ESTIMATES OF THE DEMAND FOR FOOD FROM CONSUMER PANEL DATA Thesis for the Degree of Ph. D, MICHIGAN STATE UNIVERSITY W'iIIard Robert Sparks 1961 This is to certify that the thesis entitled ESTIEL'ETES OF THE DEE-'IAND FOR FOOD '7'ROM CONSIDER PANEL DATA presented by Willard Robert Sparks has been accepted towards fulfillment of the requirements for M—degree inAgnicnltnral Economics Date m0 t 0-169 H3 3‘39; * W * 9129262005 T071999 20%9.9, ABSTRACT ESTIMATING THE DEMAND FOR FOOD FROM CONSUMER PANEL DATA By Willard R. Sparks This study focused on estimating parameters determining the demand for food using combined time series and cross section data. The major emphasis dealt with the specification of variables and functional form.of the demand relationship for food. The primary data used in the study were taken from.the MUchigan State University Consumer Food Panel; only families classified as wage earners were examined. The demand for food was assumed to depend on the income and the size and composition of the family as well as the prices of fOod and nonfood items. Three alternative functional forms were considered as representing the demand relationship for food. They were: (1) linear, (2) quadratic, and (3) logarithmic. The logarithmic relationships seemed to be most plausible of the alternative forms considered to represent the demand for food. The income specification that best represented the behavior of the family with regard to food purchases was the income received in the current period plus one-third of the income received in the previous twelve weeks. Size of the family was represented by the number of adult male equivalent units in the family. Analysis of covariance was used to estimate the demand relations. from.cembined time series and cross section data. In this model family Willard Sparks constants were estimated to represent the unobservable characteristics of the family which were not reflected in the independent variables. These constants vary among families, but are constant over time. The possibility of seasonality or trends in the effect of income 'was investigated but rejected on the basis of present evidence. The estimated elasticities of the family size variable suggest that a ten percent increase in the size of the family, in terms of adult male equivalent units, would result in approximately a five percent increase in food expenditures. The estimate of the price elasticity of the demand for food was approximately’0.59 whereas the nonfood price elasticity of the demand for food was approximately 0.10. The most interesting result of the study was the effect of introducing family constants on the estimated income elasticity. When family constants were estimated the coefficient of determination was thirtybfive percent greater than the coefficient obtained from the relationships that did not estimate these constants. In the relation- ships that did not estimate family constants, the income elasticity of the demand for food was approximately 0.25. However, when constants for each family were obtained, the importance of income was drastically reduced, the estimated elasticity of income being approximately 0.10. Two hypotheses were presented as possible explanations of this relationship, i.e., the changing importance of the income variable when family constants were estimated. Since estimating these constants removes all the variation among families, these estimates of the constants for each family were interpreted as giving a numerical value to the Willard Sparks unobservable characteristics that were particular to a family. Thus, it is hypothesized that these constants capture most of the effects of the family's income with respect to food purchases. The second hypothesis concerns the "permanent" income hypothesis developed by .Milton.Friedman. In this situation it was suggested that the family constants included the effect of permanent income and that calculated income elasticity was an estimate of effect of "transitory” income on the demand for food. ESTIMATES OF THE DEMAND FOR FOOD FROM CONSUMER PANEL DATA BY Willard Robert Sparks A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 1961 AC KNOWLEDGEMENTS The author wishes to express his appreciation to his wife and many friends who have given both encouragement and help during the course of this study. The opportunity to work with Dr. Clifford Hildreth has been greatly appreciated, and many of his ideas appear throughout the thesis. Through his guidance the author has a much deeper appreciation of econometrics and related areas. Special thanks are due to Dr. Lester Mandersheid who assisted in preparing the final draft while Dr. Hildreth was at the Stanford Center For Advanced Studies in the Behavioral Sciences. Dr. Robert Gustafson gave freely of his time and experience throughout the study. His consulting and reading of the thesis has resulted in many additions, one of which was the interpretation regarding the Friedman permanent income hypothesis. Thanks are due to Drs. Anthony Koo and Kenneth Arnold, the remaining members of the guidance committee, for their suggestions and comments. Dr. William Cromarty (a former member of the Department of Agricultural Economics) was especially helpful in setting up the author's Doctoral program, as well as suggesting the study. Dr. James Shaffer deserves credit for making the Consumer Food Panel data available for this study. Comments by Ralph Loomis, Melvin Cotner, Alan Bird, and Mike Worth at different stages in the development of the study have been particularly helpful. William Ruble made many valuable cements in an earlier draft. Under the direction of Mrs. Arlene King, the many calculations necessary in the study have been performed remarkably well. Special appreciation is given to Mrs. Helen Bishop, Mrs. Laura Flanders and Mrs. Karma Deal for their help in preparing the data for the computer. The author expresses his thanks to Mrs. Elizabeth Phillips whose coments made the final draft more readable. The tedious chore of typing the final manuscript was done by Mrs. Sandy Rogers and Mrs. Judy Leach, and this has been appreciated very much. Although a number of individuals have offered suggestions concerning the study, any omissions or errors are the responsibility of the author. ii Chapter I II III TABLE OF CONTENTS PREVIEW OF THE STUDY . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . Some Considerations Concerning Time Series and CI‘OBBSOCtionDataeeeeeeeeeeeeeee SourceoftheData............... Considerations Concerning Simultaneous Equations Initial Hypothesis and Formulations. . . . . . . Initial cross section model . . . . . . . . . Final cross section model . . . . . . . . . . A Partial Summary of the Combined Model. . . . . VARIABLE SPECIFICATION AND PRELIMINARY SECTION ANALYSIS. . . . . . . . . . . Definition of the Variables. Food expenditures . . . Income. . . . . . . . . Family size . . . . . . Family classification by stage in ”61° 0 O C O O O O O O O O O O 0 Results of Cross Section Analysis. . Analysis of period 3, 1957. . . . Analysis of eight cross sections. Summary of Chapter II. . . . . . . . COMBINED ANALYSIS. . . . . . . . . . . CROSS Int ”duet ion 0 O O O O O O O O O O O O O O O O 0 Definition of the Models and Statistical models. . . Independent variables . Dependent variables . . Quadratic Model. . . . . . Specification including the Variables . . . the O O O expection variable. Further consideration of the quadratic model. . hgafltmc MOdel O O O O O O O O O O O O O O O O 0 iii Ii ‘00) (kW-FWD H H H 0 l6 16 16 17 18 19 25 38 Al Al hh ha 50 So 52 61 TABLE OF CONTENTS continued Chapter Implications and Comparisons of the Combined Models O O O O O O O O O O O O O O O O O O O O Estimated coefficients . . . . . . . . . . . Appropriate price index. 0 e e e e e e e e 0 Form Of the demand relationShipe e e e e e 0 One income versus fifty-two income variables Test of the homogenity concept concerning monetary variables of the demand relation- BhipSoeeeeeeeeeeeeeeeeee Summary of the Combined Model . . . . . . . . . IV EVALUATION OF THE STUDY . . . . . . . . . . . . Evaluation and Implications e e e e e e e e 0 Implications of the family size coefficients Implications of the income coefficients. . Implications of the price elasticities . . Areas for’Future Study. . . . . . . . . . . . Model development. . . . . . . . . . . . . Other considerations . . . . . . . . . . . BIBLIOGRAPHY O O O O O O O O O O O O O O O O O O iv 66 66 69 7O 70 71 72 75 75 80 81 8h 87 87 88 91 table 10 11 12 13 14. LIST OF TABLES Estimated demand relationships comparing the estimates from single equations and limited information methods . . . . . Estimated coefficients of the combined model, and elastici- ties at the means for the quadratic relationship, using fifty-two income variables and the Detroit BLS Price Index. Classifications of families based on the age and number of children and the marital status of the head of the house- balderasesoseeeeeeeeeeeeeeeeeeee Results from using classifications of families by stages in the life cycle as variables in the demand relationship for fOOd O O O O O O O O O O O O O O O O O O O O O O O O O O O Description of the eight periods selected for the cross section analyses O O O O O O O O O O O O O O O O O O O O O Least squares fits of food expenditures on current income lagged twelve weeks and the size of the family as a linear and logarithmic model, for eight cross sections . . . . . . Estimated relationships with expenditures and income de- flated by the family size variable (linear and logarithmic) for eight cross sections . . . . . . . . . . . . . . . . . A linear combination of the coefficients of current income and one-third of lagged income as computed from.Table 6 . . . Estimates of food expenditures regressed on a combination of current income and family size logarithmic and quadratic in the family size variable . . . . . . . . . . . . . . . . . . Results of the tests of homogeneity of variances of the regressions of the eight cross sections . . . . . . . . . . Tests of the stability of coefficients in quadratic and logarithmic models for eight cross sections . . . . . . . . . A comparison of the income coefficients and the constant terms for the eight cross sections . . . . . . . . . . . . . A table of the actual days and holidays included in the four weeks of each period for the four years of the study . . . . Tests of the null hypothesis that the coefficient of the ex- pectation variable is zero . . . . . .,.‘. . . . . . . . . . £239 12 20 26 27 28 3O 31 32 34 36 37 46 52 Table 15 16 17 18 19 20 21 22 23 24 25 26 .2389 Sum of squares of the dependent variables in the quadratic mde 18 O O O O O O O O O O O O O O O O O O O O O O 53 Results of the quadratic model using one income variable ‘ --excluding the expectation variable -- model A(1) and 3(1) . 54 Estimates of the quadratic model estimating one constant term and with the alternative of thirteen and fifty-two in- come variables --excluding the expectation variable -- models A(13)OndA(52) seeteeeeeeeeeeeeeeeeeee55 Estimated relationships of the quadratic model with 113 family constants estimated and the alternative of thirteen and fifty- two income coefficients é-excluding the expectation variable-- “31.3(13)Ondn(52)eeeeeeeeeeeeeeeeee056 Tests to determine the number of income coefficients that should enter the model and tests of the hypothesis that the family constants are equal . . . . . . . . . . . . . . . . . 58 Sum of squares of the logarithms of the dependent variable. . 61 Estimated relationships of the logarithmic model for models A'(1) and B'(1) O O O O O O O O O O O O O O O O O O O O O O 63 Estimated relationships of the logarithmic model for models A'(52) and B'(52) O O O O O O O O O O O O O O O O O O O O O O 64 Tests concerning the significance of fifty-two income coeffi- cients and tests of the hypothesis that the family constants are equal, in the logarithmic relationships . . . . . . . . . 65 Demand elasticities calculated from model A(l), A'(l), 3(1), md B'(1) O O O O O O O O O O O O O O O O O O O O O O O O O O 73 A summary of the estimated income coefficients and elasticities under the alternative specifications of the demand relation- Ships O O O O O O O O O O O O O O O O O O O O O O O O O O O 79 A comparison of the estimated demand relationships of food" for a number of different studies, with the estimated rela- tionships of this study from models A'(l) and B'(l) . . . . . 86 vi LIST OF APPENDICES Appendix Page A DERIVATIONS RELATING TO THE ESTIMATION OF PARAMETERS . In In cmBINED MODEL O O O O O O O O O O O O O O O O 94 B STATISTICAL TESTS O O O O O O O O O O O O O O O O O O 98 C CMUTATIONS OF THE ADULT MALE EQUIVALENT UNITS IMICBS O O O O O O O O O O O O O O O O O O O O O O O 104 D DEMONSTRATION OF THE EQUIVALENCE OF DEFLATION OF A LOG RELATIONSHIP AND FORCING THE SUM OF THE COEFFICIENTS OF AN EQUIVALENT LINEAR RELATIONSHIP TO BE EQUAL To ONE . . . . . . . . . . . . . . . . . 106 E ESTIMATED RELATIONSHIPS FOR.THE COMBINED OUADRATIC MODEL INCLUDING THE ExPECTATION VARIABLE AND THE FAMILY CONSTANTS FOR THE MODELS PRESENTED IN THE TExT . . . . . . . . . . . . . . . . . . . . . . 107 F SPECIFIED SOURCE OF THE DATA AND REMARKS CONCERNING COMPUTING . . . . . . . . . . . . . . . . . . . . . . 115 G DATA OF THE MEANS OVER BOTH FAMILIES AND TIME FOR INCOME, FAMILY SIZE AND FOOD EXPENDITURES AND A TABLE OF THE PRICE INDICES . . . . . . . . . . . . . 117 vii CHAPTER I PREVIEW OF THE STUDY Introduction Major emphasis in this study was placed on the specification of relevant variables and functional forms of the demand relation- ship for food when the demand relationship is estimated from.combined time series and cross section data. A major part of the study dealt with some of the technical prdblems that arose in specifying and estimating the variables to be included in the demand relationships and estimating the relationships among the included variables. Chapter I contains a discussion of (1) the nature of the prdblems arising when either time series or cross section data are used, (2) the source of the data and the initial model, (3) develop- ment of the combined model (using both time series and cross section data), and (h) a brief summary of the results of the final combined time series and cross section relationships. A detailed discussion of the preliminary analyses leading to the final model is presented in Chapter II. Further discussion of the construction and selection of the relevant variables is also included in Chapter II. Chapter III contains the results of combining the time series and cross section data. An evaluation of the various models and discussion of future research possibilities appears in Chapter IV. The statistical model used in the combined analyses and a discussion of some of the statistical tests appear in the Appendices. Some Considerations Concerning Time Series and Cross Section Data There are a number of reasons for basing quantitative demand analysescn1combined time series and cross section data. Since aggregate consumption and income are the sum of consumption and income of house- holds, any estimates of the response of consumption to Changes in income should be consistent with both types of data. Similarly, hypotheses concerning aggregate consumption and income should be consistent with hypotheses at the micro level (households). Time series data reflect a number of variables that do not enter relation- ships using Cross section data; thus estimates of similar parameters may be drastically different if based only on one of these types of data. For example, Changes in the distribution of families within different income groups are reflected in time series data but are not present in cross section data. Time series data reflect both technological changes and changes in the market structures of the economy, whereas cross section data are usually representative of a given market structure and technology but do not reflect any changes in these variables. With reference to statistical inference from.combined time series and cross section data, Marschak notes that increasing the scope of Observations used in demand analyses increases the power of the statistical tests and the accuracy of the estimated parameters.1 1M'arschak, J., "Review of Schultz, Theory and Measurement of Demand," Economic Journal, Vol. #9, 1939, p.fih87. This improvement is a result of adding data and using data of a different nature. To give dimension to the differences that can exist between the estimates using either time series or cross section data, results of various studies of the demand for food are presented. USing Swedish data, weld and Jureen estimated income elasticities for food of 0.28 using time series data for 1921-39 and 0.51 for 1933 using cross section data.2 Marguerite Burk estimated income elasticities with respect to food for the United States of 0.68 for 1929-hl and 0.38 for 19h8-57. From cross section data, she Obtained income elasticities of 0.30 and 0.25 for 1912 and 1955, respectively.3 Realizing that any analysis of consumer behavior should be consistent with both time series and cross section data, Tdbin con- ducted a study of the demand for food in the United States using both types of data.h Time series data were reflected by national aggregates of consumption,income and expenditures. A number of different surveys of households represented the cross section data. Two demand functions were estimated: a family demand function, and an aggregate demand function estimated from combined time series and cross section data. The variables determining the family food demand were current family disposable income, income of the family for the 2weld, H. and Jureen, L., Demand Analysis, John Wiley and Sons, New York, 1953, p. 302 and p. 262. 3Burk, M., "Income Food Relationships", Journal of American Statistical Association, Octdber, 1958, p. 919. qubin, J., "Statistical Demand Function for’Food in the United States", Royal Statistical Society Journal, Series A, 1950, pp. 112-lhl. preceding year, family size, food price and the price of other consumer goods. The form of the demand relationship was assumed to be loga- rithmic. The aggregate demand function was derived from.the family demand function using an assumption concerning the aggregate distri- bution of families by income and size. From the combined data TObin. obtained an estimate of 0.27 for the income elasticity of food. I The data used in the present study were Observations on the expenditure, income, size and composition of households. The statisti- cal model used to combine the two types of data was one of several suggested by Hildreth,5 Source of the Data The basic data used in this study were taken from the Michigan State University Food Panel. The MSU Panel was composed of approxi- mately 250 households selected to be representative of Lansing, Michigan, a city of 100,000 population. Each household reported weekly all food purchased for home use, giving the quantity, price and expenditure for each item. In addition, each family reported weekly the number of and expenditures for meals away from home, the number of meals served to guests, Changes in the household composition and income received after’Federal Income Tax. Data.also were obtained concerning the occupation.of the head of the household, ages ‘of the members of the household, and education of both the homemaker and head of household.6 5Hildreth, c., "Combining Cross Section Data and Time Series Data," Cowles Commission Discussion Paper: Statistics No. 3&7, May 1950. 