43" a .. 43 fl ‘ i ,p, t.fi;{} -r A." V E m h ?~ A 'ct-M i- _ ,_\l‘_.'J,':.:.:,3 ‘0; €33 " J “i” H ‘ r ‘3" ‘ ., ‘3 13’ an? ‘77 a . a V ‘2 This is to certify that the thesis entitled FORECASTING THE MONEY STOCK presented by Philip Pfaff has been accepted towards fulfillment of the requirements for Ph.D. degree in Economlcs \ Major professor Date I 1-6-73 0-7 639 . ,‘ g 343‘; ..~.- g- .h- - armfi- ABSTRACT FORCASTING THE MONEY STOCK by Philip Pfaff A number of economic models exist which have the money stock as the endogenous variable. However, these models have not been systematically exposed to what many maintain is the ultimate test of an economic model: predictive per- formance. The subject of this dissertation is the examin- ation of the predictive performance of a cross-section of money stock models. The models examined ranged from mechanistic models to two equation models estimated with TSLS estimation techni- ques. Some models examined had explanatory variables which primarily reflected the behavior of the banking system, e.g. reserves and the discount rate, while other models had explanatory variables which primarily reflected the private non-bank sector, e.g. interest rates and income. The mech- anistic models were either autoregressive models of the money stock or money multiplier models which assumed that the multiplier either did not change or changed at a con- stant rate over the-forecast period. Philip Pfaff Predictive performance of the models for the 1961—1970 period was examined. Where parameter estimation was necessary, 1947-1960 data was used. In order to reduce the role of judgment in using the models, all models forecast ex post, i.e. the actual values of the exogenous variables were used. The root mean square error (RMSE) statistic was used as the measure of predictive performance. Other pre- diction evaluation statistics were used, but their ranking of predicitve performance differed little from the RMSE ranking. The autoregressive seasonally adjusted money stock and the no-change money multiplier model were the two models which forecast with the lowest RMSE. The strong time trend of the money stock data was one explanation of the good predictive performance of the autoregressive model while the relative stability over time of the multiplier con- tributed to the low RMSE of the multiplier model. The best performing economic models, i.e. models with expanatory variables in addition to the lagged values of the dependent variable, were a number of single equation models. One had a short-term interest rate, income (or permanent income), and the lagged dependent variable as explanatory variables with all quantity variables expressed in nominal or real terms. Another had total reserves plus reserves released through changes in the reserve require- ment, the short-term interest rate, and the discount rate as explanatory variables. No two-equation model performed Philip Pfaff better than these single equation models just mentioned. Predictive performance was improved in this dissertation a number of ways. Including the lagged dependent variable in a forecasting equation almost without exception improved prediction performance. Linear combinations of the resi- duals of prior period(s) when added to the constant term of the equation also lowered the RMSE. Correction for first-order autocorrelation also improved predictive per- formance. Where possible, the models were tested for the existence of structural shift. In this study it was observed that structural stability was neither a necessary or a sufficient condition for a low RMSE forecast, i.e. in some cases where the hypothesis that structural shift had occurred could not be rejected, the model forecast with a low RMSE; and in other cases where the hypothesis was rejected the model forecast with a relatively high RMSE. The impressive performance of the mechanistic models vis- a-vis economic models in forecasting the money stock should serve as a challenge to the econometric model builder. It also indicates that such mechanistic models provide a tough standard of comparison for conventional economic models. FORECASTING THE MONEY STOCK BY Philip Pfaff A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1973 ACKNOWLEDGMENTS The following people must be thanked for their help with this dissertation. Foremost is Professor Robert Rasche for being the ideal dissertation director. He was always available with cogent advice and criticism. What merit this dissertation has is due in large measure to his active interest in the project. The other two members of my dissertation committee, Professor Maurice Weinrobe and Professor Carl Gambs, must also be thanked for their assistance and guidance. The standard proviso clause for a dissertation--i.e. all short- comings of the dissertation are mine--is particularly true for this one. My wife Kristen must be thanked for her patience and support during the dissertation ordeal. Finally my MSU colleagues--both in and outside the Department of Economics --must be thanked for helping make the dissertation exper- ience a humane one. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES Chapter I. INTRODUCTION II. FORECASTING WITH AN ECONOMETRIC MODEL Forecasting models Non-econometric forecasting models Econometric forecasting models Judgment and forecasting with an econometric model Forecasting errors in econometric models Sources of forecasting error Reducing forecasting error Evaluation of the performance of a forecasting model Conventional test statistics Other test statistics Studies of forecasting performance Comparison with mechanistic models Other prediction evaluation statistics Summary Appendix to Chapter II III. ECONOMETRIC MODELS OF THE MONEY STOCK Money multiplier models Money supply models Money demand models Simultaneous equation models Money supply and demand models The Teigen model Large econometric models Summary iii Page ii vi 14 22 27 28 32 32 36 42 47 Chapter Page IV. PREDICTIVE PERFORMANCE OF NAIVE MONEY STOCK MODELS 48 The models 48 Mechanistic models of the money stock The money multiplier models The decomposed money multiplier models Summary Evaluating prediction performance 54 Results 55 Autoregressive models Evaluating predictive performance The Burger multiplier model Summary 70 Appendix to Chapter IV 72 V. FORECASTING WITH SINGLE EQUATION MODELS OF THE MONEY STOCK 73 The models 73 The conventional money demand function The per capita money demand function The real money demand function Evaluation of the money demand models Money supply: the Brunner model Money supply: the Teigen and Gibson models Evaluation of predictive performance 101 General evaluation Constant adjustments The Chow test . Predictive performance 107 Structural stability of single equation models The conventional prediction evaluation statistics of the money demand models The Brunner model The Teigen and Gibson models Adjustments of the constant term Summary 120 iv Chapter VI. TWO-EQUATION MODELS Introduction The models The money multiplier model The Brunner—Meltzer model The Teigen-Gibson models The estimated models Evaluation of prediction performance The results Forecasting the money stock with recursive models Forecasting the money stock with non-recursive models Forecasting the interest rate Summary VII. CONCLUSION Forecasting performance Improvement of forecasting performance Two-equation models ’ Evaluation of prediction Income and interest elasticities Summary BIBLIOGRAPHY NOTATION APPENDIX DATA APPENDIX Page 121 121 123 134 136 145 146 146 147 149 150 151 152 153 161 164 Table 2.1 4.1 4.2 LIST OF TABLES Ranking of prediction test statistics Estimated coefficients of seasonal dummies for Equation 4.10 Standard error's of autoregressive models as the number of lagged terms vary COOper test statistics for auto- regressive money stock models COOper test statistics for money multiplier models Predictive performance of mechanistic money stock models Annual predictive performance of mechanistic money stock models Ranking of the predictive performance of mechanistic models Annual predictive performance of selected models Summary of notation for Chapter V Estimated coefficients of the money demand model Estimated coefficients of the per capita money demand model Estimated coefficients of the real money demand model Selected estimated money demand models Estimated coefficients of annual money demand models without lagged variables vi Page 26 60 60 63 63 64 66 68 68 74 76 78 80 82 Table 5.7 5.20 5.21 Interest and income elasticity of quarterly money demand models Annual money demand models with lagged variables Interest and income elasticity of annual money demand models Quarterly nominal money demand model fitted over three time periods Estimated coefficients of the Brunner model Brunner's regressions Estimated coefficients of the Teigen model Comparison of the Teigen model Estimated coefficients of the Gibson model Comparison of the Gibson model Selected F-ratios for (n + m) = 100 Chow test statistics Predictive performance of money demand models Annual predictive performance of quarterly money demand models Predictive performance of the Brunner model Annual predictive performance of quarterly money supply models Predictive performance of the Teigen model Predictive performance of the Gibson model vii Page 87 88 90 91 93 94 96 98 100 102 .106 108 110 112 114 115 116 116 6.'7 Predictive performance of selected models with constant adjustment terms Estimated coefficients Of the C-series models Estimated coefficients of the money supply models equations of the D-series Estimated coefficients of the money demand models Predictive models Predictive models Comparison equations of the E-series performance of two-equation of the money stock performance of two-equation of the interest rate of the prediction of the money stock and the interest rate Annual predictive performance of selected models viii Page 118 128 132 133 138 138 139 144 CHAPTER I INTRODUCTION Many economists view the predictive performance of an economic model as its ultimate test. In light of this, in recent years increased attention has been paid to the pre- dictive performance of econometric models. Most of this in- terest, however, has been focused on forecasts of GNP, or its principle components, while little attention has been paid to models that can be used to forecast the money stock. This dissertation is an attempt to begin filling in this lucuna. This project will inevitably go beyond the question of which money stock model forecasts best. For example, it will have implications for the question of the importance of the role of money in the economy. The dissertation will lalso briefly discuss a variety of prediction evaluation statistics, and examine how well they perform in actual practice. But the basic question will remain: how satisfied can the econometric model builder be with the current models of the money stock. frhe dissertation will be organized in the following fashion. Econometric forecasting and its evaluation will be surveyed in Chapter II. An introduction to the various technixgues of evaluating the performance of predictions will also be part of this chapter. Various money stock models which can be used for forecasting are then described in Chapter III. The next three chapters discuss the appli- cation of these models to forecasts of the money stock for the 1961-1970 period. Data from the 1947-1960 period are used to estimate the parameters of the models. Chapter IV will deal with meChanistic money stock models, Chapter V with single-equation models and Chapter VI with simple simultaneous equation models. Chapter VII will tie to- gether the results derived from these chapters. CHAPTER II FORECASTING WITH AN ECONOMETRIC MODEL Until recently economic forecasting has been an "artistic, subjective, and personal" endeavor [Klein, 1968, p 9], but with the develOpment of theory--both economic and econometric--and the availability of reasonably reliable and extensive data and machines capable of manipulating this data, economic forecasting has become more objective. Most forecasting models used by economists still require the use of the fore— caster's judgment in order to make forecasts with small errors. Nevertheless the results of mOst forecasting models can be replicated and the fore- cast analyzed. It may, in fact, be possible to as- certain, at least partially, the source of a forecast error. The develOpment of a number of different models each explaining the same economic variable (e.g. GNP) has allowed us to compare the forecasting ability of particular models (as Opposed to comparing economic forecasters who would usually have difficulty con- sistently replicating their forecasts). In this chapter econometric forecasting and its eval- uation will be discussed. In Chapter III econometric forecasting of the money stock will be discussed. 3 A. FORECASTING MODELS l. Non-econometric forecasting models The business cycle attracted the early economic fore- casters. For example, Wesley Mitchell and the National Bureau of Economic Research (NBER) attempted to forecast the behavior of economic aggregates such as GNP and the national income components by tracking leading indicators: economic events which usually preceeded turning points in the business cycle.l' Since World War II, forecasts based on data from surveys of the buying intentions of consumers have been made. These forecasts using anticipations data have been used to predict either consumption (usually of durables) or investment. While on the whole such fore- casts are unreliable [Evans, 1969, p 494], they are fre- quently more reliable than mechanistic forecasts,3 and so . . . . 4 this forecasting technique continues to be used. 1. Use of the leading indicator methodology was not limited to the NBER. For an example a bit removed from the NBER mainstream, see Roos, 1955, pp 369-379. 2. For a review of early attempts at economic forecasting, see Zarnowitz, 1968, 1968; R003, 1955. For a recent dis- cussion of leading indicators and NBER methodology, see Evans, 1969, pp 445-460. 3. Mechanistic forecasts depend solely on the statistical characteristics of the data series, e.g. an autoregressive model. A good example of such a mechanistic model is Jus- ter's, an autoregressive model with up to 8 lagged values of the dependent variable. Juster, 1969, pp 226-229. 4. This grossly oversimplifies the case for forecasting with anticipations data. It is, however, an example of a widely used forecasting technique which forecasts {(‘t. [{(l.(.((l([‘ [l.[( [I [2" (.[11 [.‘ I ill! [.[Z'lll 4" (III 6". l ( (I [I Economists (usually along with businessmen and govern- ment officials) have been frequently surveyed to determine what they feel certain economic aggregates will be in the future. Since the forecast aggregate depends upon the judgment of the participants in the survey, this forecast is frequently called a judgmental forecast. Zarnowitz [1967] examined the prediction performance of these fore- casts and found that judgmental forecasts performed better than did no-change and same-change extrapolations of the series being forecast. Other studies have confirmed that judgmental forecasts predict better than mechanistic fore- casts.5 Of course the success of recent judgmental fore- casts must be viewed with caution since the judgments of many individual economists (and others) may have been influenced by forecasts made with econometric models. 2. Econometric forecasting models The econometric model has greatly increased the ob- jectivity Of the econometric forecast.6 While there were variables of interest to the economist, but which makes little use of econometric techniques or the more conven- tional economic data series. 5. For example, see Stekler, 1970, pp 74-91. Smyth, 1966 summarizes the performance of Australian judgmental fore- casts and compares them with judgmental forecasts for other countries. 6. Forecasting it seems will never be completely objective. Much judgment is required to successfully specify, estimate, and use even (or especially?) the largest econometric fore- casting models. earlier macro-econometric models, the Klein-Goldburger model is the water-shed in the use of macro models by the economist.7 Christ's review article of the Klein-Gold- burger model [Christ, 1956] included one of the first eval- uations of the predictive performance of an econometric model. The shift in attitude towards econometric models which occurred around this time can be seen in two articles which discussed econometric methods of forecasting. Bassie, for example, saw too many inflexibilities introduced by the econometric approach to forecasting, and maintained that such an approach would remain impractical. [Bassie, 1955, p 33]. R003 [1955], on the other hand, gave an optimistic prediction of the future role of econometric models in forecasting. Econometric models have come a long way since the Klein- Goldburger model. Many are quite small, consisting of one or two equations; while others are quite large.9 Some have 10 been designed explicitly for forecasting, while others have been designed to examine the behavior of a certain 7. The review article by Fox, 1956 brings this out most strongly. 