Sons, 1958. Bennion, E.G., "Forecasting Yt = f(Zt) +-zt vs Yt = f(Yt) + Zt or Yt = f(Zt) +-Zt vs Yt = f(Yt - 1)," Econometrica, Vol. 31, No. 4, (OctOber 1963), pp. 727-732. Box, G.E.P. and G.M. Jenkins, Time Series Analysis, Forecast- ing, and Control; San Francisco: Holden-Day Publishers, 1971. Bullock, J. 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Zellner, Arnold, "Decision Rules for Economic Forecasting," Econometrics, Vol. 31, No.'s 1-21, (January-April 1963), pp. 111-130. Zellner, Robert E. and Charles L. Cramer, "Traders' Expectations and Cattle Futures Market Prices" (Mimeographed Paper), Missouri Agricultural Experiment Station Journal Series No. 7174. APPENDIX APPENDIX OBSERVATIONS REGARDING THE ANALYSIS OF AND THE DATA FOR THE ECONOMETRIC MODELS This appendix to the thesis consists of two parts. First, the general procedure for analyzing the econometric models is des- cribed, and the departures from this procedure which were neces- sitated by the particular models are noted. Then, the data which were used in the analyses are described in terms of their char- acteristics and shortcomings: in a sense, how they were generated, manipulated and fabricated. Analysis of the Econometric Models In this section of the appendix, the general method of analysis of the econometric models is discussed, after which the three special cases in which this general method was altered are each described. The three cases which require some degree of explanation were the models of Myers, Trierweiler and Hassler, and Crom. The analysis of the models deve10ped by Hayenga and Hacklander and by Unger followed strictly the general procedure. An econometric model in the sense used in this thesis is construed to be a multi-equation system which represents the structure and behavior within a segment of the economy (in this case the beef cattle industry). In the compact form facilitated by matrix notation, such models may typically be represented as A Y(t) = BY(t-l) + C X(t) + U(t), where A and B are n X n 174 175 matrices of (constant) coefficients, C is an n X m matrix of co- efficients, X(t) is an m X 1 vector representing one observation of the set of exogenous variables, and Y(t) and U(t) are n X 1 vectors, representing the observation of the set of endogenous variables (Y) or the realization of the multivariate unobservable disturbance process (U) which occurred at time t. Unless this system is recursive,1 the structural form of the system of equations cannot be used as it stands to generate forecasts (or evaluate policies, or do anything other than test structural hypotheses). In order to generate forecasts from the structure of the system, incorporating all the information embodied in that structure, one must invert the structure to obtain the reduced from for that system. In the sense that I am using the expression, the reduced form of the original 1C X(t) + structure is the vector equation: Y(t) = A-lB Y(t-l) + A- A-1U(t). For purposes of generating forecasts more than one time period into the future, the general solution of the difference equa- tion should be derived. Myers' model presented several difficulties in its analysis. The matrix of structural coefficients for the endogenous variables was not in fact constant, but a function of population, an exogenous variable. The procedure used to solve this problem was described in Chapter III, and involved inverting the matrix by partitions and representing the inverse matrix as a matrix polynomial in the popula- tion variable. A system of equations is recursive if the rows and columns of the A matrix of the structural system can be permuted to form a matrix which is triangular. That is, it is recursive if all the elements above or below the diagonal of the A matrix are identically zero. 176 The stage one equations of the Myers model were at the same time the simplest and the most troublesome of the methods of fore- cast generation. They were simple from the standpoint that no matrix inversion was necessary to generate the forecasts. The troubles involved in these equations began when the particular exogenous variables used in these equations differed from one sub- set of equations to another, as well as from the form of the exogenous variables in the structural model. The structural model incorporated the income and broiler consumption as per capita variables and the pork storage holdsing variable as a total quantity variable. In the stage one equations explaining per capita pork consumption, per capita beef consumption and retail broiler price, the pork storage variable entered as a per capita variable, with the income and broiler consumption also as per capita variables. In all the other equations the income and the broiler consumption entered as total (as opposed to per capita) variables and the pork storage hold- ings were as a total variable. No discussion of the rationale for this deviation from the standard procedure of using the same exogenous variables in the "reduced form" model as in the structural model was given, by Myers, although it was apparently done in an attempt to offset the nonlinearity of the structural form equations. In fact, there was no discussion of the stage one equations in the bulletin or the thesis, except to say that "these forecasts were generated by them". In addition to these problems of a substantive nature, several typographical errors were uncovered when the table of co- efficients in the bulletin, was compared with the corresponding thesis table. 177 The coefficients of the dummy variables for June in the commercial hog slaughter equation and for August in the per capita pork consumption equation to be corrected: in the first case, the coefficient of -64.53 in the bulletin should have been -264.53; and in the second case, the published coefficient of -2.53 should have been -0.253. The coefficient of cattle on feed in the per capita beef consumption equation was reported to be 3.41 when it should have been 3.041. The performance of the slaughter steer price equation in- dicates that this equation apparently has an error in it which was consistent between the Myers thesis and the bulletin. My suspicions concerning the error center on two coefficients, based upon a com- parison of the coefficients from this equation with those of the retail beef price equation: The coefficient of the hog cycle variable in the retail price equation is positive and negative in the slaughter price equation leading me to snapect an incorrect sign; and the absolute magnitude of all coefficients in the slaughter price equation is smaller than the magnitude of the corresponding coefficient of the retail price equation, except in the case of a dummy variable and for the interest rate, whose coefficient is given to be -0.403. If the coefficient of the interest rate were -O.103, the amount of error observed between the forecasted and realized price would be significantly reduced (the possibility of mistaking a one for a four in a manuscript is not remote). It is this latter discrepancy that I have the higher degree of belief in, although I have not empirically verified the actual truth or falsity of the suspicion. 178 The Trierweiler-Hassler model varied from the standard model in that it contained both monthly and quarterly relationships. Although the monthly relationships contained only one exogenous variable which possessed monthly variation, one would be hard pressed to accept the joint determination of both monthly and quarterly variables at any point in time. Consequently, this model was analyzed by determining the quarterly variables separately from the monthly ones and evaluating those forecasts, and then analyzing the monthly relations with the realized values of the retail prices (the quarterly endogenous variables) read into the model as if they were predetermined in the analysis. There was one relationship which was specified in two alternative ways in the Trierweiler- Hassler bulletin. Both forms were included in this analysis by assuming that the calculated values from each of the forms were representing separate variables (although they had the same series of realized values). The recursive nature of the model facilitated the separate consideration of the quarterly from the monthly rela- tions, but this property was not utilized further in the analysis. The Crom simulation model was a dynamic recursive system of equations which included a number of nonlinear equations, as well as many (147) operating rules constraining system behavior in various circumstances. These nonlinearities and constraints pre- cluded the matrix inversion method of generating the forecasts, although they were easily incorporated in the computer program which he used to simulate the beef and pork sector. While verifying the program that was included in the Gram publications, I noted that there were a number of places where there was a degree of 179 disagreement between the model as discussed in the bulletin's text and the model as operationalized in his computer program. Such things as extending the Operation of binary variables beyond the period mentioned during the estimation, and thresholds for the operating rules not being the same in both the discussion and the program are typical of the discrepancies uncovered. In this analysis, whenever there was any amount of disagreement between the text and the program, the latter was given precedence in resolving the issue. As regards the empirical results of the Gram model during his validation period (1955-1970), with the help of a duplicate deck of data cards provided by Mr. Crom, I was able to duplicate all but 28 of the more than 1600 computed values of the endogenous variables, and these 28 errors included those which were simply rounding errors. However, 13 of the 28 errors occurred in the second quarter of 1969, with virtual agreement on all the calculated values of the endogenous variables in subsequent quarters. This would lead one to suspect that the results for that quarter may have been in- correctly transcribed, or perhaps altered after computation - al- though only 7 of the 13 were closer to the realized values in the publication than in my results. In my analysis of the Gram model all of the actual data in his bulletin were assumed to be correct. There was some attempt to verify the values of some of the series but not all were completely checked. As was indicated earlier, both the Hayenga-Hacklander and the Unger models were analyzed according to the standard procedure 8 indicated at the beginning of this section. Now I will describe 180 some of the difficulties encountered in the preparation of the data for use in the analysis of the models. Data Generation,,Manipulation, and Fabrication The conclusions regarding the evaluation of the econometric models are conditional upon the data upon which the analyses are based. If the data used in the period of comparison are not defined the same as the data used in the estimation period, then a major portion of the error of forecast based on those models must be ascribed to the noncomparability of the data. With data revision, correction and occasionally elflmination, there is a problem the g§,pggt_analyst must solve prior to his evaluative research, and that is the problem of data comparability. The publication by the original investigator of the data used in his estimation can be a very significant assist in the g§_ggg£_evaluation of the model. In all of the econometric models which were considered, there were some changes required in order to obtain comparability in the data series. In the HayengaéHacklander there were two vari- ables for which some additional work had to be done to obtain com- parability, the price of barrows and gilts at Chicago and the cattle on feed variables by weight groups. When the Chicago terminal ceased handling hogs in 1970, a substitute price variable had to be found. The price at Peoria, 111., is the variable that re- placed Chicago in the Livestock and Meat Statistics and was the primary choice to replace the Chicago price. However, for January through May of 1970, no price was reported for the Peoria market. For these months the adjusted price was taken to be the price re- ported at South Saint Paul plus a 30¢ per hundredweight adjustment. 