6Quacken‘bush, G. and Shaffer, J., "Collecting Food Purchases Data by Consumer Panel," Michigan Agricultural Experiment Station, Tech. Bul. 279, East Lansing, 1960. The panel started operation in February 1951 and continued through December 1958. It was not until 1955 that data on all food items were tabulated. The period 1955 through 1958 was used as a basis for this study. The data were aggregated into four week periods and the combined data were taken from families who were in the panel all four years (1955-58) and if the occupation of the head of the household was wage earner. Data for the cross section were chosen on the basis of the maximum.number of wage earner families that were in the panel for that particular period. Consideration Concerning Simultaneous Equations In this analysis of consumer demand for food, no attempts were made to work with systems of equations. The theory of consumer behavior is fairly well developed in terms of single demand relation- ships, but there is comparatively little concerning the system.within which these relationships fit. Therefore, the consideration of simultaneous equations would have required a much more extensive theoretical investigation concerning the system of relationships and model construction, thus reducing the time that could be given to the specification of the variables and form of the demand relationship. Furthermore, a priori the bias due to simultaneity are pre- sumably less important when estimating a demand function for a very small part of the total population than when estimating aggregate demand. That is, Lansing prices and Lansing incomes, etc., are prObably somewhat more exogenous with respect to Lansing food con- sumption than United States prices and incomes are with respect to United States feed consumption. It should also be mentioned that studies which have compared the estimated parameters under single equation and simultaneous systems have not found the coefficients to be too different (Table 1). An excellent discussion of the question of single equations versus simultaneous equations is given in Econometrics.7 Initial Hypotheses and Formulations This study deals entirely with families classified as wage earners since an investigation of the effect of occupations on food consumption could be a study within itself. This restriction reduces the number of exogenous variables considered, thereby allowing greater emphasis to be placed on other aspects of the model. As a preliminary step to the combined analysis, alternative ways of representing income and family composition were considered using only cross section data. Initially only one cross section was investigated but the study was later expanded to include eight different cross sections. The selection of the variables that enter subsequent analyses was based on the expanded cross section analyses. The latter analyses also were used to test the homogeneity of variances and stability of coefficients over time. These tests are described in some detail in Chapter II. 7A Symposium on Simultaneous Equation Estimation, Econometrics, Vol. 28, Oct. 1960, pp. 835-871. Christ, C., "Simultaneous Equation Estimation: Any Verdict Yet?", p. 835. Eildreth, C., "Simultaneous Equations: Any Verdict Yet?", p.-8k6. Liué T., "Uhderidentification, Structural Estimation, and Forecasting," Po 55. Klein, L. R., "Single Equation VS. Equation System Methods of Estimation in Econometrics," p. 866. Table 1. ,Estimated demand relationships comparing the esti tea from single equations and limited information methods Retail Level AFarm Level _ Income Commodity Price Current Legged Price Income Food Girshick and Eaavelmoa (LI) -0.25 0.2h 0.05 -0.08 (LS) -0.3h 0.27 0.06 Pork Nordin, Jud e, and wahby (LI) -0.91 0.76 0.29 (LS) -0.78 0.h3 0.22 Eggs Judgec (LI) -O.58 o.hh 0.29 (LS) -0.55 0.hl 0.22 Livestock Products Eildreth and Jarrett (LI)-1 -0089 0e78 (LS)-1 -0.95 0.83 (101)-2 “'Oe76 0075 (LS)-2 -0.76 0 75 aGrishick, M., and Haavelmo, T., "Statistical Analysis of the Demand for Food: Equations," Econometrics, V01. 15, l9h7, pp. Examples of Simultaneous Equations of Structural 79-110. bNordin, J., Judge, 0., and Wahby, 0., "Application of Econo- metric Procedures to the Demands for Agricultural Products," Iowa Agricultural Experiment Station, Research Bulletin #10, 195%, p. 1018. cJudge, C., "Econometric Analysis of the Demand and Supply Rela- tionships for Eggs," Storrs Agricultural Experiment Station, Storrs, Connecticut, Bulletin #56, 195h, pp. 22-2h. daiidreth, C., _ and Jarrett, F., A Statistical Study of Livestock Production.and Marketing, Cowles Commission Monograph No.-15, John‘ Wiley and Sons, New York, 1955, pp. 11h-116. The elasticities for 1 are from a linear relationship and are Those for 2 are from a logarithmic relationship. * Maximum likelihood limited information estimates are denoted by (LI) and least squares estimates are represented by (LS). - calculated at the mean. Initial cross section model The initial model deals with a particular foureueek.periqd in which family food expenditures were assumed to be dependent on a linear combination of income and size and composition of the family. Income was reflected in three variables -- current income and two lagged income variables. Food expenditures were adjusted to correct for differences in the number of meals served in the household and the number that would have been served if every member of the family had eaten twenty-one meals at home each week (page 17, Chapter II). A single variable representing size and composition of the family was computed by applying weld and Jureen's index of male equivalent units of various age-sex groups to the data for each family.8 Several modifications of the initial model were considered. A new family size variable was computed based on USDA data; as a basis for comparison, the number of individuals in the family was used as an indication of the size of the family: the variable repre- senting the greatest lag in income was deleted (in all cases estimates of the standard error were at least ten times the estimated coefficient); and a logarithmic relationship was assumed to represent the demand for food. Relationships were estimated with the families classified into different groups based on their marital status and the age and number 8Wo1d and Jureen, op. cit., p. 233. of Children in the family (see Table 3, Chapter II). Two such classi- fications were considered; however, no observable changes in the estimated relationships were evident. Final cross section model The final relationship analyzed in the single period was also estimated for a series of eightfldifferent_crgss_sections. (See Table 5, Chapter II, for a description of these periods.) Based on the results of this relationship, income and food expenditures were deflated by the family size variable. However, the result3433re_not ”promising, and no further consideration was given to deflation. Information Obtained from the analysis of these cross sections suggested defining a new income variable. The income variable chosen as the indicator of the family's basis for making decisions concern- ing food expenditures in a given four-week period was current income plus one-third of the income received in the previous twelve weeks (lagged income). The final functional forms Chosen on the basis of the cross section analyses of the eight_perigds were logarithmic and linear in all variables except the family size variable which was quadratic (hereafter, the entire function will be referred to as quadratic). These forms and the variables as described above were used in the combined time series and cross section model. With ‘these two forms and the variables specification dis- cussed above, several tests were performed to determinegthe_§tgbility of the coefficients othhe demand relationship_overwtime. The homo- W —~-r‘---- geneity of the variance of the estimated residuals for each of the 10 cross sections was also tested. These tests indicated that the coef- ficient of the family size variable did not vary significantly over time. Similar results were Obtained for the estimated variance of the residuals in the eight periods. Analysis of these eight cross sections also suggests the possibility,9fieither seasonality or . changesfiin the income coefficients over time. A Partial Summary of the Combined Model The use of time series data requires the introduction of other variables. For example, the price of the product under study and the prices of competing products must be considered. In addition, a variable was specified that represented the effects of current aspects of the economy on the purchasing habits of the family. This variable is referred to as an expectation variable. Two alternative price indices were used to represent the price of food. An_index of food prices based on informationnreported in the M§prqnsumer'rood Panel was considered as one possible index to representwthemprice,oflfopd.9 Since this index cannot be reproduced outside the panel, the Bureau of Labor Statistics Food Price Index for Detroit, Michigan was used as an alternative (hereafter referred to as Detroit BLS). These price indices were used to remove the influence of food prices from the United States BLS Consumer Price Index to Obtain two nonfood price indices. 9Wang, 3., "Retail Food Price Index Based on M.S.U. Consumer Panel," Unpublished Ph.D. Thesis, Michigan State University, 1960, Table 10. 11 The statistical model used in the combined analysis estimates a parameter for each family that is constant over time but varies among families. These parameters were viewed as measuring the effect of unObservable characteristics that were particular to a family over time. Tests were performed to determine the significance of estimat- ing a constant for each family. The logarithmic form has some advantages over the quadratic in that constant elasticities are estimated, and a diminishing marginal propensity to consume food with respect to changes in income and/or size of the family is enforced. The zero sum elasticity concept (concerning the monetary variables) is also possible in the logs- rithmic model. That is, complete rationality of a consumer unit (family) requires that the sum.of the elasticities of demand with respect to monetary variables be zero. This does not suggest that the zero sum concept be imposed, but the logarithmic form does allow such results. The results of the combined model using the Detroit BLS price index and fifty-two income variables are presented in this section. The estimated coefficients of the logarithmic and the quadratic relationships along with the estimated elasticities at the mean of the quadratic relationship are presented in Table 2. An index of physical quantity of food purchases of the families was computed from food expenditures deflated by the Detroit price index.£9' The {I 10A discussion of the procedure used in computing an index of quantity from expenditures can be found in:._Stone, R., Measurement and Consumer4Expenditures and Behavior in the United Kingdom, CTmbridge, England, 1953, pp. 9-13. 12 .oaooaa axes: e>Hesu mama one we cease oco mafia common use» ucmuuso one :« eo>fiooou oaoo:a one neuoaoo H tNm co women mecca Deana eoowaoc one mouoaee NU eUOOh MO XUGCH OUHHN Sm .vouuoa_xemsuuoow a you xawamm emu uwouuen can no peace Nomad Dogma one noncomo mm .m we chosen one monocoo mm mm moan: amoam>anve one no woman: one mouoamo mnoumsz « coco. Hwoo. neoo. nmoo. mace. onoc. unoo. «boo. awao. aooc. mode. Node. aaao. «woo. oHHo. emoo. Ammo. emoo. neoo. nnoo. onoo. coco. mode. «moo. Hmoo. mwoo. Hem. n mm ammo. mNHo. Shoo. moHo. «moo. choc. nnoo. omoo. mwoo. nmao. «ego. “Hoe. nooo. NNHo. oHHo. ooHo. «aoo. enoo. aaoo. auoo. area. 0500. whoa. wnoo. whoo. mafia. mama. amsm.u wan.u me.oH Aeuaeuuawmmou omumaaueoV owumuomso mono. ammo. Hweo. ammo. emoH. memes wwmo. mooo. mama. «mus. mafia. eodH. coma. momo. mafia. mmmo. ommfi. coma. mama. name. ammo. mono. oeHa. ammo. mood. ammo. aomfi. mus“. wdwo. wNHH. ammo. mono. mono. Noeo. ammo. cama. mafia. ergo. oeoo. owed. o~m~. omfio. meag. ammo. onH. «mac. omao. muse. omno. ammo. memo. mead. mama. somm.u _ noon. Asses one as eowuwuwueeaev owumuomoo oeao. nmao. saga. mama. ammo. demo. anoa. mafia. anNH. decfi. muoH. nmao. nmoo. nwmo. ammo. «000. Name. mmeo. some. amao. mHao. ammo. ammo. woke. unwo. Nome. new. N am umeo. mmeo.,nnmo..auao. name. nn-..~oo~. emao. Name. mmeo. amno. mono. ammo. . fined. Hcoo. anac. Humc. mmmo. anoH. HooH. fined. ammo. demo. memo. ammo. fiucfl. anoa. weam.- seam. . .35»:st n2 2 .:_ S we. a A Le.“ n e n a a «a «a an a flea mesh some we ovoquem xeearunou ma one me Home new NHV mamman ammoaa.uo«ua mam uuouuen one one moanewue> anoaa oauuhuwum wages .a«mec0wueaeu oaueumenc one we came emu us neuuuuaueefie one .uoeoa mesunsou ecu we mucouoammeoo oeueauumm .N anew 13 size of the family was represented by the number of adult male equiva- lent units in the household for each four-week period. The income variable was the combination of current and lagged income previously discussed. . As can be seen from Table 2, there was very little difference in the estimated elasticities under the two alternative models. However, the coefficient of determination (R2) was slightly larger for the logarithmic relationship. The signs of the coefficients agreed with their a priori expectation and the magnitude of the coefficients does not differ greatly between models. The price elasticities are somewhat larger in absolute value than.price elasticities Obtained in other studies.11 Since this study covered only a short period of time and was concentrated in one particular geographic area, comparisons with studies conducted on a larger scale may not be highly relevant. The coefficient of the family size variable seems reasonable in that a 10.0 percent increase in the number of adult male equivalent units in the family would be associated with approximately a 5.0 percent increase in the quantity of food. The negative elasticity of the square of the family size variable was consistent with the postulate of economies of scale for food purchases. The elasticity of demand for food with respect to the price of nonfood items was higher than the elasticities Obtained in the TObin 12 study. Although standard errors of the coefficients were not 11TObin, J., op. cit. 12mm. 1h calculated for the model with fifty-two income coefficients, the standard error for the coefficients of the nonfood price variables in the models with one and thirteen income variables were statistically different from zero at a higher confidence level than those Obtained by TObin. The income elasticities seemed to offer the most striking results. These estimates were generally much smaller than income elasticities Obtained in other studies.13 Although standard errors were not estimated, it is believed that, based on the results of the models with one and thirteen income variables, the standard errors in the quadratic relationship were approximately 0.001h and in the logarithmic relationship approximately 0.0016. In the quadratic the ratio of the mean of the dependent variable to the mean of each income variable was approximately 10: and the average estimated coefficient, 0.010. The estimated elasticities at the mean did not differ greatly from the logarithmic estimates. USing these approxi- mations, the ordinary t test would certainly indicate that the estimates were significantly different from zero. In the models estimating only one constant, income elasticities of the demand for food were approximately 0.25 (see Table 21, Chapter III). Two alternative hypotheses are offered as possible explanations of the dramatic difference in the income elasticities Obtained with 13W'old, R., and Jureen, L., op. cit. Stone, R., op. cit. TObin, op. Cit. Grishick and Havelmo, op. cit. Burk, op. Cit. 15 and without family constants. Estimating a numerical value for the effects of characteristics particular to each family reduced the importance of the family's income as an explanatory variable in the demand relationship. Alternatively, estimating family constants eliminated variation between families; thus, the family constants captured the effects of "permanent" income on the demand for food. This being the case, the income elasticities Obtained in the model with family constants were largely estimates of the effects of "transitory" income. The estimated parameters of the 113 families for thenmndelm . discussed above are presented in Table; E-5 and E-7 of Appendix E. In the logarithmic relationship, these constants represent the percen- tags that each family was of the mean predicted value of the dependent variable. However, in the quadratic relationship they represent the numerical value that must be added to the mean predicted value of the dependent variable. Intuitively, it seemed that a parameter represent- ing a constant percentage rather than a constant numerical value of the predicted mean value of feed expenditures was more plausible. A preview of the study has been presented in this Chapter and a further discussion of the alternative models is presented in Chapters II and III. Chapter II contains a discussion of the cross section models, whereas the combined models are presented in Chapter III. CHAPTER II VARIABLE SPECIFICATION AND PRELIMINARY CROSS SECTION ANALYSES m One i887 week period was chosen as a basis for analyzing alter- native specifications of the relevant variables and functional forms of t e dapapdmrelationships for food. A detailed analysis of this cross section was carried out to determine the appropriate representation of the demand relationship to be used in the combined time series and cross section model. Definitions of the Variables Food expenditures Each week the family reported the quantity, price, and expendi- ture for every food item purchased the previous week. The diary, furnished'by Michigan State University, in which the family reported the above mentioned items contained some 900 different food groups. The data were totaled into total expenditures for all food by families for each week they were in the panel. In addition each family was asked to record the number of meals served in the house- hold each week during which they completed a diary. Food expenditures of the family were adjusted to eliminate biases introduced by the variation in the number of meals served each week. That is, food expenditures of families serving a large number of guests during any one week would bias the expenditures compared to another week. Similarly, if the family had eaten a number of meals away from.home, the reported food expenditures would understate their 16 17 weekly food expenditures. To correct for these types of biases, the food expenditures of the family were changed to a twenty-one meal per week equivalent for each member of the family.-1 Families that had eaten half or more of their meals away from.home were not included in this study. 5222 Each week the family reported the total income received that week. This figure represented the disposable income (after federal income taxes). The income variables were postulated and studied in order to determine the combination of current and lagged income that best represented a family's behavior with respect to food purchases. It seems probable that changes in current income were immediately reflected in food expenditures. Hywever, since the adjustment of food consumption to family income was not likely to be completely instantaneous, some sort of lagged relationship would exist. With these thoughts in mind, three income variables were defined initially. They were current income and two lagged incomes. The lagged income variables were the income received in the three previous four week periods and the income received in the previous fourth, fifth, and sixth four week periods. As an alternative the income received in the two four week periods prior to the observation of current income was used to represent the lagged income. 1The adjustment was made by multiplying the family's weekly food expenditures by the ratio of the number of meals that would have been eaten at home if every member of the family had eaten every meal at home, to the actual number of meals served. 18 Family size Since the panel data were family data, the size and composition of the family was an important factor in determining the family's food expenditures. Attempts were made to reflect both of these considers- tions in a single variable called family size. Several different measures of the relative importance of the size of the family were considered. Initially, Hold and Jurren's index which accounts for the size and age structure of the family was used to represent the size of the family.2 In this index, the adult male is given a weight of one and all other age-sex groups were given equivalent weights based on their relative requirements for food as a percentage of the require- ments of an adult male. Also, a per capita measure was employed to test the significance of using adult male equivalent units to represent the size and composition of the family. Two indices similar to the one presented by Hold and Jureen were computed;because it was believed that their index underestimated the relative consumption of particular groups. (The importance of the teen-agers and very young children seemed to be underestimated.) Initially, an index was computed from USDA data concerning the relative importance of different food items in the diet of a U. 8. family with an average income by an age-sex classification and data of the persona tags each of these food items were of the 1955-57 average weekly expenditures for all members of the consumer panel. A second index was computed from similar data, except national data of the relative aflold and Jureen, op. cit., p. 233. l9 importance of each food item were used. A description of these indices, the method of calculating and the indices themselves are given in Appendix C. Family classification by stage in the life cycle Family size and composition influence consumption of food and other produce in a variety of complex ways. Whether these can be summarized reasonably well in a single variable was a matter that was hard to Judge a priori. Thus, an alternative method of handling these influences was considered. Families were divided into relatively homogeneous groups accord- ing to their size and composition. Each group was assigned a different constant term in the consumption relationship which reflected a shift in the relationship due to the size and composition of families. The separate constants were estimated by the use of dummy variables. Fortunately, a number of previous studies offer some guidance as to reasonably effective classifications for this purpose.3 Based primarily on Lippitt's study and the Life Study, two classifications of families were defined. Each '0: these classifications contained six categories, the first consisting of households with a single member; the other five married households differing with respect to the age and number of children. The two classifications are 3Listed below are several studies that have considered classic fications of families in a manner similar to the classifications dis- cussed. Lippitt,‘V., Determinants of Consumer Demand, Harvard Economic Series, Cambridge, 1966. Polite, Alfred, Research Inc., Life Studies of Consumer Expenditures, Vol. 1, 1958. Houthakker, H. and Praia, J., The Analysis of Family Budgets, MbGrawa Hill, new York, 1955. 