8. R005 stated that "the modern econometric forecaster, viewing the economy as an elastic membrane in disequilibrium, is forever on the altert to identify new impressed forces that might reinforece or negate his forecasts." Roos, p 395. 9. For example, The Brookings model. Duesenberry, 1965. 10. For example, the Wharton-EFU model. Evans, 1968. sector of the economy.11 There are, in fact, a number of models which can be used to forecast the money stock some of which will be discussed in the next chapter. 3. Judgment and forecasting with an econometric model While the objective nature of forecasts generated by an econometric model will be emphasized in this study, judg- ment is necessary for successful econometric forecasting. For example, in the actual forecasting situation the values of the exogenous variables must be chosen. In many cases the constant terms of the equations are adjusted to re- flect the forecaster's estimation of the size and impact of structural shifts in the economy, or the behavior of the residual term of the pervious periods. Further adjustments may be made if, after making a preliminary forecast, the resultant prediction of the endogenous variables simply do not look correct to the forecaster.12 Since we want to compare the predictive ability of vari- ous econometric models, and not the various forecasters, the role of judgment in using a forecasting model must be minimized. Thus in evaluating forecasts actual values of exogenous variables for the periods forecast (henceforth, ex post values) will be used rather than the values of the 11. The FRB-MIT model, for example, emphasized the role of the financial sector of the economy. 12. For a description of this process, see Evans, 1972, pp 1126-1128, and Klein, 1968, pp 50-51. exogenous variables which are themselves predicted when making forecasts (henceforth, ex ante values), and while constant adjustment terms may be used to improve the per- formance of the models, the adjustments will be mechanical. This approach rules out evaluation of forecasts using ex ante data and equation adjustments based on the forecaster's feel of the economy.13 Forecasts using ex ante data fre- quently predict better than forecasts using ex post data even though both forecasts make use of constant adjust- ments.14 B. FORECASTING ERRORS IN ECONOMETRIC MODELS 1. Sources of forecasting error The world is stocastic, not deterministic, and therefore we will always be confronted with forecasting error. The relationship between the dependent variable (Yt) and the independent variable (xt) can be expressed in the form (assuming that the relationship between the two is linear): (2.1) Yt = Bo + let + et where et is a random disturbance term. Standard 13. Klein, 1968, p 42 maintains that such judgmental adjust- ments are an important element in any realistic forecasting situation. So a loss of realism is the price of using the ex post approach. 14. See Haitovsky, 1970 and Su, 1971. Possible explanation of why the use of ex ante data leads to better forecasts than the use of ex post data is discussed in these articles. econometric theory is that the total variance of the fore- casting error of the prediction15 (2.2) Yt = B0 + 31x0 . . . 16 is given by the expreSSlon m 2 2 2 1 (x0 ' X) (2.3) SF = s 1 + ;'+ n 2 t=l (Xt ‘ X) ,This suggests that there is a lower limit to forecasting accuracy.l7 Estimating the forecasting error for more complex situations is difficult. Klein [1968, pp 27-28] has developed a numerical technique for determining the standard error of forecast for equations with lagged endogenous variables, while Feldstein [1971] has con- structed an estimator for the interval of the forecast error when the exogenous variables themselves are stochas- 18 tic. But except for the most elementary models, A 15. §t is the predicted value of Yt given K0. Bo and B1 are estimated values of B0 and BI' (Yt - Yt) is the fore- cast error, and sF is an estimate of that error. 16. Kmenta, 1971, pp 240-241. 52 is an estimate of the variance of (Yt - (B0 + BIX)). For the multiple regression case, see Kmenta, 1971, p 375. 17. This expression tells us explicitly that the further away the independent variable is from its mean, the larger the standard error of forecast. 18. But three rather restrictive assumptions underlie Feld— stein's estimation of the interval: the stochastic distur- bance is not autocorrelated, lagged endogenous variables are not used, and only linear models are considered. 10 estimation of the forecast interval is difficult. Even when the forecasting error can be estimated, however, the error is usually greater than the accuracy requirement of the forecaster [Klein, 1968, p 40], and therefore the fore- caster is forced to resort to an activity called "fine tuning." [Evans, 1972, p 952]. This, of course, implies the application of judgment to forecasting with econometric models. The forecaster frequently must use preliminary data which is to be subsequently revised. There is a trade-off between data accuracy and reporting speed [Stekler, 1970, pp 102-121], although in recent years the quality of pre- liminary data has improved. [Zellner, 1958; Cole, 1969]. 19 While this is a problem for the forecaster, the evaluator of forecasting performance usually has the revised data available.20 An econometric model assumes that during the period of the fit, the relationship between the variables in the model remain fixed (or at least changes are allowed for by dummy variables). If the model is used for forecasting, the relationship is assumed to remain fixed. But the world l9. Klein, 1968, pp 68, 42. Klein sees it as an important research task. 20. Poor data may influence the construction and use of a econometric model. Do adjustments to the constant term compensate for poor data, or for structural shifts? This question is not dealt with here. 11 is not that neat. Changes do occur.21 Such change may be compensated for by adjustment of the constant term al- though frequently Change is either not noticed, or is difficult to quantify. COOper maintains that statisti- cally significant shifts do occur even within a short time horizon.22 Usually the constant term is adjusted in the short run and the model is periodically reestimated. Klein [1968, p 51], however, has pointed out that a freshly esti- mated system needs adjustment as much after a year or two as it does after four or five years. This adjustments, unfortunately, are usually based on the forecaster's feel for the economy and for how expected changes will appear in the model. Since we are interested in examining the behavior of the forecasting model and not the forecaster, as mentioned above, adjustments of this type will not be considered. A forecasting model may be misspecified, and if the misspecification biases the disturbance variance term, forecasting error is unnecessarily high. The three mis- specifications which do increase forecasting error are omitted variables, incorrect functional forms, and 21. Another possibility is that the model was misspecified. In this case, while the world may have remained fixed, the hypothesis that this was the case could be rejected by a test for structural stability. 22. Cooper, 1972, pp 900-915. Cooper fitted his models over the period 1949-1 to 1960-4, and found that there was a significant shift of many of the equations in the models during the 1961-1 to 1965-4 period. l2 heteroskedasticity of the disturbance terms. Careful examination for misspecification is usually not part of the model estimation procedure. Another source of forecast error is incorrect estimation of the values Of the exogenous variables. Evans [1972, pp 954-956] maintains that the only correct way of evalu- ating the forecasting ability of a model is by examining its ex ante forecasting ability. But as Cooper [1972, p 816] has pointed out, only by comparing the accuracy of ex post forecasts is the judgmental element held con- 23 stant. In this study, therefore, the ex post values Of the exogenous variables will be used. 2. Reducing forecasting error When a forecasting model has autocorrelated residuals, adjustment Of the constant term is explicitly specified by the model. The adjustment for first order autocorrelation for a T period forecast is T (2.4) A = p e t+T t where A is the constant term correction factor, p is the autocorrelation coefficient, t is the period when the forecast was made, T the length of the forecast, and e is the observed residual for that period.24 A refinement 23. Stekler [Evans, 1972, p 1141] in his discussion of the Evans approach makes the same ObServation. 24. Goldberger, 1962. See also Klein, 1968, pp 68, 51-55. 13 of this adjustment i525 (2 5) A: =IpT et + pet-l ' t+T 2 The over-adjustment that results from exceptionally large residuals in the latest Observed quarter is thereby reduced [Evans, 1972, p 966], and, where this adjustment has been used, the forecasting performance of the model has been improved. Predictive performance of models not corrected for autocorrelation is frequently improved by introducing adjustments based on the residuals of prior periods. One such correction factor was used by Haitovsky [1970, 2 pp 504-505]: 6 u e + e (2.6) A = t'1 t'2 t 2 The desired properties of estimated models, e.g. minimum variance of the forecast endogenous variable, depend upon error specification of the estimated model meeting the .assumptions of the classical linear regression model. Otherwise the model is said to be misspecified. The restrictive nature of these assumptions is usually acknow- ledged in the course of estimating the model, but whether the residuals actually meet these assumptions is rarely 25. This adjustment is frequently called the Goldberger- Green adjustment. Evans, 1972, p 966; Haitovsky, 19728, pp 319-320. 26. Such a correction term, of course, implies that the model is autocorrelated, and assumes that the best cor- rection technique is a second-order correction scheme with p1 = .5 and p2 = .5. 14 asked. Four tests for specification error have been develOped by Ramsey [1971] and used by Gilbert [1969] to determine the prOper specification for the demand for money function. C. EVALUATION OF THE PERFORMANCE OF A FORECASTING MODEL When a number Of econometric models have the same dependent variable, comparison of the different models is inevitable. There are a number of approaches to making such comparisons.27 For example, a model can be evaluated on its use of a priori knowledge, or its use of appropri- ate estimation techniques,28 but the ultimate performance criterion is frequently seen to be the model's forecasting ability.29 It is this aspect of the performance of econometric models of the money stock which is the focal point of this Study. Goodness-Of-fit statistics have sometimes served as a proxy for direct examination of forecasting performance. The forecaster usually wants maximum mileage from his data, and so is reluctant to reduce his sample size. This is 27. See Naylor, 1966, pp 311-315; and 1971, pp 154, 158 for a discussion of the four methodological positions regarding this question. See also Zecher, 1971. 28. McCarthy in Cooper, 1972, p 934. 29. Naylor, 1966, p 318 maintains that prediction is the ultimate test of a model. 15 30 But since a model particularly true with large models. may perform quite differently outside its estimation period, goodness-of-fit statistics do not necessarily indicate good forecasting ability. Nelson [1972], for example, has shown that within the estimation period the consumption-invest- ment block of the Penn-FRB-MIT model explains a high pro- portion of the variation of the block's endogenous variables, while outside the estimation period a significantly higher prOportion of that variation is explained by a mechanistic model. Such results are not uncommon. The goodness-of-fit criteria may also promote "data mining," i.e. estimating a wide range of possible models and using the model with the best fit statistics. In such a case goodness-of-fit cri- teria provides little indication of the validity of the model.31 It is generally agreed, therefore, that the more rigor- ous test of a model is to forecast with the model, and evaluate its predictive performance. [Jorgenson, 1970, p 214]. Using either ex ante [Evans, 1972] or ex post [COOper, 1972] values for the exogenous variables, the endogenous variables are predicted and then compared to actual values. Occasionally prediction evaluation stOps 30. Of course once a model has been estimated and used for forecasting, after a while it is possible to study its pre- dictive performance. This has been done in Evans [1972] for the Wharton-EFU model and the OBE model. 31. Jorgenson, 1970, p 214 elaborates on this problem. 16 32 but usually summary statistics are calculated to there aid in the evaluation of the prediction performance of a particular model. These statistics are the subject of this section. 1. Conventional test statistics The two principal prediction evaluation statistics are the root mean square error (RMSE) statistic 1 n A 2 (2.7) RMSE _ ///n Zt=1 ( t t) and the mean absolute error (MAE) statistic (2.8) MAE = l n The MAE statistic is preferred by Klein because of its simplicity and ease of understanding [Klein, 1968, p 40], while others prefer the RMSE because it has a quadratic , 33 . . . . loss function. Two variations of these two statistics are the mean squared error statistic (which, however, does not have the same dimension as the error term), and the mean percent absolute error statistic. Occasionally the 32. For example, Christ, 1956, where the predictive per- formance of the Klein-Goldburger model is tabulated. 33. The loss function Of the MAE statistic in linear. Pro- ponents of the RMSE statistic assume, at least implicitly, that they become more concerned about an error as it in- creases in magnitude. For more on this, see Fromm, 1964. 17 rnage of the prediction error is used [Fromm, 1964, p14]. Naylor [1971, pp 159-160] suggests the possible use of Spectral analysis and various non-parametric tests. 2. Other test statistics 34 but rarely used, Another approach, frequently prOposed is to regress the forecast values on the actual values of the prediction model, and to examine the test statistics . . . 2 assoc1ated With the regreSSion. In other words, the R of the regression . A=A+A + (2 9) Yt Bo BlYt et would be examined along with the standard error of the regression and the closeness of B0 to zero, B1 to one. This approach will be used in Chapter IV. The model can also be examined to determine whether structural shift has occurred between the estimation period and the forecast period.35 This is the most powerful of the univariant tests with the same level of Type I error (the incorrect rejection of the null hypothesis).36 The 34. See Klein, 1968, p 39; Mincer, 1969, pp 9-10; and Naylor, 1971, pp 159-160. 35. An implicit assumption of any test for structural stability is that the model is correctly specified. Re- jection of the hypothesis that no structural shift has occured may indicate that the model is in fact misspeci- fied. Ramsey, 1969, has developed tests of misspecifica- tion. 36. Jorgenson, 1970, p 218. The test is from Chow, 1960. 18 structural shift statistic as described by Chow37 has an F distribution. COOper described a less powerful alternative structural shift statistic, i.e., that of the adjusted ratio of the sum of squared residuals over the forecast period to the reduced form mean squared error over the fitted period, which has a Chi-square distribution. While stability may be desirable in a forecasting model, a Clear-cut relationship does not necessarily exist between tests for stability and the forecasting ability of a model. [COOper, 1972, p 919]. The techniques described ignore the interdependence in simultaneous equation models. Also coefficients may change over time in such a way that forecasting performance of the model improves. 'And, of course, the test for structural change may fail (i.e. a Type I error is made).38 Another approach to the evaluation of predictive per- formance is to ask whether the model forecasts better than a naive or mechanistic model. The simplest naive models are the no-change model 2.1 i? =Y ( 0) t t-l and the same change model (2.11) Yt = 2 yt_1 - yt_2. 