181 The cattle on feed variables did have a degree of noncom- parability during the sample period, as well as outside the sample. In the sample the variables weret1u228 state totals for the period April 1, 1963 to April 1, 1964 and for April 1, 1964 through April 1, 1969 the 32 state total was used (the sample period was through June 1968). In July 1969, the U.S.D.A. began reporting cattle on feed numbers for 22 states on a continuous basis. These inventory numbers were adjusted to achieve comparability with the 32 state numbers as follows: 32 state 700-899# steers - 22 state 700-899# steers + 110 (1000 head) 32 state 900-1099# steers = 22 state 900-1099# steers +’69 (1000 head) 32 state 500-699# heifers = 22 state 500-699# heifers + 62 (1000 head) 32 state 700-899# heifers = 22 state 700-899# heifers + 39 (1000 head) It it is true that the Myers econometric models gave me the greatest amount of trouble in the generation of the forecasts themselves, it is likewise true that the Myers data created the most problems in obtaining comparability. Most of the associated problems involved the exogenous variables, although there were two points of potential disagreement in the set of endogenous variables. In the set of endogenous variables, the least troublesome of the questions involves the choice of weights in forming the weighted average steer price from three markets. Myers indicates that the weights he used were the "relative value share based on total yearly sales for the three markets". The prices in my analysis were weighted by the yearly salable receipts of cattle for the respective markets in the previous year divided by the total receipts by those three markets that year. The effect of the different weighting schemes should be minimal, especially since the "value share" should be highly correlated with the salable receipts. 182 The other possible problem in the set of endogenous variables involves the retail price variables for both beef and pork. In 1969, the retail price series for beef and pork were revised by the U.S.D.A. to reflect coverage of more cities and giving different weightings to the individual cuts of meat, reflecting changes in consumer tastes, through a revised "market basket". Both series were revised back through January 1966, based on the newer definitions. Because the old series is the series used in the estimation, it was decided to retain the old series for as long a period as is possible to main- tain comparability. In the set of exogenous variables, there was no problem involved in duplicating the interest rate, price of corn, pork storage holdings and time variables. In the cases of the wage index, per capita disposable income1 and the cattle and hog cycle variables, I was satisfied that the variables I used were dis- tributed randomly about the variables Myers used with a finite variance and no apparent biases. The per capita broiler consumption in month t was calculated based on the broiler chick placements two months earlier adjusted for mortality, average liveweight per bird, and dressing yield as well as exports and changes in storage stocks. The problem here was to aggregate thw weekly chick place- ments to monthly totals when some months have five weeks and some have four. The solution arrived at was to adjust all months to a 1 An adjustment of $2.00 was subtracted from the income variable to obtain agreement of this data with Myers data in the sample period. 183 four and one-third week basis, based upon the average weekly place- ments within the month in question. The resulting per capita broiler consumption was distributed with mean equal approximately to Myers series for that month and some variance greater than zero. The exogenous variable representing the number of hogs six months old and older was one series in which there was a noncomparability of this post-sample observations with his sample observations, apparently because the sample series was not calculated according to the same method as outlined in the appendix to the bulletin (his post-sample values agreed with this method). The problem then became one of how to commit the same error as was committed in the process of obtaining the estimates. For months other than December, the procedure was to take the beginning hog inventory, add the pigs born seven months earlier adjusted to reflect survival rates, sub- tract off the month's portion of the breeding herd adjustment and farm slaughter, and also subtract off the number of hogs slaughtered in the previous month. For each December the inventory of all hogs on farms weighing 180 lbs. or more plus 25 percent of the number of hogs weighing between 120 and 180 lbs., plus the number of pigs born seven months earlier expected to survive to marketing (sows farrowing times average litter size times one minus the death loss as a percentage of the pig crop), were added to form the value of this variable (i.e. it is reinitialized each December). (In the bulletin, Myers suggests using the unadjusted December inventory.) The resulting variable appeared to have Myers variable as its mean during his sample period, but did show variability about this mean. The twenty-six state cattle on feed series had to be interpolated 184 in the latter months of the test period, as a result of the U.S.D.A. decision to report cattle on feed in twenty-two states after 1969, except for the January inventory. The Trierweiler-Hassler bulletin was perhaps one of the less specific papers in terms of the descriptions of the data and the sources. In addition to this, no tabulation of the sample data was published. Notwithstanding this drawback, there were only two vari- ables about which there was any question, the slaughter hog price and the beef carcass price. The prdblem associated with the hog price was that I was unable to locate an Omaha price of hogs for observations early in the sample period (for use in obtaining the trend approximations). As the way around the difficulty, I sub- stituted the corresponding National Stockyards price minus 15¢ per hundredweight to fill the gaps in the data series. The problem with the carcass price was that the bulletin referred to it only as the price of "Choice 600-700 lb. figg§,0aracsses at Chicago", with- out specifying whether it referred to steer carcasses, heifer car- casses, or an average of the prices of both types of beef. For my analysis, I assumed that it referred to the price of steer carcasses. The Crom model also was rather non-specific as to the origin of its data, although it did include a listing of the data used in the sample period, as well as in the validation period. I did have some difficulty in locating some of the data series used in his model, as well as determining the content of several variables which he created specifically for the model. In the former category were such variables as the byproduct credits allowed for both beef and pork, and the marketings of nonfed beef, the two variables in 185 the latter category are the commercial beef cow slaughter and the inventory of heifers one to two years old, not on feed. Because the bulletin contained data through the second quarter of 1970, it was decided that the marginal gain of another couple quarters' data, would be less than the marginal cost of obtaining the data. Con- sequently, the test of the Gram model covered the time period from the third quarter of 1964 to the second quarter of 1970, rather than the period from the first of 1965 to the last of 1970, as in the case of the other models. The Unger thesis was the most specific as regards the data sources and content. Not only did he include the sources of the data and a listing of the data, he also included the table number of the most recent year's observation of the data together with the source. Even with this there were some problems that needed to be solved. One of the data series which he used in the estimation process was discontinued from publication in the Agricultural Statistics handbook, and there was some discrepancy involving the construction of one of the data series. The publication of the "cost of marketing meat" variable was discontinued in the 1965 issue of Agricultural Statistics which presented for data for 1964. How- ever it did retain the "meat products marketing spread" which repre- sented the marketing costs included in the typical "market basket" of meats for a family. To obtain a variable comparable in content to Unger's original variable, I divided this "meat products marketing spread" by the per capita consumption of all metas multiplied by the average family size. To adjust this series to the value of Unger's original variable in his last Observation, 1.10 cents per 186 pound was added to this ratio. His price of protein feed variables was the other problem. In his discussion of the data, he implied that the price of COpra meal was included in his index, while Hildreth and Jarrett (36), his model in the construction of this series, did not include the price of Copra meal in their index. Since he stated that Hildreth and Jarrett's method is the way that 118 obtained the variable, that was the method I chose to resolve this problem. Also, there was the necessity of converting the feed prices from dollars per ton to dollars per hundredweight of total digestible nutrietns. Whether or not that particular price variable was included in this index should probably not substantially affect the values of this variable since there were eleven other price variables entering this index. The final set of data which was used in this thesis involved the prices of the live cattle futures contracts on the Chicago Mercantile Exchange. As has been mentioned in the main part of the thesis (Chapter III) these data were assembled into weekly average prices by Keith Holaday Lacy in his Master's thesis. I further combined these weekly averages into monthly averages. In determining to which month a particular week belongs, the following rule applied: A week is classified into the same month as the Wednesday of the week. This means that a week.which ends on Saturday the third, would be classified with the preceding month's observations. When the futures market price was deflated to make it comparable with the deflated slaughter steer price, the deflating price index for the month in which the futures price was observed was used rather than try to account for any potential changes in the price index between the month of forecast and the month of the realization. 