20 presented in Table 3; they differ in only one respect, the age of the children. Table 3. Classifications of families based on the age and number of children and the marital status of the head of the household Classification I __._ 7......___7 _... ._M-.- _..____-__ ._ _ ._. a. _._ ___—._ -. ,., Classification II (1) Single (1) Single (2) Married and no children (2) Married and no children (3) Married and l or 2 children (3) Married and 1 or 2 children under 16 under 10 (h) Married and more than 2 (h) Married and more than 2 children under 16 children under 10 (5) Married and l or 2 children (5) Married and l or 2 children and the oldest 16 or over and the oldest 10 or over (6) Married and more than 2 (6) Married and more than 2 children and the oldest 16 children and the oldest 10 or over or over Results of Cross Section Analysis Analysis of period 31 1257 One hundred and forty-one families were studied for this partic- ular period. The model was assumed to be a linear function of food expenditures, current income, income received in the three previous four week intervals, income received in the three intervals preceding the first lagged income variable and the number of adult male equiva- lent units in the family (based on the index published'by‘wold and Jureen).h hThe term."interval" will be used to refer to a four week period; since there are thirteen periods within each year, the use of the term "period" without a year designation would be meaningless. The estimated relationship was: E = .021911 + .015h12 + .0021 + 1h.h9591rl + 1.595 (2.1) (.017) (.006) (.005? (1.76) (3.89) 3‘8 .62h S = 20.2 where: E = the family expenditures for food, adjusted for the number of meals served at home, period 3, 1957 (current dollars per four weeks) 11 = disposable income of the family for period 3, 1957 (current dollars) 12 2 family disposable income for the three four week periods previous to period 3, 1957 (current dollars) I3 = family disposable income for the fourth, fifth and sixth four week periods prior to period 3, 1957 (current dollars) F1 = the number of adult male equivalent units in the family for period 3, 1957 based on wold and Jureen's index The following format was used in presenting the results. The standard errors of the estimated parameters were given in parentheses below the estimate. The coefficient of determination, adjusted for the number of degrees of freedom, was denoted by R . The standard error of the residuals was denoted by S (i.e., the square root of the error sum of squares divided by the number of degrees of freedom). Revisions of this relationship included using the number of individuals in the family as an indicator of the size of the family. (The index presented.by wo1d and Jureen was'believed to understate the importance of some age groups.) The results of equation (2.1) suggested the deletion of the income variable that represented the income received in the fourth, fifth and sixth four week periods prior to the period expenditures were observed. The standard error 22 of the estimated coefficient was more than ten times the estimated coefficient. Also, the magnitude of the coefficient suggested that income had very little effect on the amount of food purchased twelve to sixteen weeks later. Considering these revisions, the estimated relationship was: ‘ E = .030011 + .016312-+ 9.9173F2 + 5.0167 (2.2) (.016) (.005) (1.16) (n.38) i2 = .636 s = 19.91 where 12 = the number of individuals in the family (per capita). At this point, primary interest was in reflecting the behavior of households as accurately as possible with a simple specification of variables. The more information that can be concentrated in the specification of these variables, the greater are the number of degrees of freedom.available for statistical tests. With these thoughts in mind, a new variable representing the size of the family was constructed. This variable was represented by an index of adult male equivalent units based on USDA data pe r ta :2. ni-ng to the recommended require- ments of different food groups for particular age-sex groups and con- sumer panel data pertaining to the relative importance of these food groups in determining the family's total food.bill.5 Considering this modification the estimated relationship was: s = .026211 + .016112-+ 11.1h7r3 + 6.015 (2.3) (.015) (.005) (1.25) (“.89) 2 fi’ = .6h5 s = 19.66 5See Appendix C for a discussion of the derivation of this index. 23 'where F3 2 the number of adult male equivalent units in the family for period 3, 1957 computed from line 3, Table Cal, Appendix C. Along with the specification of variables, alternative functional forms were considered. As was discussed previously, the logarithmic relationship has properties that are often desired in this type of demand analysis. For example, the logarithmic form allows for a diminishing relationship between the family's fOOd expenditures and increases in the family's income or size. Both functional forms were considered as various other specifications of variables were used. In the logarithmic relationship the estimate was: log E = .2272 log I1 + .0955 log 12 4-.5939 log F3 + .5139 (2.h) (~093) («100) (.060) (~151) 13:2 = .601 ' s = .1377 To assist in making elasticity comparisons of the logarithmic and linear relationships, the ratios of the means of the independent variables to the dependent variable were approximately 5.8, 17.8 and 0.0h2 for 11, I2, and F3, respectively. Since the index of adult male equivalent units based on the consumer panel cannot be reproduced outside the panel, an index‘based on national data was computed. As new data become available the index can be revised, a feature the index based on the panel does not possess.6 Using the nationally based index to measure the size of 6See Appendix C for a discussion of the method of computing the index. 2h the family in adult male equivalent units, the following estimates were obtained: ‘3 = .029011 + .016112 + 11.h083F-+ 5.68 (2.5) (.015) (.005) (1.28) (h.1) _2 R 3 .6’46 8 = 19.63 and in logs: log n = .2199 log 11 + .0953 log I2‘+ .6058 log I + .690 (2.6) (.093) (.100) (.051) (.1113) 2 ‘E = .685 s = .1367 where F = the number of adult male equivalent units in the family for period 3, 1957. Computed from the index based on national data, line h, Table C-l, Appendix C. The estimated coefficients of the different variables were relatively constant as different specifications of variables represent- ing the size of the family were considered-«see equations (2.1) through (2.6). Given these relatively stable estimates, the decision rule for choosing the appropriate logarithmic and linear relationships to be studied in more detail were the set of variables with the largest coefficient of determination. ‘Using this decision rule the relation- ships represented in equations (2.5) and (2.6) were chosen as the appropriate forms to be used in future analyses. These two models were used to investigate the effect of using different classifications of families (by stages in the life cycle) on demand for food. (See page 19 for a discussion of these classifi- cations). As mentioned previously, dummy variables were used to 25 estimate constants for each classification. The estimated relationships appear in Table A. Since the use of a single size variable was more conyenient and, as can be seen by comparing the coefficients of equations (2.5) and (2.6) with those of Table h, the classification of families by stage in the life cycle did not improve the estimated relations, a single family size variable was used in subsequent analysis. The relationships presented in equations (2.5) and (2.6) were used as a basis for studying other cross sections within the four years. Expanding the investigation to other cross sections served as a basis for testing various hypotheses. For example, the stability of the estimated coefficients of the variables over time was tested. Analysis of eight cross sections Since the primary interest of the study was one of estimating the demand relationship for food using combined data, the behavior of the family with respect to food purchases for different cross sections was examined to obtain some insights into this behavior over time. Seven.additional 192531993 cross sections were chosen for this purpose. These cross sections represented several situations that were studied. For example, the stability of coefficients over time was viewed. Similarly, the postulate of homogeneity of variances was tested. The possibility of seasonality and/or variability of the income coefficient over time was considered}/’These cross section analyses were used to evaluate alternative functional forms of the demand relationship. Although these intervals do not actually represent months, they were identified with the month within which the largest number of days were observed. Table 5 contains a description of these intervals. 26 AHnH.e Auoo.e Aaua.v Aeno.v Anae.v AHeH.V Anee.e AeAH.e eama. one. name. amee. sane. meme. none. enNH.H eeee. anal. emee. ewes Aea.eae Anm.ee Ase.ahe Aa~.eae Aa~.ev Anee.v Aeae.e a~e.~e . Ha.aa nee. «ANe. aesa.a eeea. anee.o e~e~.n eeen.n mane. neNe. neae.oo assess HH Aeea.e Aeah.e Aaao.v Anna.v Amae.v Aae~.e Aeee.e Ammo.e sans. eee. ea~e. ease. Heee.-_ some. ease. neao.a keno. anon. eeee. seen 36: 36: 3.8 3.13 5.8 . Gee; 38¢ Gad mean.an eee. aen~.~ san~.e eeee.~ Heoa.a ee~e.n aneu.n gene. Heue. enee.oa season H m o n a m a a N H moaueoum Nu. «coaucoamauamno owned: nomww uoouowuuooo H H h gone: nanuaflu one o« ensue he nowadamm no amowueoumuaamao wade: scum nuanced ooou you ouneooauodeu madame any ad moanmuue> no.0Hoho mugs .e nan-e 27 Table 5. Description of the eight periods selected for the cross section analyses Interval Period Year Month Weeks included I'm” °f d 1 11 1955 Oct. 10- 9 to 11- 5 150 2 1 1955 Jan. 1- 2 to 1-29 110 3 11 1956 Oct. 10- 1 to 11- 3 1H9 h 13 1955 Dec. 12- 2 to 12-29 1h? 5 11 1957 Oct. 10- 6 to 11- 2 170 6 3* 1957 March 2-2h to 3-23 1&1 7 11 1958 Oct. 10- 5 to 11- 1 160 8 8 1958 July 7-13 to 8- 9 159 *The initial period selected. Least squares estimates of the eight cross sections using the relationships described in (2.5) and (2.6) are presented in Table 6. Examination of the logarithmic relations in Table 6 suggests that the sum of the coefficients does not differ significantly from one. Since deflation of a linear relationship by one of the independent variables has the effect of forcing the sum of the coefficients of the relation in logarithms to be one, deflation.by the family size variable was considered. This deflation can be viewed as converting the food expenditures and income variables to a per adult male unit basis. An ordinary t test was employed to test the hypothesis that the sum of the coefficients of the logarithmic relations presented in Table 6 was equal to one. The hypothesis was rejected at the five percent level of Table 6. 28 Least squares fits of food expenditures on current income lagged 12 weeks and the size of the family as a linear and logarithmic model, for eight cross sections Model Linear Loga- rithmic IfistimatedhCoefficients Interval 11 12 F A* §2 S 1 .03135 .00125 11.7280 7.2017 .65h5 18.7310 (.01h) (.005) (1.h12) (n.0u) 2 .032h3 .01277 12.2933 8.2h88 .5390 22.7h5h (.020) (.007) (2.037) (5.96) 3 .02073 .01h979 12.5526 8.3955 .6137 19.8972 (.015) (.006) (1.h36) (n.32) a .01786 .011325 15.2733 9.h392 .6h32 20.6710 (.007) (.00h) (1.hh0) (h.h1) 5 .03120 .00930 12.0271 12.8h11 .5536 21.0860 (.017) (.006) (1.285) (h.19) 6 .02897 .01610 11.h083 5.6813 .6h60 19.6317 (.015) (.005) (1.279) (h.11) 7 .03360 .007h7 10.3830 11.7267 .5618 21.595h (.013) (.005) (1.310) (n.1u) 8 .03616 .00390 13.058h 1h.3939 .5571 21.5291 (.015) (.006) (1.272) (n.5u) 1 .u1h8 .0059 .5hh5 .5039 .706 .1287 (.007) (.069) (.061) (.138) 2 .3321 -.0h03 .6021 .8535 .60h .15h6 (.108) (.125) (.09h) (.23h) 3 .0h71 .2801 .5851 .5937 .6h8 .1392 (.016) (.095) (.066) (.187) h .0618 .136h .6319 .9983 .6h5 .1331 (.071) (.079) (.062) (.15h) 5 .1389 .16h9 .517h .7399 .570 .153h (.093) (.10u) (.063) (.166) 6 .2199 .0953 .6058 .6909 .685 .1367 (.093) (.100) (.061) (.1h3) 7 .2289 .1281 .5030 .63h7 .637 .1h91 (.051) (.071) (.061) (.223) 8 .3298 -.0768 .5676 .9660 .598 .1h83 (.092) (.107) (.060) (.261) *A denotes the constant term of the relationship. confidence for intervals iour, five and eight. The estimated relations using deflated variables appear in Table 7. In order to assess the effect of the deflation;multiple partial coefficients of determination were computed. The partial multiple coefficient of determination denoted by 3%, represents the proportion of variance in the dependent variable explained'by the addition of the income variables, given that the variance due to the family size variable has been taken into account.9 A comparison of the coefficient of’multiple determination of the deflated relations with the parital coefficient gave some hints as to both the appropriate functional form and specification of variable. In the linear relations six of the eight intervals had coefficients of determination smaller than the partial coefficients of determination, while all the logarithmic relationships had coefficients of determina- tion larger than the computed partial coefficient. Since deflation of the linear model has the effect of forcing a constant relationship between the per adult male unit expenditures and income, a larger partial coefficient of determination would tend to suggest a linear relationship between the unexplained expenditures (after considering the size of the family) and income. The partial coefficient of determination's being smaller than the estimated coefficient of determination for the loga- rithmic model suggests the possibility of a linear relationship between 9 -2 2_ 1.12211 RF” 1 -12“. where rgEF denotes the square of the simple correlation of E on.F. Table 7. 30 Estimated relationships with the expenditures and income variables deflated by the family size variable (linear and logarithmic), for eight cross sections Winnie Model Interval Il/F Ig/F A g 82F 3 Linear 1 .0650 .0007 16.0687 .222 .258 8.78h7 (.015) (.00h) (1.717) 2 .0655 .00h8 1h.8222 .1h5 .1h6 10.0659 (.022) (.007) (2.731) 3 -.0005 .0182 18.1255 .157 .187 8.978h (.015) (.006) (1.7h2) h .011h .0088 22.156h .087 .150 9.6918 (.009) (.ooh) (1.795) 5 .0071 .0105 20.9223 .091 .19h 10.1317 (.017) (.006) (1.60h) 6 .0263 .0079 17.7666 .155 .291 9.5521 (.197) (.007) (1.781) 7 .0h70 .0080 16.0976 .292 .212 8.7521 (.011) (.ooh) (1.h76) 8 .0h38 .0015 19.h728 .153 .091 9.937h (.017) (.006) (1.633) Loga- 1 .h2h5 .0087 .h625 .322 .283 .1285 rithmic (.076) (.069) (.1231) 2 .3137 .0160 .68h0 .1h5 .122 .1558 (.108) (.120) (.2096) 3 .029u _.3388 .hh36 .20h .1u6 .1h00 (.060) (.087) (.1609) h .0737 .1899 .7666 .125 .058 .1377 (.073) (.080) (.1h20) 5 .10h8 .2697 .h692 .20» .128 .157h (.095) (.101) (.1h60) 6 .2223 .127h .5710 .23h .169 .1372 (.093) (.097) (.1376) 7 .22oh .1972 .h17u .299 .237 .1518 (.015) (.068) (.1389) 8 .3319 .0129 .658h .226 .137 .1525 (.095) (.105) (.1h81) 31 income and expenditures and a nonlinear relationship between the size of the family and expenditures. An examination of the estimated coefficients of the income variables presented in Table 6 suggests that a linear combination of the income variables would produce essentially the same results. A reasonably constant series can be obtained by combining the coeffi- cients of current income with one-third of the coefficients of lagged income (Table 8). On this basis a new income variable was defined as current income plus one-third of the lagged income (this variable is comparable to the amount of income received in two four week periods). Table 8. A linear combination of the coefficients of current income and one-third of lagged income* iinear Model _ Lo arithmic Model Interval 11 12 11 + 17312 Interval 11 I2 Ill+ 17312 1 .031h .0013 .0317 1 .h1h8 .0059 .h167 2 .032h .0128 .036h 2 .3321 -.0h03 ..3188 3 .0207 .01h9 .02h3 3 .0h71 ,.2801 .1395 h .0179 .0113 .0202 h .0618 .130h .1068 5 .0312 .0093 .0336 5 .1389 .16h9 .1933 6 .0290 .0161 .0329 6 .2199 .0953 .2513 7 .0336 .0075 .035h 7 .2289 .1281 .2711 8 .0362 .0039 .0370 8 .3298 -.0768 .3ohh *Computed from.Table 6. To incorporate nonlinearity of the family size variable, a model was defined in which the independent variables consisted of current income, size of the family (F) and the square of the family size variable (12) -- this relationship was denoted as the quadratic model. The estimates using one income variable and the quadratic model appear in Table 9. Table 9. 32 Estimates of food expenditures regressed on a combination of current and lagged income and family size--quadratic in the family size variable and logarithmic models Form Model Quadratic Logarithmic Estimated Coefficients Interval I F F‘ A 712—‘3 1 .0361 1h.6022 -.3h99 3.897 .633h 19.16h2 (.005) (h.71) (.652) (6.73) 2 .033h0 17.005 -.6601 2.9h2 .5h11 22.6923 (.009) (6.8h) (.910) (9.hh) 3 .03h1h 16.6256 -.5519 h.h02 .61h6 19.87h1 (.005) (h.h60) (.585) (5.h1) h .0230h 15.659h -.h170 10.519 .6h09 20.7362 (.005) (h.395) (.570) (5.968) 5 .02597 20.0679 -1.0959 3.857 .5651 20.8121 (.005) (n.0h2) (.522) (5.969) 6 .03718 11.5832 -.0325 5.5701 .6hh5 19.6733 (.005) (3.811) (-h67) (5.875) 7 .0353h 18.6615 -1.1626 ‘1.h73“ .5673 21.h596 (.006) (n.5u) (.605) (6.52h) 8 .02369 17.2098 -.5623 8.772 .5588 21.h866 (.005) (h.317) (.509) (6.3h6) 1 .3850 .5732 .h676 .6829 .1337 (.057) (.06h) (.1786) 2 .2976 .6206 .6957 .5861 .1581 (.087) (.096) (.2202) 3 .2997 .5971 .7021 .6u73 .139t (.058) (.065) (.1506) h .210h .62h1 .9696 .6513 .1320 (.056) (.061) (.1h80) 5 .3032 .5202 .7266 .5699 .1531 (.058) (.06h) (.1517) 6 .3163 .6099 .63h1 .6828 .1371 (.058) (.061) (.1523) 7 .h039 .h855 .h5ho .6332 .1h99 (.058) (.061) (.1h83) 8 .2809 .55h3 .7788 .5865 .1503 (.060) (.061) (.1586) 33 Since the use of a single income variable was more convenient and the results are essentially unchanged -- as can be seen from Tables 6 and 9 -- the income of the family was represented by one variable in subsequent analysis. Supporting the hypothesis of economies of scale in food purchases as the size of the family increases was the negative sign of the squared term in the quadratic relationship. The size of the family may increase either with the addition of members to the family or with changes in the structure of the family. Both of these have identical effects on the number of adult male equivalent units in the family. At this point in the study, the statistical model was suffi- ciently developed that the emphasis of the study was focused on test- ing the stability of estimated coefficients over the cross sections. From.a statistical and computational standpoint, tests of other hypotheses were much simpler if the population from which the dependent variable was drawn had a constant variance. Thus, Bartlett's test for homogeneity of variances was employed to test the homogeneity of the estimated variances of the eight regression equations for each of the two functional forms.10 The tests for the quadratic and logarithmic models appear in the following table. 