37. This test is described by Kmenta, 1971, p 370. 38. See p 26 where we evaluate the actual performance of the test. 19 There are, Of course, other forms for naive models. In recent years a growing use has been made of the auto- regressive model39 A = p (2.12) yt {i=1 Bth_i . . 40 as a standard of comparison for econometric models. A more SOphisticated mechanistic model is the Box-Jenkins model. If the process being specified is stationary, i.e. the process repeatedly returns to the neighborhood of the mean, then the process can be discribed by a combination of an autoregressive equation (such as just described) and a moving average equation:41 (2.13) § = A + p B Y 0 2i=1 - " q ' o o t 1 t-i + ut 2j=1 AJut- J The criteria for proper identification (i.e. what are the optimum values of p and q) are minimization of the number Of parameters and minimization of the mean squared resi- duals. When the process is not stationary (e.g. GNP, prices), successive differences are taken until the model exhibits stationarity. First differences are usually suf- ficient to bring about a stationary situation with economic data [Nelson, 1973, chapter IV, section 1]. A model whose 39. For a description of such models, see Klein, 1968, p 43; or COOper, 1972, pp 830-831. 40. p is uSually-chosen so that standard error of the model is minimized. 41. ut-j = Yt-j - Yt-j’ A0 is the mean of Y. 20 values of p and q are both 1, and where stationarity results from taking the first difference would be (2.14) Y = BlAYt—l + A0 + u ,- Alu t t t-l' Identification and estimation of the model requires itera- tive techniques for which computer programs exist. [Nel- son, 1973].42 Another statistic used to evaluate the predictive per- formance of a model is the Theil inequality coefficient [Theil, 1966, pp 27-28]--the ratio Of the MSE statistic c>f the model to the MSE of the no-change naive model43 n A Zt=1(Yt' Yt)2 n 2 t=l Yt ° (2.15) 02 = 2 frlie value of U will be a positive number. The model ‘thich predicts perfectly has U2 = 0; if the forecast per- formed only as well as the ‘no-change model, then U2 = l. 44 .A variation of this statistic is the Janus coefficient n+m (2.16) J2 = (l/m) Zt=n+1(?t - Yt)2 (1/n) 23:1(Pt - Yt)2 ¥ 433.» Box, 1970 presents this model in all its rigor. Nelson, 19 72A, pp 11-14 has a brief summary of the above while Nelson, 1973 gives a rather complete explanation of the technique. ‘413- In some write-ups of the coefficient, Yt and Yt are first differences. See Theil, 1961 and Theil, 1966. 44. Gadd, 1964. 21 This coefficient resembles the F ratio, but since time series data usually exhibit autocorrelation and the fore- cast values in the numerator are obtained by extrapolation from those in the denominator, it does not have a F distribution. The numerator of the Theil and the Janus Coefficients are sometimes decomposed and these decompositions are said to measure the degree of bias, unequal variation, and in- complete variation in the forecasting model. [Thei1, 1966]. But Jorgenson [1970] has questioned the usefulness of the decomposition. This issue will be discussed in the Appendix of this chapter. An evaluation technique for single period ex ante fore- casts made with simultaneous equation models has been out- lined by Haitovsky [1970]. Haitovskey examined the error Of the individual equations when ex ante and ex post data are used, as well as the effect of three forms of con- Staint adjustment. The effects that different data sets and different constant adjustments have on the entire System is also examined. This technique revealed that con- tiant adjustments did improve the 1966-3 Wharton-EFU fore- caSt [Haitovsky, 1970], and the 1968-3 OBE forecast per- formed better with ex ante values of the exogenous than with ex post values. [Su, 1971]- \ 45 . The three forms of constant adjustment were no adjust- ment, average residual adjustment (A = [ut-l + ut_2] / 2), and the adjustment actually used by the forecaster. 22 D. STUDIES OF FORECASTING PERFORMANCE Studies of the forecasting ability of money stock models, unfortunately, are rare. With the exception of work done by the Research Department of the Federal Reserve Bank of St. Louis,46 the predictive performance of monetary models is at best a side issue. Tobin and Swan [Tobin, 1969], for example, maintained that the forecasting superiority of a naive model and a trend model over two examples of a Fried- man-Swartz money model lessened the significance of the re- lationship between the money stock and Friedman's permanent income. Christ's [1971] comparison of models of the finan- ci a1 sector described the predictive performance of the models, but concentrated on the model's structure and theo- retical foundation rather than its actual performance. This section describes the measurement of forecasting Performance of econometric models other than money stock mOdels as a means of illustrating forecasting evaluation tect‘lniques. 1- Comparison with mechanistic models Frequently the prediction performance of an econometric model is compared to that of a naive model.47 Christ [1956] , 46 - This research will be discussed in the following chapters. 47- There are test statistics which make implicit use of Such criteria. The R2 statistic, for example, compares the explanatory power of the regression against the average Value of the dependent variable. 23 for example, compared a no-change naive model with the Klein-Goldburger model and found that the naive model fre- quently forecast certain components Of GNP better than the econometric model. Evans [1972, p 967] in his large-scale study used both a no-Change and a same-change model as standards of comparison. A variety of naive models has been used. Smyth [1966] compared three specifications of the same-change naive model to forecast the Components of Australia's GNP‘19 along with an average (for the entire period) change model. In this case, only the average change model50 occasionally outperformed the econometric model. Some studies of judgmental forecasts have also used naive models as a standard of comparison. [Zarnowitz, 19 67, pp 83-88] . The autoregressive model has come into favor as a stan- dard of comparison for forecasting models. For example, Green [1972, p 51] compared autoregressive models with 2 and 4 lagged dependent variables to various equations of #1 48- His same-change model for forecasts up to 6 months was Yt+T = Yt + T (Yt - Yt-l) Where T is the length of the forecast. 49 - These were 3 . (1/3) 21:1 %AYt_i, and (1/5) {i=1 sAYt_i. %AYt %AY t 50- This of course is an ex post concept. 51. Equation 2.12. 24 the OBE model in predicting components of GNP. The OBE model proved to be superior to the autoregressive models. Jorgenson [1970, pp 203, 208] found that a fourth order autoregressive model for different investment components predicted better than the poorer performing econometric models in his study. But this was an arbitrary way to choose the number Of lagged terms for the autoregressive model of each investment component. Cooper [1972, pp 830- 831] chose the number of variables which would minimize the RMSE of the equations and found that the autoregressive model frequently outperformed (i.e. had smaller RMSE) the equations of some large econometric models of the U. S. economy . 53 The Box-Jenkins generalization of the mechanistic model was used in Nelson's [1972A] study of the term structure of interesst rates, and again [Nelson, 1972B] in his study of the Penn-FRB-MIT model.54 In the latter case, within the k 52 ~ COOper, 1972, p 917 summarized his results by Choosing the best performing model for 33 endogenous variables. No Slngle model was superior. Of the 33 variables, the econo- metric model performed best for both the period of fit and of forecast 6 times, the autoregressive model performed best 13 times, and the autoregressive model performed best 1501‘ the period of fit 8 times, and for the forecast period times. 53- Equations 2.12 and 2.13. 54 - Two examples from the Penn-FRB-MIT model study (Nelson, l972B, p 915) are the equations for gross national product GNPt = GNPt_l + .615(GNPt_1 - GNPt_2) + 2.76 + ut and non-farm inventory investment 25 estimation period the econometric model itself contributed much towards minimizing forecasting error; but outside the estimation period the Box-Jenkins model tended to perform 55 It seems probable that this model will super- better. cede the other mechanistic models in those cases where time or expense is not a constraint. 2. Other prediction evaluation statistics The Theil inequality statistic has been widely used for prediction evaluation. For an example, see much of the work of Stekler [1970]. Use of the decomposition of the Theil inequality statistic is also widespread. Theil has used the decomposition on a range of projects [Theil, 1961, 1966] as had Kuh [1963] in his study of capital investment, Smyth [1966] in his examination of Australian judgmental fore- casts, and Evans [1972] in his analysis of large scale models of the U.S. economy. But nowhere in these investi- gations was an attempt made to discuss the Jorgenson cri- tique of the approach. The test for structural shift has been used, but the I = . I + 1.69 + + . + . . t 581 t-l ut 0013 ut-l 742 ut_2 TPhe first equation illustrates the difference and auto- Jregressive forms of the model, the second the autoregres- SSive and moving average forms of the model. 555. Nelson also used a composit prediction using both ItKbdels. The composit model was better than the Penn-FRB- DTIT model for 12 out of 14 variables while it was better tfllan the Box-Jenkins model for only 7 out of 14 cases. Nelson, 19728, p 915. 26 test results at times conflict with other prediction evalu- ation statistics. For example, the MSE for real total con- sumer expenditures over the forecast period in COOper [1972, p 875] study is lowest for the OBE model yet this model ex- hibited the greatest degree of structural shift [p 902]. This pattern, however is not consistent. Current plant and equipment expenditure, for example, in the Fromm model has the lowest MSE [p 877] and least structural shift [p 903]. Such inconsistancy is also seen in Stekler's [1970, pp 65, 69] comparison of inventOry forecasts. Ranking forecasts by the Theil inequality statistic differs from ranking by the structural shift test statistic. For example, See Table 2.1 'where ten models are ranked by relative lack of structural shift56 and the inequality statistic for two different time periods. While structural stability is a desirable char- acteristic for a forecasting model, it is obviously not necessary for good forecasting performance. TABLE 2.157 RANKING OF PREDICTION TEST STATISTICS lLack of Shift 1 2 3 4 5 6 7 8 9 10 I31 1 9 6 3 5 2 4 7 8 10 I12 1 10 2 4 9 3 5 9 8 6 ¥ 556. Since the degrees of freedom associated with each of the F ratios are different, it is the significance level which is ranked. 557. From Stekler, 1970, p 65, Table 3-1, and p 69, Table 3-4. Tfiie first row is the ranking of each model based on the Chow twist for structural shift. The other two rows give the rank (*5 the model cited in the first row based on the Theil 27 E. SUMMARY This chapter has presented a range of statistics which can be used to measure the predictive performance of econometric models. These statistics will be used to eval- uate the forecasts made by the money stock models which are discussed in the next chapter. ‘ jJmequality statistic. U1 is for the period 1948-3 to 1964- ‘4; U2 is for the 1953-1 to 1964-4 period. APPENDIX TO CHAPTER II THE THEIL INEQUALITY COEFFICIENT Theil attempted to evaluate the predictive performance of an econometric model by means of what he called an inequal- ity coefficient: the scaled1 mean squared error2 (2A.l) (l/n) 22:1 (Pi - Ai)2 The most useful characteristic of the coefficient, Theil maintained, was the decomposed form of the coefficient which measured three attributes of the forecasting model. The usual form of the decomposition was 2 ._ 1— 2 2 (2A.2) (l/n){‘i‘___l(Pi - Ai) = (P -,A) + (SP - SA) + (1 - r2) SP SA where s: = (l/n) [2:1 (Pi - P)2. s: = (1/n) 2231 (A1 - 32, and ‘— .1. Theil used at least two different scaling factors: ( 7'7I7HTEP§'+ /‘TI7HT§K§')2 (Theil, 1961, p 32), and (l/n)ZA: (Theil, 1966, p 28). 2. Pi is the predicted value and Ai the actual value of the IPredicted variable. 28 29 (1/n) 2 (Pi - P) (Ai - A) r: S S ° P A The first term of the decomposition according to Theil measured bias, the second measured unequal variance, and the third measured unequal covariance.3 Theil maintained that if the values of the first two decomposed terms appro- ached zero, this spoke highly of the predictive performance of the model.4 Theil's only defense of this argument was graphical.5 In spite of this rather casual approach to- wards validating this method Of prediction evaluation, the inequality coefficient and its decomposition have been widely used.6 Jorgenson, however, in a recent article questioned the meaningfulness of the decomposed statistics by examining the expected values of the first two decomposed terms.7 The expected values of the first decomposed term was found to be8 3. Theil, 1961, p 34-35; Theil, 1966, pp 30-32. An alter- native decomposition is given in Theil, 1966, p 33; but this does not change the argument which follows. 4. This, of course, implied that the third term would in such a case approach the value of the scaling factor. 5. Theil, 1961, p 36; Theil, 1966, p 31. 6. For example, see Stekler, 1970; and Cooper, 1972. 7. Jorgenson, 1970. 8. inz = 2(xi - K)2 where Xi is the value of the explana- tory variable at the time of prediction and K'is the 30 ' __ _ 1 X- - X (2A03) B

.o mamv.o Hz SDCOE NH zucoE m Samoa m Space N SDCOE H HOUOZ pesos e Ashamedhom Eases NH 0» a mo mmzmc mdmooz spasm smzoz OHBMHzamomE so mozazmoamma m>HeoHomma m .v mamas“ 65 be known over the forecasting period. The fact that the autoregressive model performed as well as it did over the full twelve months shows the forecasting power of such a model. Since seasonal adjustment of the money stock series is done after-the-fact, even the SA autoregressive money stock model is partially ex post. The only model using no information from the forecast period itself is the auto- regressive NSA money stock model, and even this model per- formed well. Its RMSE for the one month forecast was three times that of the M1 mOdel, but less than twice the M1 model for the 12 month forecast. As the estimated models predict further away from the 1947-1960 estimation period, it would be expected that pre- diction performance would decline. The annual RMSE sta- tistics of 12 month forecasts, displayed in Table 4.6, show that such is the case. The autoregressive multiplier model, however, for forecasts of 3 or more months had a uniform RMSE over the entire 1961 to 1969 period. The performance of the money multiplier models whose forecasts depended 28 deteriorated in the late only upon immediately prior data 1960's. While this was not unexpected in light of the results of the COOper test, it also suggested that the money crunches and the impact of Regulation Q increased the vari- ance of the time trend of the money stock in the late 1960's. 28. Use of the no-change and the same-change multiplier models does not require the estimation of coefficients which are based on data from prior periods. 66 womv.m Nvmm.a hth.N ammo.H «Nvo.m momm.m momm.~ Nth.N von.m ommo.H o2 Nomv.oa oaom.h mmva.h Hmoa.m mmhm.h momm.m movo.m moma.va Nmma.v mbHH.m m2 oomm.o Nth.NH mmva.m ommo.oH onwn.oa mmmh.m oaam.h mamm.h mhva.m mwmm.h 0v: mmwo.m ooam.m mamv.m Hde.m mamo.m mmom.o mmom.H mmmv.a «Hmb.m vmmm.H mvz loH.o oNoH.m mhoa.h mmoH.m mhmm.h omom.o mmbo.m mmmm.o mHm~.v omom.v .m mmom.m mamm.m momh.m mmam.m mmmm.N mmmm.~ oovo.m NS Noao.H ommn.o Hhhv.a ommm.H mmmH.H Hmmm.o mvmv.o Hmwm.o Hohh.o mmmm.o H2 mqmooz MUOBm wmzoz UHBmHZ¢mUMS m0 m02¢2m0mmmm m>HBUHQmmm AdDZZd m.v mqmdfi moma Iaoma mood mood hood coma mood vwma mood mood Hood Ham» 67 This could account for the poor prediction performance of most of our models during this period. The ranking of the RMSE, the MAE, and the MPAE Of the different models are the same for one month forecasts and change little for 12 month forecasts. (The rankings of the monthly models are shown in Table 4.7). The rankings re- mained the same for quarterly models over the entire fore- cast period. The differences were due to the fact that the mean error measures have a linear loss function while the RMSE statistic has a quadratic loss function and so is more strongly influenced by outliers. The regression of predicted values on actual values gave little statistically meaningful information. The intercept term of the regression did at times differ from zero, but the difference was rarely statistically significant. The slope in most cases remained close to one, and R2 close to .99. Only the standard error of estimate seemed to indicate anything, and that was not much different than what the more conventional measures suggested. Therefore, this prediction evaluation method will not be used in subsequent chapters. The ranking of the standard error of these regressions for the various models is shown in Table 4.7. 