187 I am including listings of the exogenous and realized endogenous variables for each of the models, as well as the fore- casted endogenous variables correponding to each form of the models as an assit to the reader, or subsequent user of this thesis. Also included is a copy of the performance statistics on all endogenous variables corresponding to each run of the models. Finally the re- duced form equations which correspond to Myers' Structural Model are presented. 7200 NOI'N UKDIVS 1905 1905 1905 1909 1905 1905 1905 1905 1905 1965 1905 1905 1900 1960 1900 1900 1900 1960 1900 1900 1960 1906 1906 1900 1907 1967 1907 1967 1907 1907 1967 1967 1907 1907 1907 1907 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 IUIIOIIO'ICHI07000’ no.» OII‘0’\ICPHIUn '01-‘0CNO4'IIQIP ”0." ~00- .0090.\I0“UOUUD NC’IIO1IQIOQIC'U1UI- D” "09.10.90'IIO0DNC. 220020003 000100LES 0520 10 YNE 0010060 000 "002100022 00001 22.2 21.3 20.3 23.3 22.2 23.3 23.2 23.3 22.0 22.7 22.0 20.0 22.3 21.3 20.3 22.7 22.0 23.3 22.2 20.3 22.0 22.7 22.0 23.2 22.0 21.3 20.3 21.7 23.0 23.3 22.2 20.3 22.2 23.3 22.0 22.2 23.0 22.3 22.7 23.3 23.0 21.7 23.0 23.7 21.0 20.3 22.2 22.0 23.0 21.3 22.7 23.3 23.2 22.3 23.0 22.7 22.0 20.3 21.2 23.0 21.2 21.1 23.3 23.3 22.3 23.3 20.0 22.7 22.0 23.7 21.0 23.0 23.1 21.0 20.0 20.0 21.5 20.1 25.9 ‘26.1 27.1 20.7 30.0 20.3 20.0 20.0 22.9 22.0 20.5 20.0 21.1 21.2 21.5 23.0 20.1 22.0 21.9 20.0 19.0 17.0 17.0 10.3 19.2 20.0 20.3 21.3 22.9 21.0 20.5 10.9 10.2 17.5 17.0 10.2 19.5 19.9 20.1 21.0 22.2 20.0 19.9 19.2 10.9 10.5 10.3 20.1 20.2 19.9 19.7 20.0 20.0 19.0 10.9 17.3 10.0 10.9 10.1 10.5 17.0 17.1 10.9 10.1 19.2 10.0 188 Table A1 3.003 3.100 3.122 3.106 3.103 3.102 3.193 3.190 3.220 3.202 3.271 3.309 3.302 3.352 3.377 3.002 3.020 3.007 3.072 3.000 3.500 3.530 3.550 3.577 3.000 3.027 3.007 3.070 3.097 3.720 3.730 3.700 3.703 3.770 3.000 3.021 3.90. 3.935 3.901 3.09. 3.911 3.922 3.905 3.931 3.900 3.900 3 7 1799. 2097. 2097. 2097. 1970. 1970. 1970. 2059. 2059. 2059. 1910. 1910. 1910. 2339. 2339. 2339. 2312. 2312. 2112. 2590. 2590. 2590. 2070. 2070. 2070. 2033. 2033. 2033. 2300. 2300. 2300. 2002. 2002. 2002. 2029. 2029. 2029. 2090. 2090. 2090. 2390. 2390. 2390. 2055. 2055. 2055. 2350. 2350. 2350. 2702. 2702. 2702. 2532. 2532. 2532. 3221. 3221. 3221. 2700. 2700. 2700. 2053. 2053. 2053. 2093. 2093. 2093. 3073. 3073. 3073. 2700. 2700. 1051. 1707. 1767. 176'. 1029. 1030. 107.. 1651. 1051. 1051. 2005. 2003. 2001. 1790. 1730. 1790. 1951. 19u5. 1951. 1901. 1901. 1901. 2175. 2173. 2171. 2°00. 2000. 200.. 2071. 2071. 2073. 2050. 2059. 205". 2211. 2213. 2211. 2011. 2011. 2011. 2°03. 2009. 2005. 2°99. 2°09. 2000. 225?. 2252. 2250. 2261. 2201. 2201. 2209. 2203. 2293. 2252. 2253. 229’. 2551. 2551. as“. 23”.. 2351. 2311. 23" O. 7 3" I. 2390. 237%. 2375. 21". 2031. 2011. 010. 072. 072. 072. 1337. 1337. 1337. 005. 005. 005. 750. 750. 750. 1091. 1091. 1091. 1030. 1010. 1030. 905. 905. 965. 755. 755. 755. 1031. 1031. 1031. 1000. 1000. 1000. 071. 071. 071. 020. 020. 1000. 1301. 1710. 1190. 907. 020. 032. 032. 032. 710. 710. 710. 1192. 1192. 1192. 1020. 1020. 1020. 975. 975. 975. 903. 903. 903. 1319. 1319. 1319. 1000. 1000. 1000. 1007. 1007. 1007. 095. 095. 095. 1251. 1251. 1251. 1030. 1030. 1030. , 1:02. 1102. 1102. 901. 901. 901. 1037. 1037. 1037. 1100. 1100. 1100. 1177. 1177. 1177. 955. 905. 955. 1520. 1520. 1520. 1319. 1319. 1319. 1200. 1200. 1200. 1090. 1010. 1010. 1590. 1590. 1590. 1205. 1205. P 00 10312. 10312. 10312. 0005. 0005. 0005. 0013. 0033. 0011. 10030. 10010. 10030. 0000. 0050. 0060. 0107. 0107. 0107. 0352. 0352. 0352. 10903. 10903. 10903. 9930. 9910. 9930. 0750. 0750. 0750. 0970. 0970. 0970. 10332. 10332. 10332. 10000. 10000. 10000. 0900. 0900. 0900. 0095. 0095. 0095. 10710. 10730. 10730. 10055. 10055. 10005. 7109. 7109. 7109. 0970. 0970. 0970. 9000. 9600. 9000. 9592. 9092. 9992. 7100. 7100. 7130. 9053. 9051. 9051. 11220. 11220. 11220. P 120 0200. 0729. 7050. 9502. P 100 5022. 5130. 720! "ON?” 1965 1905 1905 1905 1965 1905 1965 1965 1965 1965 1965 1905 1900 1900 1900 1900 1960 1966 1960 1960 1900 1960 1960 1960 1967 1967 1967 1907 1907 1907 1907 1907 1967 1967 1907 1907 1970 1970 1970 1970 1070 1970 1970 1970 1970 1970 ”00’ Nt-OdOO'IO\IOG‘N0‘ IDO‘IOGI'UHU” OCDHCIOIPU7OI u IDOIIfl0090’IGI’ 0.. ~0- P0 23.93 123.00 20.30 25.20 26.50 26.97 26.10 26.60 26.00 26.30 26.21 20.10 26.70 27.63 20.99 27.90 26.72 25.19 25.20 25.01 25.70 25.20 20.75 20.50 25.30 20.90 20.53 20.29 25.05 25.05 25.99 20.00 20.90 26.02 26.31 20.00 20.90 27.01 20.05 27.79 27.37 26.00 27.01 27.70 27.90 20.10 20.00 20.00 29.12 29.20 30.30 31.11 33.00 30.07 31.50 30.00 29.33 29.79 20.97 20.00 29.00 30.01 31.06 3!... 30.92 10.77 11.17 30.50 30.71 29.90 20.10 27.79. 237 '0 23.03 22.10 23.57 22.03 21.50 25.02 22.33 23.00 22.70 20.00 20.29 23.07 27.12 2~Ion 26.50 25.20 22.60 20.17 20.70 22.25 21.61 20.00 23.59 21.05 20.90 22.10 23.22 21.00 20.66 20.39 20.96 20.07 22.06 25.13 20.57 23.09 25.03 23.09 25.00 20.50 22.00 25.00 22.10 20.50 22.01 26.50 20.10 23.50 26.03 20.10 25.00 25.12 20.90 29.69 26.96 25.73 25.13 26.51 25.50 25.19 27.09 20.31 20.00 29.00 27.09 20.00 20.10 27.30 20.00 20.39 20.7. 22.93 189 003001 000 057100100 (000560305 000100155 09.320 00.005 00.527 01.073 00.059 05.065 05.517 07.253 71.220 70.017 00.000 00.002 70.215 00.779 00.000 00.300 09.250 72.910 70.090 71.317 75.132 73.300 72.237 70.003 75.709 72.250 09.712 73.500 73.992 75.000 72.207 71.560 70.230 70.103 70.077 71.757 75.002 73.229 71.050 70.300 75.500 75.053 75.210 75.907 79.266 79.095 70.955 72.325 70.025 76.573 73.120 71.505 72.629 70.709 70.160 76.300 91.360 02.099 77.006 75.790 00.090 77.103 75.530 76.520 77.000 77.550 75.313 'flg39. 01.910 00.717 77.790 77.773 Table A2 “092060 IND NIC(L0~OEI 00001 033 00 59.523 00.572 00.993 01.291 00.009 63.052 50.716 62.003 62.002 63.000 00.132 01.000 60.609 66.910 66.616 62.393 60.365 60.037 65.766 66.005 67.37. 69.016 60.011 60.137 07.231 60.000 73.159 09.020 09.315 70.037 00.969 00.207 05.129 07.990 03.050 00.370 05.010 07.100 70.032 00.396 69.060 73.536 69.923 69.160 07.070 70.323 00.560 05.000 07.506 70.260 73.765 70.116 70.200 72.399 69.075 75.730 70.209 00.171 70.107 70.301 77.073 70.057 01.015 77.052 75.300 79.050 75.025 70.203 75.791 75.700 70.705 72.230 PM 19.53 20.30 19.90 20.05 20.09 21.92 22.59 20.67 20.67 19.10 10.90 19.70 20.00 10.00 21.20 21.11 20.13 35.92 29.01 27.02 20.01 25.97 10.51 17.01 10.11 10.99 10.00 20.79 20.70 15.00 75.79 12.00 20.01 10.35 10.00 10.30 031 PM 23.15 10.02 10.09 31.30 17.97 10.22 29.03 10.30 10.05 3.00 23.70 15.29 15.11 15.66 16.10 25.00 10.90 13.70 30.93 22.90 25.22 21.17 30.90 33.05 05.070 00.939 00.239 01.717 30.171 30.506 32.503 30.592 00.219 00.529 01.009 35.950 37.130 37.930 01.399 00.017 30.377 30.090 33.009 30.173 03.021 05.330 00.377 07.710 00.509 05.775 00.700 07.051 39.030 39.005 37.700 01.000 07.117 09.010 90.000 09.320 00.319 00.215 00.700 07.031 00.723 01.290 39.022 01.903 00.020 51.000 51.532 51.100 09.100 09.290 09.020 09.102 00.310 03.229 00.750 01.502 07.061 00.009 07.311 00.597 05.302 03.015 00.100 00.710 05.910 02.103 01.250 00.005 50.702 53.920 57.509 57.007 £37 0" 30.109 20.001 11.530 32.570 00.750 0.052 33.235 25.60. 03.000 22.915 20.920 21.705 12.071 11.760 .200 25.306 36.102 15.925 39.71. 32.019 09.939 20.660 30.630 01.170 30.555 30.009 23.139 00.993 51.017 10.305 03.670 33.237 02.000 36.029 03.230 00.070 07.009 07.010 29.092 50.227 55.593 20.205 52.601 06.662 75.121 02.532 57.053 59.202 57.005 55.93. 52.769 56.152 60.107 20.006 01.700 02.212 51.353 30.017 02.905 30.051 02.712 39.090 21.510 02.102 50.970 27.023 07.200 39.503 70.700 00.100 73.590 79.201 3100 309. 319. 335. 335. 292. 220. 170. 135. 120. 120. 102. 152. 155. 331. 300. 330. 293. 239. 199. 203. 250. 279. 200. 209. 291. 300. 355. 300. 320. 205. 197. 197. 222. 237. 256. 209. 200. 270. 320. 299. 206. 195. 100. 170. 202. 221. 211. 210. 237. 209. 323. 351. 300. 255. 210. 213. 200. 300. 330. 051 3100 200. 262. 170. 319. 333. 111. 170. 03. 102. 07. 77. 00. 33. 52. 12. 109. 260. 120. 190. 113. 173. 71. 105. 200. 203. 203. 200. 371. 020. 107. 270. 100. 201. 109. 200. 270. 309. 329. 230. 370. 012. 270. 303. 232. 330. 191. 295. 312. 320. 337. 277. 377. 000. 100. 235. 107. 191. 130. 197. 171. 215. 219. 190. 302. 302. 230. 293. 197. 305. 202. 300. 057. 153C) ~«~5.:« eso:oc cam:o anew. Moan. taco. ooooo~oout.0o:¢m-d ~9.m~:~ao eo.«ecc moomm «econ .ou unsoow human 0:.manm n~.n4 «coon ”mo: 5’. Hana «moow ”mod“ on: .«0003 «due. g:onoN 900m 9dom ‘3. gum. :0 u wthDOMonm udm ma zenbcdaoacu wt» 2H.mzowhq>¢mmmo no 02 ~¢~0.dl «dam. comm. cocoannd ouoomo «noon ocoo an» comm-m N¢:oo NJomo m9.“ «bod on. no. In muudoc node. cams. mwosoou oooowu 00.: .NM onoooa Nnoo. mwoun wao on. m?« «a. 06 noon.» come. Owns. scooen ~:o0m m:.«« -on gun Launcr «5N0. oc.N« mm. on. me. «a. on mucmEHOuuom wcfiummoouOm ”Juno: m4 mans“ .uoaau uu¢¢u>¢. «gum macaw zo~»«4wavou hzutoz Ibczou .zwxoz wbaaom¢< o¢.:» coauu u>~b¢aua uu¢au,¢ «exam u»=4omm« uu¢uu>¢ mmoxau bznoa uzuzma» N a .d : mmmamcm: >h~Ju¢wo hmou duozcaxuqx DZQ cuzwraz 1J9]. Table A4 troutuous v1n11uti§ 93th In avgus.noosgs .\ YkAfl Manta 05p "u n Oncnu Cu 196” 1 2N4 16716 4 n 19“., 2 5'94 . 70 o a 102.5 10099 951?' 1 127 1'20 5 ’7; I ‘ ‘ , iizi‘: :1: {'53: '25" ms 122;; in: ,: 3:151 33°22 :23': 1 45 4 3:5 13.;4 4' ' ° ° 3° 1 1.11! 12?. 91 1'0 I ' I 9 ' 3 1’ 1 Jnd 275 075 - 4 0 ‘02! 1 5o’1 109 0 1°65 3 315' 4756). 4" ' ' 61‘ ’1° ‘27 2‘2 t’v. ’4 2 4 v ' 4 ’ 4’ 1.532 °¢4 04 7: ? o 10 1 5.9“ 109.: 1905 o 293' !.qq.' " . 3 1,101 122 '30 113-or 55 , ' 25 J l 959 ~v . o 3. J 19‘.1’ 109. 6 1965 7 (34' 13 51 " 1'.” . ” 7: 1..90 121'52 4 no . .22 1 293 n¢q lion; ~. 2 ‘ °-' 2.61’ 1°4.J7 11a 1 :905 a 1’4. ‘2~“- ‘ - . 1..’1 121 "2 132'42 2 72 9 9 ° _ ~ .14 1 244 949 7 v . 9 . . ° 1 4.’ 11a. 2 avas 4 1.2' -5,< ' 4 ' ' ° °! 1-‘03 121 41 4: >3 4 t o . ‘5 5‘3 139 76 '_. '- 1 ' 2'6 0 10‘062 11° 0 1”’ 1“e. :71;-° .' ’° - ‘3 1. 5‘ 124 1e ‘04 1s 51, 9 . I ..o .‘, ‘5‘ 933 75 - Q I ‘0 ' 1 5.06 11° ‘ 1455 a. ' 1' . ’1 1. 53 124.u4 4a so ‘14 ' 1035 :41: :g:;:- 14;; 1.:23 :17 :«oo. 1§rqa 121,7: :11:13 §:;,1 :::':g i::!: o .1 2.1! 1"‘1 :22254 ‘1d'l. 2."): :95... ‘1::° 1°¢° 1 1"(. 1) 9': 4 I, ' 19:4 2 as 4‘ n’ ' 1"" °h7° 1“! 6! 1 428 122. so '2 9. 19.6 , i~;: ;S;g,- :-3: :o::; 3;; :3: 7 1: 4: 121233 t.u:.1 31:3: 122':: :1:-: 13,5 . 217 ‘7-‘1'. 500 o, .- 12 2 1.3.14 421 33 ‘1’ l. ‘ ’5', 0 4 6 o ‘ ’ v 4. . a 1.;13 .66 9901f). , - O 1 .01 112.0 19“ ’ (’2 16'6‘ 1 , ° 2 1v.£ 121.21 140 15 2 as o ” - ' 1“ 1 :12 coo 95 - - . 3 1 6.43 ‘13,, 1°96 6 2nd' 422‘ 5'3, ' ‘ . ’1! 10535 121.13 1‘5 42 2 71¢ 05 52 1796 7 2“" ;.7_~o 5'5d 1.4‘9 .5144 912?;1gci5 121;?5 1"V"O 2 uq‘ 'q.', '12.: 1°06 a 1/9' 1;, . 1.4'6 .49? «n9;1-u,q 122» , 42‘ . 1 . 1 112, . 1- 5 67 $86 599 5 9- 2 4 1 o“ 3 '7’ 1°0 91 113 3 1906 914 -»9 ‘ ' 1' 0 ¢ 139),. 121, 1 '45-). :d' 9 ' ' 1966 10 13:4 $214:. 2.1: 1. 3‘0 .930 8370; 1:961 .21;a¢ :49' ~, g: 1;: :.;o:: 1::.‘ 19°° *1 1 1: Ik'37' 5'071":9 .217 :?13: 1242’ 125. be 1°a.*s 2. 51 192'97 ' 4?; 3°°‘ *2 2'° 17424: 5:40 1:341 0;; 93~7 "ub’ "°;1’4-" 3.43 1°7I’7 1:434 - 1 1:034 12'£14 101.61 .ss: 347.90 11..; 1°07 1 75¢. 16N5' 5 25 19c? 2 'ac a, " ' ' 4°51° 3'01“ 10951: 1&09‘ 220.36 a .90 , in: L»: :02: :12: :°"‘.’: ‘22:: 14.1: 2:2: 33% fizz; 1 47 4 35 a; ' -. .3 1 J. 1 J44 125.97 , o . ' 1907 5 3»§' is; ;;3 1'2: "'92 "111 ’°2"4 ‘§°‘° 12’-°° 1’3 9: 2'2“: izg'25 113': 1967 0 31¢: 152:3 4:.0 1':;: i4i§2 31:3; i504; 1:;365 1-4.c4 2:679 198.76 g1, 5 1c.7 7 9 - ' . :34 1 :45 an a: 9 ‘ - 1967 a ngo 13€¢4 4.98 1.453 1,129 $66 16‘59 127.7. 