1oBartlett, Mt, "Some examples of Statistical Methods of Research in Agriculture and Applied Biology”, SupplementoJournal of Royal Statistical Society, Vbl. 5, 1937. See Appendix.A for the formulas used in the test. 5 3h Table 10. Results of the tests of homogeneity of variances of the regressions of the eight cross sections Significance szel Form. Test Statistic Computed Value Percent* Quadratic 2(7) 5.56 n Logarithmic 2 (7) 9.71 22.0 *The numbers in this column represent the smallest significance level at which the null hypothesis could be rejected and the n denotes a significance level greater than 50 percent. The results presented in Table 10 indicate that the null hypoth- esis (equal variances over different cross sections) would not be rejected for a confidence interval of less than twenty-two.percent. Assuming that the variance of the estimated relationship was constant, a test of the stability of a regression coefficient over time was made by using the ratio of the differences of the sum of squares of residuals under the alternative hypothesis (the regression coefficients for a particular variable are not equal for different time periods) and the null hypothesis (equality of regression coeffi- cients for a particular variable over different time periods) to the sum of squares of the residuals under the alternative hypothesis. Let SSR denote the sum of squares of residuals; then the test statistic is F, given by the following formula: SSEHa: - SSRHo: x T a K SSRHa: N F (N,T-K) = 35 where: N denotes the number of restrictions (i.e., the number of coeffie cients assumed to be equal), T denotes the total number of observations (i.e., the sum of the number of observations in each cross section), .K denotes the total number of regression coefficients estimated in the unrestricted case.11 The null hypothesis and alternative hypothesis were represented ‘by He: and Haz, respectively. The test of the stability of the regression coefficients for the Quadratic and logarithmic models appears in Table 11. These tests indicate that the coefficients of the family size variable and the square of the family size variable were relatively stable over time. The hypothesis that the coefficient of the square of the family size variable was zero would be rejected for any con- fidence interval greater than 9 percent. The relatively low signifi» cance levels concerning the a1 = 0 indicates the necessity of a constant term, i.e., not forcing the relationship to pass through the origin. The significance level associated with the stability of the income variable indicates that the hypothesis of stability over time would be infrequently rejected. The statistical tests presented in Table 11 give tentative answers to some of the questions posed at the beginning of this section. a 11See Appendix A for a discussion of the computational tech- niques and procedures for computating a test of stability of regression coefficients in different regression equations. 36 Table 11. Tests of the stability of coefficients in quadratic and logarithmic models for eight cross sections Significance Form. Null Test Computed level othesis statistic value Percent Quadratic (Q) a1 I a: F(7,ll5h) 2.17 3.0 Logarithmic (L) 1,3 = 1,...,8 F(7,1162) 1.08 39.0 (Q) a1 = 0 F(8,115h) 1.95 5.0 (L) i = 1,...,8 r(8,1162) 20.67 0.05 (Q) bli . bli F(7,115h) 1.25 26.0 (L) 1,3 8 1,...,8 F(7,ll62) 1.0h h1.0 (Q) bFi : b1.J F(7,115h) 0.h8 n (L) 1,: . 1,...,8 r(7,1162) 0.61 n (Q) hr? = brfi 177.1150 0.39 n (L) 1,3 = 1,eee,8 2 (Q) bF1 . 0 F(8,115h) 1.72 9.0 (L) 1 8 1,eee’8 These particular cross sections were chosen to indicate any seasonality of the income variable. Intervals one, three, five and seven were chosen to represent a constant period within each year, whereas the other four intervals were chosen to give an indication of any changes in the effect of income during the year. An examination of Table 7 does not provide a conclusive answer concerning a definite relationship among different seasons of the year. 37 An examination of the intervals that represented a constant period does not give any clear indication of a trend in the income coefficients. Alternatively, the relationship of the constant estimated for each time period and the estimated coefficients of the income. variables were compared. Table 12 presents a pairing of the estimated constant terms and the estimated coefficients of the income variables for each interval. The income coefficients are ranked highest to lowest and the number in parenthesis denotes the ranking of the constant term from.lowest to highest. It was interesting to note that in the loge arithmic model only the fourth and sixth ranking do notgagree with a perfectly inverse relationship. This seems to lend some support to the hypothesis that the income coefficient should be allowed to vary over time. The ranking of the quadratic model was not as striking, but there were indications of an inverse relationship between the constants estimated for each cross section and the corresponding income coefficients. Table 12. A comparison of the income coefficients and the constant terms for the eight cross sections “Quadratic Model Logagithmic Model Income Constant Income Constant Interval Coefficient Eerm ~ Interval coefficient term 6 (1) .0372 (6) 5.570 7 (1) .h039 (1) .h539 1 (2) .0361 (2) 3.h97 1 (2) .3850 (2) .h676 7 (3) .0353 (1) l.h73 6 (3) .3163 (3) .63h1 3 (h) .03hl (5) h.h03 5 (h) .3031 (6) .7266 2 (5) .033h (3) 2.9h2 3 (5) .2997 (5) .7021 5 (6) .0260 (h) 3.857 2 (6) .2976 (h) .6957 8 (7) .0237 (7) 8.771 8 (7) .2808 (7) .7787 h (8) .0231 (8)10.519 u (8) .210h (8) .9695 38 Summary of Chapter II The cross section analysis of eight different time periods served as a basis for selecting the models to be used in the combined cross section and time series analysis. Several different specifica- tions of variables were studied to determine the specification that most accurately represented the behavior of the family as it related to food purchases. This led to the selection of an income variable (I) that was believed to be a reasonable estimate of the permanent income of the family. The final variable selected to represent the size of the family was based on an index of adult male equivalent units computed from US DA . d aha giving both requirements of different food groups for different ageosex groups and relative importance of these food groups in the total retail food bill for the United States. The dependent variable was the family's four week food expendi- tures, adjusted to a twenty-one meal per week equivalent for all mem- bers of the family. Two alternative functional forms of the demand relationship were selected for use in the combined time series and cross section model. The quadratic model was linear in the income variable and quadratic in the family size variable. The logarithmic model was linear in the logs for'both the income and family size variables. Tests performed on the estimated coefficients indicated stability of the coefficients of the family size variables over time. Also, a test of the hypothesis concerning the homogeneity of the variance of 39 the estimated relationship indicated stability. Although the F test indicated stability of the income coefficient, other tests suggested that the coefficient of the income variable varied over time. CHAPTER III COMBINED ANALYSIS Introduction Up to this point, the analysis of the demand for food has been limited to consideration of either one or several cross sections. These cross section analyses serve as the base for specification and estima- tion of the combined models. Both time series and cross section data were used in the estimation of the combined models. For the combined analysis 5876 observations were obtained by selecting data on 113 families for four years (fifty-two four week periods). Since only seventyaseven of these families reported their expenditures every week during the four years, the missing data were estimated for the remaining thirty-six families. If a week's data were not available, the preceding and following weeks were averaged to obtain an estimate of the family's food expenditures and income for that week. Once all the missing data were estimated, the data were aggregated into four week periods. 0f the thirty-six families who ' did not report every week, eighteen missed only one or two of the 208 weeks. Thirty of the thirty-six families missed less than seven weeks in the four years. Since only a relatively small amount of estimation was done, it was believed that the bias introduced by this estimation was more than offset by the addition of more families. When combined data are used, a number of situations arise that are not generally faced when either time series or cross section data are used separately. The use of combined data usually adds to the ho hl choice of models available to the researcher. Also, the larger number of observations tends to make statistical tests more sensitive. The contents of this chapter are divided into four sections. The first section contains both a discussion of the statistical model used to estimate the demand relationships from combined data and the variables entering the relationships. Secondly, the results of using the quadratic form of the relationship and alternative specific cationszof the variables are presented. The third section contains the results based on a logarithmic relation. Lastly, there is a discussion of the implications of the combined model and a summary of the combined analysis. Definition of the Models and the Variables Statistical models In this section the various models postulated as representing the demand relationship for food are discussed. Model A (l) (3.1) The demand for food is assumed to be represented by: it it t it Y - u + 2; Zr + “3 X8 + u where: , ... I, , eee T, ’ see S, ’ 0.. R h‘hlhih' and it I is the observed value of the dependent variable for the it_h_ family sampled in time t. 2}} is the rth independent variable that varies over both i and t. h2 XE is the sth indep ent variable that is constant over families-Ti) but varies over time (t) -- such variables as prices and outlook information. u is a random.disturbance for the ith family at time t. It is assumed that the disturbance is independent for any change in either superscript, i.e., the variance covariance matrix of the random variable is given by Crefi, where H is an IT x IT identity matrix. a is the general constant term. if is the coefficient of the rth independent variable that varies 1‘ over both i and t. n is the coefficient of the sth independent variable that varies over time. Results of the purely cross section analysis suggested defining an income variable that allows for trends and/or seasonality in the income effects. This specification is included in the next two models. Model A (13) (3.2) it it 2 “1+er + + at: + where: k = l, ... 13 and Iit represents the income of family i in time t, if t corresponds with the hth_period of the year and is zero otherwise. it. Ck is the coefficient of Ik The other variables were defined previously, except lit no longer includes I. This particular model estimates the coefficient of the income variable for each of the thirteen periods. For example, I1 represents the income data obtained from period one in 1955, 1956, 1957, and 1958. “3 Model A (52) (3.3) Yit 7r .1t it it where: m = l, ... 52 and Ii? is the income of family i in period t if m.= t, and zero otherwise. 4m is the coefficient of lit. The other variables are as defined in (3.2) For this model an income coefficient was estimated for each of the four week periods for each year, i.e., there were fifty-two income coefficients. The models having only one constant term were denoted as Model A; the number in parenthesis represents the number of income variables. As an alternative to the models discussed above, a model was defined that makes greater use of the combined time series and cross section data. This model estimates constants that vary over i‘but are constant over t. Model B (l) (3.h) it i it t it I - a + ‘7; 2 + fig XS + u where a 1 is a constant which allows for persistent differences in the behavior of particular families (1) over time (t). The other variables were as defined in (3.1). Model B (13) (3.5) it i it it it I =a “*er + “BK: +Cka +u The variables were defined in (3.2) and (3.h). hh Model B (52) (3.6) Yit : 01 -i- £2111: 4- Its X: + Cm Iit + uit The variables were as defined in (3.3) and (3.h). Model B denotes the relationships that estimated 113 family con» stants and the number in parentheses represents the number of income variables. The equations (3.1), (3.2) and (3.3) were estimated.by ordinary least squares. Although the coefficients of (3.h) through (3.6) were Obtained'by least squares, the large number of variables tends to make computations by conventional methods quite cumbersome. Estimates of xi were obtained in a manner similar to the method of estimating the constant term.in an ordinary regression'by defining the moments of the normal equations as deviations from the mean of each time period. A definition of the moments and the normal equations are given in Appendix A . Specification of the variables in the demand relationships was described in the purely cross section analysis. The form of the demand relationship was assumed to be quadratic (denoted as A and B in the statistical models) or logarithmic (denoted by primes on.A and B). .Independent variables The prices of‘both food and competing products (nonfood) became tin important variable in determining the demand for food when combined 'time series and cross section data were used. In the cross section (analysis it was not necessary to consider the price of food, since all <2onsumers were assumed to be confronted by the same price. There were, 1&5 of course, small differences in price applicable to different families, but it seemed impractical to try to observe them. Thus, if the price index for any particular cross section were assumed to be one, then the expenditure variable was identical with the quantity of food purchased. Two price indices were used to represent the food price faced by the members of the panel. These indices were (1) a price index of food taken from a study of retail food price indices based on the consumer panel,1 and (2) the BLS food price index at Detroit. The My _BLS price 333m converted toga thirteen period” index _i__n_o_rder to identify it with the thirteen four week periods used in,_,thi~_smstlidy. The monthly BLS data (collected on the fifteenth of each month) were identified with the four week period including the fifteenth of each month; thus in this study the first six periods were identified with the first six months and the last five periods with the last five months for each of the four years. (See Table 13 for a definition of each period.) In 1955, the seventh month was identified with the seventh period and the eighth period was the average of the seventh and eighth months. The eighth period was estimated by the seventh month and the seventh period by an average of the sixth and seventh months for the other three years. Using these two food price indices, the BLS Consumer Price Index was converted to an index of nonfood prices by removing the lwang, op. cit., Table 10. 46 .heeueuem e no mode one heomom oesoeeeoo e~-~n oo om-nH an oesoeeeao e~-~n on «-Nn an mseanmoxseaa e~-nn on ~-~H NH menaamoxseae H-~n on e-nn NH H-nn so n-en he m-nn on e-oh nn e-on oo e-e on o-o~ on e-e on see noose he-e on on-» e hso noose e-e on NH-e e a-» on mn-h e He-» so nn-e e anon no one ~n-h on an-e a seen he the en-e so an-e a see hotness: en-e oe eH-m e nee neaeosox eH-e so o~-m e an-n so o~-e n on-n on --e m noose» en-e so n~-m e nooaem n~-e on n~-m e --m on m~-~ m e~-n oo e~-~ m . --~ on e~-H e n~-~ on e~-~ N ease» as: n~-n on e~-~n n ,mmeH;. eoeoe soz mN-n on H-H a amen seasonoao ewteh oo H-~H an eesoeeaeo Hm-~n on e-~n ma maa>amexseaa on-HH on m-nn NH msaaamoxaeaa m-~d on e-nn NH ~-Hn so e-en an n-nn on e-on he n-en on e-e on e-o~ on en-e on sea sense e-e on he-» e use none; e~-e so e~-m e on-» on on-“ a n~-e on an-“ a mean no see nn-e on on-o a neon no one on-e oo on-e a as: henna-oz. mn.e on an-n .e nee newness: an-e on --m e noose» e~-m so n~-e n Hw-m on e~-e n o~-e on e~-n e nooosm n~-e on e~-m e m~-m on e~-~ m .e~.nfl on e~-~ m MN-" so e~-H N ‘ euuu so om-~ ~ ones» so: e«-n on en-- n amen ease» so: e~-n on «-n n amen eheewwom euev one.nunoz eowuem. nee» ehevuaom euee one nunoz powwow Meow e no seamen xeea seems «hesue emu nu amended“ eueeM.unom enu now weaken Aoee mo exeea noon emu a“ eevnuonu ehepwaon one emev aeauue enu ee>ww esseu easy .MH manea relative importance of food.2 Thus, for each food price index there was a comparable index of nonfood items. Considerable thought was given to the possibility of specifying a variable that would represent the effect of current aspects of the economy on the demand for food. It was believed that the willingness of the consumer to spend during the current period depends on his outlook for future income possibilities. In other words, a variable was desired that would reflect the current expectations of the family. These expectations were represented by a variable based on the percena tage of the total workforce unemployed. A priori, it was believed that as this variable decreased individuals would become more optimistic and increase food purchases. This increase was either the result of a greater quantity or an increase in the quality of food purchased. In particular, the expectation variable was defined as a simple average of the percentage of the total work force unemployed in Michigan and Lansing for each time period in the four years.3 The price indices and the unemployment variable are represented by x: in the models presented in equations (3.1) through (3.6). 2The relative importance of food items in the Consumer Price Index (CPI) was taken from The Handbook of Basic Economic Statistics, July 15, 1958, Vol. XII, No. 7, Economic Statistics Bureau, Washington, D. C. Let A denote the relative importance of food in the CPI, Pt represents the price index of food, (Ct) represents the CPI, then the nonfood price index, (qt) was computed from the following formula, Whom gt 3 Ct - Apt 'Tf‘:‘}f" }k = 0.285. 3The data were obtained from.Michigan Labor Market, Michigan Security Commission, Detroit 2, Michigan. and h8 The family size variable was as defined in the cross section analysis and was represented by one of the zit 's in equations (3.1) through (3.6). The income for the 122 individual and the tth time period was as defined in the cross section models. However, study of the effects of seasonality and trends in the income coefficient required three alternative specifications of the income variable. The first alterna- tive estimates one income coefficient and was represented by one of it the Zr '8 in equations (3.1) and (3.h). The second specification of income estimates a coefficient for each of the thirteen periods; i.e., the observations on income for the four years were used to estimate coefficients for the thirteen income variables. This particular specification of the income variables was given in equations (3.2) and (3.5), i.e., Iit. The third specification of the income variable estimates a coefficient for each of the periods for each of the four years, i.e., fifty-two income variables. Equations (3.3) and (3.6) represent this particular alternative, i.e., If. Dependent variables In the cross section models the dependent variables were repre- sented'by the family expenditures for food. Since price was constant over the cross sections, expenditures were proportional to the physical quantity of food. The prices varied in the combined analysis, there= fore expenditures must be converted to quantity in order to have a relationship associated with a theoretical demand relation. Thus, two quantity variables representing food purchases were computed by deflating expenditures by the food price indices. In this manner, the 1*9 identity of price times quantity equals expenditures was preserved. To have estimates of the coefficients that were comparable with the cross section results -- in terms of magnitude -- the quantity variable was multiplied by 100. The quantity index was computed in the follow- ing manner: Q” = Eit / Pt times 100 where: B11: is food expenditure of individual i and time 1:. Pt is the price index of food for time t. Certainly a family's food purchases are influenced by the various holidays of the year; therefore, adjustments were made to reduce the bias that could enter the relationship because of holidays. The following special seasons of the year were considered: (1) New rears, (2) Easter, (3) Memorial Day, (’4) Independence Day, (5) Labor Day, (6) Thanksgiving, and (7) Christmas (see Table 13). An examina- tion of these indicated the necessity for three adjustments. Period one of 1955 and 1956 included January 1, but did not include the weeks in which food purchases were made in preparation for New Iears' Holiday (see Table 13). The family's food expenditures in the week prior to January 1, was added to the expenditures of the first period of the year and then multiplied by four-fifths to adjust for this bias. As a result of this adjustment, the food expenditures for period 13 of 1955 included the expenses for food used on January 1, 1956. To adjust the expenditures in period 13, 1955, the last week's expenditures of period 13, 1955, were subtracted from the total food expenditures and the result was then multiplied by four-thirds. Quadratic Model Specification includLnLthe expectation variable Throughout the combined analysis both of the price indices were used, thus requiring that two relationships be estimated for each specification of the variables and form of the demand relations. Pres liminary specification of the combined model was limited to the quadratic model since the large number of both observations and inde- pendent variables required a tremendous amount of calculation in order to estimate the demand relationships. Initially, three alternative definitions of the demand relationship were considered. Each of the relationships contained a variable denoting the size of the family, the square of the family size variable, the price index of both food and nonfood, the expectation variable and the income of the family. The income variables appearing in the three alternatives