3. The Burger multiplier model In recent years economists at the Federal Reserve Bank of St. Louis have studied:the possible use of multiplier 7 Rank 01¢le- OKDGJNICD Year Burger M1 M2 M3A M6 68 TABLE 4.7 RANKING OF THE PREDICTIVE PERFORMANCE OF MECHANISTIC MODELS (Rankings based on 12 month forecasts) RMSE Value M3A 2.5513 M6 3.9897 M3C 4.2155 M1 4.3810 M3B 4.4373 M2 7.4997 M4B 14.6002 M4A 23.1651 .M5 25.6654 M4C 35.8358 MPAE Value M3A 1.1131 M3B 1.6453 M1 1.9361 M6 1.9412 M3C 2.06.8 M2 3.5001 M4B 4.8557 M5 10.2336 M4A 10.4939 M4C 15.2384 TABLE 4.8 A Y SE of on Y M3A M6 M2 M1 M3C M3B M4B M4A M5 M4C Value 2.2335 2.2594 2.8916 2.9884 4.2013 4.4440 14.7280 23.2814 25.9279 36.0477 ANNUAL PREDICTIVE PERFORMANCE OF SELECTED MODELS (RMSE of 1 month forecasts) 1964 1965 0.6890 0.6602 0.2842 0.3263 1.3784 1.2369 1.5968 2.0097 1.6116 2.0221 1966 1.4993 0.5798 1.5459 2.2468 2.0221 1967 1.6307 0.8581 1.8937 1.5754 2.2851 1968 0.9333 0.5324 1.5273 2.1761 1.5784 1969 0.0456 0.3517 1.4578 2.5995 2.6253 69 models for forecasting.29 The most successful multiplier forecasting model was constructed by Burger [1972] who / 3 used the following monthly model: _ 3 (4.16) m _ 80 +151 [(1/3) Zi=lmt-i] + 82 TB The coefficients of the regression used to forecast each month's multiplier were estimated by OLS using the previous 36 months' observations. Each forecast, therefore, depended only on the data of the preceeding 3 year period.31 The multiplier was then used to forecast the SA money stock ex- actly as shown in Equation 4.8. The use of the Treasury bill rate removes this model from the mechanistic model catagory. However, of the vari- ous models considered, it would seem most appropriate to compare the Burger model to the multiplier models. Table 4.8 lists the annual RMSE statistics for this model along with the RMSE statistics for the no-change multiplier model and the three monthly autoregressive models. In comparing the models we find that the RMSE of the autoregressive SA money stock model is consistently quite 29. A review of this work can be found in the Burger, 1972 article. 30. TB is the lagged percentage change in the Treasury bill rate, the Di's are seasonal dummies. 31. In this study, the forecasting equation was eStimated once. Burger's approach would have required the eStimation of 108 foreCasting equations if forecasts were made for the 1961-1970 period. 70 smaller than the RMSE of the Burger model. The Burger model, however, except for 1967, outperformed the other models considered in this Chapter. Considering the com- plexity of the Burger model (especially when compared to the autoregressive money stock model), and its inability to outperform a mechanistic model, it is difficult to be- come particularly enthusiastic about its forecasting abilities.32 D. SUMMARY The results of this chapter can be summarized briefly. (l). The autoregressive SA money stock model forecast best for time periods up to 6 months. For 12 month forecasts the no-change multiplier model forecast with smaller error. For all models, the RMSE usually increased as the period of the forecast lengthened. (2). The performance of the auto- regressive NSA money stock model, the only model which did not to some degree use ex post information, forecast poorer than the two models mentioned above, but its relative fore- casting ability did improve as the length of the forecast increased. (3). The minimum standard error of estimate is an appropriate statistic for selecting the proper number of lagged terms for the autoregressive forecasting model. 32. In his article, Burger showed that the use of the RPD base gives inferior predictions in comparison to the use of the monetary base. Our data above (compare Model M33 and M3C to MBA) confirms Burger's conclusion. 71 (4). The difference which exists between the various pre- diction evaluation statistics (RMSE, MAE, and MPAE) can be accounted for by the loss function implicit in each sta- tistic. The test statistics of the regression of predicted on actual values tended to statistically insignificant. APPENDIX TO CHAPTER IV THE RATIO MULTIPLIER MODELS 33 A. THE RATIOS a = (DTM - TDM) / DDP DDO / DDP 9) ll " TD/ TDM DJ ll 0 ll (CURN - CURR - CURB) / CURR k = CURR / DDP r = RSU / (DTM - TDM + TD) r' = RSRPD / (DDO + TD) t = TDM / DDP B. DEFINITIONS M = CURR + DDP B l RSU + (CURN - CURR - CURB) + CURR r (a*DDP + t*DDP) + (1 + C) k*DDP B 2 RSRPD = r' (a'*DDP + a"*t*DDP) C. THE DECOMPOSED MULTIPLIER m = M/B (1 + k) / [r(a + t) + (1 + C)k] 1 m = M/B (1 + k) / [r'(a' + a"t)] 2 33. Notation is listed in the Notation Appendix. 72 l 8 l CHAPTER V FORECASTING WITH SINGLE EQUATION MODELS OF THE MONEY STOCK In this chapter a number of single equation models that can be used to forecast the money stock will be evaluated. In the first section the different models will be described and the coefficients, estimated over the period 1947-3 to 1960-4, will be presented. The various prediction evalua- tion statistics and the use of the constant adjustment term will be discussed in the second section. The fore- casting performance Of the various models over the 1961- 1970 decade will be the subject of the third section. A. THE MODELS The coefficients of a number of version of six different money stock models will be estimated. Three of these six models can be described as money supply models; the other three will be called money demand models.1 The money demand function will be first specified in conventional form: the nominal money stock regressed on nominal values of income or wealth and either a short or a long term interest rate. 1. See pp 36-37 where the meaning of money demand and money supply models within the forecasting context is discussed. 73 74 The other two money demand functions will have the quantity variables scaled by either population or the GNP price deflator. These models are in log-log form. The money supply models will include three versions of the Brunner linear money supply model, the Teigen free reserves equa- tion, and the equation Gibson prOposed as an alternative to the Teigen equation. The notation used in this chapter is summarized in Table 5.1. Each of the six models will be identified by a letter. The variables included in each model will be identified by the letters which follow the model identifier. TABLE 5.1 SUMMARY OF NOTATION FOR CHAPTER V Log-log nominal money demand model. Log-log per capita money demand model. Log-log real money demand model. The Teigen money supply model. The Brunner linear money supply models. The Gibson model. "113.10 000:» Yield on commercial paper. Yield on long-term Aaa bonds. The Federal Reserve discount rate. The difference between S and F. Y Gross National Product. W Permanent income (defined in footnote 2 below). RRSL Total reserves plus reserves released through changes in the reserve requirements. ><"11L"'U) BA, BB, BC The three versions of the Brunner model. D Dummy variables in the Teigen model. S Seasonal dummy variables. -L Model includes a lagged dependent variable. -A Model corrected for first order autocorrelation. 75 The letters which follow a hyphen will indicate whether the model had either a lagged dependent variable and/or has been corrected for first order autocorrelation. l. The conventional money demand function The conventional single equation money demand function is (5.1) 1n M = B + B + 0 1n R + 82 1n Y e 1 where R is either the commercial paper rate or the long- term bond rate, and Y is either GNP or permanent income. The estimated coefficients for different versions of this ~model are given in Table 5.2.3 (The t-ratios are given underneath the coefficients). The signs of the estimated coefficients conformed to conventional economic theory.4 The value of R2 was high with the lowest value being .96. But at the same time the values of the Durbin-Watson statistics were low, e.g. for regressions not corrected for autocorrelation or with a lagged dependent variable, they were between 0.17 and 0.21. 2. The following expression was used to compute the per- manent income: 19 i YP = 0.114 0.9 Y t+l Zi=0 ( ) t-i This expression was developed by deLeeuw, 1969. 3. Notation heading the columns are given in the Notation Appendix. 4. The only exceptions were models ALW-A and Blw-A, where the estimated coefficient for the interest rate variables were positive. However these coefficients were not signi- ficantly different than zero. 76 TABLE 5.2 ESTIMATED COEFFICIENTS OF THE MONEY DEMAND MODEL 2 Model RMCP RMLB Y Y? Lag Const Rho R /SE D.W. ASY —0.0345 .4271 2.3616 .9700 .2140 A -3 4041 23.9443 24.0774 .0153 zAsw -0.0049 .3591 2.7760 .9667 .1654 9 -0.5083 22.6158 32.4503 .0161 EALY -0.0344 .3982 2.5442 .9643 .1907 f -1.2302 18.6082 25.8986 .0167 l I .ALW -0.0293 .3710 2.7374 .9673 .1815 ( -1.1105 19.5880 32.7377 .0160 lASY-L —0.0180 .0833 .8462 .2733 .9967 .6531 -5.2234 4.6825 20.4712 2.5536 .0050 iASW-L -0.0108 .0281 .9553 .0673 .9955 .5378 . -3.0381 1.4644 18.0959 .4400 .0059 TALY-L -0.0417 .0624 .9045 .1482 .9965 .7555 * -4.7865 3.7263 21.8499 1.3016 .0052 'ALW-L -0.0367 .0390 .9488 .0697 .9959 .6584 -3.9434 2.0815 18.9722 .4854 .0056 i ;Asy-A -0.0153 .2490 .00 .7701 i -2.4751 6.2567 .0059 (Asw-A -0.0095 .1742 .00 .5139 7 -1.3166 3.7022 .0069 ALY-A -0.0219 .2249 .00 .7750 -0.8305 5.0995 .0062 ALW-A 0.0235 .1356 .00 .5481 0.8759 2.9865 .0070 ASY-LA -0.0189 .1013 .8039 .3719 .65 .9982 1.7774 -4.8237 3.7748 12.2279 2.1111 .0037 Asw-LA -0.0179 .0663 .8812 .2045 .75 .9980 1.7468 -4.2946 2.5712 13.3468 1.0566 .0039 ALY-LA -0.0557 .0999 .8394 .2581 .65 .9980 1.8220 -4.0003 3.4974 12.1638 1.3514 .0039 ALW-LA -0.0459 .0523 .9368 .0573 .65 .9978 1.6683 -3.1931 2.0956 14.2495 .2929 .0042 77 When these regressions were corrected for first order autocorrelation, it was found that a first difference equation minimized the standard error for those regres- sions without a lagged dependent variable.5 Some individual coefficients of the various models were not statistically significant, but there is little pattern to this Situation. Only those money demand models which had GNP and the short- term interest rates as explanatory variables had t-ratios of above 2.0 for all the estimated coefficients. This was the case whether or not the model had a lagged dependent variable, and whether or not it was corrected for auto- correlation. 2. The per capita money demand function The quantity variables (as opposed to the interest rate variables) Of the money demand function can be expressed in per capita form. This transformation in some cases caused a considerable difference in the values of the estimated coefficients. The estimated parameters for the per capita regressions (and t-ratios for the coefficients) are given in Table 5.3. The values of R2 for the models without a lagged variable and not corrected for autocorrelation ranged from 0.51 to 5. A scanning technique was used to estimate the value of p, i.e. various values of 0 were used to estimate the parameters, and that value of p which gave the lowest stand- ard error was choosen as the estimate for p. See Hildreth, 1960. 78 TABLE 5.3 ESTIMATED COEFFICIENTS OF THE PER CAPITA MONEY DEMAND MODEL Model RMCP RMLB Y Y? Lag Const Rho Rz/SE D.W. BSY -0.0276 .1790 5.3098 .5496 .1256 -2.2311 5.8328 23.2958 .0182 BSW -0.0225 .1581 5.4716 .5053 .1328 -1.7407 5.1392 24.0815 .0190 BLY -0.0882 .2024 5.2102 .5892 .1486 -3.2198 6.8483 25.9745 .0174 BLW -0.0763 .1804 5.3711 .5388 .1707 -2.6370 6.0177 26.6514 .0184 BSY-L -0.0172 .0445 .9270 .1574 .9573 .7638 -4.4814 3.9600 22.1001 .6466 .0056 BSW-L -0.0143 .0321 .9523 .0832 .9519 .6829 -3.5378 2.8676 21.7761 .3231 .0059 BLY-L -0.0443 .0521 .9022 .3027 .9605 .8671 ' -5.0769 4.5511 21.9321 1.3035 .0054 BLW-L -0.0366 .0378 .9328 .2004 .9540 .7668 -3.9314 3.2712 21.4870 .8053 .0058 BSY-A -0.0121 .1341 1.00 .7901 -l.8759 2.8067 .0061 BSW-A -0.0071 .0723 6.1061 0.95 .9426 .6178 -l.0188 1.2283 13.2266 .0065 BLY-A -0.0194 .1122 1.00 .8304 -0.7232 2.1872 .0063 BLW-A 0.0074 .0437 6.3132 0.95 .9415 .7010 0.2967 .7757 14.4380 .0065 BSY-LA -0.0183 .0586 .8247 .7310 0.70 .9741 2.0811 -3.8337 3.1042 10.2235 1.5259 .0044 BSW-LA -0.0180 .0529 .8380 .6876 0.75 .9738 1.9778 -3.7200 2.8275 10.2018 1.3480 .0044 BLY-LA -0.0636 .0856 .8062 .7063 0.65 .9754 2.0359 -4.3107 4.0893 11.0888 1.6640 .0042 BLW-LA -0.0527 .0623 .8594 .5200 0.65 .9733 1.8090 -3.6763 3.3968 12.0509 1.1956 .0044 79 0.55. These values, relatively low for time series data, are partly due to the value of the per capita money stock essentially remaining unchanged over the estimation period. The natural logarithm of the per capita money stock was 6.66, at both the beginning and the end of the estimation period, and ranged between 6.61 and 6.70. The estimated coefficients of three of the four regres- sions with neither a lagged variable nor corrected for auto- correlation had t-ratios above 2.0.6 If the estimated con- stand term was ignored, all the regressions with a lagged _ dependent variable whether or not corrected for autocor- relation had coefficients with t-ratios above 2.0. The Durbin-Watson statistics behaved in the per capita money demand function as they did for the nominal money demand functions, i.e. the statistic was close to 2.0 only when the regression was corrected for autocorrelation and included a lagged dependent variable. 3. The real money demand function A money demand function with the quantity variables divided by the implicit GNP deflator was also estimated. The estimated coefficients and their t-ratios are shown in Table 5.4. The R2 values of the regressions ranged from 0.15 to 0.32 for regressions without a lagged dependent 6. The exception was model BSW. 7. The inclusion of the lagged dependent variable, of course biases the Durbin-Watson statistic towards 2.0. 8C) TABLE 5.4 ESTIMATED COEFFICIENTS OF THE REAL MONEY DEMAND MODEL Model RMCP csy -o.0487 -3.3595 csw -0.0433 -2.9488 CLY CLW CSY-L -0.0216 -3.8014 csw-L -0.0235 -4.3652 CLY-L CLW-L CSY-A -0.0229 -2.4371 csw-A -0.0173 -l.8009 CLY-A CLw-A CSY-LA -0 0217 -2.9993 csw-LA -0.0234 -3.3452 CLY-LA CLw-LA RMLB -0.1549 -5.3836 -0.1436 -4.8387 -0.0588 -4.4175 -0.0613 -4.9437 -0.0898 -2.3998 -0.0605 ~1.8149 -0.0748 -3.6082 -0.0703 -3.9486 Y 0.0900 2.1226 0.1353 3.6969 0.0539 3.3383 0.0613 3.8869 0.2290 2.5418 0.2129 2.5698 0.0553 2.2135 0.0824 3.0400 YP 0.0683 1.6676 0.1128 3.1344 0.0582 3.9667 0.0633 4.4675 0.0937 1.3262 0.0754 1.2601 0.0615 2.6663 0.0751 3.4304 Lag 0.8495 17.5519 0.8633 18.6359 0.7979 15.8350 0.8127 17.1339 0.7428 9.4154 0.7691 9.8693 0.6835 8.8146 0.7353 10.4954 Const 4.4592 18.2401 4.5870 19.5556 4.3290 22.5537 4.4541 23.8945 0.4393 1.7790 0.3472 1.4316 0.7013 2.8938 0.6212 2.6441 3.5387 6.4242 4.3915 10.3446 3.7605 7.8712 4.5682 13.3659 0.9592 2.2988 0.7942 1.8700 1.1604 3.0694 0.9440 2.6514 Rho 0.95 0.90 0.85 0.60 0.60 0.55 0.55 2 R /SE .1769 .0222 .1505 .0226 .3590 .0196 .3185 .0202 .8828 .0084 .8910 .0081 .8913 .0081 .8988 .0078 .8710 .0088 .8619 .0091 .8715 .0088 .8588 .0092 .9190 .0070 .9222 .0068 .9232 .0068 .9251 .0067 D.W. .2194 .2085 .2740 .2770 .8960 .9223 .9966 1.0446 .8429_ .7331 .9263 .8106 1.7687 1.8191 1.7842 1.7877 81 variable, but when the lagged dependent variable was included in the regression, the value of R2 approached .90. The value of the dependent variable declined over the estimation period and the range of the natural loga- rithm of the real money stock was small, from 4.91 to 5.03. The latter fact especially may partially explain the low value of the R2 term. The coefficients of most models had t-ratios of above 2.0.8 The autocorrelation coefficient was less than 1.0 in all cases, and the pattern of the Durbin-Watson sta- tistic was similar to that of our previous two cases. 