3’. . 3. 31 19fl,90 11“: l . 13-1! 4 7’ $ ‘5 3o ! ." 2 l 1 ‘4 ’ 206‘? 199 11 11., '2'; ' 1'9. 13577 4:16 1.12: 1.142 :333! ""' '2"" "" 2‘ 2° ' ' 1'°:°’ “‘:' 1" 10 2J . 7 7 ' ° : 1:325 130.75 I: d 1:2; 11 2,3, 19339 2:32 1'35: 1'3: :3"! 12°" who iuzei 5321 133:3: :33: 1 12 279016C“ 5' ‘3 ' . 1. 104.2 131,1; 13° " 2. O . o 3 036 . 39 ° - 44 2 9 £99 91 117 n ' ‘ ’°1”a 1:94 131." 1’1; :2 2.242 200.0411232 1900 1 286. 1756‘ 5 4o 1964 2 26°. 16247' 5'2: :':§' "'3' ":"' "°‘3 "'*“ 4"i" 3.454 200.24 111;. 190° 5 292 170919 5:, 0 10° 1 1’ 0?. 1iC1. 1’1n‘o 1";V2 2. 62' 2'. J, ;" . 19on a auo' 19°24' 5 '7; "'2; "°" 1"27 ‘l3" 159-3‘ 3 6‘4 2.9°9 zco:’o 11", 1463 5 555 18734' . 5. "° "°" 4°"11 1100’ 131£35 1' ;'l 2.41? zoo.09 119:! :90» o 387.1537r' , ,, *°°7“ ‘-3‘° 1941“: taro. 132.44 213.44 2;750 zoo. 41 12. J 1966 ? sze' 13131' s'u 1.040 1.;55 '5“! ‘=°°’ 1’? ’9 3’1.°J 2.$9> 230.94 120:4 1908 l 2C5'12824' 5'52 1'838 1'0‘7 91"! .997 132. JC 1'°.’l 2.4“? 201 15 12 3 10°" 9 lqe' .‘13;- 5.6 .y 5 1.047 9103: 1It45 152;79 go;,vv 2,14 20": 5 121'! 1°os 10 117' 17.7“ S'l: '»;2 "°'5 '2’“! 1;‘°“ 15° 5° 2:3"1 2.67 241:59 372.2 1°ea 11 222' ’(332' 5'97 2 1,332 93" ‘gii’ ‘3°;56 2JJ:£u 2,25: 2J1 la 22' 19°” 13 297' 17~24 0'2. 1. :2; i'ggz §'§"’ 1’ '3 ";'° 23°"’ 2'2" ‘°‘ " ‘25:‘ . . 1 an. 1,.1 134,04 2:1,14 2.501 202.10 123.7 1969 5 7 1¢oo §,§' i§,g,: 4:, {03:3 058 fi146; 12.24 1 6.4) 19.9 .529 393 5 ‘3 - 1°69 204 17; 71 0'00 0 . '2 19‘2“ 51° 330,4 3.5“? 202,123'1 196° :19 ,Lzla- 6‘.. * °,° '-°’3 115‘”: 1:623 13‘ ‘9 2::;°‘ 2.474 2oz as 125'. 1'67 3)‘o ‘9‘, o ’0’. 1.0‘2 1.644 1171‘. 1:129 155! 76 233.10 a.abo 302'); 32"4 196° 2v9' 1¢14:' ’0’, 1.152 1.3371900}. 12328 1J5 '37 4J5'7c 5‘92‘ 202.43 12" 1°00 (46' 13:7: 0'4 1.144 1.315 10731; 1,221 134.59 299.6] ;,1z¢ 203'05 127'. 1962 1vo' 12:31' 3'1: ""5 "123 ‘°""' 1&34’ 13’304 292.55 3.123 '205'22 12u'2 196° 10c. 15 93' 9' . 1.12 1.015 10447; 1.1117 156:59 232.01 3.1:: 2°3.¢o 12..) 1464 170' 7: .' o"2 1’°" '3‘? “3“2 1,111 13°$°9 2:2;°4 2.912 203.01 129:: 1960 2.2. 'B1%5' 6"! 1.015 1.313 10226: gérgl xsngao g's.go 2.19; 20"'2 2, .-:. ‘5” W m *ut" . o . . 011 24, 11394 149:31 232.50 2.674 2:4.14 131.: 1°70 1 :13: 12:3:: 2:3: i':;2 ';3: 1:22;! :13:: 1352': $33.49 2.96 204.54 133.0 1‘70 ' J 247, 1610‘ I o. ' ' I i H .14 2 4‘4 204 21 132: ’ .044 °5° 2:19 ‘ a I ' ' ' §°’° ’ §?°' {3::§- 3’33 "°" "’° “7"! 11L4’ 14° ’2 zol'a' 52373 3:.°.. iii' : 1°70 4 391' 14~ss 7'10 "'3' "" “"" 11‘14 1‘7 3’ 36$ 64 J I"? aci'aa 154'. 1970 7 504' 12171 2'. "'79 "" u“’“I ‘1°‘ “7¢‘° 314: 5’ 4:440 205'lo 835'! 1970 | 255' 1145!' I'z' "'5: '9': 10714, 15°3‘ “‘:1’ (a4. " 4.574 209'40 159'? 1°’“ ’ 217' 14244' 1'“: "¥; '9" ‘°“'7' 1,a24 3‘9213 102.24 4636: 205:00 140's 1“70 10 210' 16440. 4'34 "'ag "'° ’°"‘! *i3z‘ 3":’1 264.44 2.9°4 205 I1 131:. 191° “ ‘..' 2 "2" 34” ‘4‘“. o"‘ 1050?. 1,029 146 10 293.!1 (,93q 2..:.; ‘37:‘ 1'70 12 Jo4 ; ¢ssz go; i'év. 1':;: 1‘ 3°! 3é?4: 4‘;:g' 44 . 4.240 zoo. 44 13154 o o l I; 1301 14 I 30 ;’2 2.}4; aggzq; "i’ HF; YEAR 1665 1905 1965 1965 1965 1965 1965 1965 1965 196$ 1965 1969 1966 1966 1966 1966 1966 1966 1966 1966 1966 1966 1956 1966 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 1967 1960 1965 1966 1966 1965 1960 1968 1968 1960 1966 1968 1969 1969 1969 1969 1969 1969 1969 1969 1959 1969 1969 1969 1969 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 1970 192 Table A5 HVERS SYRUCYURQL NOOEL ICIUIL 030 1511H11£0 ENSOGENOJS V60110L£1 (1) ION'H OH £51 0" PM £11 7“ P? [17 7' 1 1679. 1620. 19.90 12.62 91.70 01.00 2 1619. 1627. 16.01 21.17 92.16 62.79 1 1756. 1150. 16.76 16.00 92.19 00.69 b 1599. 1325. 17.1. 20.05 52.92 76.61 9 1127. 1110. 19.07 11.50 92.01 01.96 6 1160. 1217. 22.76 12.27 97.06 01.11 7 1231. 969. 21.99 10.90 61.90 92.69 0 1102. 601. 21.97 60.61 61.02 109.11 9 1605. 160. 22.29 99.16 61.00 120.19 10 1696. 019. 22.66 60.02 61.06 100.96 11 1515. 791.- 23.91 91.96 62.97 119.06 12 1611. 609. 26.96 61.10 69.90 112.19 1 1130. 600. 26.70 60.69 69.02 130.69 2 1265. 706. 26.10 60.90 70.61 100.61 1 1503. 926. 21.16 99.10 69.66 127.20 6 1691. 795. 21.10 67.29 66.90 106.26 9 1610. 757. 21.91 69.21 61.61 109.30 6 1569. 999. 21.19 60.01 61.99 91.72 7 1200. 1001. 21.50 17.09 66.16 00.16 0 1615. 1121. 26.11 11.76 66.61 02.91 9 1692. 1061. 21.69 16.60 66.60 06.66 10 1654. 1270. 20.11 11.96 61.21 ‘ 00.97 11 1755. 1100. 10.76 12.10 60.69 01.90 12 1772. 1519. 10.97 11.29 99.02 00.17 1 1779. 1266. 19.32 91.96 50.09 99.19 2 1561. 1102. 10.20 19.91 30.10 07.11 1 1009. 601. 17.66 60.26 97.10 100.06 6 1615. 906. 16.73 62.21 56.11 90.99 9 1510. 965. 20.61 63.71 99.19 100.07 6 1681. 099. 20.97 65.99 90.71 101.99 7 1360. 667. 21.22 90.02 59.60 112.50 0 1503. 929. 19.01 95.06 99.71 120.01 9 1666. 669. 10.12 99.10 99.01 120.91 10 1067. 921. 17.10 66.91 97.79 102.26 11 1521. 061. 16.19 90.79 96.11 111.79 12 1739. 001. 16.17 91.11 99.11 110.09 1 1011. 001. 17.00 96.00 99.60 129.11 .2 1550. 979. 17.97 91.00 59.66 110.09 '1 1671. 192. 17.62 69.22 99.71 117.91 6 1765. 950. 17.96 97.26 99.21 126.16 9 1771. 907. 17.60 97.19 99.20 126.66 6 1636. 969. 19.79 56.60 99.09 121.19 7 1500. 690. 19.69 55.32 59.09 120.00 0 1500. 190. 10.67 60.69 96.11 129.19 9 1676.. 113. 10.27 60.11 99.09 129.66 10 1979. 913. 16.76 69.20 99.66 106.16 11 1001. 1096. 16.15 60.50 59.29 101.72 12 1019. 1106. 17.09 65.33 99.16 106.99 1 1037. 1276. 17.06 19.99 96.39 99.66 2 1612. 1090. 19.37 39.05 59.91 '09009 1 1755. 910. 10.92 66.57 91.90 109.11 6 1791. 1266. 10.21 11.19 91.00 02.19 9 1607. 1610. 20.51 20.69 96.62 76.62 6 1512. 1591. 22.21 20.09 56.97 60.91 7 1529. 1662. 22.99 22.66 97.00 61.16 0 1676. 1250. 21.71 29.16 90.20 79.97 9 1699. 1177. 22.01 10.91 99.70 70.96 10 1066. 1916. 22.19 21.69 60.61 66.09 11 1577. 1736. 22.67 21.21 99.09 61.02 12 1720. 1611. 21.60 25.19 60.70 69.66 1 1666. 1611. 21.62 27.01 62.29 71.26 2 1.91. 1.5.. 26.29 21.99 61.7. 61.90 1 1663. 1220. 72.29 19.23 61.11 06.01 6 1761. 1192. 20.60 19.60 99.60 06.66 9 1566. 1676. 20.19 29.66 99.66 69.09 6 1511. 1671. 20.59 17.66 99.17 99.99 7 1619. 1699. 21.39 12.17 99.60 69.12 0 1567. 1610. 10.07 26.00 90.60 67.60 9 1006. 1369. 17.20 26.21 56.19 71.09 10 1997. 1766. 19.20 10.29 96.29 90.10 11 1969. 1711. 10.00 29.99 91.00 70.07 12 2111. .1762. 11.10 26.00 69.19 72.71 0P0 9.091 9.011 9.362 9.612 9.691 6.096 5.195 5.260 9.169 6.122 5.611 9.710 9.010 5.160 9.702 9.696 9.219 9.007 9.162 6.030 9.109 9.726 6.072 9.901 9.207 6.962 9.262 9.196 6.969 9.112 9.190 9.100 9.790 6.110 9.901 6.901 [3! 0'0 6.611 9.659 6.266 6.166 9.170 6.199 0.696 1.157 2.016 1.760 1.621 1.006 0.116 2.909 2.970 0.269 0.129 1.776 1.790 6.167 6.103 6.612 6.959 6.001 6.260 1.066 1.760 1.796 1.009 1.716 1.331 1.279 1.260 6.162 1.015 1.012 1.900 1.011 2.955 1.057 1.202 1.221 1.109 2.971 2.965 6.106 6.161 6.162 6.190 1.912 1.066 6.267 6.669 6.961 6.710 6.962 6.676 9.210 9.160 9.121 6.979 6.929 6.170 6.122 6.716 9.106 6.716 6.011 6.790 9.690 9.166 9.260 POH 16.66 15.61 15.12 15.32 06.60 17.06 30.75 16.16 16.30 06.07 16.90 16.60 19.09 10.19 11.02 10.76 17.12 17.11 17.91 15.99 16.61 39.76 11.11 11.06 12.17 11.00 11.97 11.19 11.10 12.07 11.22 11.11 11.19 11.76 11.61 11.66 12.06 11.91 36.73 11.06 11.00 31.00 12.66 33.59 33.00 12.67 31.77 11.69 11.91 12.10 12.72 11.19 12.10 01.66 16.61 16.90 06.97 12.76 12.00 12.06 12.67 11.92 11.16 10.92 10.19 10.67 10.90 29.61 20.06 20.91 27.72 29.91 (St POM .J.02 36.00 «6.3. .1.07 62.21 39.75 .1.69 «5.69 .9.~7 61.99 .7.92 51.50 90.16 62.60 69.66 66.19 65.20 36.26 10.90 16.09 19.67 16.91 19.16 19.61 51.09 30.23 .6.62 61.09 .1... .1.~5 .5.9. 67.76 £9.07 19.67 66.57 60.1. .I.§5 .6.79 53.12 51.21 90.27 ...91 ...16 .9.77 «9.33 .1.13 66.32 66.06 10.31 19.67 66.09 16.75 11.10 27.69 29.97 12.61 11.22 27.61 27.19 29.10 27.7. 25.5. 31... 11.99 27.00 22:00 86.19 2.... 80.2! 21.92 12.69 33.92 193 Table A6 I'ERS SYRW'UIHL HOOEL OCVUIL IMO CSIIMIEO ENOOGENOJS V‘RIIBLES (2) 7:09 mourn 000 057 can 20 007 75 20 250 29 925 257 925 no 257 0c 0905 0 7.955 5.500 22.52 09.50 72.00 000.57 990. 090. 2090. 2200. 0905 2 7.050 5.009 22.05 27.92 70.00 05.70 000. 050. 2090. 2002. 0955 0 0.090 5.905 22.50 09.55 70.90 000.95 0070. 020. 2757. 2005. 0905 5 7.005 0.050 20.50 05.57 70.55 002.00 995. 050. 2590. 2055. 0955 5 7.070 5.705 25.05 52.50 72.00 007.00 059. 000. 2500. 2020. 0905 0 7.072 5.920 25.90 50.09 70.20 000.97 090. 005. 2000. 2070. 0905 7 7.000 5.500 25.50 55.90 70.95 020.90 020. 709. 2077. 0090. 0905 0 0.050 5.955 25.55 55.29 70.09 050.00 005. 055. 2759. 0700. 0905 9 0.025 5.295 25.57 00.70 75.00 050.09 955. 550. 2009. 0505. 0905 00 0.200 5.000 25.70 55.75 75.09 050.09 900. 700. 2050. 0795. 0955 00 0.000 5.509 25.00 57.00 75.02 052.05 902. 009. 2002. 0555. 0905 02 0.000 0.902 25.05 00.20 75.50 000.79 070. 500. 2020. 0077. 0900 0 0.570 5.000 25.55 00.99 75.77 055.57 050. 500. 2905. 0509. 0900 2 7.509 5.005 25.50 50.02 75.05 000.02 005. 505. 2597. 0527. 0900 0 0.007 5.000 25.50 59.00 70.00 055.50 0002. 500. 2025. 0520. 0900 5 7.000 5.905 25.02 50.05 70.00 000.92 095. 052. 2005. 0752. 0950 5 0.055 5.070 25.02 55.07 75.50 055.09 090. 550. 2790. 0725. 0900 0 0.552 5.000 20.50 57.05 75.50 020.50 905. 750. 2950. 0992. 0900 7 7.907 0.007 20.50 55.05 75.05 009.99 000. 750. 2702. 2055. 0900 0 0.007 0.702 20.92 50.00 70.00 000.52 929. 020. 0020. 2097. 0955 9 0.005 5.900 20.97 00.25 70.00 000.09 997. 000. 2909. 2555. 0905 00 0.550 5.900 20.02 50.00 70.55 000.05 0027. 905. 2095. 2570. 0950 00 0.050 0.020 22.75 09.05 72.95 009.22 0070. 907. 2000.’ 2055. 0905 02 0.295 5.555 22.50 09.00 72.09 009.07 0095. 950. 2550. 2055. 0907 0 0.722 5.907 20.00 50.05 72.09 005.20 0000. 055. 0005. 2005. 0907 2 7.757 5.002 22.70 50.05 72.50 020.00 900. 705. 2009. 2000. 0907 0 0.502 5.000 22.50 50.50 70.90 050.70 0025. 752. 2905. 0025. 0907 5 0.000 5.000 22.50 50.52 70.20 000.00 900. 755. 2707. 0790. 0907 5 0.070 5.009 20.27 50.07 70.07 050.00 0000. 772. 0020. 0720. 0907 0 0.707 5.705 20.02 57.00 72.25 050.70 909. 750. 2909. 0500. 0907 7 0.050 5.005 25.52 00.95 72.09 050.90 905. 550. 2750. 0500. 0907 0 0.720 5.500 25.00 00.02 72.50 000.05 0009. 050. 2990. 0502. 0957 9 0.255 5.009 25.29 50.00 70.55 050.09 0055. 550. 2059. 0005. 0957 00 0.052 5.000 25.55 55.75 70.09 050.52 0020. 000. 2999. 0905. 0907 00 0.005 5.597 25.07 50.00 70.00 052.95 0029. 700. 2025. 0009. 0907 02 7.902 5.590 20.05 50.20 72.05 057.75 0005. 707. 2770. 0509. 1909 0 0.979 5.500 . 0900 2 0.050 “'0‘9 :1 g: :2’:: $2.77 000.52 0072. 700. 0007. 0597. 0905 0 0.000 0.705 25.00 09 05 '2.59 050.05 0000. °°’° 2005. - “"- 0900 5 0.000 0.900 25.55 05.00 72.30 077.50 ‘°"' ’92' 2"" ""- 0950 5 0.909 5.000 25.25 05.00 7i'°9 ‘55055 5096- 603. 2000. 0502. 195! 6 .5155 350‘.‘ 25.25 '6.~° '2015 ‘72.,5 “°‘o 6‘3. 3069. 1‘56. ‘96‘ , 6.43“ “.250 25.00 :bo" 7‘9;9 :IIOOZ 98". 65’. 2,029 1562- 0909 0 0.900 5.200 25.3, 00 95 ’0. 7 000.50 0055. 025. 0005. 0500. 196d 9 65559 50509 25.1, 63. 63 7292 1720’, 10559 5959 30720 15.30 0900 00 9.502 5.705 25.09 55.99 7&26 005.55 0070. ‘02' 2962' ‘663' 0:2: 00 0.220 5.005 25.00 59.59 70.3; 003.0: iii?‘ :26' J’°~° 2000. 12 6. . ° ' ' 9 50. . 1“; 5 69' 25‘“9 5‘7‘3 ’1-32 137-59 0055. 050. ;::;: 50:2. 0959 0 9.092 0.505 . ' 0959 2 0.05.9 5.055 2.30: 3.2: 7:.” 029.05 “75' "6' 32°" 23““ 1.59 3 0.090 5.752 25 9“ 50 50 7 .20 000.20 0052. 792. 2795. 2207. 0909 5 0.225 5.000 27.09 52 20 ’:.50 ’35'9’ 0055. 700. 2050. 2092. 0909 5 0.005 5.095 29.05 50°05 7 '7. ‘1’°3' “‘”' °°" 2‘59' 2297' 0909 5 0.205 7.090 29.02 02 50 7“. 