were as previously defined. The demand relation was estimated with one general constant ten: -- Model A, and with aparameter estimated for each family -- Model B. Both of these specifications were considered for the three alternative income variables. The statistical“. models were given by equations (3.1) through (3.6). Also, for each of these relations two alterna- tive specifications of the price variables were used, the prices being represented by the price index based on the consumer panel and the Detroit BLS price index of food. The estimates of these twelve relations are presented in Tables 1-1, 3-2, and 3-3 of Appendix I. Table Eel contains the estimates based on one income variable Model A (1). The estimates 51 using Mbdel A (13) and.Model A (52) are given in Table E-2. The relationships obtained when 113 family constants, Model B (13) and .Model B (52), are presented in Table Es3. The performance of the expectation variable was of primary interest in analyzing these estimated demand relationships. Did the expectation variable respond as a priori expected? That is, did the expectation variable, represented by the percentage of the workforce unemployed act as an indicator of a wage earner's expectations regard- ing future income? A priori it was believed that as this variable increased, the quantity of food purchased would decrease. However, in eight of the twelve relationships estimated this was not the case. Only the model estimating thirteen income variables -- Models A (13) and B (13) -- had expectation variables with negative coefficients. However, there was consistence for the two price variables considered; i.e., similar results were obtained for the consumer panel and Detroit BLS food price indices. Also, for the relationships in which standard errors of the estimated coefficients were computed, the standard errors were greater than the estimated coefficients of the expectation variable. The I test was used to test the significance of deleting the expectation variable from.the demand relationship (Table 1h). In each instance the test statistic is F with one and approximately infinite degrees of freedom. The general format used in the previous tables presenting results of statistical tests was followed. The results of the tests concerning the significance of the expectation variable suggested that the variable be deleted from.the 52 Table 1h. Tests of the null hypothesis that the coefficients of the expectation variables were zero —Consumerpane1 price index Detroitprice index Computed Significance Computed Significance Model value of the level value of the level statistic ,e . WTW" «statistic “ __--rcent I. One constant term Model A (1) .196 n 1.98 82 Model A (13) .060 n .967 67 Medal A (52) .685 62 1.h32 79 II. Estimating 113 constants Model B (l) .101» n .563 53 Model B (13) .379 11 $06 n Model B (5%) .h82 51 .399 n relationship. Therefore, all models discussed in the remainder of the thesis do not contain the expectation variable. Further consideration of the quadratic model Since the tests concerning the significance of the different models were based on the sum.of the squares of error, the error sum of squares of the dependent variable under the two alternative definitions of the price of food appear in Table 15. This table also contains the sum of squares after obtaining deviations from the mean and the sum of squares after accounting for the family constants. The twelve quadratic relationships previously estimated were re-estimated without the expectation variable. The estimates using Models A.(l) and B (1) appear in Table 16. Table 17 contains estimates using Models.A (13) and (52). Table 18 contains the results when family constants were considered, Models B (13) and (52). The estimated family 53 Table 15. Sum.of squares of the dependent variables in the quadratic model Sum of squares after Dependent variable Total sum Sum of squares accounting for the - of aqua. 1‘08 abt ”Ch. fem-11 counts _-.___,_.. “~___—__,___ _.— Expenditures deflated by the index based on the consumer panel (quantity) Qit' 38,399,87l.66 7,856,h08 l7 l,h21,665.92 Expenditures deflated by the Detroit BIS it food price index, Q2 .38,h60,906.63 7,888,hh0.22 l,hh7,h65.28 constants of Models A (1), A (52), B (l) and B (52) are given in.Tables E-h, and E-5 of Appendix E. Standard errors were obtained for the coefficients in models with one and thirteen income coefficients. Since there was very little change in the standard errors between these two models, standard errors for the larger relationships were not computed. Mereover, standard errors of the estimated parameters of each family were not estimated. The primary reason for not computing these statistics was the inadequate computing facilities. As was expected, the estimated relationships excluding the ex- pectation variable differed very little when compared with estimates obtained when the expectation variable was included in the demand relationship. This is pointed out by the similarity of the coefficients of determination. The significance of estimating more than one income coefficient and the significance of using combined data to estimate coefficients for the different families were of primary interest. The alternative .euneuenou hawaem enu mo eweuepe emu meuonenn AHHeo.v AeeH.V AHNH.V AeMN.v Ase.Ho N eN.noH.eeN.H Heme. neoo. one. eee.- ome.- eo.NH see.Nn eHo AHHoo.V AmoH.v AmMH.v AeoN.v Ams.Hv H “m me.Nos.neN.H ammo. eeoe. NeH. men.- NNe.- Ne.eH sem.em one “Hon Hoes: Aeeoo.v AoeH.v AeeN.v aNee.V AmmN.v aNn.NHv N ee.emn.mee.n nMHe. eeNo. HHN. see.- eee.- em.mH mm.ee sHo . Aeooo.v nemH.V AeoN.v aNeo.v AenN.v Amm.nHv H eN.eee.eme m AmHe. eeNo. one. 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Ammv< Hose: {mace I N“ em.eee.ene.n n ems AeHoo.oAmHoo.VAmHoe.VAmHoo.vAaHoo.vAmHoo.VAeHoe.vAoHoo.vAeHoe.VAeHoe.VAnHoo.vAoHoe.VAeHoo.v AwH.VAeH.V Aeo.vaee.a He.NHV eeNe. aANe. meNo. ,oNNo. NeNo. NMNe. eNNe. mNNe. enNo. eeNe. enNo. NeNe. oeNo. NNN. NHe.- mme.- me.en mo.sm .s 3H0. I NM ee.eHN.eNo.m a mom AmHoo.VAnHoo.VAmHee.vAeHoe.vAmHoo.vAmHoe.VAeHee.VAeHoo.vAeHeo.vAeHoe.VAeHee.VAeHoo.VANHeo.v AeH.VAHN.v Aeo.VAmN.v AN.eHV eeNo. NnNo. oeNe. enNo. HeNo. HmNo. nmNe. nmNe. NnNo. enNo. emNo. enNo. NnNe. NNN. eHn.- Nee.- Nn.eH NH.Hn H . eHo 3H: Hose: mH NH HH oH e m N e n _e . n N H oz on NAsHmV one 4 Aum eaoonH Amnv< one AmHv< eHeoQZuueHneuuen noHueuoenxe emu wnuonaoxeateuneauammeoo eaoona oauthumau one neeunnmu mo e>nuenueuae emu muHm one seen uneuenoo Henenew eno mnwueaHuee Heooa oaueuoena on» no meueaaumm .NH oneH 6 5 oooo. «moo. ammo. huao. Onoo. «moo. naao. noao. ouoo. a Hooo.. neoo. oaao. osoo. «Nao. «moo. oaao. .ooao. nooo. hooo. «ado. dooo. naao. enoo. mooo. «moo. mooo. nooo. nnoo. Hugo. noao. «moo. oooo. nefio. maao. «woo. «woo. oooo. oooo. ohoo. ouoo. Amado. Nooo. Nooo. «coo. mooo. .onoe.. neeo. onoo. mooo. nooo. «woo. omoo. omoo. Hooo. unoo. nnoo. nsoo. «moo. Hnoo. «moo. eooo. nooo. «ooo. onoo. oooo. oooo. enoo. enoo. Hsoo. «soo. onoo. Hugo. oooo. oooo. osoo. muno. osoo. oooo. gooo. nNoe. mooo. noHo. nuao. «moo. nooo. nooo. ouao. eooo. nmoo. ooHo. «woo. eoHo. onoo. ouao. oooo. ooHo. nooo. «moo. Noao. Hooo. naoo. «moo. NeHe. eeeot . 28. .i 38. snoo. @500. 1 HHHo. mooo. nooo. naao. nHHo. oooo. «Odo. AmHoe.vAnHeo.VAnHoe.vAeHoe.vaeHoo.vAeHoo.VAeHoe.vAeHeo.VAeHee.oAeHoe.VHeHoe.vAeHoo.VAnHoo.v Heoo. omoo. fiNHee.vaeHoo.VAeHoo.vAeHoo.VAeHoo.vaeHeo.vAeHoo.vAmHoe.erHoe.VAeHoo.vAeHeo.vaeHoo.ofieHoe.v . HHon. I.NM Hn.eoe.nnN.H u mmm neH.. eNn.- een.- ne.oH N oHa moon. I Nd Nm.eNn.omN.H u now oeH. eem.- eNe.- NN.eH H oHa aNnvm Hose: name. a m oe.eem.NeN.H u m m aNH.V Aeo.v AHN.VAeN.HV HeN. eoe.- Nan.- nN.eH No , nH comm. [.mm ee.nNH.noN.H - mom aNH.V aNo.v AHN.VAmo.HV eaoo. Neoe. eeeo. ease. eNeo. eeoo. Nmee. “moo. eNoe. "Noe. eNeo. eeoe. eNoo. eHe.- emm.- eHn.- eo.eH Ho oH NmflqmiHsemm. eH NH HH eH e e .N e n e n N H ”oz on _NAeHmo on 8 33.: Aumvn one Aman eHeooxaaeHmegue> nouueuoenxe one wnaonmoxet- euneHufluueoo eaoonu onuthumwm one neeuuHmu e>HuenueuHe son no“: one oeueanuee muneuenoo mHaaem nHH new: Heooa oaueuoeso en» mo mnnnonowueHeu oeueanueu .oH eHneH 57 definitions of the income variable were important because of the extra computations required when the number of variables was increased. The consideration of constants particular to individual families were particularly important because of the definition of the model. These constants cannot be estimated unless both time series and cross section data are available. If these constants produce a significant differ- ence in explained sum of squares, then one would expect "better" estimates of the parameters of the observable variables. A brief examination of the error sum of squares obtained in estimating the 113 family coefficients and the error sum.of squares when only one general constant term.was estimated,suggests that the difference in these models was significant (see Table 11). However, this large difference was not present when alternative definitions of the income variable were considered. Table 19 contains the statistical test of the above mentioned hypotheses. The test statistic of interest was I, defined in a fashion similar to that described in Chapter II, page 31!. Perhaps the most conclusive results were obtained when tests were made to determine the significance of estimating a coefficient for each family. In all instances the significance level at which the hypothesis of no difference could be rejected was less than 0.05 percent. From this result alone, it would seem that the combined model used in the study should be considered quite seriously whenever adequate data are available. In general, the coefficient of determination increased by approximately thirty-five percent when coefficients were estimated for each family. not only were better fits obtained but also, the .NHHV aNnvm no. e.HN no. n.HN AeoNn Hoes: no. e.oN no. H.NN AeoNn .NHHV AmHVm Hoes: no. m.HN no. m.HN AneNn .NHHV Ava Hose: 00 5 mudwundoo .AHHENN 05» HO hufifimfivw my: wdwummfi m.o ee.n H.o wN.H AeoNn .emv soseHonomooo osoosH Nn e> «H O.“ mm.H o.e mm.H .AeONn .Hmv ooseHoHeeooo osoosH Nn e> H s an. s eeH. AneNn .NHV oesoHoHeeooo osoocH mH e> H m Hose: s so. n. oe.H AeHmn .emv sesoHonmmooo osoosn Nn e> mH o.me mo.H o.e Ne.H Aonm .Hmv oosenoHeeooo esoosn Nm e> H m. Nm.N s nNe. Amman .NHV moseHonmeooo oaoosH mH e> H < Hoes: He>eH en~e> He>e~ en~e> sooeeuw eweemuowxm .t .Heoo: euneowmanwfim oeunosoo euneu«wwnmqm oeunnsoo mo neon“ ocean) eeeuwen eoaun uqouueo Henen nesnenoo Hence the nuneumnoo hawsew one gene eweenuonms enw mo women one Heoos eon nouns oHnone uenu euneaogwweoo eaouna mo Henson enu enHSmeueo cu.mumey .oH wanes 59 estimated coefficients of the observable variables should be better estimates of the true parameters in that consideration has been given to both time series and cross section data. In general there was a significant difference between the estimated relations using one income and those with an income coefficient for each time period, Models A (l) and A (52). The exception was Model A (1) when the Detroit BLS price index was used as the price of food measure. There was more difference between the relationships with one and fifty-two income coefficients than between those estimating one and thirteen income coefficients or those with thirteen and fifty-two income coefficients, in most instances. The differences between thirteen income coefficients and the other two definitions of income imply that further consideration of an income coefficient for thirteen periods would be of little value. In effect, this elimination rules out any possibility of seasonality of the income variable. Persistent seasonal effects on the estimated coefficients of income were not indicated in Tables 17 or 18. Also, an examination of the estimated coefficients for each of the feur years in the Models A (52) and B (52) failed to indicate any persis- tent seasonal variation. The signs of the estimated parameters agreed with their theoretical expectation, with the exception of coefficient of the nonfood price variable of’Model.B (13) using the consumer panel price index. Since the nonfood price index represents the price of all items except food, a two commodity world has been assumed. This being the case, the two commodities (food and nonfood):must‘be 6O substitutes. Therefore, the coefficients of the price variables should have different signs.h The greatest divergence of magnitude among the models was in the estimated coefficients of the price variables. For example, in.Model A (l) the estimate of the coefficient of the price index for food was oO.88l; for A (52) it was -O.h37. Similar differences were found in the models estimating family constants. For'Model.B (l) the estimate was -O.5h3 and forIB (52) it was -O.3h9. For'both models the estimates were based on the consumer panel price index. The estimated coeffi- cients tended to be smaller in magnitude as the number of income variables was increased, as was pointed out above. The estimated coefficients can be coverted to elasticities at the mean by using the associated ratio of the mean of the independent variable to the mean of the dependent variable.5 In order to convert the food price index, the nonfood price index and the income variables to elasticities one may use the ratios 1.38, 1.38, and 10.0 respec- tively. The elasticity of the family size variable is equal to 0.038 “A mathematical statement of this proposition is given by: Hicks, J., Value and Capital, Oxford, 1939, Appendix to Chapter I, section 10, p. 31. Henderson, J. and Quandt, R., Microeconomic Theory, MoGraw Hill, New'Iork, 1959, p. 30. 5The elasticity of a dependent variable, say Q, with respect to an independent variable (X) at the mean-is given by: %% a coefficient of x times i/é where the bar above the variable denotes its mean value and where the coefficient of X is taken from an equation where Q is the dependent variable. 61 times the coefficient of the square of the family size variable.6 These ratios were approximately the same for both definitions of the quantity of food purchased. Logarithmic Model On the basis of results frbm.the quadratic relation, a loga- rithmic relation was defined. The variables entering the relation were: (1) the size of the family, (2) price of food, (3) price of nonfood, and (h) income of the family. Again, both definitions of the price of food were considered. The income variable was defined for one and fifty-two income coefficients, Models A' (1), A' (52), B' (l), and B' (52). The primes denote a logarithmic relationship. The relevant sums of squares of the dependent variable (quantity of food purchased) appear in Table 20. Table 20. Sum.of squares of the logarithms of the dependent variable Total Sum. of Sum.of squares after Dependent variable sum of squares about accounting for the t“ .,M1._° H .m Homo: o.oH «N.H n.e we.H HoNQn .an eosoHoHumoeo eaoosH Nn e> H . .< Homo: .Ho>oa o=Ho> Ho>oH on~s> Eovooum naaonuomwm Homo: oosmowwwame wounaaoo oocanwHome wounaaou mo novoH Nomaa ooaua moouwon ouauo UHouuoa Hogan unannooo mannoouuoaou oHEAuHuoon on» ma .Hosvo one mucouaaoo kHHauw onu uosu awuonuomhn on» mo munou one nucowoammooo oaooaH cauuauuHu mo ooaoonuaan onu wnwcuoouoo munch .mN edema 66 In one respect there was more consistence among the logarithmic models. That is, there was less variation in the coefficients of the price variables. In these relations the coefficients differ by less than 0.10, whereas in the quadratic models the difference was 0.h5. A discussion of the relative merits of the different forms and the various models is included in the next section. Implications and Comparisons of the Combined Models Thus far, the demand relation for food has been represented by a quadratic and logarithmic relationship. There were no attempts to specify which of these forms represented the appropriate demand rela- tionship. Similarly, there was little evaluation of the usefulness of considering more than one income variable. Also, the question of the appropriate price index was not resolved. Only general state- ments were made concerning the estimation of family constants. For the most part, implications of the estimated coefficients were ignored. This section contains a discussion of the implications of the various alternative models. Estimated coefficients The average estimated elasticity of income with respect to food purchases was approximately 0.10 and 0.25 for models A.and 3, respec- tively. That is, the consideration of family constants reduced the estimates of the income elasticities by approximately 60 percent. Therefore, it seems that defining models to estimate the effects particular to individual families drastically reduced the importance of the observable income of the family. A discussion of the implications 67 of this situation appears in Chapter IV. It was interesting to note that estimating family constants had very little effect on the magnitude of the coefficients of either the family size or price variables. What were the consequences of considering more than one income variable? In the logarithmic relationship the estimated coefficients varied from 0.0689 to 0.1201 for lodels B (l) and B (52), respectively (Tables 21 and 22); whereas in Iodels A (l) and A (52) the range was from 0.2232 to 0.2873. Converting the coefficients of the quadratic models to elasticities gave results similar to those obtained in the logarithmic models. The estimated elasticity of the family size variable was approximately 0.52 and 0.6h for Models B (l) and A (1), respectively. These estimates were reasonably constant for alternative definitions of the models. The estimated elasticities of the family size variables at the mean in the quadratic models were approximately 0.50 and 0.60 for Iodels B (l) and A (1), respectively. for a given fucticnal fern there was very little difference among theralterh‘itiVG-‘formulatio‘h‘s of statistical models‘with respect to the family size variable. Comparisons of the elasticities of family size variables were not relevant, since most studies of the demand for food did not estimate elasticities for the size of the family. Ioreour, the use of a special index to compute the size of the family mahes comparisons difficult. In the Tobin study a per capita measure of the size of the family gave an elasticity of 0.27.7 This estimate seemed low; it 7Tobin, o . cit., p. 119. 68 implies that doubling of the size of the familvaould only increase food expenditures by 27 percent. From.a.practical standpoint, the estimates obtained in this study seem.more reasonable: the addition of an adult male to the average family in the study would result in an increase of twenty percent in food expenditures.8’Ilurthermere, the addition of a new baby to the average family results in.an increase of ll percent in food purchases. In terms of expenditures, this would mean about a ten dollar increase in.toed expenditures, based on a 1955-58 3 100, price index. Although it was not mentioned, the reader should keep in.mind that all statements concerning changes in food purchases as the result of changes in the size of the family iere ceteris paribus conditional statements. Table C-1 of Appendix C can'be used to compute other responses for different size families. The greatest divergence of magnitude in the coefficients among alternative specifications of the demand relation was in the price elasticities of the quadratic and logarithmic relations. In general, the estimates obtained in the quadratic were larger than those obtained in the logarithmic; and, the computed.price elasticities of the demand for food were larger than those obtained by Tobin's study of the demand fer food. The estimated price elasticities of the logarithmic relation- ships ranged from.0.72 to 0.52 with the smaller'estimates‘being obtained from Model B. There was an.inverse relationship between the nunber of income variables and the estimated coefficients of the price variables; i.e., as more income variables were considered, the estimated 8The average size family is 2.7h adult male equivalent units. 