4. Evaluation of money demand models The estimated interest rate and economic activity co- efficients of the money demand models discussed above are low. The highest interest rate coefficient is 0.15; the highest income coefficient is 0.43, and the highest per- manent income coefficient is 0.36. Table 5.5 lists the results of some other studies of the money demand function. The estimated coefficients of other studies listed in this table are higher than those obtained in this study with similar variables. These studies, however, were all esti- mated over time periods which were different than that used in this study, and none of the other studies use the same 8. The exceptions were Model CSY-L, Model CLW-A, and all Inodels which have both the short-term interest rate and ‘permanent income as eXplanatory variables. 82 ..me a .amma .uumnafloi .ocoomm mzu ca Hmmu “muflmmo mom pom Hmmu mum cofimmmummu uwuwm 0:» ca wmwuflucmsv Had .mump Hmsccm "pumnaflw .Hmmm a .woma .umacflmq. .m» new mom: ma mmflnmm meoocw Hmmu Umuoomxm muflmmo Mom m.cmEUmwum .mump Hmsccd "umapflmq ..HmHH m .ommH .3OLUH .uospoum Hmcoflum: um: we wEoocH can .mmflumm pcwuso mcu ma Came» been one thmp magmafim>m ummummc Ho .cuom wash mo ma xooum xmcoz .mmpmum pmuHCD mzu m0 moflumwumum Hmofluoumflm Eouw sumo .mump Hmscca «30:0 .Hmoma .umaammi .pmms Hwaamm mmfluwm xooum MmCOE :0 ucmEEOU How uxwu mmm .mamp >HH0#HMSO "Hmaam: Ac.ucooc m.m mamas 83 vom. mmm. mwo. hmh. mmo. vvm. womaoo. mmoo. mnamoo. mmmm. mum. mmm. mm\ m N mmm.o NmHH.0I mvhh.o ov.vl ovmv.01 mvH.m omo.m mmam.o Hmmm.01 mmm.m MNH.N umcoo Ho.ma mh.va mmvh.o Hmo.a mmm.o ~ma.o evo.o mmao.ea mmom.o mmq mesa mqmooz vm.N thH.o mv.m HmmH.o mhmH.m ommm.o vmmm.mh Hmmm.o ooowNH omHN.o oom.o mvHN.mH who.a wad m.m mmo.o mqmdfi mm.vl vaN.OI om.ml NomN.OI mwm.¢a mam.OI vmmm.ml moam.0I mmmm.ml movm.ot OAZMCH mQZmCH www.ml th.oI bmwv.ml vOH.OI mUSmGH 0242mm wmzoz omefiEHBmm QMBUMAmm mmmHImHmH ommalmvma mmmalhmma mmlhwma ooflumm mafia gymnafiu umaoflmq 3050 umHHmm HmUOZ 84 series as was used here. Heller's regressions were closest to the regressions whose estimates appear in Table 5.2, but he apparently used an unusual money stock series to obtain his estimates.9 Even so, his interest rate coefficients were close to those in Table 5.2. All the models other than Heller's model listed in Table 5.5 were estimated with annual data. In order to determine what impact this would have on the values of the coefficients in Tables 5.2, 5.3, and 5.4, the money demand equations were reestimated with annual data. Table 5.6 gives the coefficients and the t-ratios for the annual money demand functions without a lagged dependent vari- 10 able. There is little difference between the annual and quarterly models. The coefficients for both the interest rate variable and the economic activity variable continue to 11 be low. The largest interest rate coefficient is -.15 12 while the highest economic activity coefficient was .43. 9. It is impossible to determine from Heller's description of his data what money stock series he used. He cited as the source of his data The International Monetary Fund, Intennatéonaz Financiaz Statibticb. It would appear that the data series he used from there was derived from Flow- of-funds data. He never eXplained why he used this source for his data. Interestingly, he did use the Federal Reserve Bu££eth--the best source for money stock data--for his interest rate data. 10. These regressions were estimated with average data for the year. ' ll. This was for Model CLY. Model CLW had an interest rate coefficient of -.14. The next highest coefficient was -.08. 12. This was for Model ASY. The two models cited in foot- note 11 had estimated coefficients of .097 and .086 for the MODEL ASY ASW ALY ALW BSY BSW BLY BLW CSY CSW CLY CLW ESTIMATED RMCP -0.0452 -1.6360 -0.0406 -1.4070 -0.0367 -1.1320 -0.0325 -0.9958 -0.0684 -1.8225 -0.0622 -1.6434 The top values Values are the 85 TABLE 5.6 COEFFICIENTS OF ANNUAL MONEY DEMAND MODELS WITHOUT LAGGED VARIABLES RMLB -0.0152 -.02325 -0.0168 -0.2490 -0.0780 -1.2647 -0.0773 -1.2235 -0.1458 -2.0908 -0.1416 -1.9758 Y .4345 9.3172 .3745 7.9122 .1879 2.4364 .1777 2.8243 .1195 1.1265 .0968 1.1435 YP .4151 8.7557 .3647 7.6523 .1706 2.2880 .1696 2.7348 .0959 .9387 .0864 1.0392 Const 2.3282 9.1637 2.4500 9.5531 2.6627 12.5292 2.7317 12.9572 5.2502 9.2127 5.3843 9.8207 5.3901 12.7647 5.4563 13.2465 4.3005 7.0689 4.4390 7.6210 4.5514 10.3668 4.6111 10.8368 R2/SE .9672 .0168 .9634 .0177 .9594 .0187 .9570 .0192 .4846 .0191 .4623 .0195 .4976 .0189 .4840 .0191 .1865 .0236 .1599 :0240 .2420 .0228 .2277 .0230 D.W. 1.2577 1.5552 1.0668 1.3391 1.1017 1.1863 .9885 1.1397 1.5809 1.5663 1.3719 1.4344 are the estimated coefficients, the bottom t-ratios. 86 Log-log models with a lagged dependent variable yield not only an impact elasticity but also the steady state elasticity. Table 5.7 gives these elasticities for the interest and activity variables for the quarterly money demand functions. Only one model has a steady state elas- ticity greater than 1.0; the nominal money demand regression with the long-term rate and permanent income. But it takes 25 quarters before the elasticity reaches one-half its steady state value.13 The adjustment time is also long for the rest of the nominal money demand models and the per capita money demand models. The real money demand models, however, in all cases reach at least one-half of their steady state elasticities by the fifth quarter. Correction for autocorrelation of all three sets of models lowers the value of the steady state elasticity and shortens the half-life of the adjustment process. The estimated interest rate and economic activity coef- ficients of the annula money demand models with lagged coefficients were approximately the same size as the quar- terly models. The coefficients for the annual money stock models with a lagged dependent variable are given in Table 5.8. As would be expected, the coefficients of the lagged activity coefficient. The latter was the lowest of the economic activity coefficients. 13. The first and second quarter elasticities are also given in Table 5.7 as well as the half-life of the adjustment, i.e. the length of time necessary to reach one-half of the steady state elasticity. 87 TABLE 5.7 INTEREST AND INCOME ELASTICITY OF QUARTERLY MONEY DEMAND MODELS Model B l a s t i c i t y Half Impact Quarter 1 Quarter 2 Steady State Life ASY-L -.0180 -.0332 -.0461 -.1170 5 .0833 .1538 .2134 .5416 ASW-L -.0145 -.0279 -.0404 -.1989 10 .0452 .0873 .1262 .6214 ALY-L -.0417 -.0794 -.1135 -.4366 7 .0624 .1188 .1699 .6534 ALW-L -.0369 -.0728 -.1076 -1.3132 25 .0319 .0629 .0930 1.1352 BSY-L -.0172 -.0331 -.0479 -.2356 10 .0445 .0858 .1240 .6096 BSW-L ~.Ol43 -.0279 -.0409 -.2998 15 .0321 .0627 .0918 .6730 BLY-L -.0443 -.0843 -.1203 -.4530 7 .0521 .0991 .1415 .5327 .0378 .0731 .1060 .5625 CSY-L -.0216 -.0399 -.0555 -.l435 5 .0539 .0997 .1386 .3581 CSW-L -.0235 -.0438 -.0613 -.1719 5 .0582 .1084 .1518 .4257 CLY-L -.0588 -.1057 -.1432 -.2909 4 .0613 '.1102 .1492 .3033 CLW-L -.0613 -.1111 -.1516 -.3273 4 .0633 .1147 .1566 .3380 Trhe top quantity is the interest rate elasticity, and the Ibottom line the economic activity variable elasticity. Half- life is the time (in quarters) it takes for the elasticity to reach at least one-half its steady-state value. Model ASY-L ASW-L ALY-L ALW‘L BSY-L BSW-L BLY-L BLW-L CSY-L CSW-L CLY-L CLW-L The top values are RMCP -0.0306 -1.4733 -0.0271 -1.0790 -0.0282 -0.9891 -0.0204 -0.6714 -0.0586 -1.7455 -0.0544 -1.6014 88 TABLE 5.8 ANNUAL MONEY DEMAND MODELS WITH LAGGED VARIABLES RMLB -0.1024 -2.5324 -o.0944 -1.8527 -0.1346 -3.1794 -0.1273 -2.5683 -0.1904 -4.0815 -0.1918 -3.9408 Y 0.3277 4.1586 0.3078 5.2593 0.1451 1.8544 .2161 .0589 #0 0.1471 1.5532 0.2161 3.5414 Y? .3436 .0967 WC .3219 .7385 (NO .1221 .4279 P‘O .2075 .2377 660 .1288 .3976 H0 0.2084 3.4044 the estimated coefficients, Lag 0.2760 1.6652 0.1949 0.8032 0.3983 2.8129 0.3214 1.5192 0.4011 1.5570 0.3835 1.2919 0.4326 2.3335 0.3583 1.5736 0.3293 1.2932 0.3398 1.3027 0.1966 1.1051 0.2154 1.1854 2 Const R /SE 1.6097 .9808 3.7471 .0123 1.9151 .9729 3.1548 .0146 1.2307 .9861 3.0004 .0105 1.5176 .9778 2.5634 .0132 2.8968 .6353 1.9220 .0167 3.1885 .5891 1.9115 .0177 2.2736 .8096 2.0572 .0120 2.8320 .7510 2.1677 .0138 2.4881 .1347 1.7552 .0209 2.5460 .0985 1.7503 .0213 2.9073 .5937 3.1374 .0143 2.8673 .5750 3.0102 .0146 D.VJ. 1.5537 2.0671 2.1536 2.6712 1.2224 1.3007 2.0529 2.3594 1.6851 1.7075 2.3538 2.8383 the bottom values are the t-ratios. 89 dependent variables of the annual models are lower than in the corresponding quarterly model. However when the steady state elasticities for the annual models (given in Table 5.9) are examined, the elasticities are found to always be at least one-half the steady—state value within one year, and the steady state elasticities to be lower than the quarterly steady-state elasticity. The use of annual data - merely exacerbates the problem of low coefficients. Thus the money demand models seem to imply a world where a change in income or a weighted average of past income had little initial effect on the money stock, and even in those cases where the effect is sizable, the impact on the money stock occurs slowly. The interest rate elasticity also specifies a world where a change in the interest rate does not have a sizable impact on the money stock. It should be noted that the period over which the money demand equation was estimated was a time of slow money stock growth. The natural logarithm of the SA money stock in .1947 was 4.72; in 1960, 4.95 while by 1970 it was 5.43. TTLis situation can be observed in comparing the regressions ftxr the nominal money stock for the 1947-1960 period, the 1947-1970 period, and the 1961-1970 period. Table 5.10 Shows low coefficients (and therefore low elasticities) —¥ l4u. The annual money stock models were estimated with both t1“; average of daily data for the entire year and the aver- age of daily data for the two months around the first quar- ter. There was little difference between the two sets of eStimates . 90 TABLE 5.9 INTEREST AND INCOME ELASTICITY OF ANNUAL MONEY DEMAND MODELS Model B l a s t i c i t y Half Impact Quarter 1 Quarter 2 Steady State Life ASY-L -.0306 -.0390 -.0414 -.0423 1 .3277 .4181 .4431 .4526 ASW-L -.0271 -.0324 -.0334 -.0336 1 .3436 .4105 .4235 .4266 ALY-L -.1024 -.1432 -.1594 -.1702 1 .3078 .4304 .4792 .5116 ALW-L -.0944 -.1247 -.1345 -.1391 1 .3219 .4254 .4586 .4744 BSY-L -.0282 -.0395 -.0440 -.0471 l .1451 .2033 .2266 .2423 BSW—L -.0204 -.0282 -.0312 -.0331 l .1221 .1689 .1869 .1981 BLY-L -.1346 -.l928 -.2180 -.2372 1 .2161 .3096 .3500 .3809 BLW-L -.1273 -.l729 -.1893 -.1984 l .2075 .2818 .3085 .3234 CSY-L -.0584 -.0779 -.0843 -.0874 l .1471 .1955 .2115 .2193 CSW-L -.0544 -.0729 -.0792 -.0824 1 .1288 .1726 .1874 .1951 CLY—L -.l904 -.2278 -.2352 -.2370 l .2161 .2586 .2669 .2690 .2084 .2533 .2630 .2656 The top quantity is the interest rate elasticity, and the .bottom line the economic activity variable elasticity. Half— life is the time (in quarters) it takes for the elasticity to [reach at least one-half its steady-state value. EstimationModel Period 1947- 1960 1961- 1970 1947- 1970 ASY ASH ALY ALW ASY ASW ALY ALW ASY ASW ALY ALW RMCP -0.0345 -3.4041 -0.0049 -0.5083 -0.0654 -3.8911 -0.0692 -4.3214 -0.0537 -2.9739 -0.0554 -2.9012 QUA -O -1 -0. -1. #0 NO 50 U10 91 TABLE 5.10 RTERLY NOMINAL MONEY DEMAND MODEL FITTED OVER THREE TIME PERIODS RMLB Y 0.4271 23.9443 .0344 0.3982 .2302 18.6084 0293 1105 0.7151 30.7192 .1437 0.4987 .4681 15.3069 .1232 .4884 0.5151 22.9416 .1873 0.3147 .8724 10.6747 .1998 .0017 YP 0.3591 22.6158 0.3710 19.5880 0.7237 32.4332 0.5205 14.4815 0.5163 21.7542 0.3046 9.9442 Const 2.3616 24.0774 2.7760 32.4503 2.5442 24.8986 2.7374 32.7377 .5441 .1381 .50 0.5158 4.1126 .6291 .8670 \Dr—I .5362 .4926 (DP 1.8543 15.3878 1.8636 14.7142 2.7750 21.0393 2.8288 20.7770 2 R /SE .9700 .0153 .9667 .0161 .9643 .0167 .9673 .0160 .9866 .0171 .9879 .0162 .9877 .0164 .9865 .0172 .9678 .0356 .9648 .0372 .9718 .0333 .9696 .0345 D.W. .2140 .1654 .1907 .1815 .2088 .3226 .2604 .2869 .0502 .0659 .0974 .0980 The top values are the estimated coefficients, the bottom values are the t-ratios. 92 for the early period, and considerably higher coefficients for the latter period.15 This situation should go a long way towards eXplaining the difference between the money demand regressions and those of other studies. 5. Money supply: the Brunner model The estimated regressions of the Brunner linear money supply models in all cases had high R2 values, and in two of the three cases (EBA and EBB) Durbin-Watson statistics which rejected the hypothesis of statistically significant autocorrelation. The results of these regressions are shown in Table 5.11. The coefficients in some cases were not significant. The best performing base concept (Model EBA) was currency in circulation plus member bank's total reserves, but even this model had an estimated coefficient which was not significantly different than zero. Compen— sating for reserves liberated through changes in the re- quired reserve ratio (Model EBB) and subtracting out excess reserves (Model EBC) did not improve the goodness-of-fit statistics for the 1947-1960 estimation period. These results are quite close to the results Brunner obtained when he estimated his model. Brunner's results 15. This data will be used below in the Chow test for struc- tural stability. The interest rate coefficients for the 1947-1970 and the 1961-1970 regressions were all significant (while only one of the four interest rate coefficients for the 1947-1960 regressions was significant). However the sign of the long run interest rate coefficients was positive in all cases. 93 mvm>.H owam.a momm.H Nwmo.H ommH.H mmoH.H mamm. mmme.d mmmm.H .B.Q Hmmm. mhmm. om.c mmvm. memo. mm.o mnmm. sham. mm.o vvmm. Nmmm. comm. mwmm. mmmo. mmmm. mvvm.m rhea. Nmmw.H poem. hhmv.a wmmm. mm\~m 9E mavm.m Hm¢N.om mmoa.m Nvmv.wm mama.~ eevm.m~ omem.e mmmm.m~ mmam.v mmmh.hm HNmN.v momm.om wwmv.m onmn.maa hmmm.m momm.vm Nmmh.m bmmv.mm umcoo mm¢m.oa Howh.o mmmv.m mNo>.o momN.h mmmm.o movm.m~ Hmom-o N¢NN.©H maom.o mohH.MH mumm.o mmq oonm.al N0MH.H¢I ¢mh¢.0I Noam.NHI mmhm.hl mHom.mwNI .H mmhm.N HmmN.mH momm.m ommw.NN NHmm.m mmmH.oN mmmo.o Nmm~.o mmHN.o mmmm.o onmm.o mmvm.a mmmv.o vam.m mmav.~ oonm.MH ommm.o Hamm.¢ u Hth.HI ewmm.~¢l mach.HI meh.bml mom0.HI mmmm.mMI homm.wl mmo~.HmI omhm.¢l mawh.awr avmm.¢l mmhw.mwl memo.vl mmmv.moml mwmb.v| NHNH.mvHI oamm.ml vam.NmHI x Ammo: mmzzsmm mmB m0 wBZMHUHmmmOU QMBfiZHBmm HH.m mam¢fi vwmm.o mome.o Hmmh.H momm.o hmm¢.m ommm.o meH.o mmHo.o hamm.o bvNH.o hwmm.o omva.o ommm.m oowv.H Namm.HH movm.~ ammo.vH mHFN.N mmmm 4QIUmH 4AImmm éfllfimm Alumm Almmm Qlfimm 0mm mmm 0m0.0 NO ma.mmqm<8 NN00.0 m000.0 mmw0.0 0000.0 H050.0 5000.0 0mv0.0 hm00.0 NhMH.N m0a0.0 mmmH.N m0H0.0 HQ AMQOZ szHmB MSE m0 mBZNHUHmmMOU OmfidiHBmm mowN.H MNN0.0 0N00.0 50N0.0 mm0~.0l mm00.0| mvH0.H hHN0.0 aux“ mm00.0l 0000.0I HhN0.H HmH0.0 HON0.H mm~0.0 00m0.0 hMH0.0 Quid 000N.0I 0m00.0l mmmm.m m0v0.0 v0v~.v vmm0.0 amha.m mmv0.0 hQZfl Almamma momma mmmn Qme AIWDXD moxo me 0x0 Hmpn 97 demand equations, but statistically significant autocorre- lation still existed. The interest rate coefficient was statistically significant only when the difference of the interest rates was used as an eXplanatory variable. The dummy variables in the equation where the interest rate co- efficient is not constrained were also not significant. The results obtained in all cases fell between Teigen's estimates and Gibson's reestimation of the Teigen model. This comparison, shown in Table 5.14, includes a reestimate of the Teigen model for the 1947—1 to 1958-4 period—~the period over which Gibson estimated the Teigen regression. Gibson maintained that the difference between the Teigen coefficients and his reestimated coefficients were due to the difference in the data series used. Teigen used call Report data which is for one day each quarter while Gibson used quarterly averages of daily figures. 8 Data for the reestimated regressions was the average of monthly averages of daily figures of the two months that straddle the quar- ters. This concept is quite close to Gibson's, and the similarity of the reestimated coefficients to Gibson's co- efficients would tend to confirm the hypothesis that the dif- :ference between the various estimates of this model is due t1) differences in the data series. ¥ 123. Since banks almost always know when they must submit ‘tfle Call Report data, window dressing of the data can occur. NCDr is the Call Report data for the same day each year. MOreover neither December 31 or June 30, two common Call Report dates, are typical days for the financial sector. 98 Hmom. vmhv.a 0mm.H vmmo. Hmom. 0mmo. mmom. mooo. mmwm. own. mm\ m N momm.vma vm>0.H mn¢¢.m¢a VH50.H vmmm.oma moon.0 mmmm.mma mmmm.0 umcou mmhw.m hom0.o hmwa.h vmma.0 ND mmmH.N mma0.0 0050.H mmao.o mmmm.m NHH0.0 wmmv.m mmm0.o HQ va.m mqmflB mwha.m mmvo.0 >0N0.N 0000.0 m¢v0.v mmm0.0 mmh.