3 ‘12'23 ‘062' 9‘2‘ 2"6' 35‘5- 0909 7 0.00; 7.790 27.59 32.90 ’0.0 95.07 0000. 0000. 2057. 2009. 196q 0 0.525 ,.55’ 20.70 37.05 ’;.00 95.00 0055. 900. 0005. 2057. 1969 9 q.“‘ ’06:, 25.~‘ 36.52 ' 01° 1°65~1 96“. 92“. 2,6,. 2,630 0959 00 9.700 0.000 25.59 05.50 7"“J ‘07-05 1060- 900. 0052. 2792. :22: .. 12 0.005 . ' ' ' . 00 1. 2009. 7 000 25.00 02.00 70.00 95.50 0050. 0055. 0070. 313:2 ”70 1 9.000 7.909 . 19'0 2 5.009 7.000 25.2: g: g: 7"“. 100.52 396“- 0007. 0095. 2900. ‘9'“ 2 0.795 0.250 20.00 55.20 ,5.52 ““"° ‘°"- 090. 2900. 2520. ‘,,o 5 0.507 'on' 25 00 07.90 5.55 025.00 0005. 055. 0005. 2002. .97, 0 0.000 ,.‘,~ 25.00 ’050‘ 70.05 000.09 0000. 970. 2925. 2009. 1970 7 0.002 7.557 20.75 05.30 2"“ ’2-‘3 1053. 0057. 0050. 2900. 0970 0 0.505 7.055 25.05 05.95 75.20 °"’° ‘°°5- 970. 0050. 2029. “'“ *' 9-255 0.157 25.50 00°50 72°25 9°"' “55- 970. 0055. 2950. 0970 00 0.220 7.555 20.20 05.05 7‘°2’ 90.00 0250. ""' "“° ’°9’0 0970 02 0.907 7.259 22 77 05° °‘°’ ‘°°'““ ‘217- 0005. 2075. 2799. . .22 59.50 005.55 0039. 0005. 0009. 2095. 1514 om:«.n- oa~ao o~:n. mo.~o~oom-ms“moosnmoNo.an«o~nnn.mmnnus oownomNN mn~sommo nmmn. «ss:. :«uaoo ssmuoc :smmowwu wmaooo 0:59.- coonounmnoso.oo:m~m009.unsnm~ oaoaoo- ~¢oo~ No.5uu own nonwoc 03am. :o.ssom~o :m.onooo :a.:-« 0:.350 «0.50 onodn oc Noou~ om.s- .sn n:mo.~ Nums. «cocoa om.:m« «oowa :non and «does -.mm .nn cacooom coco. 50.0oon No.0» vn.on as.» on. on coonm mo.m~ .un mweoocn «nuo. o~.ooo ~:.~a mecca coca s». on mNomoN mac“. n«o~. ososaa -.c~ :ooww am.~ .«n «mm:.: unwo. «o.~ om. am. u«. no. can hm~a.mn smwwo 0n~cu oaowmosa :oomwwu oson~ coon .sn n~oo.o :ooo. @n.«@ as.“ :m.« nw. mo. You no~«.« s-~o oson. oo.m he.» ~e.«~ saga .nn ommoow nmms. :~.« ma. ea. no. no. one u quhzommbm «cm no onhcqaoaao u!» «03noono ocno. a:~«. odoom:oouo-.mnnmmo an.~mo«o« «m.ssso« asowc :4.mn .:n mooaosw mNom. whowcou oo.o~ m:.s« m~.~ 0:. an n adso.o~I «39o. ”sac. m~.mou owoow .cn :cno.~« mNmo. No.05m m~.«« nq.e« omoa on. In moooo3ma ~«:«. :oo~. muoacnm¢wn snoowamMco onocn oo.mmm .on coon.: mo~o. o~.:o~«ma o~.-«a om.co¢ u:.:~ a:.»« to ucmEuOMHwa wcflummuouom ” 4 2H MZOub<>¢ummo no 02 auomdu w9¢ww>¢~ mcnm macaw onpcquxxoo hZutOt thdaou hzwtoz wb340mm¢ oxuxh «comm u>up¢4m¢ wo<¢w>< moamw ubaaomoc uu¢¢w>< muoumw hzucn cauzaap N a a 3 mwaamdmt >bHJ¢DGwZH .cw.m.t. econ I anoe ~n.e Naoo soon WMOJ ow>umwo hwou woo: Addahonmbm new»: n< m ;mu 1595 Table A8 Ac 'VER ' .rqaL nun ‘5':H:v£ge I ,uu"‘° I s‘°n°‘”°03~:aut ‘ULts (1) vi.“ "o HY " On I! ' Du PH :33? 1 ’° 19 2 ‘°75 '2' Pp °’ 3 1‘39' sarg q. :36, ‘ 1’56. ‘5‘3' 19:50 O '3' Or 19:; ’ ‘58”: ‘°’1: 1°25: 17.25 ’0 RC- 1965 . 13270 ‘°.‘o $.17. :9"1 ’11, . .!' '6“ 1965 1 13‘0. 163‘? 1’.3‘ 10.05 92'1 d7. . 1905 a 1231. 19in, 1°,07 17.69 ?2,39 50;,o 9. 32 1965 . 1502' ,‘/0' 2227‘ 17.9“ ,2"2 51": ‘.§.. 3.9.. 1905 ‘° “‘“. "". 25159 19.59 ’zg's ‘°-76 ’c’sz 1710 i‘ 4 . 1‘ ' 1‘9d I40: 23.97 20 65 ’7.fl° ‘9 d ’ ‘3, 5 2 5‘5: 3 -965 ‘2 1535. :30”. 22. 21. .1 52- 2 .u ,1 6 J 3 $99 1‘1 ' 17 ’ 22"25 21'72 ov'9' s '9‘ c"2‘ 5‘0 ‘ ’33: :2: 1. x.§‘° 25:66 a .33 53.0; ,;.50 .:74 ,ago 49;: 3¢.;o ‘96. ‘ 6' 1.251 21.75 5 .89 ’ .‘3 ‘.(}i {C}! ’4 4' 3‘0‘2 19 2 1310. 14.33 02.57 56..‘ ‘.3qa ‘ a“. {5.7' 3‘-.‘ :72: 3 ‘zaso 5". 9:150 ’5-33 ‘u!°s ‘:7>J ‘6i3; 33:.- 196. 4 1°03. 353, 26,73 ’I;44 ‘.?ol 9'1}; 36‘3. 3‘... 19¢. 5 i:93. 32‘5. 2°!3 2:075 ., ‘t‘*’ 9‘2 1 3‘16: 3¢.5c 1960 ; H3. ;‘32. 31'1“ ’6.50 ,3.33 ,9 ’.1 s ::[:' 303°. 13:: - is“: 3:4 25°? 2‘13: 22M “1‘? . ” H3? 9 15 5 3° 5.64 - 9O 62.9 18’ 1 1956 :0 1592' ’4!2' 25:5. 24 3, 05.4 .1 3 ’01 5 ‘.2:: 35 1:00 ‘: 1653- 1351' 2‘§11 3‘:11 95:93 53:3, ‘:,2' 4:73. 35:33 3 :3‘ °‘ :2 1755: 19’2: 21.69 3‘.¢¢ 9‘,3¢ ’9... ‘.!50 ‘a513 '9103 3 ~53 1772' 17.60 2093; 2‘062 9‘0“ ’9'“. ‘ .5, "7QJ ‘al7. 3,;.‘ 1957 10". 3:2’6 25-31 :‘-69 :1-50 .196. : 79‘ :;[13 39:92 19., 1 , 1 ,57 31.35 .4.2; 5.39 ‘.’14 1‘23 )1: 3 ox: 1907 2 ‘57‘. , 30.65 °°,69 °0.so ’w05l ‘t'Vt ‘715‘ 3 ‘2’ 19a; 3 1 61c 1"00 ’9132 5,-55 ’Il°‘ “3°93 '5198 346.1 1957 ‘ i207. 1237. 12:32 9 33:3, ’.4¢; ’.140 4656; 34:9; 19., 5 1516, ‘5“). ,928 g .75 ’0 ")33 5'299 6‘17. 39;5. 1967 g 14:0. ',:5. :69‘4 2°-32 ,6205 5 ’gSIs ‘31:; 29:94 1967 . ;3.§' 1093' 202;: 1g':, 51". 92", g , ‘31.! 2'33. 13:; o 1:05: 33": 20:97 17:2: 99:2: 913;: 4;;3; 5...: "3' 1967 i: {523. 15::' 2:23: :;.:g ::.;9 :;.$2 3.3:, ;;:?s 3331. a ‘9', 12 1:21: 1393: *;;32 1’21A ;:;¢: 17:5: ':03: 9:53; os§39 3:::: 39. 19’5. 16910 13.24 ,’o7x ‘7240- :-97; ’1‘:J ‘35:: 3 :92 :a' 1 ,ss 1 .34 .o; ‘194 '!'$ 5:520 ‘3isl 3 an? 16‘17 15.63 ”,79 ,10‘1 ’c'OI '.1Yo ‘2!°( 3§‘°. 15 65 9° 91 9 1° 9 3 la 3 . . .3; .50 o J n 31 123 V.Ia ?’.3; ‘5940 ’.£13 ’gldo 331:3 30tt: ‘9’. “.94 :.:4¢ :xgot :iigv ::832 o ’J x 0" 3:: 3 up. ”'3' “i" "m 19 a 167 ' 1555' 37, , 1‘09: 19:: 5 176;: ‘600' t, 87 15.73 19¢. o 177:, '°’o, 17,9; 17,9. 96,4. 19.. ’ 1‘3" 17.5. {7'5‘ :90‘2 ," ‘. “030 196. o 15:0. .702 17“ 15"? ’r.” “79“ ’0.,3 1965 9 153.. 1557. 10.78 15.51 9.:21 ’9.o ,'.3l ’h' ‘ 1960 10 167‘. 1517. 19:6‘ I6.‘3 ’5’2‘ ‘ ; , ,Ii“ ‘.. ‘ '2‘. 11 197o I‘ve :U.c7 17.59 ’” as “'93 ,0 !3 50317 "l°‘ 2 '7‘ ’9‘. ‘2 1503: 36’s: 15.27 19,3 ’“139 ‘5:29 9.‘*: 5:312 “170 2 88’ £619. ‘5“2, l:.7o 15.4% :f.; ‘7.54 ‘.000 9.092 ‘31“. 3 :5. 195° )avlo 1 '35 10.14 3 p :o,1. ’.go, 5'44, 83". 2 .7. 1969 1 ‘IQO, $6.60 3",6Q ‘09“ 5.3“! 5"“, ‘310' 2 ‘1‘ ‘9 2 1037, 17,71 ",29 50.65 ’ 446 9.1YI 32‘s, 1 301 1:2: 3 1°32. 3”“6. ’“.1o ‘°s’2 ‘ol’? 90‘1‘ " 5’ 2“" 1°69 ‘ 17S'J. ’S’J' 1,006 ‘7'}. ,.°.‘1 :.‘U° .5 0' 2 3 I m. 2 1:31. 333;. {gm :z-ga ,. ’c’w gig“ 19;: 3 ‘ : I . O . 3 . :2‘9 7 19x2' !’*3' 1",;? 19,a: ";a: "-00 ' ' 41:9' 3.:30 5' . abzo' ioV¢' 2c 5‘ :o a DJ q ‘9 a 5 I 3! 99 :36: 9 147r' lbvg. 22.2; 16.5% 359g: 30,5; ’11:: 5.7.3 0 6 '0 0 ° 0 .f o go :2on 12:. ::-:v :g-u :3.» 3;". ,, ‘9., 1 ‘5759 'IL 3 22": $300. ,,|" ‘90,} 5".“ 5.3,. J it. 22:12 - 82 . so~1' 22. 3 .‘u ’;,a. .90 .23: g?94 2i78 .5; $725, —°. 22:59 }g.?s 9.20 ;V.J; ’.01I ’.517 43139 3|;2; 10" 1042, .¢7 -,.¢~ : .7! ,ZoJa ’o1¢z ’n"7 “zit! 7 34' 197 1 23,.; 1 .33 .t,Aa 2.93 ‘.Jsa 1.225 33g¢g 3?.o0 19,: 2 $64A, 19.45 ",65 34.09 5.309 5.299 3“6‘ 3’35; "’ 3 t‘Cg. 1°50. 9(‘7. 6.91 ’,)30 $.22, 34“, 2 .5, 1‘7. ‘ M‘3. "". 23.52 ?°y’o "371 ’t°12 “‘t5l 3 '71 m: : 3:1. 1:19 :33: am .— "“4 9:“ :3‘3’ 33'}: l ‘c t 70 7 a "’0 7 15$J' t7..' :9... 22:3: 91:7. 50.0‘ ' ' 32:0: 34.0: t:;: o :5!~: ;’96: 30.15 13.72 3:,1; :4.a¢ :. a: ’ 24.7‘ >67 0;) 9 $9 1 e ‘ 3 6; 6,9 ‘0, . ' . g ‘ o ‘ 0 ‘ ' I 197° 10 1714: 36:9, 31.35 1'. 2 ".44 ’0.30 ’03“! ‘1‘36 ’znil 1. 9 3g *‘7’ ‘bzr I'.~1 10.95 ".t( ‘0.97 ’.09‘ ’s%¢- ’*1°¢ 1 :50 7' :2 *'69: 3"6: *'.2o l°.x~ ".4u ‘°o24 ‘.‘A: 9.6!: ‘*nt9 ’ 3'0 2:33. ;::;’ ii'i' ::°:: :;'°' :3"' :'::‘ ;'3°‘ :2"‘ :3"‘ 1 J o 01’ 0.3 o 0 I‘72 15’ 9|, ' 15.3. 19,3‘ ".;o :V.v. $.x-n ’.1q2 ‘Ogll 30.3! l‘.63 91.3. 9,93 9,75, 5'3‘3 {0.5. 29.3: 9‘.!! ‘2... “.1‘0 9.340 ‘9164 30.09 C. ,. Q .0: ’ 7‘9 3! 4. 2"‘3 ' 5', s'o «8‘s 2 ° . 9} .g Q. 37"; 2 03. .‘Y’ .73 3,. 39139 :v.vo .633 Vina "ONYN 96 19. 1965 1965 1965 1965 1965 1905 1965 1965 1955 1965 1966 1966 1966 1966 1966 1966 1966 1966 1966 19(6 1966 1960 1907 1967 1967 1967 1967 1967 1967 1907 1967 1967 1967 1967 1968 1966 1960 1968 196! 1968 1960 1968 1960 1966 1966 1966 1909 1909 1959 1969 1999 I959 1969 19A9 1969 1959 1969 1969 1970 1970 1970 1979 19,0 197° 1v7o 1979 )9’9 19’: 1970 1979 oat-u oovounuun ~uaooflo¢uunfi Mod“ ”HO NF aOfiflOUOUW ”tau guano-canny: 0H0 (in on V!) H015”! ‘1‘ oouv~mxu‘uo.u NONUOOVOO-‘NvN .ggtxr.’ WOOHVUIVVLOM -DJ¢£ C‘CJ‘OOQQ.‘ :3 ac VP JO Alb-NP ”P 6 ‘ afil" 'yL‘ V‘n‘..‘bk J-§¢1 or CCCGG.CICO CC 057”“ be -O\"o" "oc"‘-ho E3! 090 0.090 '.511 r.559 9.31a 1,523 P.'39 9"0fl r v]? 1‘.IV" 9 10 9.0! *.931 9.359 {.757 P.III ACIUAL AND isvlnatiu gNUCu$n0us vAnLAuLts 'C 196 Table A9 uvens $1595 I suuallonsv £37 '6 19.70 90.29 no.18 21.59 23.22 (2) [I Is;a1 22,69 I:,75 99,60 I?! vs 68.86 05,29 60,94 70910 72.79 ’°.1J 75.16 74.95 60.95 69.50 ’2.16 12,92 7I.47 72,6. 72.55 77.03 79.05 79.5. ’0.11 ’9.11 74.72 '1.51 74,31 74,15 72,54 11.46 70,66 [1,49 15.93 76.3: 74.75 72.92 [1,79 12,39 73.09 ’1993 3" 3901 3J°29 5’5 "’0 014' ‘51. ago, 997, 132'. ‘3I50 3:95. “331 ’05. 81“. ‘03. 1’01. ’9 '1‘. 399°. ‘3’60 ‘13‘1 1&39. 89351 [.7 0g! 1 6 m: 1000. 95., 979. 99;. 9oq, 92!. 909. 11”. QC 591 a; 1997 noecnuoc woos. anua» away.» cyan.» “A... «oom.6 cosm. n~or. ~o.ammmeo.sy.mo«ooa.oo.nn~ od.~o«o«.c«=.unnoon ”a.am no.v o~.nu. «vw ~00.“ sac”, °~.vd~o~ nu.n~ o~.on o~.« ow. u, c».w o=.an .«~ ouca. 0V5». ya) hm.» no.~ .av scan.» cvas. o~.cau an.» on.” s¢.~ o°-mr m a ”an once“ at n~s«.. ovmom ononq .0. No. an.» .NN «asmm snanw a“. “a. do. on. no. n~v9.n n~o~. «moanam co.oc~ -.od en.n .mn «afio.u o~.a. «moo.. a.ao.a ano~. no. 0.. ”V. ow. n .ow sack. “won. "a. no. no. ”a. ago . uz.»:oxnam :«w to za~h<4204«o mm» a. mzo_k«>zwm=c ha a: mono.s ~c~h. ~ono. so.oco¢ “a.aan «o.~u ~o.n .«N soco.m cauo. na.so co. ..~«. no. at cosmm “~55“ na~om so.no no.0“ an.o «o.~ ao~ machaa coon” o~.m ea. o“. no. L; ~¢«o.mvc Haas. coco. annouos~dm -.~.nanom cams «amend .on anso. -qn. «awoconw uo.cu oouan mn.~ 0'. Lo wucmEquuwm wCaumwumuom o~< wdnwa .zsazm mot¢w>«. a«.c waer ro_»44uerau hzmro: Lhijju pxuso: uthmmn‘ a;_x» mnxau u>.»«4aa xnxzm »»34omw< muoaaw ~z~oa m:«z¢.< m:.;¢>< nn_;u:» u 3 a a . mmn3mp_J.:w= .mou ara_»<:cm — mo‘»n aza>r 198 Table A11 (IOGEIOUS vnnmcs usco In no: Humans: mo NhiSLER Run 9502 NORTH 1965 1965 1965 1065 1965 1365 1965 [965 L965 106% 1965 $765 1066 0966 1966 1966 1966 1966 1766 1966 1966 1166 1306 1366 1567 1067 1967 1967 0967 1967 1967 £967 1967 1967 1967 1067 1960 1960 0060 1960 1960 8960 $960 1966 1960 1966 1960 196a 1960 1969 1960 1960 1069 1969 1969 1960 1969 0969 1960 0069 1970 [070 8970 0170 0070 1070 0970 0970 0970 0970 0070 0970 ‘7C‘IOHDC'UIUH FDIF Nt-GDJIO‘VG'W.Pllhln 100‘000ll‘900” .- GIDC‘i'II'OINI. h” QI— 000 29.10 25.70 25.90 25.?0 25.20 25.20 26.10 26.00 26.10 26.)“ 26.30 26.00 26.50 26.50 26.50 26.70 26.70 26.70 20.00 20.J0 20.00 z’I’o 27.30 27.50 27.20 27.20 27.30 27.70 27.70 27.70 27.90 27.70 27.40 26.70 26.90 26.00 27.00 27.00 27.00 27.70 27.70 27.70 29.30 29.20 20.20 26.30 20.90 20.30 20.00 20.00 20.30 27.00 2,000 270‘0 20.00 29.30 29.00 29.10 29.10 29.10 29.00 20.l0 20.10 20.30 2.050 2000. 29.70 21.70 230'“ 20.20 20.30 20.20 -.00 -000 “0‘0 -.70 ’u’fl '0’. 1.60 1.60 0.60 .30 .10 .10 ¢.60 -05“ ‘0‘“ .20 .20 .20 1.30 1.30 1.30 -070 -.70 -.70 -02. -010 -020 .50 .50 .50 .20 .20 .20 -1000 ‘1.00 -100. -030 -030 0.30 -062 .060 .060 1.90 0.30 0.90 -020 .020 .020 0.3) 0.00 0.00 .05. .05. .05. 0.80 0.80 0.10 0.50 ‘0’. '05. 02) [5.00 15.00 15.0) 19.00 09.00 15.90 00.73 03.00 13.7J 1.052 13.50 19.50 03.5) 10.50 05.50 15.30 16.2] 16.20 16.10 06.10 03002 86.20 0902’ 16.20 16.!) 16.2) 15.20 15.10 15.10 {5.3) £5.50 15.50 15.50 15.00 15.00 00.00 16.00 [6.30 16.30 16.00 13.10 16.10 15.00 16.)0 13.0) 17.50 17.50 17.60 17.00 17.00 l7.JJ 15.03 15.90 15.90 05.30 05.52 06..) 06.33 13.51 00052 13.51 00.30 99.53 0,000 |§.50 05050 16.90 16.10 [6.90 10.10 19.10 87.0) I0 0.26 8.27 2205. 2:05. 2255. 2200. 2519. 2600. 2627. 2632. 25310 2631. 2531. 2633. 2533. 2033. 25570 25370 2357. 25530 2350. 2650. 256.- 25650 25690 20050 25050 25050 2530. 25020 2500. 25600 2060. 25600 7E0? MON!” 195% 19'“; 096. 1965 1965 1965 19'!) 1905 190) 1””9 1°”) 193.9 1306 1956 1936 0966 19.5 1°96 1966 1966 1906 1950 0966 1°33 1967 1967 0987 [057 0907 19u7 1957 1957 1967 0957 1967 1967 1960 1954 1961 0°19 1961 0965 1960 1963 19.3 1961 1963 1963 1959 19h9 0°61 196) 1069 1969 1951 196) 1933 11.1 1531 1419 16’; taln 1070 1°71 1970 1070 1071 1573 ‘u[) I07: 1070 .071 $.D‘63.Al'ufivw ”fin ‘ Ni’tilnb‘IO.l€'a'du 430'6c\;: .»3n n.