69 coefficients of the price variables decreased in mapitude (see Tables l6, 17, 18, 21 , and 22). The explanation of this phenomenon may be partly the result of having had only fifty-two observations on the price variable. This variable was repeated for each family; therefore, any correlation between price variables and the alternative definitions of the income variables would be magnified. It is quite possible that an association of this type could influence the estimates in the manner described above. In general the estimated price elasticities for the logarithmic relations were more plausible than those obtained in the quadratic relation, in that they were smaller and there was less fluctuation among models. The average nonfood price elasticity in the logarithmic relation- ships was approximately 0.10. The estimated coefficient of nonfood price was larger than that obtained by Tobin, and the ordinary tctest indicated a higher confidence level for testing the hypothesis that the coefficient was zero than was obtained in the Tobin study. As was the case for the price elasticities for the demand for food, the smaller estimates were obtained for the logarithmic relations. The coefficients of the price variables in the quadratic models can be changed to approximate elasticities at the means if multiplied by 1.38. Appropriate_price index Throughout the analysis of the combined data two indices of the price of food were considered, an index based on the consumer panel and the Detroit BIB food price index. The United States Consumer Price Index was adjusted to remove the influence of food prices in order to obtain indices of nonfood prices by using each 70 of the food price indices. An examination of the tables of the estimated coefficients presented in this chapter suggests that varia- tion'between the relationships using the alternative definitions of the prices was negligible, given that the other specifications were constant. This being the case, the relationships with the larger coefficient of determination were chosen as the more plausible. However, the real question was not what definition of the price variable gave the relationship with the larger coefficient of deter- mination‘but what were the implications of constructing a quantity index'based on a price index obtained outside the panel? It seems quite significant that an index of food prices computed from data obtained in another city performs as well or better than an.index computed from.the original panel data. Form of the demand relationship Based on the results previously presented, indications were that the logarithmic relationships were perhaps the more plausible of the two. However, it is emphasized that the preference over the quadratic was only slight. In most instances the coefficients of determination were larger for the logarithmic relations. In addition, it seems that the estimated elasticities of the family size variable and the price of food and nonfood were more plausible in the logarithmic relations. One income versus fiftyotwo income variables The introduction of the large number of independent variables (extra income variables) brought with it a number of computing problems. The resulting moment matrices were quite large, thereby requiring a large computer to invert the matrices and solve the normal equations. To assess considerations of more than one income variable, the extra computations required must be compared with the improved fit of the demand relation. In essence, the extra cost must be compared with the additional accuracy of the estimated relationships. For this particular situation it seems that the improvement in accuracy was outweighed by the additional cost. This conclusion was based on the relatively small range in the estimated income elasticities when fifty-two income variables were considered. The relatively small increase in the percentage of total deviations explained lends further support to this conclusion. Test of the homogenity concept concerning monetary variables of the demand relationships . If the families studied were acting in a rational manner, doubling all prices and incomes would not affect food nxpenditures. Given this situation, the sum of the coefficients of the monetary variables in the logarithmic relationships should be zero. ‘Using the consumer Panel Price Index and both Models A (l) and B (l), the sum of these coefficients «rs 0.3079 and -0.3380, respectively. For the Detroit BLS Price Index the sums were -O.3329 and -O.3576 for Models A (1) and B (1), respectively. A‘t’test indicated that the sum of these coefficients was significantly different from zero at the 0.05 percent level. One possible explanation of this result is the presence of money illusion. Since this only can be true in the short run and this study covers a limited period, it is quite possible that money illusion was a reasonable postulate. Tobin obtained sums that 72 were much nearer zero, and his study covered a much longer time period.9 Modifications of demand theory currently under way may provide another explanation.lo Summary of the Combined Model In this chapter, several alternative specifications of the variables and functional forms of the demand relationships for food were considered. Statistical tests performed indicated that the models with family constants were superior to models that estimated only one constant term. In most situations the logarithmic relation seemed more plausible than the quadratic. The significance of con- sidering fiftyatwo income variables over one income variable was not settled by formal statistical tests. The decision regarding how many income variables were to be used depended on equating the additional cost of considering more than one income variable to the benefits of the additional accuracy of the estimated relationship. The Detroit BLS price index performed at least as well as the price index‘based on the MSU consumer panel. Table 2h contains a summary of the estimated elasticities using one income variable under the alternative definitions of the demand relationship for food. 9Tobin, o . cit., p. 132. 1OManderscheid, L., "Quasi-Fixed Assets in Consumption.Theory", Econometrics workshop, Michigan State University, 1961, (mimeo). Koo, A., ”Emperical Tests of the Revealed Preference Theory", Econometrics workshop, Michigan State University, 1961, (mimeo). 73 Table 2h. Demand elasticities calculated from Models A (1), A'(l), B (1), and B'(l) j Elasticitigs Model fanny reed Nonfood size price index price index Income Consumer Panel Price Model A (1) .606 4.210 .633 .225 Model A'(l) .6h5 - .732 .169 .2116 node1 B (1) .509 - .752 .266 .1068 Model B'(l) .511 - .600 .157 .10h8 Detroit BIB Price Index node1 A (1) .606 - .833 . .291 .22h Made]. A'(1) e6h5 " e699 e121 e2“ Nada]. B .(1) e508 " eghl .583 e1°18 lbdol B'(1) e521 - e599 e138 e103 An examination of Table 2h indicated that the estimated elasti- cities of the family size and income variables were quite similar under altemative models. However, this was not the case with the price elasticities. In general the income elasticities obtained when family constants were estimated were approximately two-fifths those obtained when one constant ten was estimated. Thus, an increase in the family income of ten percent would result in a one percent increase in food expenditures. This value was much smaller than estinates obtained in most other Studies of the demand for food; however, these studies did not estimate constants for each family. The greatest difference in the logarithmic and quadratic rela- tionships was found in the estimates of the price elasticity of demand. 7h However, it seems that estimates obtained by the logarithmic relation- ships were more reasonable, which supports the logarithmic relationship as being more plausible. For example, a ten percent increase in the price of food would decrease food expenditures by approximately six percent. Similarly, a ten percent increase in the price of nonfood items would be expected to increase food expenditures by approximately one percent . CHAPTER IV EVALUATION OF THE STUDY ‘ The contents of this chapter are divided into two sections. The first section contains a general evaluation and implications of the study. In the second, there is a discussion of problems encountered that were not pursued and some possible areas that should be considered for future research. ‘ Evaluation and Implications It should be pointed out that {he most time consuming part of the study was the specification of the variables, rather than estima- tion of the demand relationships. ror this reason the number of alter- native techniques that were considered to combine time series and cross section data was quite limited. This being the case, more attention was given to analyzing the demand relationship for food rather than model development . A major part of the specification of the variables dealt with the family size and income variables. That is, how should the size and composition of the family have been represented in the demand relationship? How were food expenditures affected by the income of the family; should lagged relations have been investigated? These ”re Just a few of the questions that were considered .in developing this study. How valid were the results.of the study? Certainly, the results 01’ the study can be no better than the data used in the analysis. Since the total food bill was computed by totaling the expenditures of 7S 76 teach food item.purchased.by the family, there were undoubtedly some xnistakes due to the omission of small purchases that were not recorded. {The panel data were mailed to the University each week and tabulated as they arrived. Shaffer and Quackenbush have a discussion of the relative merits of collecting data by different methods, as well as a discussion of the MSU consumerpanel.1 They concluded that the panel ‘was representative of Lansing, Michigan'but did not attempt to claflh representativeness for any larger area. For the purposes of this ' study, it was not necessary to assume that the panel was representative of any population other than the sample itself. This follows from the fact that the panel furnished the only available data for testing the combined model. If, however, the panel is representative of a larger :population than itself the conclusions are of even greater importance. There may'be some biases introduced because of the storage of :food during the year. For example, food can.be stored in food freezers, allowing families to capitalize on specials of certain food items. ramilies may also plant gardens, thus affecting the purchases of different food items. There were no attempts in this study to correct :for the storage of food during the year. One of the major limitations of the data was the information that was not collected, rather than that collected. Data were avail- able on disposable income of the family and food expenditures, but there was no information pertaining to how the remainder of the family's income was spent. What part of the residual income, income after food 1Quackenbush and Shaffer, op. cit. 77 purchases, was saved or invested? It seemed quite reasonable that the income determining consumer expenditures for food was determined in part by the amount that remained after payments for rent, utilities, purchases of durables, educational expenses, etc. Except for a questionnaire that was mailed to the members of the panel in April, 1957, asking how much of their present income was comitted, not including food expenses, information was not available concerning their residual incomes. The demand relationship for the initial cross section was estimated with uncommitted income instead of current income. The estimated relationship was: s a .0307 Iu + .0160 12 + .0012 13 + 1h.85 :1 + 1.239 (M1) (.013) (.005) (.005) (1.71) fig = .633 s = 19.9 where: In is the income of the family that was uncommitted for period 3, 1957; that is, income after all monthly payments exclud- ing food purchases (current dollars). 12, I3, F1 and E are as defined in (2.1) of Chapter II. This relationship is comparable with equation (2.1) of Chapter II. The estimates of the coefficients do not differ greatly, but the coefficient of determination is numerically larger with uncomitted income as the indicator of current income. Since these data were only available for one month, any additional investigation along these lines was not possible. An examination of the estimated coefficients of the income Variables suggests that the superiority of using fifty-two income ‘4 "D variables rather than one was perhaps lost, due to the increase in the number of calculations that were necessary to estimate the relevant parameters. The changes in the estimated coefficients were rather small considering the additional computations. The similarity of the estimates under alternative definitions of the demand relation is pointed out in Table 25. Although there was variation among estimated coefficients in the different models, the average of the elasticities for the fifty-two income variable model was approximately equal to the estimated elasticity for the model with one income variable. Certainly one of the most promising parts of the study seemed to be the use of dummy variables to represent the unobservable charac- teristics of the family that were not reflected in the independent variables. The sample was small enough so that each family was represented by a durm variable. Classifications of families into homogeneous groups could be considered if a large number of families were observed. However, in terms of computations required, the use of a large number of dummy variables did not present the computational problem that was presented by the large number of other independent Variables. The estimated coefficients of the dum variables were obtained in a manner similar to the procedure used to estimate the constant term of an ordinary regression.2 The procedure used to estimate the constant terms for each family from combined time series and cross section data can equally Yell be applied to other areas. For example, a procedure similar to 28ee Appendix A. 79 ..d . «use. “use. none. once. sumo. assess an -- -- uses. «see. once. mace. assess an asset anon. mace. . _ssesaa a Noemi eouum mam uaouuon «one. dose. some. sense once. «one. .aseosa an -- 4. case. one“. once. «ace. assess no wood. «can. ease. assess a. sheen“ eoaum demon Holmemoo a means «ass. seas. been. some. some. case. assess an. -- .. mesa. meow. sumo. same. assess as scan. has". «sue. . assess a . noose sense an: uaouuoe some. nuns. mass. ses~.. some. acne. assess an -- -- menu. «ecu. ammo. anus. assess ma. anew. mean. a. .meoso. assess a lemma eowum demon nonsense. . . e 4 seed: rem dawn. seat has: sou t . mum: t, it , assesses-sea sassaoacasmu saaeaoouueoo ocean 033.:qu sausneeoa . . . Ly Q munemowoeueu unwise on» no emowurowuwoome epwuerheune emu noeoo.eo«uuo«ueeao emu euueuouuueoo oloomu eon-lanes emu we shellac d. .nN smash 80 the one used in this study was used by Koch to study the behavior of firms from combined time series and cross section data.3 He was primarily interested in estimating production functions for. farm firms and testing the economic efficiency of the different firms. Implications of the family size coefficients Elasticity estimates of the family size variable points out some very interesting aspects of the behavior of the families with regard to food expenditures. There was essentially no difference in the elasticities obtained from Model A and Model D. That is, family size affects food expenditures in essentially the same manner, whether or not family constants were estimated (Table 2h, Chapter III). In the purely cross section analysis the elasticity of family size was approximately 0.68 (Table 9, Chapter II) which compares with 0.65 obtained in Model A (Table at, Chapter III). However, when variation among families was removed the estimate was approximately 0.52, thereby lending some support to the hypothesis that the size of the family acts in essentially the same manner whether observed in time series or cross section. It is noted that these conditions do not hold for the income variable which are discussed in the next section. Not only are the coefficients of the family size variable relatively stable for alternative models, but also they seem to give reasonable results for computing changes in food expenditures as the size ofthe family changes. Using Model B (l), for example, given a . 3‘Eoch, L. , "Estimation of Agricultural Resource Productivities Combining Time Series and Cross Section Data,” Unpublished Ph.D.. Thesis, University of Chicago, Department of Economics, 1957. 81 family consisting of a man, his wife and a child less than one year, the addition of a child in each of the next four years would increase food expenditures from.$63.22 to $95.12 by the fourth year. It should be pointed out that these conclusions are based on elasticities and are only ceteris paribus conditional statements. Implications of the income coefficients As a consequence of using dummy variables, one of the more interesting results of this study was the behavior of the income elasticities. In the purely cross section analyses the average income elasticity was approximately 0.32 (see Table 9) When models were defined that considered both time series and cross section data -- Model A -- the estimated elasticity of income with respect to food purchases was approximately 0.2h (Table 25). However, the largest difference was observed when family constants were estimated -- Model B. For this model the average income elasticity was approximately 0.10 (Table 25). Two hypotheses are presented as possible explanations of this phenomenon. One relates the implications to Friedmanfis permanent income hypothesis.h The second points out other inferences that can be drawn from the model estimating family constants. The estimated family constants were interpreted as giving a numerical value to all the unobservable characteristics of the family that were not reflected by the independent variables of the demand relationship. Therefore, a reduction of the income elasticity from “Friedman, 14., A Theorz of the Consmtion Function, National Bureau of Economic Research, Princeton Press, Princeton, 1957. 82 0.2% to 0.10 would suggest that some of the unobservable characteristics of the family were falsely attributed to family income in the earlier models. Alternatively, the permanent income hypothesis may explain the differences in the income elasticities with respect to food when alternative models were used to specify the demand relationships for food. Friedman has demonstrated that the elasticity of permanent consumption of food with respect to measured income separates into two elasticities: the elasticity of permanent food consumption on permanent income and the elasticity of permanent income with respect to measured income.5 He demonstrates the equivalence of the elasticity 5rhe following discussion is based on Section 2 of Chapter VIII of the Theory of the Consumption, o . cit., p. 206. Let Cf denote the observed food consumption of a family with a given measured income, and assume that the transitory component of food expenditures is uncorrelated with permanent or transitory components of income and it averages to zero. Thus, Cf can be regarded as the mean permanent consumption of food. Then the elasticity of Cf with respect to measured income is dc dc d 72‘ch = _3- e y_. 