¢ Hmh0.0 mazm ammo: meHmB mmB m0 ZOmHm000.mm ammo. ~0HH.O eemm.o omao. mmmm.mm ammo. mmmm.m «men. mnvm.a omvm.mm noee.au mmmm. mmmm.m mmmm.o mmem.ou eomm.m mamm.m mmmv.a mmmm. mmmm.mm mamm.a memo. «Hem.ou emmm.0~ Noam. mvmm.Hu Hamm.o ommo.m mmflm.oa maem. mmmm.ae mm\mm cam umcoo mmq mozm 0000.HI 0000H01 Hmh0.0 0BH0.0 hh0o.aa 500040! hN00.0 0000.0 mmzm H00N.0I 0H00.0| >0h0.0 00H0.0 5000.01 00H0.0I 0000.0 H000.0 muzm AMQOE zommHU mmB m0 mBZMHUHmmmOU DmfidzHEmm mH.m mqm¢e 0000.0 0000.0 H000.H H500.0 00h0.~ N0h0.0 0000.5H 0000.0 00HN.N mmm0.0 0500.0N 0000.0 000>.H 0000.0 0005.0H 0000.0 Qmm Alhmm.m mmm.m Almmmm mmmm H0002 101 As can be seen from Table 5.16, the reestimation of the Gibson model was quite close to Gibson's results. The coefficients which were insignificant in the reestimated regressions also tended to be insignificant in Gibson's regressions. The standard error of his regressions were larger and his R2 smaller, but Gibson used NSA money stock while the reestimated regressions used the SA money stock. The incorporation of the reserves variable into the money supply equation according to the t-ratios for this variable was a significant addition, but this was not the case where the interest difference term was replaced by the individual interest rates. The Gibson model with the Teigen difference of interest rates variable appears to be the most promising forecasting model examined in this section. B. EVALUATION OF PREDICTIVE PERFORMANCE While there is a wide range of possible prediction eval- uation statistics, actually there is little difference be- tween most of them. In this section will be described those statistics which will be used to evaluate the predic- tive performance of single equation models used to predict the money stock. 1. General evaluation The primary prediction evaluation statistic will continue to be the root mean squared error for the entire forecast iperiod and annually. The period of prediction will be up 102 A... pHH-::IIIw 0000.0 0000.00 0000.0 0000.0I 000H.H 0000.00 0I00 I HI00 0000. 0000.00 0000.0 0H0~.0I 0000.0 0~H0.0 000 00000 0000.0 0000.0 0000.0 0000.0 0000.00 0I00 I HI00 0000. 0000.00 0000.0 0000.0 0000.0 040 000 00.00 H000.m HOH0.H 0000.0 0000.0 0I00 I HI00 0000. 00.00 000.0 000.0 000.0 0009 acmbflw 00.HH 0000.0 0000.0 0000.0 0000.0 0000.0 0I00 I HI00 0000. 00.00 000.0 0000.0 000.0 000.0 0009 cowbflw ooflumm \0onumz mm\mm umcoo Ho 002% 0020 0020 £00 cofiumsflumm 00002 me02 ZOmmMO mmB m0 ZOmHmHEUHQmmm H42224 00.0 mqmfifi 0000 0000 000a 000a 000A 000a 0000 0000 0000 H000 Haw» 000a 000a 000a 000a 000a 0000 000a 000a 000a H000 Ham» 113 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 QIZAU 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 01000 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0I300 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 Alwmo 0000.00 0000.00 0000.00 0000.00 0000.00 0000.0 0000.0 0000.0 0000.0 0000.0 300 0000.00 0000.00 0000.00 0000.00 0000.0 0000.0 0000.0 0000.0 0000.0 0000.0 MAO Ac.u:oov o~.m mqm.o mmmm.o mmmm.o Hbmm.a Nmmm.o Alxmm mmo¢.mm momH.mN mmvo.ma Hmvm.oa memo.h Hamm.v mmhm.a omwm.m omno.¢ moam.m 0mm mamm.~ moam.a omvm.~ mmaw.a mmmo.H mmvH.H Nmmv.o vmha.o mmvH.H mmmm.o Almmmm hmmH.m Hmmh.a whom.v ommo.m mmvm.m momo.b NHHm.m boom.w «wa.m mmvm.m mmm mmvh.ma Hmbo.va mwmh.m mmmm.a vmvm.v hmhb.m mm¢m.h Mbom.m mmmH.HH NNmH.OH mmm.m mmvv.m mvom.m Navo.v oomw.m Nmmv.m mvam.v vmao.¢ mmhm.m Hmao.m mmmm.a émm HHNo.HH hmmm.o mmvm.m mahm.m vmmm.m hmmm.m mmHH.0H smo>.HH mvma.ma mmvm.aa xmm mmmm.vm mmhm.m mmbm.m mvwm.m mmam.v vhmm.m vmvm.a ommm.H mmnv.H onao.m mmmo mmom.NH mmh>.b momm.~ mmNH.v on~.b mhmb.h mHvN.a HMOF.OH mwbm.HH mmmm.0H mmmm vmnm.nm hvmh.oa mmmm.H mmom.m mvmv.b mmmo.m mmmo.a wmmm.N voma.m vah.m mDXQ mem.HH NhHh.m omva.m mHHv.h homm.v mmvc.m wmmm.m mhmv.N momH.H hmmn.o Alumm vvmm.mm NNmH.mH mhho.m omhm.v vmam.m mmvm.¢ maam.~ mmma.m mhhm.H monv.d me NNNN.OH wvom.b moma.m mvom.w mHHH.v vHH¢.v mme.m hmmo.m mmmm.o hhaw.o Almmm onav.hm mmvm.m mmmm.o Nomm.m comm.h mmmm.m «mam.o wome.m vmwa.~ omvw.m DXD mqmooz wqmmDm NMZOZ wqmmflmdbO m0 mOZ<2m0mmmm m>HBUHQmmm AdDZZé NN.m mqm<8 osma some mead head mead mama ¢mmH mwma mama Hmma Ham» onma mwma mead hwmd mmmd mwma voma mood mama mea How» 116 TABLE 5.23 PREDICTIVE PERFORMANCE OF THE TEIGEN MODEL (1 quarter forecasts) Statistic DXD DXS DXDS DSFS RMSE 18.7372 19.2451 18.7875 17.5586 MAE 6.6685 7.1941 6.8172 6.2040 SE of MAE 17.7334 18.0773 17.7300 16.6353 billion, and in 1971 approximately $4 billion. No other model displayed such erratic behavior. The standard deviation of the MAE statistic confirms this observation --it is about three times the value of the MAE.28 The Gibson alternative to the Teigen model outperformed the Teigen model in the area of prediction. (In the summary of the predictive performance of this model--Table TABLE 5.24 PREDICTIVE PERFORMANCE OF THE GIBSON MODEL Model RMSE(1) MAE(l) RMSE(6) MAE(6) FRSF 8.9383 8.1890 FRSF-L 1.5010 1.1715 7.5353 6.1326 FRX 9.4493 8.6965 FRX-L 1.8129 1.4588 8.9923 7.0113 F'RSF 10.1771 8.8360 F'RSF-L 1.4752 1.2118 i 28. Except in this case, the standard error of the MAE is less than the actual value of the MAE itself. 117 5.24--the prime indicates a log-log version of the Gibson 29 In model, D the Teigen structural shift dummy variable). all cases the decade RMSE statistids and the MAE statistics are quite low. Within the decade (Table 5.22), no strong trend stood out. Predictive performance declined in 1970 as it did for most models, but not as radically as it did in the Teigen model. 5. Adjustments of the constant term In order to examine the effect on a forecast that a constant adjustment term based on the residuals generated by the regression has, three different constant adjustment terms were studied. Table 5.25 tabulates the RMSE for the first quarter forecasts of the 1962-1970 period when no constant term was used, and when one of the three constant adjustment terms described above was used.30 In all cases the constant adjustment term reduced the RMSE over the entire period. The larger the error without constant adjustment, the greater the proportionate reduction of the RMSE when the constant adjustment term is used. The constant adjust- ment term also reduced the RMSE of models which included 29. Gibson included the Teigen dummy which marked the end of the Korean War, but drOpped the dummy indicating the begin- ning of the U.S. balance of payments problems. Gibson did this because his estimation period was shorter. ' 30. See p 104-105. The 1962-1970 period was used so that RMSE statistics could also be computed for the triannual periods 1962-1964, 1965-1967, and 1968-1970. These results will be discussed in the next paragraph. 118 TABLE 5.25 PREDICTIVE PERFORMANCE OF SELECTED MODELS WITH CONSTANT ADJUSTMENT TERMS (RMSE of 1 quarter forecasts) Model . No Adjustment Adjustment Adjustment Correction A . B C ASY 9.2670 1.2569 1.7192 2.5782 ASW 9.6991 1.3688 1.9096 2.8481 ALY 10.4144 1.3647 1.8571 2.7130 ALW 10.0731 1.4192 1.9780 2.9523 ASY-L 2.2223 .8874 .9536 1.0123 ASW-L 1.6318 .9342 1.0069 1.0418 ALY-L 2.7690 .9664 1.0576 1.1367 ALW-L 2.3403 .9646 1.0517 1.2070 CSY-L 1.4758 .9155 1.0087 1.0332 CSW-L 1.4305 .9030 .9946 1.0287 CLY-L 3.5105 .9835 1.1097 1.2126 CLW-L 3.3847 .9769 1.1066 1.2260 EBA 4.2089 2.3467 2.2196 1.7566 EBB 6.0818 2.2883 2.1137 1.9122 EBC 11.8320 2.3325 2.5647 3.0148 EBA-L 4.6764 .9871 1.1036 1.1364 EBB-L 4.9675 .9887 1.1076 1.1541 EBC-L 5.6262 .0023 1.1206 1.2026 FRX 9.2581 2.6416 2.6044 2.7177 FRSF 8.4490 2.4738 2.3790 2.5218 F'RSF 8.7192 2.3507 2.3954 2.8458 FRX-L 1.5330 .9122 .9676 .9600 FRSF—L 1.3788 .9370 .9970 .9890 F'RSF-L 1.2968 .9535 1.0318 1.0298 119 a lagged dependent variable. The constant adjustment term which worked best for all the money demand models was the error of the previous period's prediction. The average of the previous period, however, reduced the forecasting error of two (EBA and EBB) of the three Brunner non-lagged models better than did the previous period's residual. The latter adjustment worked best in the remaining of the Brunner models. The triannual RMSE statistics showed that constant ad- justment did not always improve the prediction performance of the model. In many money demand models, especially those where the lagged dependent variable was included in the model, the RMSE for the 1965-1967 period was lower when no adjustment term was used. This was not the case in the Brunner models where in all cases constant adjust- ment reduced the RMSE. The triannual data also showed that the error terms for the 1968-1970 triannual period was the highest of the three periods. Longer forecasts increased the RMSE and lessened the effect of the constant adjustment. However the relative relationships described above held constant, i.e. if a particular constant adjuStment term improved the predictive performance of the model in a one quarter forecast; then the predictive performance of the model in a six quarter forecast would also be improved, but to a lesser degree. This relationship held whether the constant term improved or worsened the performance of the model. The superiority 120 of the single period adjustment term was not reported in other studies where this approach was tried, but there is no instrinsic reason why one particular constant adjustment term should usually work better than another. D . SUMMARY In this chapter a set of money demand and money supply models were estimated over the 1947-1960 period and then used to forecast the 1961-1970 money stock. The estimated coefficients of the various models were compared to esti- mates made by others of similar models, and where the re- sults were different an attempt was made to eXplain the differences. The principle observations of this chapter were as fol- lows. (l) The Gibson models with the lagged dependent vari- able predicted with lowest RMSE. The real money demand fun- ction with the short-term interest rate and the lagged de- pendent variable was a close runner-up. (2) The hypothesis that structural shift occurred could be rejected by the Chow test for the models mentioned above. But this hypothesis was also rejected by the Chow test for a number of Brunner money supply models which had inferior RMSE statistics. (3) Adjustment of the constant term improved the predictive performance of the models. The previous period's error usually worked best. (4) The impact and steady state elas- ticities of the money demand model were low relative to other estimates of the money demand model. CHAPTER VI TWO-EQUATION MODELS A. INTRODUCTION In Chapter IV the predictive performance of naive and mechanistic money stock models were examined while in Chapter V the predictive performance of a number of single equation money stock forecasting models were examined. In this chapter the forecasting performance of some two-equa- tion money stock forecasting models will be studie . The selection of models to be examined is based to a great extent on the results of the previous two chapters. Using the convention of the previous chapter, each model will consist of a money supply equation and a money demand equation.1 It is unlikely, of course, that two equations adequately specify the complexities of the money stock determination process. The primary concern, however, is the predictive ability of various money stock models, and not this more basic question. The criteria of correct specification and good forecasting ability are not in con- flict, and ideally a model should have both characteristics. If the equations estimated in Chapter V are incorrectly 1. See pp 36-37. 121 122 specified reduced-form equations, and more than one endo- genous variable of the equation system is included in an equation, unless the system is recursive, the equations estimated in Chapter V are biased and inconsistent. There- fore, to the extent that the two-equation models used in this chapter reflect the actual structure of the money stock determination process, the used of system estimation techniques such as two-stage least squares (TSLS) will give estimated equation parameters which are at least consistent. The coefficients of the simultaneous equation models will be estimated using the appropriate estimation techniques for the 1947—2 to 1960-4 period.2 While statistics such as the canonical correlation coefficient measure the goodness-of- fit of the simultaneous system; if the structure of the system changes, such statistics are of little value. In- stead, consistent with the approach used in the earlier chapters, the estimated models will be used to generate six quarter simulations for the 1961-1 to 1972-2 period, and the forecasting ability of the models will be evaluated based on these simulations. Two equation models imply that there are two endogenous variables in the system. Obviously one of these variables is the money stock. In this chapter the commercial paper rate shall also be treated as the other endogenous 2. Only quarterly forecasts will be generated since the data for many of the exogenous variables are available only in quarterly form, e.g. GNP. 123 variable.3 While the primary focus of this study remains the forecasting of the money stock, in the simultaneous equation context, the forecasts of the other endogenous variables can not be ignored. This situation will be discussed in more detail in Section D of this chapter. B . THE MODELS Three different classes of money stock models will be examined in this chapter. Each has a money supply equation and, where warrented, three different money demand equa- tions. That is, there may be as many as three versions of a single model. The better performing versions of the var- ious types of models which were examined in Chapter IV and Chapter V will be considered as components for the models to be examined in this chapter. 1. The money multiplier model The supply side of the multiplier model is specified by the money multiplier identity discussed and evaluated in Chapter IV. There it was observed that the multiplier identity which forecast with the lowest RMSE was the 3. The conventional second endogenous variable is the interest rate, but prior to the Treasury-Federal Reserve Accord, the Federal Reserve attempted to fix the rate at a low level. This convention, then, results in speci- fication error. But this difficulty would exist for any variable the Federal Reserve viewed as a target variable. Therefore, if the interest rate is used as- an endogenous variable in the 1947-1951 period, the ques- tion is not whether specification error exists, but how serious the problem really is. Wu._'i Ir: 124 no—change naive version of the multiplier.4 This version of the multiplier, i.e. the value of the multiplier is assumed to be fixed at its current value over the period of the forecast, will be used in the multiplier model. The demand side of this model will be Specified by the nominal money demand equation with income, the commercial paper rate, and the lagged dependent variable as explan- atory variables. Since this equation is the only equation which contains both endogenous variables--the money stock and the commercial paper rate--the system is recursive, and, therefore, ordinary least square estimates of the coefficients of the money demand equation will be consis- tent. Since the money supply equation is an identity, the OLS coefficients of the money demand equation will also be asymptotically efficient. 2. The Brunner-Meltzer model The money supply equation in the second simultaneous equation model will be the first of the Brunner-Meltzer linear money supply equations.5 Since the predictive per- formance of most models is improved by including a lagged dependent variable in the model, the equation will also include the lagged variable. However, as observed in Chapter IV, the predictive performance of this particular 4. Equation 4.8, p 54. 5. Equation 3.10a, p 38. ">1 .‘hl‘o- s i 1 125 equation did not improve when the lagged variable was used.6 This situation suggests that the Brunner-Meltzer money supply equation without the lagged variable could also be considered as an alternative formulation of the model. However as we shall bring out in Section D, since this equation does not contain the endogenous interest rate variable, and is, therefore, part of a recursive equation “1 system, both versions of this model can be evaluated with data from only one version of the simultaneous system. As with the multiplier model, the nominal money demand model with the short-term interest rate, income, and the lagged dependent variable will be used as the money demand equation. From the recursive nature of the system, the OLS estimates of the coefficients of the two equations will be consistent, but asymptotically efficient only if the variance-covariance matrix is diagonal. 