- r-u p N --u fur-0.0..‘00.l0'u70u a.au Nolh.i.“0.fl‘\d~tfi pt- 0 —.:_.I~6,wfl6'acor ” J PB: 37.16 36.30 36.7] 30.71 “1.36 63.63 60.95 No.16 39.). 3". '7~ 39.99 ‘A.:s ~00 ,-~ “10" 62.75 61.13 19.95 3l.33 30.2. 30.57 35.53 37.10 17.39 17.65 10.1: 39.19 37.31 "01' 39.59 .6.41 .1.7. 52.09 62.60 61.11 60.3. 61.00 .1035 .263. 62.30 62.20 62.30 62.70 “‘0 ‘* 53.50 62.35 62.00 53.00 ...13 55.79 “b.19 .6.Qh 69.05 52.15 53.15 59.30 66.16 66.93 .1.J1 52.35 63.90 99.92 50.12 .7.12 “'0‘, 09.30 “(.31 60.09 67.15 “‘61, .I.73 61.00 61.19 251 ’00 37.63 07.53 37.30 39.53 00.90 39.66 60.27 “0653 .0.1. 50.21 39.95 33.57 60.07 50.07 60.52 50.56 .2066 90.56 39.31 39.69 300'“ 60.15 39.39 33.50 11.00 37.57 57.31 36.69 35.39 35.52 33.42 33.05 30.9? “. 3~ 39.95 33.69 35.19 35.06 37.93 35.62 35.16 39.16 19.22 39.67 ". as 19.70 39.27 13.01 39.66 19.56 19.20 65.36 5..93 .5.06 65..6 .§.9l 55.39 63.2. 9209‘ ~2033 63.10 53010 “20“ .5.74 ~3.?6 9303‘ 62.75 63.13 93.2: “3.35 60.76 50.3% P051 22.90 22.53 23.17 2.... 26.00 2h.u9 26..6 26.20 26.19 2i.31 2..Q! 25.30 25.91 2,030 25.25 26.0. .2509“ 25.25 25.37 25.75 25.5“ 2..70 23.92 23.92 2..9. Z..JZ 21.92 23.09 2..76 ZS..5 36.15 26.57 26.61 25.90 25.3. 25.60 25.69 26.37 26.60 26.50 26.30 26.39 27.17 27.5. 27.37 27.06 27.10 27.9. 27.76 27.50 Z‘O“ 30.16 32.79 33.53 31.29 30.0. 23095 27.00 27..“ 27.73 20.30 29.30 30.09 10.79 29.“, 30016 31.12 30.09 23.21 20657 27.22 20.02 199 30020021120 0N0 N053L£R 90021 Table A12 001001 0N3 £510l01£0 2300320333 060160LES £57 P051 23.07 23.17 23.00 23.93 230.9 26.02 25.03 25.30 25.06 25.03 26.91 26.66 25.76 25.75 25.56 25.59 25.59 25.59 26.65 26.92 25.7“ 29.20 25.06 26.79 23.95 2.2066 (A... 21.0. 23.0. 23.13 2..50 26.67 2..50 25.56 25.29 25.11 26.26 26.07 26.00 Z~O~J 26.25 2.025 26.20 2...6 (“.19 25.36 25.01 2.0" 25.33 25.33 25.15 29.19 29.10 29.09 29.67 29.03 21.56 2703’ 27.57 27.31 20.03 20.03 27.06 20.50 20.15 20.23 27.79 (“.95 27.62 36.7% 76.60 26.06 '052 22.90 22.53 23.07 26.30 26.00 26.69 26.65 26.20 26.19 25.33 26.93 25.30 25.91 27.16 20.25 26.96 25.6. 25.25 25.27 2‘07. 25.5. 26.70 23.92 23.92 26.96 26.32 23.92 23.09 26.75 25.65 26.10 26.57 26.63 25.90 25.36 25.60 25.69 26.37 26.60 26.50 26.30 26.39 27.37 27.56 27.27 27.05 27.30 27.96 27.76 27.50 20.00 30.15 32.79 33.63 31.29 30.06 20.66 27.60 27o~~ 27.73 20.30 29.30 30.99 30.79 29.57 30.36 31.12 30.09 29.21 2.097 27.22 26.02 657 2002 22.09 22.09 22.03 23.76 23.7. 23.03 23.05 22.02 25.03 25.66 29.66 25.5. 5.00 25.-o 25.50 25.37 23.96 25..) 35.11 2.... 2..79 290.. 2..02 23.09 23.56 23.56 22.62 Z..9. 25.00 26.9. 25.65 25.27 25.16 26.06 2'.97 26.91 ('0.9 21.77 2h.77 26.66 2505‘ 29.39 29.59 2‘.36 21.22 26.90 25.91 25.70 29.01 21.75 29.01 33.25 1.50 1.31 21.60 27.36 ‘50., 2%.09 21.96 23.67 2é.22 21.20 29.53 29.72 23.20 27.92 27.67 27.60 1...! 17.25 17.39 17.91 20.57 23.70 26.76 25.0. £3.02 £3.27 2..23 23.66 20.01 23.59 25.93 22.01 23.00 25.60 25.53 23.97 23.09 21.60 23.20 20.63 20.66 19.96 "036 10.00 22.39 22.01 22.93 21.30 13.60 05.65 07.76 17.06 10.92 19.90 19.33 19.39 19.37 20.96 21.59 20.21 23.15 10.61 13.01 10.79 20.33 20.0. 23.65 11.70 23.0! 25.73 Zé..7 27.10 25.1. 25.73 25... 27.70 20.66 23.90 26.53 25.06 26.61 25.02 25.96 22.66 20.61 13.00 16.22 03.66 (51 2'5 15.51 15.51 15.50 07.09 17.00 07.11 22.69 23.71 22.67 23.26 23.22 23.00 27.26 27.26 27.23 25.06 2...~ 29005 26.72 26.76 2..73 22.13 22.30 22.76 20.70 23.77 23.76 20.05 20.05 20.06 22.26 22.25 22.26 20.61 20.57 20.55 20.26 23.25 20.23 23.37 20.36 20.36 21.25 21.27 21.33 23.36 20.39 20.36 21.36 21.36 21.16 23.21 23.20 23.21 25.36 26.19 26.35 27.06 27.32 26.96 20.52 20.52 25.69 27.67 77000 27090 25.06 26.30 26.01 22.53 22.57 22093 70.60 75.60 70.60 00.50 00.50 50.50 36.20 06.20 06.20 02.90 02.90 02.90 06.60 06.60 06.60 05.50 55.50 05.50 06.60 05.60 06.50 03.90 03.90 03.93 02.90 02.90 02.90 02.50 32.53 02.50 06.90 05.90 06.90 06.00 35.00 05.00 06.60 56.50 06.60 06.60 05.6) 06.60 37.00 07.00 07.03 05.30 09.30 05.30 90.10 ,UQ30 90.13 97.90 97.90 37.90 131.03 100.00 101.00 ,,033 35.30 95.30 90.20 19.20 39.20 33..) 31.6) a...o 100.00 100.00 100.03 17..) 17..0 I7..0 257 P00 05.63 05.63 05.63 .6." 00.07 00.07 09.73~ 03.73 01.73 09.67 00.67 05.67 03.53 00.53 00.53 00.66 00.66 00.66 04.52 01.52 03.52 09.05 03.05 01.05 00.92 00.92 00.92 Oo..2 00..2 0...: 91.07 91.07 91.07 92.27 92.27 92.27 09.61 09.61 09.61 91.30 90.30 91.30 09.90 03.90 09.92 90.67 30.“, 90.67 09.01 09.01 09.01 92.03 92.03 92.03 90.75 90.75 90.75 09.26 09.26 09.26 07.99 07.99 07.99 90.66 90.66 90.66 90.06 90.06 90.06 03.35 03.35 09.35 PPR 56.00 56.00 56.00 59.70 59.70 59.70 .907. 69.70 69.70 70.70 70.70 70.70 70.10 70.10 70.00 72.60 72.60 72.60 73.60 73.60 73.60 70.20 70.20 70.20 06.70 56.70 66.70 55.50 65.50 65.50 69.60 69.50 69.60 66.50 66.50 66.50 66.10 66.10 .66.10 66.60 66.60 66.60 60.00 50.00 60.30 57.50 67.50 ‘70,. 50.50 60.50 60.50 71.90 70.90 71.90 77.60 77.60 77.60 70.90 70.90 70.90 31.00 01.00 01.00 00.00 00.00 00.00 79.00 79.00 79.00 71.30 70.30 71.30 (51 PPR 60.20 61.20 61.20 63.77 63.77 63.77 69.97 69.97 69.97 76.01 76.01 76.01 76.90 76.90 76.90 67.17 67.17 67.17 60.55 63.55 60.55 60.02 60.02 60.02 6109~ 61.96 61.96 61.03 61.03 61.03 63.26 63.26 63.26 66.61 66.61 66.01 62.13 62.03 62.03 59.02 59.02 59.02 60.39 60.39 60.39 62.33 62.33 62.33 50.02 50.02 50.02 61.22 61.22 61.22 66.03 66.03 66.03 67.29 67.29 67.29 .5... 65.90 65.90 5:.2. 52.2. .z.2o 51.14 s..73 59.13 s..31 ...33 56.33 ZCHD no::.o od~n. uomw. No.0:mou onoa:~u 0:.aa no.5 mona.: noes. «an sowaoo nnud.~ om:,. oo.-:~ Ono—an ~:.o o~.m .0 :ema.. mane. :m.mn us. up. he. «a. ”ma noomol ammo. aces. o~.o~u amonm soon wood .ou Pam; muas. ~m.m Kw. mu. mg. «a. man u m2~>36¢oam 14% no rauhaaauqao m!» 2” monha>mwmzo uo.oz unns. ncao. sumo. n:.«« n:.: do.” no." .ou o-0.u ammo. ow." ma. 3:. do. so. Nana moanou o:ma.« s«:o. so.o~ 03.0 o~.m @:.« .cn savoou :Ooso m~.n cc. so. «c. as. «man noa0.« wave.u mums. naoomm maos: n~.m amo~ .~n ownn.~ ~a~s. o~oo o«. 0‘. we. do. wocmsgomumm mew muomuou .xoxau uoqmu,¢. wean macaw zo_»<4ux~ou .zflto: :bu:Om azurOt m»:41mc< Cfiuth 2022M w>~.«dmx moqxu>a acxmm u.:49mr¢ unaau>¢ myc;mu .zhca o:~z«=~ N 2 u 3 mmaavcmt yhuaaacuz~ ..u.m.t. ea.“ u z .n.n u z ~n.o u z ~«.a u r 36.5 u.7 Mme; cw>~dwc bMOU “Jamar ~u4wm¢;.n.« wu4~m2yu~a_ m~¢ oflnnH 22(311 Table A14 61*4 ’INULAV'DI IOOIL AC'UIL liq I‘YIIIIIU :hflObEhOuS VIUI‘CLES (1.) Au?! :stc It! "\fc Alta? I'll II I! '6!!! Oath! C"! I! I! ll PCP! VIA- oil 15‘} ‘57. 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L 90 30 8!... 0’ I {232 ”In: 3373 269; on non «:7 n u:- u.’ «27 an n u 30.: scum. 15;, iii; 1!: £613 833 1322 m 3! 13:9 m: 3383.111: m 3! 11:343...‘ :31; - $3 3313 32:! 33% § 3: 1; ti 1::3 3:3 3:2; 33%! it? 33 3:2 .23...‘ 1323 §5§1 1 i! 23?i :3} :32 31% :3 13:3 3:) 3:!3 ::53 1:: 3 13:1 .£¢;.L' 13:3 252i 3%? §§Z3 333 §§§3 3}} a} ?:3 ‘Izl 3312 38%: g}! :1 {3:3 .23..L' 1‘"’ :"° :23! 3:25 :23 }:{2 333 :3 13'; ,2:5 3:33 333: 33 3: {3:3 .2?..L‘ 182: 71:3 3 II 313: :23 1335 332 :3 18:: 3:8 3353 33!! 33 33 13:1 .23.-g‘ §?€: 11;: €£§3 3:1: :23 i€2 138 {t 38:: 3:3 333? 32;: ;3: 33 13:3 .23.." {33: 3523 133: 5:33 333 {335 :13 }: {3:3 {3:3 1;I§ 3:33 :3 2: {3:3 I::g‘L' i323 :3§3 2213 34:3 353 £23 333 $3 3:! '3:§ 3:33 3 33 1:3 33 13:3 -3¥.a3‘ 1333 $823 :33: ?:3: 333 1333 93! 1! liz: :tt 3:35 1:1! 113 $5 {3:3 .E$..s' léii 5333 :31 53!: 33! 113i 3?: 5% 33:3. 3:3 33:: 3:12 11: 3| l3:l .Z!.a" FR'IE J’.tl ¢loil 5. 14 40:14 38:3! ilzig 45.4 go.s 39:33 22'}: ‘39,. 4.013 33:: 33:33 «0.90 47.0. 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Quezon pzo000:000304090.07 80.10 '1331100 2070.030! 01317‘1‘ 10§EIIV 0000016 9000070 78030: 01190. 81900. 701.1: 30017. 30073. 70700: 20700. 21'... 2.000: 31517. (1’32. 74091: 330'). 210.16 07090: 311.9. 31040. 70001: 22050. 33420: 0351!: 21000. 32020. 70:23: 28083. 38000. 05000: 33117. 33300. 7'350: 13700. 38000. 00005: 31072. 20100. 01301: 30002. 34010: 0007!: 3000:. 3000s; POII0IV 1050'0' 000001: 004001? 319 19.79 :01 .00 3“ 1250’. 20" 20.7 3502 3701.03 01.00 00.77 0:00 9127,50 00.00 07.00 100.50.103810.05 209090.00 205200.00 00072 .7007 .0300 .3030 1.0010 0.5000 .0009 .0125 ll ‘0 '0 00 10309 $012.00 930310 130.10 10:00 7.09 13'. 1.0‘ 0002:00.0030:0.00.104090:00.01419:.00 00007.000009000.00021’9103010717091.01 0"“ 0".. 0".’ 9.1., 1.737 00.33 .9100 .0‘0' 0120045300030.0003 090.0000 :91.00.9 0NtLY$IS 2 0- uuncn ZSLS '60! AC! 05 £9? 03 AC? 00 Est 04 ACT 65 E<1 65 act 66 ES? 66 AC? 67 :81 67 AC? 0! EST 68 AC! 09 E§Y 09 ANALYSIS 2 '° ONCE? 7$L§ I COST DERIVED LOSQ 0.04 0.17 0.32 0.30 10.0 (3.3.50) 3"::: .I... INEQUALIYY MEASURES U 1 U 2 runulna roiwr Funons AanAGE AusoLute tuunu IVFRAGE “ELAYIVE Epunu vulnn AusoLuvt ”PHONT rank!» hnm N1 couufiLAIION «Love 0:05 IAVKHAhr ENNUO) no or auscuvatsous I" in! 209 Table A22 000:0 NOBEL 0:!00; 000 08710010: eunocuuous 000000000 roasts 000070 0000010 0050'! 0906'? 3000'! Poueavr 71.01 10004. 09.04 10.09 00429. 0512. 10.70 02.30 10370. 00.02 39:02 17020. 9300. 20.80 74.01 19020. 93.00 10.10 00499. 10110. 17.00 9A.01 10092. 77.09 20.00 11705. 00095 29.59 74.:4 20109. 00.00 10.47 19000. 1003:; 10.11 116.75 17,02. 90.55 31.10 11257. 90173 80.5! 12.3; 20710. 00.00 10.10 20100. 10970. 00.0? 121.1: 30042. 00.00 03.90 :1300. 30572. 30.00 72.03 ?1022. 09.5! 19.3: 20040; 12:40. 10.70 122.00 10052. 92.01 05.90 01070. 00000. 03.00 75.0; 22009. 00.80 20.90 21120. 0277'. 10.00 110.70 19040. 04.23 03.30 19700. 12130; 00.00 730:2 72.250 ‘90,‘ 2.03. 2“’:. ‘23‘,. :.0.. 11..10 20700. 00.41 31.00 10030. 12799. 20.3! Forecasting Pevformance Voriffi onetrc vowiatn 0020'! 0000'! I030?! .87 7.9! .42 '31! 0:24 1.07 1.71 50.00 1.01 .00 00:03 0.09 13.04 0202.90 10.00 3.02 1:10:02 000.10 .01 1000.04 10.00 0.40 1729.00 200.79 1519.069030080.01 0290.09 100.075080080301 939100.5' 099‘0 0'00, 0,... 0,0}, .902. 3.23. 19.0000 0.2031 32.7771 13.0400 0.0903 031300 30 30 30 30 30 33 40.00 0007.00 00.00 10592 2007300 040.04 no.2; 0.10 07.00 00.21 10:13 0.0! 043.0.70.010030.01 00000.90 1070.00.900030:010200007.10 3000799.37o000000.392182749.00 039., 0.300 07.07 1010’. 08757 03600.70 30"03007 034.195! CALCHLIYIUI or 000 sunnouVINE 0 2030'.35070ssaszsoo4100xl.70 '1']. 08.0!!!“ 2.6701607 00%.. 39.08 8001‘. 8033' 0003300: luilflf 0006016 000001! 73.3.: 3‘854. IIVU'b 70000: 10001. 804’25 7474.: 2.7.,. 31...0 7‘5'.: 20017. 81130. 740.8: 2‘J.J0 3"‘10 .‘52': 313". 8‘O‘J0 79,33: 32.53. 3342.0 02.0‘: .1.‘.. 23"C0 7.}2’: 23083. 33.... .Q’?‘: 3.31,. 33.... 70:00: 20700. .30044; .307‘: 33.73. 343... .‘3’3: 1‘..’. 8.97.0 .5C57: 3‘433. .‘UU'; POHIAV' '.I".' Dental: 00“." 3‘7 :301' 841 09. :7! 1.4.7. 30" 2007 ’32: 319‘..' 5‘30. .‘0” 0.00 4304.72 00:00 07.90 t";.5'4.3‘330.! 1.9.0...‘ "9”...‘ 09‘24 0...? 093.. 03563 '00333 .0,fi’7 .0... 0.129 30 30 30 ‘0 I367. 5...033 43.5}. ,5'08. 0'66, .0.‘ ‘3'. 30" I'JIoSIVISOOG'éllott“9150.018.151.05 48631i9907.404....Ott7036.619317'35oll 013.3 0734‘ l".' 0‘... 083873...’...031.. .9’02 890...?! 19106'!’ QNlLY$I> 3 '° UHCFR "H“ I YE 53? AC? ES? AC! 60? ACT., ER? 361 ES? ANfLYSIS J '0 UNGIR HNK COST DERIVED LOSS zarzfl II... o 1.00 (".S.5.I INEQUALITY "5000055 u 1 u 2 VURNING POINY FH"O"S AVFnAGI ABSOLuth Luann IVENAbf 90007106 Funny tutun AucoLutf "fivfwl rounru nnnfint CORNFLAYIOM SLflPE fills (avruaor £000»; .0 03 03 ‘0 .0 65 69 66 66 67 67 68 69 69 121() Table A23 “I053 ”00 l ACIUAL AID I’Y‘NAItfl ENDOGkIOU3 VlIIIULEI 039‘. 0'3Qfl .3503033 3"3.3’|C .300. 