0 —E ...!P— ' z— ‘ {E 8 ”cm nypy 6! or dip fly is 0r * where y denotes the measured income and, the p subscript denotes the permanent component of income and denotes elasticity; But one of the primary hypotheses of the theory is: yr 8 cp/k, where cp is permanent consumption of all commodities and k is a constant. Then ncfy ‘3 Ucfyp ncpy. That is, the elasticity of permanent food consumption or measured income is equal to the elasticity of permanent food consumption with respect to permanent income times the elasticity of permanent consumption with respect to measured income. Thus, the difference between families is not solely represented by the permanent income components but also has some transitory components. of permanent income with respect to measured income and the elasticity of permanent consumption on measured income, given the assumptions of the theory of the permanent income hypothesis. Thus, it would seem that the income elasticity estimated from purely cross section data was a reasonable approximation of the elasticity of the permanent income. As more time series data were introduced the permanent income component of the measured income was reduced/ This was essentially the situation confronted in this study. When family constants were estimated the variation in the income variable among families was taken out , and only the variation over time remained. Thus, the estimated income coefficients of Model B capture only the variation of the income variables over time. Based on the permanent income lupothesis the family constants capture or remove most of the effects of pemnent income on the consumption of food. Furthermore, these estimates remove the effects of other differences among families. The income elasticities obtained in Model B would be mainly estimates of the effects of transitory income. An examination of the results of this study in the context discussed above seems to support the permanent income hypothesis. However, it was not possible to obtain estimates of the elasticities of the permanent and transitory effects of income on the consumption of food. It is usually argued that the income elasticities obtained from cross section data are better approximations of the permanent income elasticity than estimates obtained from time series data. For this study the average income elasticity for the eight cross sections studied is 0.32. When the mean effect over both time and families 84 has been accounted for the estimated elasticity is approximately 0.2h -- Model A. Separately, the variation among families was accounted for by estimating constants for each family; in this model the estimated income elasticity is approximately 0.10 -- Model B. These results suggest that the estimates obtained for'Model A contain a combination of the effects of permanent and transitory income, although less of the permanent income component as compared with the purely cross section analysis. However, when the deviation among families has been removed, the permanent income effect is less important, and the coefficient is almost entirely an estimate of the transitory effects. It is believed that most of the permanent income effect is included in the estimated family constant. This argument seems to be plausible and quite reasonable. Although it was thought that the effects of permanent inchme were captured in the family constants, the dramatic differences in the sum of squares of error for Models A and B suggests the presence of a number of other characteristics that have been measured'by the family constants. Therefore, it seems that neither of these hypotheses can{be completely rejected or accepted. Implications of the price elasticities The estimated price elasticity of the demand for food is reasonably constant within either the logarithmic or quadratic form; however, there is a large difference between functional forms. One of the primary reasons for suggesting that the logarithmic form.was more appropriate, was the greater plausibility of the price elasticities in the logarithmic compared to the quadratic models. Then, given the logarithmic as the more plausible, the price elasticities are compared 85 with estimated price elasticities obtained in other studies of the. demand for food (Table 26). In general, the estimates of price elasticity of the demand for food obtained in this study were much larger than those from other studies. From.annual U. S. data, Tobin and HaavelmouGrishick obtained food price elasticities of approximately -o.25 (Table 26). word and Jureen obtained a similar estimate from annual Swedish data, whereas Stone obtained an estimate of the food price elasticity in the United Kingdom of «0.35 from.snnual data (Table 26). Data for this study were based on a four week period rather than annual data, and*the food price elasticity estimates ranged from -0.70 to -0.59 for MOdel A'(l) and B'(l), respectively. Part of the difference in estimated elasticities can be attri- hated to the difference in the length of observation period. Elastici- ties estimated by Tobin and others relate year to year changes in food purchases to equivalent changes in prices, whereas the elasticities computed from this study were based on month to month(technically four weeks) changes. It is expected that the shorter period would result in a higher elasticity because consumers could purchase in a period of low prices and store for use in a period of higher prices, thus avoiding seasonal price changes. Also, consumers may be more willing to make _short run adjustments in consumption, but be reluctant to change their total annual consumption pattern. 0n the other hand, in the long run,say ten years, consumers may be quite willing to change their consumption habits,.thus resulting in a higher long run elasticity than year to year elasticity. Another part of the difference between the presented elas- ticities and those computed by Tobin and others can be attributed to the differences in occupations and geographic areas studied. 86 .mcn .a . «o .um..aoounfiovao=o .HeaH aw poem ”do no on~a> moans: Aaueu one we deduce. guess a so vegan one consequao sauna Auv .snuweaa .eoomyaae mo sauce uoxual , Hauou oAu we cusp aewuou alga so uses: one usualuuao sauna AHV .uu oINI.. manna .00” .n .mmm..mm..xo«na«uuoolao>aamo ehNM a“ cam-0| ONIOI ...“ QWQOUMA .hawiau onu nu auasnu>uvau no moans: emu »£ wouaoaouaou ea haaauu on» no onus onu .auouolauoa on» so uncuuoquuaou has agony“: .«na .a .uwo (NI..suaoaa .- - a~.- - an. o:uouan-umom «a. - - - ae.. a~v an. - - a. . an. age eaten - - nN.-. - «N. , cauaasauo-oaao>uam an. NH. nm.- - mm. museum ma. so. a~.- nu. Nu. «cases «a. ma. an.- an. an. Aav.m gone: «a. «a. 7 oh.- so. e~. AHV.< “use: poemsoa spoom , «a mo ooaum mo soaum seam haulih aloud“ unouuso .. hvsum an. e8 3.4 33...: :9: has: «any we moanmcoauoaou comedians any Ava: .uouvsum uaououuuv mo . woman: a you noon no saunasouuaaou unease wousaguao unu we acuaueaaoo < .oN manna 87 Areas for Future Study This study has been rather limdted in one sense; that is, only wage earners were considered, only single equation models were estimated, only one statistical model was considered in the combined analysis, and only a limited number of demographic variables were investigated. Areas in which more research would be fruitful can‘be separated into two categories: 1) the development of alternative models to combine cross section and time series data, and 2) inclusion of more demographic variables in the analysis of the demand for food. .Model development The following statistical models are presented as possibilities for future research in combining time series and cross section data.6 2“- “ Job/3" + 71, 2:. + a“ (4.2) it it t- 1 1t '2 y -11. +12: +T7éxa+6kwk+u (4.-) 1 1 t r1t= [1 +7; z:t+TT. 13+ 610%. +-v + v + u“ (4.4) Inhere: ! Yit is the observed value of the dependent variable for’indi- vidual i in time t. LL is a general constant. C}; varies over i but is constant over t. [3% varies over t but is constant over i. sit is an independent variable that varies over i and t. X’ is an independent variable that is constant over ifibut varies over I. 6The models were originally suggested by Hildreth, op. cit. 88 t Wk is an independent variable that is constant over t but varies over 1. 7;, 773’ and ékare the respective coefficients of 2“, It and W1. v“, v:1 and vt are random disturbances that vary over their indicated indices. vi is regarded as a drawing from a population with zero mean and constant variance and relat s the disturbance of the ifl individual for every time t. v is defined in a similar fashion as v1 except it relates to the tth time period for every 1. It is assumed that sit is independent of v* and and is independent with any change in its indices. uit is regarded as a drawing from an IT- ’.variate po ulation with zero mean and variance covariance matrix 0’ I. A more general model could be obtained by combining (ha?) and (11A). The least squares estimates of (h.2) and (MB) are maximum likelihood if the disturbance has a normal distribution. Although least squares estimates of (th) would be unbiased, they are not maxim likelihood because of the lack of independence of (v14. vt+u1t). The model used in this study was a combinination of (L2) and (MB). This study has dealt only with samples in which every individual was observed for every time period. A considerable amount of work is needed to investigate methods of estimating the coefficients when particular individuals do not appear in all time periods. It may be that this analysis is similar to analysis of variances with miss~ ing plots. Other considerations One result of this study suggests more use of demographic variables in demand analysis. Duncan had investigated some of the interrelationships of income, education, and occupation; a similar 89 investigation could be undertaken using consumer panel data.7 The number of meals eaten out could be considered also in a study dealing with demographic variables and there effect on food purchases. Classification of families by the stage in the life cycle (a classification depending on the age and distribution of the children and the marital status of the head of the household) should be investi- gated more completely. Perhaps a finer classification would be more rewarding than the classifications considered in this study. A study of the relationships between the estimated family constants and various demographic variables could be quite valuable. Family constants reflect particular characteristics of the family that were not reflected by the income and the size of the family. A study of the relationships among the estimated family constants and education, age of the children, and age of the head of the household could be helpful in trying to specify and quantify some of the behavior determining characteristics of a family. Data were available concerning some of the personality traits of the families that were in the panel in 1958. Spaeth has done a study using this information, but he has not considered the interrela- tionship of the these personality traits and the demand for food.8 Investigation of the relationships among personality traits and estimated family constants should prove fruitful. ¥ 7Duncan, 0. , "Occupational Components of Educational Differences in Income," Journal of American Statistical Association, Vol. 56, 1961, oC‘tober'. 88paeth, 1)., “Relationships Between Role and Self Perceptions, I"Emily Characteristics, Shopping Attitudes and Food Purchase Behavior," Unpublished 14.8. thesis, Michigan State University, 1960. 90 A few of the areas that the author feels are worthy of further investigation have been pointed out in this chapter. It is hoped that this study has provided some useful information and direction for other studies of the consumer demand for food. Further development of models similar to the one used in this study could prove quite helpful in obtaining more accurate estimates of demand relationships. The author feels that this study, limited as it was, provided some useful estimates of income elasticities and suggested some interest- - ing hypotheses with respect to the effect of estimating family constants on the estimated demand relationships. If future investigators can build on the results of this study, it will have served its purpose. BIBLIOGRAPHY IBartlett, M., "Some examples of Statistical Methods of Research in Agriculture and Applied Biology," Supplement - Journal of Royal Statistical Society, Vbl. 5, 1937. 0 IBurk, Marguerite, "Some Analyses of Income-Food Relationships," Journal of the American Statistical Association, Vol. 53, 1958. Cs Chow, G., "Tests of Equality between Sets of Coefficients in Two Linear Linear Regression," Econometrics, Vol. 28, 1960, pp. 591-606. E Christ, C., "Simultaneous Equations Estimation: Any Verdict Yet?", Econometrics, Vol. 28, 1960. ZDuncan, 0., "Occupational Components of Educational Differences in Income," Journal of American Statistical Association, Vol. 86, 1961. iEconomics Statistics Bureau, Handbook of Basic Economic Statistics, l95h-l958, washington, D. C. (e- Graybill, F., An Introduction to Linear Statistical Models, Vol. I, McGraw-Hill, New York, 1960. 15> Gustafson, R., "Testing Equality of Coefficients in Different Regressions," Michigan State University, East Lansing, 1961. (mimeo). , "Algobraic Relation.Between (X'X)'1 and M'1 in Least Squares Regressions which have a ' Constant Term," Michigan State University, East Lansing, 1960, (mimeo). Friedman, M. , A Theory of the Consgption Function, National Bureau of Economic Research, Princeton Press, Princeton, 1957. 9 Bald, A. , Statistical Tables and Formulations, John Wiley and Sens, New York, 1952. ' (Dilildreth, C., "Combining Cross Section Data and Time Series Data," Cowles Commission Discussion Paper: Statistics No. 3h7, May 1950. , "Simultaneous Equations: Any Verdict Yet?", Econometrics, J Vol . 28, 1960 . and Jarrett, F., A Statistical Study of Livestock Produc- tion and Marketing, Cowles Commission Monograph No. 15, John Wiley and Sons, New York, 1955. 91 <3 92 Hock, 1., "Estimation of Agricultural Resource Productivities Combin- ing Time Series and Cross Section Data," Unpublished Ph.D. thesis, University of Chicago, Department of Economics, 1957. Holte, F., "Introduction to discussion at a Conference on the Theoret— ical and Statistical Problems of Combining Time Series and Cross Section Data," Yale University, 1958, (mimeo). Haavelmo T., and Grishick, M., "Statistical Analysis of the Demand for Food: Examples of Simultaneous Equations of Structural Equations," Econometrics, Vol. 15, 19h7. Judge, 6., "Econometric Analysis of the Demand and Supply Relationships for Eggs," Storrs Agricultural Experiment Station, University of Connecticut, Bulletin No. 307, Storrs, 195%. Klein, L., "Single Equation Versus Equation System Methods of Estima- tion in Econometrics," Econometrics, Vol. 28, 1960. \ Kutzents, 0., "Measurement of Market Demand with Particular’Reference to Consumer Demand for Food," Journal of Farm.Economics, V01. 35; 1953. Lippitt, V}, Determinants of Consumer Demand, Harvard Economic Series, Cambridge, 1989. Liu, T., "Underidentification, Structural Estimation, and Forecasting," Econometrics, Vol. 28, 1960. Marschak, J., "Review of Schultz, Theory and Measurement of Demand," Economic Journal, Vol. h9, 1939. .Michigan Security Commission, Michigan Labor Market, Detroit, l95h-58. Nordin, J ., Judge, 6., and Wahby, 0., "Application of Econometric Procedures to the Demands for Agricultural Products," Iowa Experiment Station, Research Bulletin th, Ames, l95h. Praia, J. and Houthakker, R., The Analysis of Family Budgets, New'York, 1955. Polits Research, Inc., Life Studies of Consumer Expenditures, Vol. 1, 1958. Quackenbush, G. and Shaffer, J., "Collecting Food Purchases Data by Consumer Panel,".M1chigan Agricultural Experiment Station, Technical Bulletin 279, East Lansing, 1960. 93 Spaeth, D., "Relationships Between Role and Self Preceptions, Family Characteristics, Shopping Attitudes and Food Purchase Behavior," Unpublished M.S. thesis, Michigan State University, Department of Agricultural Economics, East Lansing, 1960. Stone, R., Meagurement and ConsumeEyExpenditures and Behavior in the United Kingdom, Cambridge, England, 1953. Theil, 3., Linear Aggregation of Economic Relations, North Holland Publishing Co., Amsterdam, 195k. Tobin, J., "Statistical Demand Function for Food in the United States," Royal Statistical Society Journal, Series A, 1950. United States Department of Agriculture, "Consumption of Food in the United States, 1909-59," Supplement for 1958 to Agriculture Handbook No. 62. , Food, The 1959 Yearbook of Agyiculture, Washington, D. C.. United States Department of Labor, Monthly Labor Review, Bureau of Labor Statistics, Washington, D. C., 195h-58. Wang, R., "Retail Food Price Index Based on M.S.U. Consumer Panel," Unpublished Ph.D. Thesis, Michigan State University, 1960. Wold, H. and Jureen, L., Demand Analysis, John Wiley and Sons, New York, 1953. APPENDIX A DERIVATIONS RELATING TO THE ESTIMATION OF PARAMETERS IN THE COMBINED MODEL Estimation of Parameters Write the combined model as: Y = Zla + Z28 + u (A-l) where: . it Y = a l 1 IT vector with elements I . Z1: a IT xI matrix of zeros and ones. Z2= a IT xK matrix whose elements are independent variables,Z:t. a = a l x I vector of unknown coefficients, a1. 6 = a 1 x K vector of unknown coefficients. u = a IT x 1 vector of distrubances. Let, 1 l O O 2:222 = l 1 z (A.2) 1 1 2 1 1 l O . l a. __ The model also can be written as: it i it it Y =01 + B' Z +u / i=l,...,I and t=l,...,T. (A.3) ‘where the it supercript denotes a drawing from individual i and time t of (A.1). i Let L(B,a ) denote the likelihood function, then use“) = z {(11152 (M) i t 94 95 then L s a1 - 412; nitsit (A.5) and “in“ (A.6) A solution for B can be obtained independently of estimates 5 of 01. The moments used to estimate B separately will be denoted by superscript 0. Then ’6 = mg? M; = ‘(mozhzbrl (.8) and mthy =§ gait ":lf :tZHtXYit _% gym). (A9) The estimates of the coefficients that are constant over t but vary over i are given by i 1 it it a = E'( Z Y - g 2: Z ) ’ 1:10,’,0Ie (A010) t t Estimation of the Standard Error of B Let (TE denote the standard error of the residuals,i.e., A o 2 M° - B'M O = W +51 (A.ll) u IT - (I+K -l A 0 then the covariance matrix of B isCfE.M . Proof: From (A.2) , let (2%) be written as : 96 wr- —- u— IZ I .1 Z1 1 Z1Z2 M11 M12 (Z'Z) = = (Ao-12) Z2Z1 Z222 I M21 MQQI _ _. l— .4 and —- i. 11 C12 -1 (2'2) = . (A.l3) C C ...21 22_ If B and o are estimated by [g]: (z'z)'1(z'r) (A.l’+) theanE 022 is the covariance matrix of the 6. Thus, showing the equiv- alence of C and Mo is sufficient. 22 22 From (A.12) and (A.l3) the following equations were obtained. 0 c .-. (‘) M11 12 + M12 22 0 (A.15) .(b) M21012 + M22622 3 I thus, 0 = (M - M '1 )‘1. (A.16) 22 22 21M11 Ml2 But '1 = (l/T) I (A.17) Mll ' [I] and the row J of M12 is composed of sum over t for each i on the Jth independent variable, i.e., 1t 1t f2 :2. . . t1 t2 £221" :42?“ . . . d1~1 de1 a”? We (A.18) Fvl be H d- df‘1 . mNH d. dTV1 KPH ' ' d- 97 ' in row h - Then M21MillM12 is a K x K matrix and the element and col umn k is 1 ,it it a; [(2% )( ZJZK ). (A.l9) t t Then C22 can be obtained by inverting a K x K matrix with elements in the hth_row and kth_column defined as: Z Z " — A020) 1 t Zh Z" T i t Zn 2; z" 0 which is equivalent to the elements of M22 ,i.e., o -1 C22 - 22 ° Estimation of the Standard Error of [011 The variance covariance matrix of the 01's can be written as cf: C11, where C is defined by (A.l3). Thus, from.(A.l2) and (A.13) 11 the following equations were obtained. (a) “11012 + Ml2C22 (b) Mllcll + Ml2c2l = O I (A.2l) then 0 ll 11 Mill + MlllMl2C22M21Mlll or (A.22) l 1 Cll ’ E'[I + T'Ml2022M2l]° The ith_diagonal element is '3: Zit" l 1 1 it it it o '1 t it -T- +6 2[§Zl 2122.2 0 . . gzk]Mzz E222 (A.23) 'it 22:. L." _ "2 . A1 which when multiplied bycfu glves the standard error of a . APPENDIX B STATISTICAL TESTS Tests of the homogeneity of variance of residuals among several regression equations Given the following set of P regressions in the same variables: = + Y X 61 u l l 1 Y2 = x2 [:32 + u2 (3.1) Y = X B + u P P P P where: Y = a T x1 vector of observed values of normal variables with i a meanx1 Bi and a variance covariance matrix of (Xi rip-452 ; i= l,2,...,p and X T1 = T. 1 X1 = a T1 x K matrix of values of Observed independent variables; i=1,...,p. Bi = a K x 1 vector of unknown coefficients; i=l,...,p. u1 = a drawing from a Ti-variate normal pOpulation with zero mean and covariance matrix (j'f I, where I is an Ti x Ti identity matrix and i= 1,...,p. the hypothesis of interes+ can he stated is: 2 2 2 2 Ho. 61 "’ 62 - a a a a 0 -6p — d g The unbiased estimate ofcff is given by A Y'Y - s' (X'Y) A of: 11 1 11. (13.2) T1 -K Bartlett has Obtained a test for homogeneity of variances when 98 99 1 there were an unequal number of Observations for each regression. The test statistic is chi square with P-l degrees of freedom. 2 . 7(2 I>ropriate. Let the number of coefficients for which equality is being tested be denoted by R; if Rwae less than half the number of equations, the test described in (3.18) is appropriate. In general, this test W1ll require a smaller inverse if KP . (3.22) R " 2(3-‘15 103 Tests of the hypothesis that the ai's are equal Let SSEu represent the sum of squares of error of the estimated relationship, (A.3) when.ai's are estimated. The sum of squares of error of the estimated relationship when only one constant term.is estimated is denoted by SSEr. The test statistic for testing that the 01's are equal is SSEr - SSEu x TI - (K+I) SSEu I-l N F[ I-l, TI-(K+I)] (3.23) where K denotes the number of coefficients excluding ai, i=l,...,I. APPENDIX C COMPUTATIONS OF THE ADULT MALE EQUIVALENT UNITS INDICES Data pertaining to the recommended weekly requirements of various fodd groups for three income levels by age-sex groups were obtained from the 1960 Yearbook of Agriculture.1 The relative importance of each food group was computed from consumer panel data.2 The food groups used to compute the index based on consumer panel data were: (1) dairy products, (2) meat, poultry and fish, (3) eggs, (h) dry'beans, (5) cereals, (6) fruits, (7) green vegetables, (8) potatoes, (9) other vegetables, (10) fats and oils, and (ll)sugars and candies. Let M 'be the measure of adult male equivalent units for the 13 13h age group and Jib sex group. Then, 11 M = W P 13 k3: k kiJ where ° the percentage that the ith age group and 332 sex group is of recommended requirements of an adult male (these requirements are in pounds of food group k required each week and are for a family with moderate U. 8. income. W H- Li. I Ak/T, k z 1, ea. 11 I: w m Imnd the l955~57 average family expenditure for all food based on the consumer panel. *3 In > W M the 1955-57 average expenditure for food group k per family'based on the consumer food panel. lunited States Department of Agriculture, Food: Yearbook of Agriculture, 1960, Washington, D. C., p. £287 2The data were taken from.Wang's Ph.D. thesis, op. cit.,Table 10. 