3. The Teigen-Gibson models Two versions of the Teigen-Gibson model will be used as money supply equations. In both cases reserves adjusted for reserve liberations due to changes in the required reserves ratios and the difference between the commercial paper rate 6. See Table 5.21, p 114. The deterioration in predictive performance was small. The RMSE for a one period forecast was 4.1393 for the equation without the lagged variable, 5.3046 when the variable was included. 7. Kmenta, 1971, p 586. The correlation coefficient of the estimated residuals of the two equations is 0.65. 126 and the Federal Reserve discount rate will be used as explanatory variables. In one version of the model the lagged dependent variable will be excluded, in the other it will be included. Three versions of the money demand equation will also be used in the model. The first will be the demand equation used in the above models. The second demand equation will_ use permanent income rather than GNP as an explanatory variable while the third equation will eXpress all quantity variables in real terms. The economic activity variable in this last equation is real income. This set of demand equations will be expanded to six by having three with the lagged dependent variable, three without the lagged variable. Since both equations of this model contain both endogen- ous variables, to obtain consistent estimates of the coeffi- cients, a simultaneous equation estimation method must be used. Therefore, the coefficients will be estimated by means of TSLS and limited information single-equation (LISE) estimation techniques. Full information estimation tech- niques are ruled out because of the probability of equation misspecification and the complexity of such estimation tech- niques for non-linear equation systems. 4. The estimated models In order to facilitate the discussion of these model, the following discriptors for the various models shall be used. 127 The multiplier model will be referred to as Model A, and the Brunner-Meltzer model as Model B. The Teigen-Gibson models will have a two letter identifier. The first letter C will identify the model with the lagged dependent vari- able in the demand equation, andwithout the lagged vari- albe in the supply equation. The letter D identifies those models where both equations have the lagged variable, and the letter E models with the lagged variable in the supply equation but not in the demand equation. The second letter specifies the money demand equation used in the model. A represents the nominal money demand equation with income as an explanatory variable, B represents the equation with permanent income, and C represents the real money demand equation. The OLS regressions used for Models A and B have been estimated in Chapter V. The estimated coefficients for the money demand equation for both Model A and Model B can be found in Table 6.1 as the OLS estimates for the demand equation of Model CA8 while the Brunner-Meltzer money supply coefficients are given in Table 5.11.9 The OLS, TSLS, and LISE coefficients for the three C models are given in Table 6.1. The demand equation coefficients for the D models are the same as the C model coefficients since the exogenous variables are the same for the two models. The 8. These coefficients are also given in Table 5.2, Model ASY-L. 9. Equation EAA-L. I“: g“ up 1" a k 128 mvvv. Nmmo. mnmm. .3.D vmhm. mmmm. ammo. mmoo. momm. mwoo. momm. mmoo. mmmm. mm m \N omoo. ammo. mmoo. vomm. omoo. homo. m mm\m thm.o HHmo.o mmmv.o mmmo.o oowv.o muoo.o umcoo Hmmm.a mmHN.o mmoo.m mHNN.o wmmm.m mmnm.o umcou mmmm.ma momm.o mmhv.ha moom.o mmmo.mH mmm¢.o mmq Hmsm.>a hovm.o momm.ma wNmm.o Nah¢.om vam.o man momm.~ hmho.o ovom.N mmoo.o vvwv.a ammo.o QM momm.m mwmo.o mmmv.m mhmo.o mmmm.w mmmo.o M mmom.mn hamo.0I mmmm.ml m5mo.on ammo.m1 omHo.0I mozm vomm.mt wwwo.ou hamm.mn vmmo.OI wmmm.ml omHo.o1 mozm mqmaoz mmHmmmIO mmB m0 mfizmHUHmmmoo H.m mqméfi mqu mqmfi mAO mmHQ mAmB mQO Gonumz OMBdSHBmm 2 CH ocmEmo 2 ca ocmEmo manmflum> coflpmeflumm unwocmmmo 83 Iapow VD IGPOW 129 mmhm. comm. momm. .3.D mmmw. ovmm. momm. m¢NH.OH mnma. omvm.m mamh. vommum mmmm. mm\mm mHNN.m hmmm.o mmvm.m moom. oomm.m mmmm. mm\mm . I .t wmmo.v momm.ah omvm.m mmam.mm mawm.m mmmm.mm uncou Homm.v tho.mo vumm.m mamm.v¢ mamm.m mmmm.mm #mcou a O N. 1'31. lic‘li mHmN.N ommm.m omvm.m vamm.m mhvamN vvvo.m Amm mmmm.m Hmmb.N mohm.ma mwmv.v mhvm.mm vevo.m 9mm A©.ucoov H.m moms.“ mmmm.om Hmmm.m omhh.ha mmmv.H mmmm.H mmzm Mbhm.m mvmm.mm mmam.N hvmo.m momv.H mmmm.H mazm mqm<8 mmHA mAmB qu mmHA mama mQO Begum: mammsm 2 mammsm manmfium> soflumswumm usmoammmo DD IQPOW 83 IGPOW 130 ammo. hqvm. momm. .3.Q momm. homo. comm. mmmm.m «mom. mHoN.v comm. oomm.m mmmm. m m m\~ mmoo. vase. omoo. mmhm. vmoo. mmmo. mm\mm Hmoh.v «www.mo mmmm.h mvmv.mv mamm.m mmmm.om pmcou oamo.o Hamm.va vmmo.o mhom.o mmam.o mmHH.mH mhao.o ooom.o omnn.a mamm.ha mmmv.o mmvm.o umcoo mod omma.m mmmm.m mmwm.m «mom.v mhvw.mm vvvo.m amm onma.o mnoo.o oaav.o mmao.o mmmm.m mmmo.o m\w mmwm.m mwhv.mm momm.N vmom.ma momv.H mmmm.H mozm moam.on mmoo.o: Nomv.OI mooo.ol vaom.MI oHNo.oI muzm Ao.ucoov H.o mamas mmHA mama mAO mmHA mAmB mAO eonpmz mammsm m\z :H panama manmflum> coaumaflumm usmocmmmo V3 Iapow DD IGPOW 131 OLS, TSLS, and the LISE estimates for the money supply equations, however, are given in Table 6.2. Likewise the estimates for the coefficients of the money demand equation without the lagged variable (Model B) are given in Table 6.3. The dependent variable of the money demand equation of the C, D, and E models is the logarithm of the money stock.10 The dependent variable of the money supply equation, the Teigen-Gibson money supply function, is the nominal money stock. The coefficients of the different estimates of the money demand equations in models CA, CB, DA, and DB are quite close whether estimated by OLS, TSLS, or LISE; and the increase in the standard error for these regressions when the coefficients are estimated by TSLS or LISE is small. The difference between the OLS estimates and the TSLS and LISE estimates of demand equations for Model CC and DC is striking. But the coefficients which change the most have t-ratios of above 2.0 when estimated by OLS, and less than 0.6 when estimated by TSLS or LISE. The decrease in the standard error is slight when the demand equation is not estimated by OLS. The coefficients for the demand equations without the lagged variable, however are affected by the estimation method used. The coefficients for the two variables are 10. For Models CA, CB, DA, DB, EA, and EB it is the loga- rithm of the nominal money stock, for Model CC, DC, and EC it is the logarithm of the real money stock. “flunk-)mm 6' _ I. ,. I I 132 mmmm. ammo. momm. boon. ooow. movm. omvo. .3.D .H.o magma CH oasom mm mEMm on» mum mucofloflmmmoo :oflumsvm ocmfimo mmcofi one ovvv.H mmmm. moom. ommm. Hoao.H mamm. Hmow. ovmm. hva.H hmmm. mmoa.a Nomm. Noon. Nmmm. mm m \N ommo.m vmwh.oa mmmn.a mmmh.m mmvm.~ Nmmo.h oovm.m mnmm.m hoaw.m wmon.m momm.m vomm.h mhvm.a Nmom.m umcou ommm.m hmmm.o HHmo.oH mmmm.o mom¢.oH Nth.o boon.ma mvmm.o oomm.NH momw.o mmma.va voom.o omvm.mm mmmm.o mod ohvo.a momo.o mmmo.m mhh¢.o omHo.m Hmam.o vaH.N mmov.o momm.H mmom.o momm.H mowm.o ovam.m mmmv.o Amm Nomm.H Hasm.m mqu mmos.o . ssao.a mama on mmmo.a momm.H mmHa ssmm.o smmm.o mama mo mmms.a mmms.~ mqu mvmm.a momm.~ mama sows.a- mmsm.o- moo am gonna: mozm COflngflumm Homo: mAWQOZ mmHmmmIQ HEB m0 N.o mqmflB mZOHfidadm MammDm NMZOZ mmB m0 mBZmHUHmmMOU OMB.m «www.ma mmmm.m| mmmm. oomm. momm.a oaom.o ommo.ou mums mmao. vsno.vm mvvm.mm avov.mn ovam. oonm. oaom.m anmv.o memo.on mqo 4m oonumz .3.o mm\mm umcoo m» Am\w Hoe w mozm nodumefiumm Howe: mamaoz mmHmmmlm mmB ho mZOHB 0 (6.1a) ln RMCP = -‘_‘ ID) 82 3 -T1nY-_lnM 1 B1 1 The other two versions of the money demand equations for MOdel C were rewritten with the apprOpriate changes. The transformation shown in Equation 6.1a was also used for the simulations of Model A and Model B. Since we have both TSLS and LISE estimates of the model, the question arises as to which should be used for fore- casting. In all cases, the standard error of estimate was lower for the TSLS regressions. This suggests the use of the TSLS coefficients. Moreover Theil maintains that when a difference in the values of the estimated coefficients is observed, that the TSLS estimates should be used in pre- ference to the LISE. [Theil, 1971, p 532]. The TSLS coefficients, therefore, will be used in the forecasting equations. D. THE RESULTS 1. Forecasting the money stock with recursive models The results of the multiplier model (Model A) and the Brunner-Meltzer model (Model B) are identical to that of 137 the single equation models.l3 Since there is no lagged term in the multiplier model, predictive ability does not decline as the forecasting period is lengthened.l4 The lagged term, however, is included in the Brunner-Meltzer equation so that forecasts longer than one quarter can be generated. The overall RMSE statistics for one through six quarter forecasts for these two models are given in Table 6.4 while the mean percent absolute error data is given in Table 6.6. The error associated with forecasting the money stock in recursive models is the same as for the single equation models with the same endogenous variables. Therefore the ability to forecast the money stock by recursive models can be determined by examining the predictive performance of the apprOpriate single equation models.15 2. Forecasting the money stock with non-recusrive models Unlike Models A and B, the Teigen-Gibson money demand Inodels are not recursive, and so the money stock and the 13. For Model B, see Table 5.21. The RMSE statistics for the multiplier model are a bit different since they were computed over the slightly different time period. Compari- son of the annual RMSE data, however, shows the identity of the results. 14. When a lagged term is not included in an ex post fore- casting model, all forecasts are essentially one period forecasts. See p 103. 15. For example, the predictive performance of the not- lagged first Brunner model can be found in Table 5.11, Model EBA. 138 TABLE 6.4 PREDICTIVE PERFORMANCE OF TWO-EQUATION MODELS OF THE MONEY STOCK Model Quarter Quarter Quarter Quarter Quarter Quarter 1 2 3 4 5 6 A 2.2380 2.2245 2.2294 2.2270 2.2489 2.2292 B 5.3047 10.1477 14.3957 18.1238 21.6497 24.9463 CA 2.6365 4.9729 6.8511 8.3591 9.7970 11.1019 CB 3.1680 6.0131 8.3720 10.3227 12.1529 13.8141 CC 1.9134 3.6563 5.1022 6.3747 7.6729 9.0093 DA 2.4253 4.6765 6.5591 8.1445 9.7159 11.2000 DB 2.0239 3.9690 5.6142 7.0363 _8.5071 9.9419 DC 1.9200 3.4675 4.7988 5.8844 6.9840 8.1252 EA 5.3809 8.6880 10.6554 11.8599 12.8755 13.7668 EB 3.6542 6.7601 ' 9.0871 11.0575 12.7402 14.2245 EC 3.2573 5.8414 7.7314 9.0897 10.2491 11.2196 TABLE 6.5 PREDICTIVE PERFORMANCE OF TWO-EQUATION MODELS OF THE INTEREST RATE Model Quarter Quarter Quarter Quarter Quarter Quarter 1 2 3 4 5 6 A 4.4144 6.6897 6.6651 6.6306 6.6508 6.6743 B 9.5640 10.0182 10.3759 10.5826 10.9316 11.3324 CA 0.5427 0.5730 0.5856 0.6088 0.6701 0.6990 CB 0.7537 0.8306 0.9070 0.9856 1.0742 1.1226 CC 0.5174 0.5664 0.5854 0.6256 0.6971 0.7410 DA 0.6010 0.6156 0.6145 0.6235 0.6836 0.7137 DB 0.5629 0.5970 0.6182 0.6541 0.7502 0.8043 DC 2.1404 1.9838 1.9504 1.9972 2.0446 2.0589 EA 1.7657 1.1202 0.8126 0.7036 0.7318 0.7642 EB 2.2249 1.8731 1.5012 1.1862 0.9570 0.8187 EC 1.5824 1.2159 0.9741 0.8668 0.9290 1.0093 139 TABLE 6.6 COMPARISON OF THE PREDICTION AND THE INTEREST RATE OF THE MONEY STOCK (Mean Percent Absolute Error of 1 to 6 quarter forecasts) Model Var- ible A M RMCP B M RMCP CA M RMCP CB M RMCP CC M RMCP DA M RMCP DB M RMCP DC M RMCP EA M RMCP EB M RMCP Ex: M RMCP Quarter Quarter Quarter Quarter Quarter Quarter 1 1.1064 2.4016 1.0942 40.4304 1.2435 62.6529 0.8220 36.1223 0.9594 41.3806 0.8170 34.7272 0.8100 101.0370 - 2.0959 129.5495 1.4267 1.3672 2 1.0855 4.5519 1.9787 41.9595 2.3235 74.8008 1.5581 39.7880 1.7912 42.6238 1.5529 37.0629 1.4396 92.5165 3.2785 80.2648 2.5643 2.4038 3 1.0937 6.4459 2.6774 43.5137 3.2548 84.1521 2.1977 42.7585 2.4899 42.5931 2.1917 39.3253 2.0023 90.5429 3.9573 58.8731 3.4482 l60.6466132.6093103.5653 3.1593 121.0287 92.2134 71.3931 4 1.0805 8.1175 3.2316 45.7172 4.0475 89.2568 2.7478 47.4642 3.0973 43.7603 2.7547 42.8063 2.4070 5 1.0850 9.6476 3.7246 49.5123 4.7703 95.4925 3.3071 51.8774 3.6489 47.6267 3.2930 48.6164 2.8334 6 0.0545 341.0984486.3145482.4153472.3153477.l770483.5724 11.0543 846.2194887.4291922.6329949.8376983.91101018.660 51.7881 5.4011 98.2608 3.8625 56.9353 4.1768 50.2393 3.8307 52.8889 4.4933 94.9068100.1531101.8735 4.3761 50.7998 4.1110 79.1343 3.6926 62.1599 4.7017 52.4478 4.6970 62.6211 4.1328 62.2277 4.9634 54.6762 5.1871 54.4618 4.4737 64.4639 140 interest rate are jointly determined. The forecast quan- tities, therefore, will be differerent than the single equation forecasts. Ignoring the inclusion or exclusion of the lagged variable for a moment, the best performing models in general had the real money demand equation.16 In two cases the models-with the money demand equation with permanent income as an explanatory variable were ranked second.17 The model with the lowest RMSE was Model DC where both equations had lagged variables and the demand equation was in real terms.18 The incorporation of the lagged dependent variable in a single equation model usually reduced the forecasting error for short forecasts. This gain in forecasting performance with the lagged variable, however, deteriorated as the forecast was lengthened since the model now included a forecast independent variable: the lagged variable. This caused error buildup to occur as the period of the fore- cast increased. Both of the equations in the D series models have lagged variables while the C series models only the money demand equations have a lagged variable. In the E series it is the money supply equations models which have the lagged variable. Single equation models without the 16. Models CC, DC, and EC. 17. Models DB and EB. 18. In the first quarter Model CC had a lower RMSE, but this was by a very small margin. 141 lagged variable tend to have a relatively high, but con- stant error term over the forecast period while single equation models with the lagged variable tend to have an initially low error which builds as the forecast is ex- tended. To what extent does this pattern carry over to the simultaneous equation models? The data presented in Tables 6.4 show the D series models which have lagged dependent variables in both equa- tions forecasting with a lower RMSE than models where at least one equation lacks the lagged variable.19 Moreover error buildup occurs in all the forecasts, and with a pattern and magnitude similar to that of the single equa- 20 The series with the highest overall RMSE tion models. statistics of the non-recursive models was the E series models which had the demand equation without the lagged variable. The annual RMSE statistics for the C, D, and E series models are given in Table 6.5. These statistics reveal that part of the lower overall RMSE associated with the D series models is due to the fact that the annual RMSE statistics for these models deteriorate less in the late 1960's than 19. The exceptions to this statement are the one-quarter forecasts of Model CC and the six quarter forecasts of Model CA. The difference in RMSE in each of these cases was small. 20. See Table 5.19 for the single equation money demand RMSE statistics. The RMSE statistics for the Teigen-Gibson model ‘with the lagged variable over the six quarters are 1.8129, 3.5459, 5.0301, 6.3308, 7.6775, and 8.9922. 142 do those of other models. In the early 1960's other models frequently outperformed the D series models, but around 1967 when the quality of the 6 quarter forecasts declined, except for Model CA, the decline was less for the D series models. Some of the RMSE statistics of the single equation model are given in Table 6.5 in order to compare the relative forecasting power of one and two equation models. Exam- ination of the overall RMSE statistics show that except for Model DC, the RMSE for the six—quarter forecast of the Teigen-Gibson model with lagged variables was lower than any of the other two-equation models. The real money de- mand equation also had a lower RMSE than any of the two equation models. When two superior forecasting models are combined to make a two-equation forecasting model, that model is characterized by low RMSE statistics although in these cases the predictive performance of the single equa- tion model is superior.21 The RMSE statistics for the two equation model tends to be the average of the two individual equation models. 