3.3199 3730000.10-011000.122000102.22 .3001 03.33 039.533. N0 0' OBSERVAIIOHS IN INF CALCULIYIU‘ 0' 8IN 800.003INI 0 "00000 0000': 0000070 000000 000000 100070 2005.7: 730.7 3‘..‘0 "0.. 1‘0“, 3"?'. ,’12§ 1.." .2071 1.3... “0,. 3.0‘, 3"3‘0 9‘23. 21.5. 74.37 19023. 33.3. 13.13 . 30099. 101333 37.6. ,,0‘. 1‘3‘20 7’0" 2‘0“, 3,9150 .5... 3,093 70.00 20100. 00.30 10.07 10000. 10031. 10.1! 116.75 11793. 333.43 30:70 17305. 9903: 30.30 72...; 20716. $.0,. 3.01. 3'1..0 1",00 3,03, 121.21 10103. 100.70 33:10 31001. 10000. 30.00 72.10 21022. 00.09 10.31 20000. 12100. 10.79 122.20 10001. 00.3: 00.0. 17700. 11020. 02.10 70.01 22000. 00.00 201'0 21120. 12770. 10.0? 314.¢1 19737. 35.32 32.09 107903 322093 33.93 73.52 22020. 00.00 80.30 21001. 12100. 10.00 109.00 20700. 01.30 31.00 19511. 120773 20.0! \ Forecasting Performance 0.0.0. 0005:: 0000010 000070 oaserr 100077 .30 7.39 .00 .31 7.03 1.20 30" 92.3. 30.3 0‘, 5.33, '0’. 13.3. 1103.¢9 13.21 2.09 1310.79 122.00 35.59 1232.32 17.00 .20 1.01:9. 234.3" .4:7.13007.077.00 1300..7 120.704.70077:09 0070.0.0' .0005 .0700 .0000 .0000 .001¢ 39°07 10,0000 0.1302 13.2010 13.1103 3.0000 1:060! 3. 30 3. 30 3. 3: 0 35.56 1’9lo‘fl 35.50 10.50 19.3340 992.06 00.00 0.02 00.01 00.02 0:70 0.20 01000.31cn02000.02 07301.30 1700.00.002000:02-001000.00 20027.02.003000.12ol70o00.29 .03.. .3733 0 .193. 303.4. 101.0 332.0 .3...000 1001.1000 000110.0 7 Intern: Oontafc 0002010 70000: 21300; 81900: 28925: 33017; 31373. 70700: 33739. 31000. 10000: 21017. 81132. 70001: 21003. 21001. 31335: 23149. 83003. 20020: 22003. 23020; 02000: 21000. 22020; 70120: 20030. 23000. 00011: 23317. 23000. 10390: 33700. 33000. .3002: 23.720 3.30.0 31303: 2400!. 34,73. 09303: 34033. 30000: 0000017 IOEifIf 0000000 “V‘E‘I' 83. 32.37 3‘! .00 070 00.00 2.02 2.07 0.00 2070.01 07.00 00.77 9;.) ..;..02 00.00 97.08 10:;.0.003002.70 200000.00 200000.00 .0100 .0007 .0304 .1010 0.1107 0.0000 .0009 ~01!” 13 30 '0 3- 13320 0020.00 010.10 130.10 03630 0.03 0493 103‘ 1000;0ac1270".000110001:00.110001..0 00107.700007012.000217.30301o217011.11 073.3 0‘77! .0202 091.9 .90.. ~10:2010-0021.7002 100.0000 101.0090 ANALVSIS 0 '0 UNCER 351$ 1 YEAR AC? 63 E57 63 AC? 00 ES! 00 0C? 00 ES? 05 ACT . 00 £07 00 AC! 07 £57 07 AC? 00 ES? 00 AC? 09 ES! 00 ANALYSIS 0 -- UNGER SSLS 1 cost nrnxvru 1000 0.00 0.12 0.32 822320 ID... 0.34 1.00 «0.0.0.; INEQUALITY HFASURES U 1 U 2 YURNING POI”? EHWOPS AkaAGf au0OLUYE knauu AVERAGF QLLAYIVt LPdHR 70100 AufioLut‘ HflMIJY 70007» mourn! ROARFLAIION SLflPf 0100 IAVIHAGi ("PM") 2151 Table A24 UNNEI NOflEL ACTUAL AID CUTINAIpt hNDUflthUS VARIA'LEI 00057! [REEF' 'ONEII' 10020. 0012. 10.70 10200. 0030. 21.02 1.697. 10116. 17.00 10200. 0010. 20.00 19590. 10031. 10.11 1023‘. 0'27. 32.30 20104. 10970. 13.1, 10091. 10677. 32.32 20.00. 12100. 10.70 10006. 110101 30.20 21120. 12770. 10.07 21100. 122205 25.60 21001. 12300. 10.07 21010. 129100 22.36 Forecasting Performance rarern 000070 7000170 700077 71.07 10000. 00.00 10:05 00.30 10010. 00.07 10:00 70.07 10020. 03.00 10.10 00.71 10000. 70.70 23:22 14.0. 20130. 00.30 10.07 112.13 10003. 03.01 27.7. 72.71 20710. 00.00 10.10 33‘0‘. 392230 .20.: 2.037 72.03 21022. 00.00 10.31 115.00 10022. 07.20 31.00 75.01 22000. 40.30 20.00 103.70 21030. 70.00 2710. 73.;2 22020. 00.00 20.30 90.01 222.‘0 .903. 2,05, 0.1070 000070 0000010 02. 0 02‘ 3‘ 1.20 13.77 1301 0.03 300.01 10.03 11.:0 072.59 12.20 1000. 31033011.22 015.50 .0047 .0130 .0307 10.1710 1.7301 10.7023 3. 10 30 20.30 1032.00 20.71 34790.010.151‘7.06 31532.00 1010;42.0'0130030.371103302.’. 03213 0.313 05.1. .010. .0720 .1700 ~70.3-01 1032.0000 020.7001 00 or 0u0cnvntlous In fut CALCULAYIUN n! 000 000000110! 0 7 000677 000077 Intuit .00 2:20 1.20 .20 10.25 0.00 13“ 367:70 371,10 1.06 403:2, 223.21 01.1'1003911122 000 77.37 .0009 .0000 :0020 0.0000 2.1007 170000 30 30 33 7.00 1032:04 001.39 30:24 0:07 0.1“ 0'2.73.010107700.020001.09 0130.000130000:30.002200.0‘ 70070 0’171 .044. .2000 .000. 10210 77.0“!“ 1.37.0360 02717077 1000700 73030: 00003: 74700: 103160: 70001: ‘100200: 29923: 100265: 76123: 10.130: 70350: 1.0924: 013.3: 1.0103: POIEAVF '21: 3137 33.8 '7.‘Q.35 00000 000070 10 13315 20720.20 70:41 0000.70 0002077 31390. 81000. 00773. 21320: 20700. 210003 21073. 21000; 213.3. 31091: 21300. 31702. 32003. 23020. 32040. 82220. 23033. 23000. 23343. 2301.. 33700. .3300.: 2‘100. 20230. 31002. 30070; 20000. 20031. 10007.: 000.110 000011. “‘0', 33’ 0’. 72180.00 01:20 00.00 00.00 70.72 200010.01 200410.01 .0010 .3020 .3177 10.0620 .0027 .0013 ‘0 .0 ‘0 001:30 007.10 37.00 1:00 1.01 19903330350063.3701330703770100273007 30702.00.010103....010000:00.010100.00 333.7 0..3. 01.19 0373’ 023333'9'.??.0355. .0335 o.." 00.7076 0.755 07.13 03.9000 10017515 0 -- 00000 13515 YEAR ACT 63 EST 03 ACT 00 EST 00 ACT 05 EST 05 ACT 06 EST 66 ACT 07 E «am. 02 mac. uo> mo.~ ti; IUHO‘HH.’ Bedgfluflghh g .>o h :00 o o o o o c o o o o t o nno.n o o o t + o o s.~nc muo. 0 mom. 0 o o o once. 9 o o o ~ooo. o o o o i o o o o d.mo~ +. o an~.o o o o l o o o o n.0n- u 3 00 02 so .mvo;_ua nouczrn Jana“ owuuu Ow .3 + x a n a < Euou ecu «0 ad «0603 on. quasnna ceauuucomoua much .u0uuo choocuuu moauu oouzu cozy uuou acouuauuuoc + .uouuo vuwchun oven» 03» easy .uu— u:o«o«uuoou * o o o o o . ceoa\cmn.u o o o _ o o o o o o . o 6868\21 o o a o + o o o "4.x- o a o~oo.- 0 «m8.- 3 o o + + o.~n~ ~.~q o 683‘ o q 3 o o 3.~m o o + o o o 36.60 o . o o o o oo.~ H k 0 o o no.n o n o 0 name. o A ~moo.- o o o mn.c o n o c ~o~3.- o a nn~o.o * . o 33.3 o n~n mm“. . o 3 o o o o o o o oo.~. a r o o o o o o i. .NN on.n ado. mead o . o o o a o c you 6 ea uncoo any 6: «a: an on use :08 o>< .~I uu .uuoxx scum ~ovoz ouuuoEOcOom mm 62669 duuauunuum en.n o C an 600403060 6:63.858» 43¢ O On~=.u o A odoo. t. o co.~ 1... a con.. 3 ¢.oo. 3 x o o e.o~ vac an a P auwucovu chaoa c -auanam “66¢ Auuucovn nausea o cadnaam xuom o camp-x moon o Aguasm .Huu.o chsua o nouwoun ~«ouom ocqaon 0 Nova dunuux vcuaua o xuom adage: Adansm ~n~.- soon "aqua: cumuu: xuom d aaaasm we: retiiiitttitie no~noqua> unocouovcm «iicateatthh so 22L6 wu.¢o ~m.mo N¢.N mm.~ Hw.¢ mo.¢ mam men awed oaHH onH HHNA nonaouoo uonEo>oz No.00 mH.Hn m¢.m mH.n mo.¢ «0.0 0mm mnn weed ¢m~H ocHH “Nod nonouuo nonEouoom .Hoooe oauuosocooo .ao uo .uuomz mnu a“ aoaamauw> heaaa onu mo oucowoumwooo oo.nn -.n m~.¢ onm omNH owed uwows< No.0“ oo.n no.0 mam omHH shod >354 n~.nm no.m mam wm- mmHH meow eoscaucoo .Nm magma mn.nn no.m o~.¢ mam amoH omoH >02 om.o~ H~.~ n~.¢ «mu ohm mooH H4084 mmwan ww.~ 00.0 can wmoH anH on.~o ~¢.~ aa.n New on» Noo~ oo.mo o¢.n “5.0 unn «cad oqma noumz humauaom humscah ocmEon uonoum panama woom vcweoo xuom hummam. xuom hammnm ofluumu Awmmsm mom cofiuoovm 217' rue no u: to acute 22 um: .01 poo emu oa< 431 2:» ya: gee cc: .ue anou aomn~.. aoavo.uca .amsm.o .ovua.ma oo=n~.mso«. .oun... oewom.m. gamun.~o oomnm.~o oomo~.r coca-.6 cocoa.mo~. oaauo.au~. aaavn.occ e.g.“.onu. canon.u~». oa=n*.~n~. coacn.mo~. ooaou.anu. anaon.~ou. uoaou.moo °o=~u.m.~. .oauw.~o-. mo ooosu.u coonm.». econ». guano.“ ca.a¢.¢« canon. oosnu.« o:~on.-~ .o»u..«n oonfiw.o ooooo. ea¢ow.a gonna may oaono.auo convo.uau occa~.o~u coorw.aoL oesom.swm coon»...- ooouw.suu. ouco«.«om oomoo.sva. cameo. 3 mxo aonwo. a.awm.». cooeo. °.~Hn.». a.a.n.~.. canon. a.a.e. geomu.~ e.gso.n e.-~.. .noo.. cocav.~ 9.9mn.n 9.9;“.v a.aun.c .6834.» c..n~.n ooos~.n .oo»~.. ..~u«.n. o.~v~... a°¢n«.n. canm«.on~ a. goody-3cm cacao. novsu.uo cacao. oouno.. °o°-.mu. ohdoo. noonu. oaovemau coons.« noun... guano. qoonn.« conga.» cocoa.» .aooau.w panno.n oo~fl..» cocoa.“ oouov. cannou. 90646.". ”0.444“. ooo.nuoo« an H owoum .nuoa: you oueouowuuooo avocado-u mm oHQuH cones. cocoa. gon«~.. acme». cad...» ~uona.u onana.o nuanu. ouoau. can“... nuooo.. anuam.. onoam.u one"... .um.«.. .cooo.. ocean.. ogoa«.6 econm.. oaoow.n aco-.. oovn~.u nounm.uun are aamno.~. gonna. canon.- .84.... .9896. eavaeu .50.“... oaona.~ eocsv.u oooon.uu vcooo.. oeu8°0~ aavnc.a ooocnua 595.6.8 .aon~.o ocean.» ocean.o ua~na.v ogouv.v .aumn.» ooa~n.a .oflao.nn Iut coon~.. o~aaa. oo°«~. oaoua. onuoa. nuooo.. noooa.. canon.“ ouNnn. ounce. cacao. comma.. ouaso.u oa~s~.o once... ounm~.. oueuc.u cons».- “no“... ooo-.. page... cahma.. ocean. qua gunma. onona.1 answu.u cpoos.- cn~u«.¢. auunn. guano. onaoa.-. ousmc. ouon~.u oeauo.- ounn~.~ eun4m.n guano.u ogo¢~.o oeavm.a cumun. cameo. ouaev. Queen.“ opmos.~ ou~n~. cuonn.~o« an oun~u. annom. oooou.c oomnv.n comma.- coma“. oooou.. nova».su0 coon“. no~no.. 9.959.. onoou.u o°°«~.. coun«.n oovoc.n canoe. nomad. once“. anon”. anon“. ocean.“ oo-o. oaona.no x6 Dancw.. coaun.n- no.s~. nooso.«q oo~«~.~n ooonn.. 9069”." onom~.o«v canon.o- a.a.o.au coonn. ononn.»n. aoovv.o~uu onoaq.ovn. .o.n~.u~.o oooe~.nfin. oouhn.u~na ononm.vo~o aaono.~o~. ooown.m~nu noouu.«nu. ooo~u.vono canoo.s~.. Io 218 Table B4 Structural Equations from Trierweiler - Hassler Model. Retail Beef Demand: Pbr* . 78.33983 - 2.72784 de + 0.03673 I - 1.45827 Winter - 0.93549 Spring (0.34372) (0.00423) (.75145) (0.74229) + 2.08688 Summer R2 . .693 (0.78934) Retail Pork Demand: Ppr* = 132.28963 - 1.54889 de - 5.02065 de + 0.02559 I - 6.96960 Winter (0.59060) (0.53225) (0.00723) (1.43948) - 11.05847 Spring - 8.43848 Summer 22 . .806 (1.66744) (1.83342) Retail to Wholesale Structure for Beef: 2 Pbc* = 9.96677 + 0.74830 mm: - 12.95590 l-Jr - 0.60680 A068 R .. .804 (0.03842) (0.82630) (0.15065) Wholesale to Slaughter Structure for Beef: 9651* . - 4.35784 + 0.68975 Pbc* + 0.19009 1 R2 = .951 (0.01482) (0.01702) 2 Pbsz* = - 3.06539 + 0.52421 Pbr* - 6.27350 Wr R = .760 (0.02768) (0.59443) Retail to Slaughter Structure for Pork: 2 Pps* a - 12.77266 + 0.55540 Ppr* - 1.34337 Wr R a .855 (0.02180) (0.72409) The numbers in parentheses are the standard errors of the coefficients. 219 Individual Equations of the Unger Model (1) Demand for Beef at Retail. (2) Marketing in the Cattle - Beef Sector. (3) Supply of Beef Cattl on Farms. (4) Inventory Demand for Beef Cattle Retained for Feeding. (5) Inventory Demand for Beef Cattle Not Retained for Farm Feeding. (6) Demand for Other Meat at Retail. (7) Marketing in the Other Livestock - Meat Sector. (8) Supply of Other Livestock at the Farm. (9) Cattle - Beef Sector Identity. (10) Other Livestock - Meat Sector Identity. The system is presented in the form A Y 2 B Z + U, Where the A matrix is above the row of vertical equality signs. 220 Table BS Estimated Coefficients of the Unger Model (Ordinary Least Squares) Variables (1) (Endogenous) Pbeefr Qbeefc Pomeetr Pbeeff Qbeefp IbeeffZ Pomesti Ibeefnf2 Qomeatc QOIeatp (Predeter-ined) Constant 37.01 (8.16) Dispy .3910 (.0336) katgn lbeeffi lbeefnfl Pprotein Arengeci Pomestil Pfeedidl Qomeetpl Nbieport Noni-port R .9364 1.24 1.0000 0 ”7“ (.0008) .e 3572 (.1179) (2) (3) (9) (5) -.1400 (.0402) 1.0000 105.70 047.697 -307.01 (26.06) (19.933) (73.10) .00261 1.0000 (.00045) 1.0000 76.000 (32.223) 1.0000 II II in it (2.90) (2057.42) (903.69) (4168.1) .1126 (.0201) -.4330 (.1191) .4534 .9204 (.2030 (.0659) .1452 1.0234 (.0273) (.0209) .56e27 (10.11) 53.50 (53.05) .9744 .9042 .9710 .9930 1.63 1.41 2.01 1.07 (6) .e 5651 (.0749) 1.0000 .00006 -(.00134) 84.99 (9.18) .2508 (.0511) .8283 1.25 The numbers in parentheses are :he estimated coefficient standard errors. (7) (a) (9) 1.0000 -.3502 (.0202) '1.0000 1.0000 .00116 1.0000 (.00037) u I N 23.44 0370.9 (5.62) (2267.6) .0308 (.0157) .05370 (.1442) 107.97 (67.60) '79.15 (20.997 .0300 '(.539) 1.0000 .9501 .9732 137 an» (10) 1.0000 -1.0000 1.0000 Variables (Endogenous) Pbeefr Qbeefc Pomeatr Pbeeff Qbeefp IbeeffZ Pomeetf Ibeefnf2 Qomeatc Qomeetp (Predetermined) Constant Dispy antzl Ibeeffl Ibeefnfi Pprotein Arangeci Pomeatfl Pfeedidl Qomeatpl lbimport Nomimport 0.". 