21. For example, the RMSE statistics for Models FRD—L and .ASW-L are close--8.9922 and 9.8346; but when combined to become Model DB, the RMSE is slightly higher--9.94l4. This, of course, would be expected unless there were a strong negative covariance between the error terms of the two equations. 143 3. Forecasting the interest rate While forecasting the interest rate is not a matter of concern in this study, nevertheless it may be worthwhile to examine these predictions. Table 6.7 gives the RMSE statistics for six quarter forecasts of the interest rate. The RMSE statistics for Models A and B are high. This situation follows from the recursive nature of the system: the money stock is first forecast, and only then is the interest rate forecast. Since the average value of the interest rate is different from the average value of the money stock, a more realistic statistic to examine is the mean percent absolute error (MPAE)for both the money stock and the interest rate fore- cast. These statistics are given in Table 6.6. They bring out dramatically the inferior quality of the recursive model forecasts of the interest rate. This table, however, also shows that the interest rate forecasts of the non-recursive simultaneous equations models are inferior to that of the forecasts of the money stockJ The forecasting performance over the six quarters remained constant for the D series models, deteriorated over the forecast period for the C series models, and actually improved over the forecast period for the E series models. It appears, therefore, that the money demand equation contributed more to the determination of the interest rate than the money supply 144 omHN.HH omHN.HH mmom.ha moom.mH vhmm.va hmhh.b ommo.N hwhm.m momv.o onmm.m ommm.m Um m¢~N.¢H omo¢.hm «Nam.mm Hamo.ma mono.oa oomH.m vmmm.~ vmav.m mvmm.o mmmm.~ Hoao.m mm mooh.ma whom.om momm.mm hmmo.ha mmmm.¢a Noom.o vovm.H omhm.H ONNN.H oovo.m oomm.m ém NmNH.m NmNH.m Nmoo.o mhmo.m mmom.ma hmom.m mamm.¢ mnmm.o omom.v momo.m hmmm.o 0Q oavm.m wovo.md «www.mH mmmo.NH mmov.ma mmmH.h coma.m Hmvh.m whoH.H omon.a momo.m ma ooom.HH mmho.HN oavm.mH mmmh.ma mmvo.va boom.h mmhm.H vmmv.m ohom.c momb.H mamm.m fin mmoo.m ONHN.oH hmoo.oa NHbv.o mvov.mH mooo.m Noom.¢ mmmm.h mono.v mmmm.~ Hooo.o UU AmummomHOM umUHMDW o mo Mmzmv Hva.MH Nmmh.mm omov.m~ hHmo.mH Hmmn.bH moom.m mahm.v mmba.o hobo.m Hmvo.H Nmoo.m m0 mAMQOZ Dmeumqmm m0 @0242m0mmmm m>HBUHDmmm AdDZZd h.o mamde mHOH.HH HHHm.om mamm.ma ommo.MH Hmoa.mH hmmo.o moom.o Namo.m mamb.o Hove.m omah.m flu omlao onma momH moma homa ooma moma vomH momH momH HomH uwmw 145 equation. The larger forecasting error for the interest rate can be explained by the fact that the interest rate series itself is a more volatile series than the money stock series. Moreover these equations were not conceived as having the interest rate as the endogenous variable. 0' D . SUMMARY The results of this chapter can be summarized very briefly. (1). The money stock forecasting error in a two- equation recursive system will be the same as for the single equation model without the system's other endogenous variable. (2) The forecasting error of a model where the two endogenous variables are jointly determined, the money stock error statistic will tend to be the average of the results of the two single equation models.23 (3). The models which were used in this study were poor forecasters of the interest rate. 22. In order for error buildup to occur in a model over the forecast period, the lagged version of the model must be used. Error build-up occurred when lagged versions of the Inoney demand equations were used, and the money supply equations had no lagged term (the C series models), and the opposite phenomenon occurred when the not-lagged version of the money demand equation and the lagged version of the money supply equations were used. 23. See footnote 18 for a caveat on this observation. CHAPTER VII CONCLUSION This chapter is an outline of the principle results of this study. Because of the nature of the study, emphasis will be on the preceeding three chapters. Of immediate interest, of course, is the answer to the question: Which of the money stock models considered in this study fore- cast best? In addition, the various means of improving predictive performance and the evaluation of predictive performance will also be discussed in this chapter. A. FORECASTING PERFORMANCE The models which forecast best in this study were two mechanistic models: the autoregressive seasonally adjusted money stock model and the no-change money multiplier model.1 The autoregressive model had the lowest RMSE statistic for short forecasts while the multiplier model had the lowest RMSE statistic for longer forecasts. The predictive performance of the autoregressive multiplier 1. Since the monetary base must be forecast when using the Inultiplier models, they are not true mechanistic models. It may, in fact, be almost as difficult to forecast the jbase as it is to forecast the money stock. The value of -the multiplier itself, however, is forecast by means of a mechanistic model . 146 147 model was also quite respectable, but examination of the estimated coefficients for this model show that it is essentially a no-change model. Three models are next in terms of forecasting performance. Two are single-equation real money demand models with the lagged dependent variable, the short-term interest rate, and either income or permanent income.3 The other mOdel is the single-equation Gibson model with the lagged dependent variable.4 The RMSE statistics for these models were quite close for the short forecasts, but the real money demand models had better prediction statistics than the Gibson models in the longer forecasts. The next level of predic- tive performance would include many of the other money de- mand equations with the lagged variable. Also included in such a list would be the better performing of the two- equation forecasting models. B. IMPROVEMENT OF FORECASTING PERFORMANCE Since time series data were used in this study, the improvement of forecasting performance which occurred when a lagged variable was included in the model was not 2. See Equation 4.11 and Equation 4.14. 3. The real money demand model with permanent income as the (explanatory variable had a slightly lower RMSE over the forecast interval. .4. The Gibson model with the interest coefficient not con- srtrained to be equal in magnitude had somewhat lower RMSE sytatistics than did the model with the constrained interest rates. 148 unexpected. Only once did forecasting performance of single-equation models deteriorate when such a variable was added.5 The Gibson model was one example where the improvement in predictive ability of a model was quite marked. One way of correcting for the temporal interrelatedness of the variables is to include the lagged variable in the model; another way is correcting for first-order auto- correlation. When the money demand equations without 1agged variables were corrected for first-order autocor- relation, the RMSE of a forecast was reduced even more than when the lagged variable was included in the equations. When the money demand equations with the lagged terms were corrected for autocorrelation, the RMSE was also usually reduced but by a smaller proportion. While no attempt was made in this study to use or evaluate judgmental corrections of the constant term, various mechanical constant adjustment techniques were used, all of which used the residual from the forecast of the previous period. The RMSE in most cases was reduced when these adjustments were made. The RMSE of models with- out the lagged term, when adjusted, was lower than that of the lagged model not adjusted. Adjustment of the lagged model, however, also improved forecasting performance. 5. This case was the first Brunner-Meltzer model, Equation 3.10a. 149 The best performing constant adjustment term in this study was the previous period's residual. C. TWO-EQUATION MODELS Two types of the two-equation forecasting models, re-. cursive and non-recursive, were examined. When the system was recursive, the money stock RMSE statistics were the same as the statistics for the one equation model which contained both endogenous variables. The other equation would forecast the other endogenous variable which in this case was the interest rate. The RMSE statistics for .the interest variable were quite high. The non-recursive two-equation models, where both endo- genous variables were in both equations, consisted of the Gibson equation and three forms of the money demand equation. The models with the real money demand equation with the lagged dependent variable and the Gibson model with or without the lagged variable had the lowest RMSE.6 These models, however, had higher RMSE's than did the better single equation models. Forecasts of the interest rate by the non-recursive models, while better than those by the recursive models, still had high RMSE statistics. Comparison of the coefficients estimated by means of OLS and TSLS reveals that the two nominal money demand equations with a lagged variable were insenstive to 6. The Gibson equation with the lagged variable had a slightly lower RMSE over most of the forecast period. 150 estimation model. But the estimated coefficients of the Gibson model and the real money demand model whether or not the models include a lagged variable were quite sensitive to estimation method. This was also the situation for the nominal demand models without the lagged variable. In practicially all cases for these models, the TSLS estimates differed from the OLS estimates, and the LISE estimates differed even more so. The sensitivity of the lagged Gibson equation to the method of estimation was particularly interesting in light of the forecasting performance of the single-equation Gibson models. D. EVALUATION OF PREDICTIONS The RMSE statistic was the main prediction evaluation statistic used in this study. When it was necessary to compare the forecasts of different variables, the mean percent absolute error statistic was used. Little if any- thing seemed to be gained by using other prediction evalu- ation statistics. Regressions of the predicted quantity on the actual quantity showed themselves incapable of being a measure of forecasting performance. Only the standard error of estimate appeared to be of interest, but it re- vealed nothing that was not already revealed in a more meaningful fashion by the RMSE statistic. Both the mechanistic and the one-equation models were tested to determine whether a significant structural shift occurred between 1947-1960 and 1961-1970. The 151 hypothesis that no structural shift occurred was rejected for four of the five mechanistic models examined. The exception was the no-change multiplier model which for forecasts of over six months was the best performing mech— anistic model. Among the single equation models, the hypothesis that structural shift had not occurred could be rejected for only two models without a lagged dependent variable: the first Brunner-Meltzer model and the Teigen model. Inclusion of the lagged term in the equations im- proved the structural stability of the models, and in- creased the number of models for which the hypothesis could be rejected. Included in a listing of these models would be the money demand models and the lagged Gibson model. It should also be noted that the Teigen model, with and without the lagged dependent variable, did not exhibit significant structural shift, but also had the worst prediction record of the single equation models examined. Structural stability, therefore, is not a suf- ficient condition nor, based on the evidence from this study, a necessary condition for forecasting with low RMSE. E. INCOME AND INTEREST ELASTICITIES The coefficients of the money demand equations indicated low impact and steady state income and interest elasti- cities.7 The elasticities derived from TSLS estimates of 7. In only one case were the elasticities above 1.0, and then it took 25 quarters for the elasticities to reach 152 the money demand equations with the lagged dependent vari- able were close to those derived from OLS estimates, but TSLS estimates of the demand equations without the lagged variables indicated elasticities of somewhat greater magnitude. Compared to other studies, however, the values of both the interest and income elasticities remained low. F. SUMMARY The results of this study can be seen as presenting a challenge to the econometric model builder. Two mechan- istic models forecast with lower forecasting error than any of the one- or two-equation economic models. 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Ratio of time deposits held by the non-bank public to time deposits of member banks. Monetary base. Usually the base is the net source base: the sum of unborrowed reserVes plus currency held by the non-bank public or not included in reserves. Ratio of borrowed reserves to total deposits sub- ject to the reserve requirement. Currency and coin held by member banks and included in reserves. Currency in circulation. Currency held by the public, i.e. currency com- ponent of the money stock series. 1. The multiplier model. Ratio of currency held by banks but not part of their reserves to currency held by the non-bank public. 2. The Teigen model. Ratio of currency held by the non-bank public to the money stock. Non-Federal government and non-bank demand deposits held at member banks subject to the reserve requirement. ‘ Demand deposits held by the non-bank public, i.e. the demand deposits component of the money stock series. Adjusted net demand deposits at all member banks. 161 DTM M'k RMCP RMDD RMDF RMFR RMLB RMLG RMTB 162 Total member bank deposits subject to the reserve requirement. Structural shift dummies for the Teigen model. The ratio of Treasury deposits to demand deposits held by the public. Exogeneous expenditure (as used in the Teigen model). Ratio of private demand deposits held by non- member banks to the money stock. 1. Multiplier models and the Brunner-Meltzer models. Ratio of currency held by the public to demand deposits held by the non-bank public. 2. The Teigen model. Ratio of private demand deposits of member banks to reserves required for those demand deposits. Money stock held by the public, i.e. currency and demand deposits held by the non-bank public. Money stock component that is based on supplied (exogeneous) reserves (as Specified in the Teigen model). Money multiplier. Population of the United States including the armed foreces overseas. Net worth (as used in the Teigen model). Implicit GNP price deflator. Commercial paper rate on 4-6 month prime paper. Implicit yield on demand deposits. Difference between the commercial paper rate and the Federal Reserve discount rate. Federal Reserve discount rate. Yield on domestic corporate bonds, Moody's Aaa rated. Yield on long-term government bonds. Yield on 90-day Treasury bills. RMTD RRD RRT RSF RSL RSR RSREL RSRPD RST RSU TD TDM YP 163 Effective yield on pass-book savings deposits at commercial banks. Implicit reserve requirement against demand deposits subject to the reserve requirement. Implicit reserve requirement against time deposits of member banks. Free reserves. Total reserves plus reserves released through changes in the reserve requirement. Required member bank reserves. Reserves released through changes in the reserve requirement. Reserves available to support private non—bank deposits (RPD's). Total member bank reserves. Unborrowed reserves. Ratio of reserves to total deposits. Seasonal dummies. Time deposits held by the public. Time deposits of member banks subject to the reserve requirement. Ratio of time deposits held by the non-bank public to demand deposits held by the public. Gross National Product. Permanent income, an exponentially weighted average of GNP. See footnote 2, p 75. DATA APPENDIX The sources of data used in this study were quite stan- dard and are listed below. Where possible seasonally ad- justed data was used. Reserves and member bank deposits data was taken from Board of Governors, Federal Reserve System, revision of "Aggregate Reserves and Member Bank Deposits," Statistical Release H.3, April 1972. Some early data (prior to 1958) was obtained from the Federal Reserve Bulletin. Reserves released by changes in the reserve requirement data series was from the Economic Research Department, Federal Reserve Bank of St. Louis. The money stock series (i.e. demand deposits and currency held by the non-bank public) was from the Federal Reserve Bulletin, December 1970, pp 895-898. Other data such as income and the interest rate series were taken from Business Statistics, 1971. In some cases, earlier volumes of this series were consulted. 164