221 Table B6 Estimated Coefficients of the unger Model (The Stage Least Squares) (1) 1.0000 .00744 (.0009) -e 3357 (.1348) H 38.40 (9.29) .3910 (.0376 .9363 1.26 (2) (3) .0109]. (.0456) 1.0000 92.08 (28.10) .00303 1.0000 (.00052) H II 20.00 7665.29 (3.07) (2084.33) .1307 (.023) .03”, (.1266) .4521 (.2046) .1458 (.0275) -55s0, (18.26) .9729 .9839 1.06 1.40 (4) -42.635 (21.161) 1.0000 60.437 (37.917) M 937.13 (1036.12) .9447 (.0727) .9706 2.85 (5) .371e83 (76.74) 1.0000 -11138.2 (4171.6) 1.0239 (.0209) 56.11 (53.45) .9937 1.07 (6) 's“19 (.0872) .00094 (.00172) 86.50 (11.38) .2787 (.0650) .8155 1.40 The numbers in parentheses are the estimated coefficient standard errors. (7) (5) (9) (10) 1.0000 -.3444 (.0290) -1.0000 1.0000 1.0000 .00120 1.0000 (.00044) in a: '1 ll 24.24 0378.9 (6.23) (2267.6) .0324 (.0180) -05“, (.1556) 107.97 (67.60) -79.15 (20.99) .0308 (.0539) 1.0000 1.0000 .9500 .9732 1.54 2.86 2122 Table B7 Estimated Coefficients of the Unger Model (Unbiased K - Class) Variable! (l) (3) (3) (‘5) (5) (6) (7) (3) (9) (10) (Endogenous) Pbeefr 1.0000 -.0966 -.6726 (.0468) (.0903) Qbeefc .00746 1.0000 (.0009) Pomeetr -.3274 1.0000 '-.3426 (.1349) (.0290) Pbeeff 1.0000 87.73 «60.310 -374.35 (28.35) (21.356) (76.77) Qbeefp .00320 1.0000 -1.0000 (.00054) Ibeeff2 1.0000 Pomeetf 52.330 1.0000 (38.266) Ibeefan 1.0000 Qomeatc .00939 1.0000 (.00178) Qomeatp .00121 1.0000 -1.0000 (.00044) (Predetermined) 1' H 11 11 1| M u u 11 11 Constant 38.93 21.30 7551.18 770.31 -11155.1 87.77 24.49 8378.9 (9.30) (3.15) (2096.57) (1045.66) (4173.3) (11.79) (6.24) (2267.6) Dispy .3914 .1381 .2935. .0329 (.0377) (.0239) (.0673) (.0188) katgm -.3586 -.5474 (.1298) (.1557) Ibeeff1 .4517 .9571 (.2058) (.0734) Ibeefnfl .1460 1.0241 (.0277) (.0209) Pprotein -54.85 (18.37) Arengeci 57.07 (53.47) Pomeatfl 187.97 (67.60) Pfeedidl -79.15 (20.99) WCCP1 .0300 (.0539) lbimport 1.0000 Iomisport 1.0000 R2 .9361 .9715 .9837 .9700 .9938 .8019 .9579 .9732 D.". 1.27 1.93 1.07 2.85 1.07 1.44 1.53 2.86 The numbers in parentheses are the estimated coefficient stanierd errors. 223 {Tamales 118 Estimated Coefficients of the unger Model (Three Stage Least Squares) Variables (1) (2) (3) (4) (5) (6) (EndOgenous) Pbflefl' 1.0000 -.1100 -065” (.0336) (.0686) Qbeefc .00676 (.00077) Pomestr -.4713 1.0000 (.1089) Pbeeff 1.0000 79.286 -36.475 -358.63 (24.074) (17.390) (67.72) QbCQfP .m304 1.W00 (.00038) IbeeffZ 1.0000 Pomestf 49.330 (30.498) Ibeefnf2 1.0000 Qomeetc .00843 (.00135) Qomeetp (Predetermined) H '1 N u u '1 Canstent 8.92 21.38 5093.26 731.67 12564.8 82.86 (7.55) (2.24) (1432.29) (850.52) (3419.8) (9.02) Dispy .3654 .1311 .2577 (.0322) (.0171) (.0511) “it“ '0‘“, (.0846) Ibeeffl .6826 .9669 (.1497) (.0599) Ibeefnfl .1364 1.0263 (.0203) (.0186) Pprotein ~32.27 (12.14) Arengeci 77.66 (43.20) Pomestfl Pfeedidl Qomeatpl Mbimport Mamimport 22 .9322 .9727 .9809 .9696 .9937 .0142 0.". 1.18 1.84 1.05 2.88 1.05 1.34 The numbers in parentheses are the estimated coefficient standard errors. (7) (a) (9) (10) 1.0000 ‘e3563 (.0246) -lem 1.0000 1.0000 .00130 1.0000 -1.0000 (.00036) 11 11 11 1' 24.00 9933.1 (4.98) (1810.1) .0364 (.0153) ’.5300 (.1185) 224.89 (51.85) '96.20 (15.94) .m: (.0451) 1.0000 1.0000 .9572 .9721 1.60 2.89 Vsriables (Endogenous) Pbeefr Qbeefc Pouestr Pbeeff Qbeefp Ibeeffz Poneatf Ibeefan Qoneatc Qonoatp (Predeteruined) Constant M 999 katgm Ibeeffl Ibeefnfl Pprotein Arangeci Pomeatfl Pfeedidl Qoneatpl Nbimport Nouimport 0.". (1) 1.0000 .00667 (.0007) ’a‘57l (.0941) 29.63 (6.51) .3621 (.0300 .9322 1.20 (2) .al751 (.0258) 1.0000 .00222 (.00028) 19.15 (1.80) .0949 (.0127) 'aS“? (.0542) .9729 1.32 .2134 (3) 65.67 (34.47) 1.0000 {renalxa 159 Estimeted Coefficients of the Unger Hodel (4) ~38.811 (17.101) 1.0000 49.584 (30.343) -974.665 677.61 (1160.22) (849.65) .7927 (.1123) .1701 (.0167) 22.74 (7.463) .9636 0.809 .9695 (.0597) .9697 2.88 (5) -362.11 (63.68) 1.0000 (6) -.6558 1.0000 .00803 (.00112) u -10803.69 80.18 (2567.72) (7.72) 1.0260 (.0182) 54.56 (29.77) .9937 1.07 .2421 (.0423) .8129 1.29 The numbers in parentheses are the estimated coefficient standard errors. (7) -.“74 (.0242) 1.0000 .00128 (.00033) 23.09 (4.53) .0349 (.0136) .e‘737 (.1057) .9562 1.66 (8) 1.0000 I 13237.9 (1815.7) 246.35 (50.68) -1zo.sw (13.36) .7487 (.0481) .9651 2.73 (Iterated Three Stage Least Squares) (9) (10) 1.0000 ‘lem 1.0000 -lam n a 1.0000 1.0000 Variables (Endogenous) Pbeefr Qbeefc Poneatr Pbeeff abs-iv Ibeeffz Poneatf Ibeefnfz Qomeatc QOI-ltv (Predetermined) Constant 010?! katgm Ibeeffl Ibeefnfl Pprotein Arangeci Pomeatfi Pfeadidl .Qoueatp1 lbieport Iona-port (1) 1.0000 .01726 (.01059) 1.2755 (1.7228) 148.04 (117.36) .7790 (.4211) .0852 1.08 11215 Table BIO Estimated Coefficients of the Unger Model (Limited Informetion Single Equation) (1) .a0281 (.0718) 1.0000 .00428 (.00001) 23.85 (4.30) .1852 (.0835) .,2190 (.1824) .9513 1.95 (3) 54.24 (33.39) 1.0000 6672.29 (2316.29) .4485 (.2251) .1474 (.0303) 052.20 (20.12) .9003 1.27 (4) (5) “7.601 .237e1’ (37.516) (94.90) 1.0000 -92.960 (92.073) 1.0000 n q I ~2140.17 -11022.0 (2354.29) (4659.6) 1.1779 (.1571) 1.0323 (.0235) 94.91 (60.71) .9237 .9922 2.01 1.00 (6) -.7798 (.1256) 1.0000 .01167 (.00247) 96.58 (15.19) .3725 (.0920) .6962 1.49 The numbers in parenthebee are the satin-tad coefficient standard errors. (8) (7) (9) '2e6212 (87.0304) -lem 1.0000 .17710 1.0000 (6.71704) \1 II M 1901.59 8378.9 (71690.65)(2267.6) 7.4940 (284.93) ~42.6475 (1607.81) 187.97 (67.60) '7’e15 (20.99) .8308 (.0539) ‘Cm .7730. e’732 2.01 2.86 (10) 1.0000 -1.0000 1.0000 Table C1 226 Reduced For. Equetione Corresponding to Hyers' Structural Model 00.000.000.000... Exogenous Qh Variables Usp -.4277 Nb 0.0060 R -10.4625 Prcrn 85.1199 Ch -313.5891 Nc 0.0000 Cc 0.0000 9 1.1615 1 0.0000 Qch 0.0000 T '2.390) Jan -480.7007 Feb -377.0397 Her -537.9164 Apr -432.0079 Hay -466.9966 Jun ~455.3714 Jul ~475.7618 Aug -576.3692 5... ~639.6354 0c; -710.3619 Nov -567.8457 Dgc -534.8552 09p ~2.4659 Rh -.0810 R 18.6413 Prcrn -649.7954 Ch -1586.0803 Nc 0.1656 CC 959.1362 0 -9.9>87 1 20.8793 Qch -780.7711 2 -9.3280 Jan 587.5373 Fob 1040.7661 {Dr 1050.9661 Apr 1400.2547 bay 1435.1041 Jun 1308.8680 Jul 1469.0279 498 1413.0975 59? 1530.8551 Oct 796.4410 NOV 724.8449 Dec 279.6737 03p 0.0168 Nh 0.0006 R 0.5554 Prrrn 7.8565 ”6 10.7953 "' 0.0000 Cc 0.0000 0 0.1072 1 -.3519 th 6.1423 T 11.10““ JO" -3.4864 Yeb -5.9s7s “GT -6.3167 Apr -8.4783 “'9 -8.5551 Jun -6.9319 Jul -7.3702 Au8 -6.6376 590 o6.8357 Oct -1.4624 “0V -1.9929 Dec c.4428 The 5(3) polynoaiel is Ph Pp de Pch 068 The following coefiicients ere multiplied by one. 0.0111 7.0279 -.0010 0.0049 0.0004 0.0005 0.0009 .,0001 0.0002 0.0000 0.5252 0.9221 0.0120 0.2937 -.1678 6.9861 13.0434 -.4181 2.9439 -.7297 10.6174 17.9225 -1.0204 3.1265 0.2646 0.0000 0.0000 0.0001 0.0002 -.0003 0.0CJO 0.0000 0.6171 1.2713 -1.7257 -.C437 0.1780 -.0064 0.0387 -.0080 0.0000 0.0000 0.0134 0.1602 0.0675 0.0000 0.0000 -.5023 ~21.5589 -.0798 0.1521 0.1816 -.0060 0.0406 -.0097 68.6859 128.6152 0.3780 67.0902 -.2145 54.1094 102.1752 0.6696 60.7960 -.4327 63.6554 119.0372 0.6762 69.6636 -.3642 56.2891 105.9622 0.9009 68.7096 -.6702 58.7703 110.3687 0.9233 71.3386 -.5144 59.7244 112.1501 0.8421 71.6275 -.6795 58.3855 109.6360 0.9451 72.7975 -.8618 62.5068 116.8079 0.9091 73.7181 -.9382 62.0672 116.8813 0.9849 69.8493 -1.1022 69.5756 129.2425 0.5124 66.9674 -1.0795 66.8865 126.8295 0.4663 63.9055 -.8225 69.5941 129.8848 0.1799 64.9062 -.3962 $nJobovous Variables tittitfiitfliififltfiti Pc 0.0000 0.0000 1.4711 7.4037 0.0000 0.0024 14.0757 0.0000 0.0000 0.0000 0.0970 35.2707 27.1096 32.3445 30.4126 34.2.53 35.4606 36.9563 40.0723 40.3216 44.1231 37.0809 33.9337 Pb 0.0000 0.0000 3.0395 15.2969 0 . 0.300 0.0050 29.0819 0.1758 0.0000 0.0000 0.2045 92.9767 74.0487 84.8647 80.8731 88.7919 91.3028 96.3931 100.8312 101.3462 109.2007 94.6506 88.1481 The following coefficients are Inltiplied by Population/g(Populstion) . 0.0927 0.0030 -.7009 24.6330 59.6383 -.0062 -36.0645 0.3745 -.7851 29.3579 0.3207 c22.0920 -39.1339 ~39.5174 ~32.6511 -53.9614 -49.2148 -55.2370 -53.1340 -57.5618 -29.9470 -27.2549 -10.5160 0.1607 0.0053 o1.2164 42.3328 103.3298 -.0108 -62.4857 0.6688 -1.3602 50.8656 0.6077 -38.2768 -67.8)37 ~68..662 -91.2237 -93.4940 -85.2700 -95.7041 -92.;603 -99.7320 -51.8865 -47.2221 -18.2202 -.0097 -.0003 0.3175 -l.3302 ~6.2451 0.0011 6.1113 -.0251 0.0072 -2.7730 -,fl;03 2.6968 4.5078 6.4373 5.8898 6.3844 5.8607 6.724. 6.6301 7.3096 4.5519 3.9039 1.6237 0.0269 0.0009 0.2998 9.609. 17.2786 -.0010 ~5.6376 0.1375 -.3821 9.1160 0.1354 -6.0228 -10.5050 -10.8337 -14.4788 -14.7401 -12.8019 o14.0663 -l3.l977 -14.0756 -5.7591 -5.7212 -l.9702 0.0038 0.0001 -.7110 -2.4426 2.37:; -.0014 -7.9952 '.0242 0.1780 0.3639 -.0317 -1.4100 -Z.7l61 o2.4417 -3.1910 '3.4048 -3.9762 -4.8728 -5 '5.9229 -5.1761 -4.0486 01.8883 -.0113 -.0003 5.2200 22.7380 -7.6206 0.0093 53.7726 0.2490 -1.4799 2.4856 0.3609 6.6833 13.4825 11 . 3:959 16.6521 16.0281 21.1749 26.6531 29.2346 34.3652 33.6330 25.6287 12.3444 -.0265 -.0009 11.7028 50.8992 -17.0588 0.0208 120.3706 0.5575 -3.3128 5.5640 0.6735 14.9607 30.1807 23.4047 32.7999 35.8791 47.4002 60.1109 65.4419 76.8820 75.2991 57.3702 27.6331 Qpe Qc 0.0000 0.0000 0.0000 0.9000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0900 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -.8821 0.3802 -.0290 0.0125 6.6682 -167.7757 -232.6396 -729.7068 o567.3604 244.5605 0.0592 -.2980 343.0948-1725.6713 -3.5623 -7.9923 7.4688 47.4928 -279.2914 -79.7680 -3.3367 -9.6557 210.1693 -214.483. 372.2948 -432.650i 375.9635 -364.2099 500.3583 -470.2139 513.3543 -514.3744 468.1961 .679.5438 525.4893 c861.7696 505.6823 -938.1955 547.6056-1102.2054 284.8967-1079.5112 259.2859 -822.4770 100.0427 -196.1576 The following coefficients are multiplied by (Population)2/g(Population) '.0006 '.0000 -,_209 '.2954 '.4059 '.0000 “.0000 ‘o0040 0.0132 -.3310 '." .01 0.1311 0.2240 0.2373 0.3180 0.3217 0.2606 0.2771 0.2496 0.2570 0.0530 0.0749 0.0167 u(I) -.0011 -.0000 ~.0362 -.5118 .,7033 —.0000 -.0000 -.0070 0.0229 -.4002 ..:drl 0.2271 0.3881 0.4115 0.5523 0.5.73 0.4516 0.4802 0.4324 0.4453 0.0953 0.1298 0.0289 0.0001 0.0000 -.0005 0.0164 0.0400 -.COOO -.0242 0.0003 -.0005 0.0197 .u. .7' '2 *."51 '.0205 -.0354 -.0362 -.0330 °.0371 -.0357 -.0386 -.0201 -.0183 '.0071 -.0002 -.0000 ~.0115 -.1155 ~.1227 -.0000 -.0499 -.0015 0.0056 -.0761 -.0016 0.0357 0.0591 0.0654 0.0883 0.0880 0.0637 0.0637 0.0527 0.0503 -.0136 0.0002 -.0061 '.0000 -.0000 0.0066 0.0286 -.0096 0.0000 0.0677 0.0003 -.0019 0.0031 0.0004 0.0084 0.0170 0.0143 0.0185 0.0202 0.0267 0.0338 0.0368 0.0433 0.0424 0.0323 0.0155 9.0000 -.0000 -.0533 '.'6 t -.0000 -.0001 -.so9a -.0031 0.Ll64 -.0667 -.0o36 -.0400 -.0660 -.0653 -.oezz -.0947 -.1544 -.2099 -.2327 -.2799 -.3091 -.2296 -.1161 936.6 - 9.9043691807 n + 0.039240698583829 .2 -.0000 °.00)Q ‘.1193 -.6003 °.0000 -.0002 ~1.1412 -.0069 0.0367 0.1448 -.0080 “.0896 '.1969 '.1462 -.1839 -.2120 0.3455 0.4595 '.5210 -.6265 -.6920 0.5141 .0255] 0.0060 0.0000 (.0002 0.0000 0.1987 1.7099 2.8104 8.0055 3.8616 0.0000 0.0000 0.0028 0.0000 16.3605 0.0383 0.0988 -.1259 .,5259 2.1972 2.0755 0.0391 0.1150 -1.2471 1.2846 -2.1311 2.8226 -2.2596 2.0963 -3.0328 2.6370 -3.0602 3.0392 -2.4796 4.9537 -2.6364 6.5872 -2.3744 7.4691 -2.4452 8.9815 -.5231 9.9204 -.7129 7.3696 -.1584 3.6607 ”17111111111111[11111111111111111‘5