104 105 Another index was computed which differs in that the weights (Wk) were computed from national data, W} denoting these weights. Let R denote the average total retail sales of food in the United States. Set 31. represent the average total retail sales for each of the follow- ing food groups; (1) dairy products, (2) meat, poultry, fish and eggs, (3) fruit and vegetables, (h) cereals and (5) fats and sugars.3 Thus, W; (Sr/R) for r : 1, see, 5 then 5 Wild ' Z w' P . r=l r rlJ where M'iJ is the index based on national data and is presented on line 3 Table C-l. The indices computed described above and the index suggested‘by Wold and Jureen are presented in the following table.h Table C-l. Relative weights for different sex-age groups in adult male equivalent units Under Girls Bo s Adult Adult Source one 1-3 h—6 -9 10-12 I -1 16-1 1 -l5 16-1§ women men Wold and Jureen .15 .15 .hO .75 .80 .90 .90 1.00 1.00 .90 .90 Panel and USDA .55 .62 .76 .88 1.0h 1.10 1.07 1.18 1.33 .80 1.00 USDA .53 .62 .75 .85 1.01 1.07 1.0h 1.15 1.30 .79 1.00 ‘ * 3These data were taken from, "Consumption of’rood in the United States, 1909-59 ," gupplementgfgr 1958 to Agriculture Handbook No. 62, IJBDA, Washington, D. C., p. 35. l"Wold and Jureen, o . cit., p. 233. APPENDIX D DEMONSTRATION OF THE EQUIVALENCE 0F DEFLATION OF A LOG RELATIoNsuTv AND FORCING THE SUliOF THE COEFFICIENTS OF AN EQUIVALENT LINEAR RELATIONSHIP TO BE EQUAL TO ONE Consider the following regression equation: log y = a + hllog x1 + bzlog x2 + . . . + bklog xk + u (D.1) If the variables of the relationship were deflated byxk the regression equation can be written as: I I log y/xk a + cllog xl/xk + . . .‘+ ck_llog xk,1/xk +iu (D.2) alternatively (D.2) can be written as: a B ' - log y log xk a + cllog x1 cllog xk,+ .1. . + ck_llog xk_1 ”ck-1103 xk +iu (D.3) or 103 y = a' + cllog x1 + c2103 x2 + . . . + ckrllog xk-l k-l ‘ + (l - Z c1)log xk + no (DA) 181 Equations (3.1) and (D.4) differ in only one respect, i.e., forcing the sum of the coefficients of (D.1) to be one would make the equations identical. Therefore, deflation of a linear regression equation has the iEffect of forcing the sum of coefficients in an equivalent logarithmic equation to be one. 106 APPENDIX E ESTIMATED RELATIONSHIPS FOR THE COMBINED QUADRATIC MODEL INCLUDING THE EXPECTATION VARIABLE AND THE FAMILY CONSTANTS FOR THE MODELS PRESENTED IN THE TEXT 107 108 Auaee.v Aeeo.v Aema.v Anoa.e Ae-.v Aoan.v on.eee.ee~.a sees. eeoe.- .oua., «an. aaa.- nee.- ea.aa emo . Auaeo.o Aaee.v Anna.v Aaea.e Aae«.v Assn.v a ea.een.ne~.a sane. eeee. nee. «as. «an.- oue.- ne.ea use Anon Hove: Aeeee.o Aena.v Anea.o Anna.v Aee~.o Aen~.o Ae.eao a ne.eon.aee.n anae. eeuo. «Heat ”nee. . «ana.- whee.-. _ an.ea eo.em nae Aeeoe.v AonH.v Aeaa.v Aua~.v Aeau.v Aena.v Ae.aao a oa.eee.eme.e ease. eeuo. ease. some. neue.- eene.- e«.ea o~.en sac Anya Heemm baa no em an «enema use 4 . a saneaue> sauna House on“. magnum onus plus» canluuob mmm N osoosH Ltouusuoemuu roomsoz voou. no oussvm hag-Uh useuouou usovseaon eaaeaoauuoeo Auvn use Aav< naeuoxuueansuue> sowusuooauo any unwosausauo sansuusb sloosa oso moans deuce sandstone any we eousluuou .«un canes 109 mafia. ammo. «mac. amuo. Home. came. nowo. «ONO. anuo. «duo. ammo. onNo. some. memo. ammo. aewo. osuc. ammo. Mano. anuo. oHNo. ceuo. «use. oeuo. hone. memo. tho. mmNc. nmwo. mnNo. oeuo. ouuo. ”mac. o¢~o. enuo. nmuo. nsno. omuo. onuo. nmao. nuuo. memo. ammo. wmuo. onuo. wmuo. neuo. oNNc. omac. anNo. onNo. nmuo. oaao. nauo. onuo. nnuo. nnuo. mafia. och. dado. nouo. mouo. ammo. nnNo. nouo. memo. memo. some. omuo. ammo. «duo. ammo. aouc. wnNo. memo. ammo. moNo. neuo. oeuo. mama. snuo. nnuo. ammo. «Nae. oawo. huuo. ammo. «sac. onNo. NmNo. Nome. «nae. memo. soda. name. name. memo. «duo. sauo. ease. @fiuo. oaNd. «Nae. Houc. 33.229223 38¢ seeing; 38¢38o33.22331333.1.53 aeeo. anue. emuo. oeue. naee. meme. ee~o. heme. enue. same. name. neue. knee. Aeao.vanao.vanae.vAaao.vanao.vAeao.vaeao.vaeao.vAeao.vaeao.vaeao.vAeao.vaeao.v memo. eeuo. meme. memo. «eeo. eeee. eaeo. meeo. enNo. mneo. memo. heme. eeNo. Ma NH . Hg OH m m s o n e m N a oaoosw AHV muaoaowuumoo “an. I & N¢.H~¢.o~o.m I m m eoH. moo. Hau.n «mm.u wo.mH a~.mm uqa ado. I w“ oe.nnu.oma.~ I umm nae. sea. eee.- emn.- ee.ea.ne.ee as flames seen: 0H0. I N“ n~.men.mmo.n I New Asa.v AeH.VAea.v Ase.vaea.o Ae.o~o one.. see. eee.- nae.- «n.ea.aa.me ‘ use . eao. I NM Ho.eea.a~e.e u use Ana.v Aa~.vaa~.v Aa~.vans.v Ae.mev eee.. eea. eem.- nee.- m~.ea aa.en .s ~maw< Ammo: +l 1'11 Aumv < was AmHv < maopozuuofinmaus> souuouoonxo onu wsaunaosaanousoaoawmooo oaoosa cauuhuwww was soouuasu mo o>auosuouao on» mugs .auou usoumsou oso sues Hopoa owuoupoov osu mo emanmso«uoaou pouoawumm .N-M m~am~ 110 nooo. Hmoo. QNHo. nNHo. mooo. onoo. MHHo. ooHo. aeoo. seeo. eeao. naoo. muse. shoe. mane. none. .eeoo. eeoe. noHo. mwoo. MHHo. nnoo. oooo. Hooo. nnoo. ooHo..omoo. Nnoo..uooo. HNHo. oooo. HoHo. NoHo. oHHo. HNHo. nooo. eooo. «moo. oeoo. anoo. ooHo. Hwoo. omoo. «woo. moHo. «moo. oooo. nmoo. mooo. owoo. «NHo. moHo. NHHo. «coo. mooo. whoo. oooo. choo. owoo. oooo. onoo. mnoo. Hmoo. oHHo. HNo. ouH. moHo. «NHo. sooo. «moo. enoo. ouHo. nooo. NNHo. moHo. MHHo. .neHo. mwoo. Hooo. wooo. onoo. onoo. ouoo. «woo. mooo. «ooo. nHHo. nwoo. whoo. nooo. anoo. «moo. «NHo. NoHo. oooo. oeoo. «woo. mooo. ooHo. mooo. Nnoo. oooo. mooo. Nooo. muoo. ooHo. NHH.u HmH. anew. u «a He.ean.nn~.a a new wom.a mmo.s nn.oH N uHU wage. I Nd Ho.om¢.on~.H I mmm mum.u omo.u mo.oH H 3d Hunom Hope: «an». n as oe.eea.ne~.a a new 8:338;39:29:33Gaoosaeifleo3:3Gaeifloifloifloo 2:; go go 5.23.: aeoo. eaoo. eaoe. shoe. ence. eeoe. neoo. Heeo. eoae. «one. eeoo. meoe. aeoo. eeo. eaa. 5: 5e; 98.229223 38.133 38.123 Afloinoifloo 28o . omo.n Hoo.u mn.wH a use ease. I «a om.oeo.ne~.a . use Aea.v Hana.v hea.v Ase.oaes.av NNoo. muoo. mmoo. smoo. omoo. unoo. mnoo. Hooo. eoHo. oooo. «ooo. eeoo. omoo. oeo. noHo.u mom.u mmn.a Hn.oH H qu asses Hoes: ma «a an on a . a a e a e m N a no oz ea «Assay use muooHunmooo Amnvm use HMHVM mHooozauoHnsHuo> soHusuoooxo sou msHpsHoanu moHnano> saoosH oauumuuHm use smouuHsu mo o>HussuouHs on» SUHs one oooesaumo museumsoo mHHsaw mHH :uHa Hopos oHumsnosv onu mo emanmsoHusHou nousawumm .m-m oases 111 Table E~4. Estimated family constants from Model 8(1) Family Consumer panel Detroit Family Consumer panel Detroit no. price - price no. rice rice 0044 61.24 I 59.63 1536 37.53 35.73! 0055 35.54 33.94 1544 43.22 41.69 0092 40.71 39.16 1566 45.39 43.82 0163 58.18 56.55 1599 56.54 54.80 0172 65.69 64.01 1653 58.22 56.29 0177 43.17 42.04 1656 72.23 70.54 0183 90.64 89.37 1662 53.23 51.58 0199 36.91 35.23 1673 49.81 47.98 0240 51.74 50.17 1674 65.81 64.24 0246 66.76 64.81 1719 40.26 38.39 0252 .63.59 61.95 1736 38.75 37.17 0253 67.45 65.64 1739 33.66 32.11 0261 44.16 42.24 1741 49.03' 47.49 0271 50.92 48.98 1766 45.5 -' 44.12 0274 47.93 46.39 1896 71.5 70.00 0280 48.42 46.34 1934 51.02 49.06. 0308 43.02 41.06 1949 77.05 75.11 0362 135.42 133.87 1951 43.08‘ 41.40 0397 65.64 63.92 1967 53.83 52.08 0456 74.86 73.14 2000 41.05 38.99 0473 48.30 46.55 2048 98.32 96.36 0513 83.98 82.10 2051 59.67 58.13 0545 54.01 52.24 2085 72.36 70.57 0554 63.10 61.12 2096 76.15 74.98 0586 54.27 52.45 2098 49.73 47.94 0611 54.24 52.54 2116 77.54 75.70 0616 116.19 114.23 2148 41.78 39.83 0634 45.85 44.19 2151 64.13 62.11 0723 81.11 79.54 2156 67.31 65.44 0763 34.51 32.79 2213 75.65 73.82 0771 52.90 51.16 2222 41.04 39.08 0775 64.97 63.50 2226 50.47 48.51 0836 68.79 67.55 2240 41.50 39.78 0840 67.77 66.01 2255 87.82 86.12 0846 45.08 43.19 3048 65.17 63.25 0847 41.10 39.54 3125 85.24 83.74 0865 45.51 43.80 3128 43.76 41.37 0970 77.24 75.70 3139 56.96 55.04 0990 38.48 36.66 3140 36.40 34.68 1015 83.09 81.42 3142 62.92 61.16 1059 51.72 50.14 3146 46.15 44.35 1061 87.41 85.81 4022 48.49 46.53 108] 107.75 106.60 5207 65.53 63.76 1138 53.08 51.15 .5241 60.13 58.43 1156 76.40 74.54 '5396 62.26 60.44 1207 37.76 36.26 5513 68.82 67.47‘ 1257 47.92 46.09 5570 64.35 62.83 1259 48.18 46.51 5624 51.37 49.59 1289 50.88 49.08 5847 49.10 47.40 1319 134.84 133.37 5924 59.42 57.40 1376 76.79 75.10 5972 30.32 28.46 1448 66.72 65.02 5994 73.13 71.32 1460 41.63 40.03 6001 78.47 76.54 1480 67.93 66.24 6165 60.44 58.89 1485 73.67 71.77 6266 95.41 93.93 1502 38.17 36.61 6843 40.61 39.34 ..1522 69.77 68.15 - Table E=5. Estimated family constants from.Mode1 8(52) Family Consumer panel Detroit Family Consumer panel Detroit no. price price _ p _ _ 7 l_l1,._ 1 1 0044 45.01 52.13 1536 21.63 28.83 0055 19.50 26.48 1544 27.26 34.30 0092 24.69 31.76 1566 30.08 37.16 0163 42.27 49.32 1599 40.68 47.98 0172 50.12 57.27 1653 42.46 49.82 0177 27.52 34.57 1656 57.58 64.75 0183 74.92 81.63 1662 37.73 44.85 0199 21.38 28.55 1673 33.98 41.11 0240 36.17 43.37 1674 50.76 57.91 0246 51.17 58.48 1719 24.17 31.41 0252 47.60 54.55 1736 22.80 29.93" . 0258 52.21 59.48 1739 17.64 24.68 0261 28.48 35.70 1741 33.13 40.13 0271 35.46 42.70 1766 30.86 38.00 0274 32.76 39.90 1896 56.33 63.41 0280 32.78 40.00 1934 35.59 42.86 0308 27.43 34.81 1949 61.45 68.82 0362 119.74 126.97 1951 27.42 34.79 0397 50.39 57.73 1967 37.97 45.20 0456 59.93 67.18 2000 25.21 32.57 0473 32.48 39.68 2048 82.86 89.89 0513 68.55 75.75 2051 44.82 52.01 0545 38.19 45.26 2085 57.19 64.49 0554 47.35 54.71 ' 2096 60.89 67.97 0586 38.35 45.53 2098 33.72 41.08 0611 38.69 45.89 2116 62.22 69.53 0616 100.74 107.95 2148 26.13 33.45 0634 30.48 37.49 2151 48.55 55.81 0723 65.75 72.84 2156 51.76 58.98 0763 18.91 26.03 2213 59.61 66.91 0771 37.18 44.39 2222 25.34 32.66 0775 49.98 57.06 2226 34.93 42.32 0836 54.56 61.90 2240 25.98 33.09 0840 52.17 59.47 2255 72.63 79.81 0846 29.11 36.26 3048 49.58 56.89 0847 25.20 32.22 3125 69.41 76.81 0865 29.66 36.97 3128 27.61 35.04 0970 61.96 69.10 3139 41.27 48.56 0990 22.47 29.68 3140 20.85 27.99 1015 67.53 74.72 3142 47.97 54.96 1059 36.35 43.48 3146 30.90 38.05 1061 72.29 79.57 4022 32.90 40.12 1087 92.71 99.85 5207 50.30 57.47 1138 37.64 44.82 5241 44.86 52.21 1156 60.77 67.92 5396 46.32 53.54 1207 22.43 29.61 5513 54.36 61.54 1257 32.02 39.20 5570 49.25 56.31 1259 32.62 39.87 5624 35.55 42.72 1289 35.62 . 42.92 5847 33.13 40.45 1319 119.66 127.06 5924 43.60 50.99 1376 61.68 68.90 5972 14.26 21.51 1448 51.30 58.30 5994 57.85 65.14 1460 26.28 33.45 6001 63.07 70.20 1480 52.29 59.46 6165 45.56 52.82 1485 58.39 65.59 6266 80.33 87.57 1502 22.19 29.22 -6843 25.40 32.25 .._1522 54.45 61.45 Table E-6. Estimated family constants from.Model B'(1) Family Consumer panel Detroit Family Consumer panel Detroit no. pprice price no. rice. rice 0044 '2293 .286 1536 .009 .022 0055 .049 .030 1544 .037” .055 0092 .108 .103 1566 .104 .075 0163 .236 .204 1599 .196 .165 0172 .261 .231 1653 3.197 .169 0177 .029 .025 1656 .274 .245 0183 .268 .239 1662 .167 .137 0199 .029 .014 1673 .136 .106 0240 .151 .122 1674 .260 .230 0246 .247 .217 1719 .030 .018 0252 .222 .193 1736 .056 .087 0258 .244 .215 1739. .191 .223 0261 .091. .061 1741 .134 .102 0271 .162 .133 11766 .142 .113 . .0274 .122 .092 1896 .301 .271 0280 .147 .118 1934 .170 .141 0308 .114 .085 1949 .283 .254 0362 .583 .553 1951 .078 .049 0397 .239 .210 1967 .176 .145 0456 .279 .250 2000 .099 .070 0473 .122 .092 2048 .399 .369 0513 .329 .300 2051 .220 .192 0545 .178 .147 2085 .263 .234 0554 .226 .196 2096 .255 .227 0586 .170 .139 2098 .144 .116 0611 .176 .147 2116 .289 .261 0616 .456 .427 2148 .103 .074 0634 .096 .066 2151 .205 .175 0723 .283 .255 2156 .253 .224 0763 .020 .051 2213 .338 .307 0771 .166 .135 2222 .102 .072 0775 .266 .237 2226 .163 .133 0836 .262 .233 2240 .069 .039 0840 .259 .229 .2255 .340 .311 0846 .089 .059 3048 .237 .207 0847 .005 .036 3125 .365 .335 0865 .100 .068 3128 .124 .094 0970 .336 .307 3139 .194 .165 0990 .002 .028 3140 .023 .003 1015 .365 .335 3142 .231 .201 1059 .160 .129 3146 .134 .105 1061 .346 .317 4022 .054 .083 1087 .347 .319 5207 .249 .220 1138 .165 .136 5241 .206 .176 1156 .306 .276 5396 .244 .213 1207 .019 .021 5513 .261 .233 1257 .123 .092 5570 .257 .228 1259 .133 .104 5624 .153 .123 1289 .157 .128 5847 .225 .198 1319 .458 .430 5924 .212 .182 1376 .279 .250 5972 .103 .134 1448 .261 .231 5994 .271 .242 1460 .073 .044 6001 .306 .277 1480 .279 .249 6165 .194 .165 1485 .280 .251 6266 .426 .396 1502 .093 .125 6843 .051 .021 ~.11522 .293 .264 114 Table E-7. Estimated family constants from Model B'(52) Family Consumer panel Detroit Family Consumer panel Detroit no. price price no. rice rice 0044 . 323 . 363 1536 .016 .054 0055 .077 . 086 1544 . 043 .084 0092 . 102 . 139 1566 . 110 . 148 0 163 . 243 . 283 1599 . 204 . 242 0172 . 268 . 306 1653 . 209 . 243 0177 .038 .078 1656 .281 .316 0 183 .280 . 312 1662 . 173 .211 0199 .008 .046 1673 . 144 . 182 0240 . 158 . 196 1674 . 267 . 304 0246 .256 . 291 1719 .040 .077 0252 . 235 . 267 1736 .049 .009 0258 . 253 . 288 1739 . 184 . 144 0261 . 100 . 137 1741 . 140 . 181 0271 . 171 .207 1766 . 149 . 185 02 74 . 128 . 166 1896 . 306 . 345 0280 . 157 . 192 1934 . 180 . 214 0308 . 124 . 159 1949 . 293 . 328 0362 .591 . 629 195 1 .087 . 123 0397 .247 . 283 1967 . 184 .222 0456 . 286 . 322 2000 . 110 . 145 0473 . 130 . 168 2048 .406 .443 . 0513 .338 . 374 205 1 .227 .264 0545 . 186 . 224 2085 . 272 . 307 0554 .236 . 271 2096 . 266 .299 0586 . 179 . 217 2098 . 155 . 191 0611 .184 .221 2116 .298 .333 0616 .464 .500 2148 . 113 . 148 0634 . 102 . 140 215 1 . 214 . 249 0723 .293 . 327 2156 .262 .298 0763 .014 .025 2213 . 347 .385 0771 .173 .211 2222 .112 .148 0775 .271 .310 2226 .172 .207 0836 . 268 . 304 2240 .076 . 114 0840 . 267 . 303 2255 . 348 . 384 0846 . 098 . 136 3048 . 246 . 281 0847 .001 .042 3125 .373 .411 0865 . 108 . 146 3128 . 138 . 172 0970 . 342 . 381 3139 .203 .239 0 990 .012 .049 3140 .073 . 114 10 15 .372 .410 3142 .238 .274 1059 . 166 . 205 3146 . 142 . 178 1061 . 353 . 389 4022 . 145 . 109 10 87 . 357 . 390 5207 .258 .294 1 1 38 . 174 .209 5241 .213 .250 1 156 .315 .351 5396 .253 .291 1207 .015 .054 5513 .268 .303 1257 .131 .169 5570 .261 .300 1259 . 141 . 178 5624 . 162 . 199 1289 .165 .201 5847 .253 .296 13 19 .467 .502 5924 .223 .257 13 76 .288 .322 5972 .093 .056 1448 . 267 . 305 5994 . 279 . 315 1460 .079 . 118 6001 .314 .351 1480 .286 .324 _ 6165 .201 .238 {-485 .288 . 324 6266 .431 .469 1 502 .087 .046 6843 .053 .095 —%‘ . 299 . 338 APPENDIX 1 ’ SPECIFIED SOURCE or m DATA AND REMARKS CONCERNING COMPUTING Source of the Data The specific source of the consumer panel data and a discussion of the primary computer program used in the analysis is presented in this Appendix. Primary data usually require a number of calculations before they are in a form amenable to analysis. The Michigan State University Consumer panel weekly masters were used to obtain a listing (881) which -’was used as the primary source of the expenditure and income data of the families. The listing contained four week sums of the number in the family, income of the family, and number of meals eaten at home. Also, the weekly average number of meals eaten in the house- hold and the ratio of the number of meals eaten at home to the number that would have been if every member of the household had eaten twenty-one meals at home each week were computed. The expenditure data were taken from the MSU Consumer Panel listing SS-5. From the SS-5 and SST listings, SST-5 was computed. This listing contains total food expenditures by families by four Wek periods for 1955-58. This listing contains the results of multiplying four week smus of food expenditures by the ratio of total I“Meals served at home to the number that would have been served if 9 ach member of the family had been there for twenty-one meals. Data pertaining to the age of the different members of the household and the occupation of the head of the household were taken 115 116 from the annual questionnaires. This information is summarized in the MSU Consumer Panel booklet SS-Q. Remarks Concerning Computing Once the sum of squares and cross products of the different variables were obtained, the primary part of the computing was done on the Michigan State University digital computer -- the MISTIC. The maJor computations were done by use of a linear equation solver and inverse program (M-l3). This is a fixed point routine which generates an idenity matrix, then operates on successive rows of the matrix to be inverted (A) so as to get zeros below the diagonal of A. After each operation the matrix is scaled so that the absolute value of the largest element is between one-half and one-fourth. If the matrix to be inverted were conditioned well, i.e., the elements were of approximately the same magnitude, than reasonable accuracy can be expected in a 50 x 50 matrix. This is the largest matrix that can be inverted at the present time on the MISTIC. However, it is possible to solve a set of 96 equations and 96 unknowns with this program. Again the accuracy depends on the conditioning of the matrix of coefficients. APPENDIX G DATA OF THE MEANS OVER BOTH FAMILIES AND TIME FOR INCOME, FAMILY SIZE AND FOOD EXPENDITURES AND A TABLE OF THE PRICE INDICES 117 118 Table G-l. A four week average over all families for each time period of the food expenditures and income for 1955-58, and the average size of the family for each time period Year Period Income Family Si to Food Expenditure 1955 1 716 . 81 2 . 75 71 .98 2 751.12 2.76 69.17 3 769.98 2.75 69.59 4 785.58 2.73 69.22 5 800.04 2.72 69.42 6 807.66 2.73 71.41 7 798.12 2.73 71.27 8 800.73 2.73 72.29 9 807.33 2.75 71.18 10 802.70 2.75 72.05 11 803.48 2.72 71.52 12 821.45 2.72 72.75 13 822.99 2.72 74.05 1956 1 777.02 2.77 68.12 2 779.40 2.77 71.57 3 777.26 2.77 71.49 4 756.46 2.77 73.53 5 759.85 2.76 71.00 6 765.67 2.72 71.52 7 755.16 2.73 71.53 8 786.02 2.73 72.07 9 767.13 2.73 71.38 10 789.96 2.73 72.74 11 793.37 2.73 69.73 12 797.92 2.73 75.31 13 846.92 2.73 75.78 .1957 1 811.39 2.75 70.49 2 803.33 2.75 72.25 3 806.42 2.76 70.55 5 775.65 2.76 72,74 5 766.93 2.78 71.36 6 770.89 2.77 69.81 7 795.74, 2.75 72.65 8 796.87 2.75 71.27 9 809.22 2.72 74.05 10 788.59 2.73 75.33 11 777.24 2.72 72.14 12 784.09 2.71 73.37 13 800.65 2.71 74.58 119 Table G-l continued Year Period Income Family Size Food Expenditure 1958 1 786 .06 2.75 74.60 2 779.79 2.75 73 .95 3 790 .69 2 . 75 75 .10 4 771.16 2.75 72.34 5 767.77 2.75 76.31 6 776.61 2.75 72.22 7 747.50 2.70 70.88 8 785.25 2.70 70.34 9 789 .49 2.70 74.04 10 778.80 2.71 71.63 11 734.70 2.72 70.35 12 808.42 2.72 71.63 13 859.00 2.72 71.19 *These variables are as defined in the final combined model. That is, income represents the current income plus one-third of the income received in the previous twelve weeks. The size of the family is given in male adult equivalent units, and is represented as the average over all families for each time period. 120 Table G-2. A four week average -- over each time period for each family -- of the food expenditures and income and the average size of each family over the four years* Family No. Income Family Size Food Expenditures 0044 673.92 .79 39.85 0055 246.69 .79 15.64 0092 255.63 .79 20.87 0163 341.13 1.00 42.50 0172 678.88 1.79 64.97 0177 144.79 .79 22.76 0183 998.44 8.10 158.45 0199 737.73 1.86 37.79 0240 741.85 1.79 51.64 0246 786.98 3.26 86.97 0252 1669.46 7.36 123.78 0258 1138.63 4.47 105.45 0261 649.23 2.50 53.05 0271 906.02 3.45 74.56 0274 1045.46 1.96 53.19 0280 736.44 3.52 71.30 0308 830.92 3.91 71.57 0362 501.60 1.79 132.56 0397 1120.21 3.90 96.86 0436 1372.20 3.83 107.88 0473 481.31 1.79 45.64 0513 854.54 3.12 103.05 0545 389.36 1.79 50.39 0554 734.29 3.62 87.13 0586 379.10 1.79 50.60 0611 759.15 2.12 59.06 0616 923.10 2.83 132.31 0634 869.13 1.79 47.08 0723 1143.61 6.06 134.85 0763 655.21 1.82 34.04 0771 561.50 1.79 51.04 0775 1167.54 1.79 69.03 0836 1855.71 3.47 101.76 0840 754.46 2.65 79.63 0846 372.23 1.79 41.43 0847 303.27 .83 22.42 0865 445.83 1.79 42.47 0970 932.29 1.79 79.00 0990 324.04 1.63 32.06 1015 712.40 1.79 82.75 1059 803.36 1.79 52.19 1061 1203.04 2.85 106.40 1087 1524.25 7.11 174.06 1138 879.29 3.32 75.01 1156 647.44 2.79 89.04 Table 0-2 continued 121 Family No. Income Family Size Food Expenditures 1207 911.85 1.79 39.44 1257 394.42 1.79 44.48 1259 806.67 2.56 59.37 1289 1100.25 3.39 75.77 1319 1284.50 4.47 173.52 1376 1286.79 5.00 121.65 1448 849.21 1.79 67.72 1460 918.77 1.77 43.31 1485 1014.36 3.44 98.31 1502 270.96 .79 18.51 1522 875.63 1.79 70.96 1536 444.81 1.79 34.54 1544 290.08 .79 23.71 1566 879.60 1.79 46.58 1599 460.63 1.79 53.66 1653 892.71 5.24 101.63 1656 1436.56 3.62 103.33 1662 783.67 1.96 55.89 1673 455.23 1.79 46.96 1674 1053.02 2.30 75.62 1719 265.15 1.79 35.51 1736 325.23 1.11 24.41 1739 245.87 .79 13.75 1741 317.08 .79 29.84 1766 1516.58 3.73 76.88 1896 1003.88 1.79 73.92 1934 969.52 4.60 88.72 1949 815.35 4.25 109.31 1951 682.85 2.61 53.55 1967 444.58 1.79 50.83 2000 621.13 3.78 65.98 2048 835.54 2.38 107.75 2051 1491.96 2.73 79.81 2085 1229.80 4.27 108.92 2096 1296.60 7.19 140.74 2098 495.61 2.60 58.03 2116 1104.27 4.14 111.49 2148 775.11 3.44 64.17 2151 810.50 3.76 90.63 2156 807.04 2.76 81.26 2213 299.21 1.79 71.20 2222 623.17 3.62 63.69 2226 838.36 3.95 79.42 2240 708.61 2.11 45.54 2255 1084.81 3.30 111.08 Table G-2 continued 122 Family No. Income Family Size Food Expenditures 3048 840.69 3.87 93.33 3125 514.17 1.79 82.57 3128 477.38 4.37 74.10 3139 670.15 3.04 73.01 3140 651.40 2.14 40.01 3142 1256.98 3.44 89.94 3146 1042.67 3.39 70.45 4022 797.75 3.56 72.16 5207 1102.21 3.20 88.13 5241 1081.21 2.76 76.80 5396 355.83 1.79 58.37 5513 1507.46 3.90 102.78 5570 1027.88 1.79 67.07 5624 499.94 1.79 48.97 5847 357.27 1.72 44.15 5924 664.60 4.39 91.72 5972 283.06 1.79 25.76 5994 1057.00 3.87 103.33 6001 906.06 3.15 98.38 6165 1361.48 3.21 84.67 6266 1078.63 1.79 98.69 6843 867.40 .79 26.79 *The variables are as defined in the combined model, i.e., income is current income plus one-third of the income received in the previous twelve weeks. The family size is denoted by the number of adult male equivalent units in the family. 123 Table G-3. The price indices and the expectation variable by periods, by years Year Period P1* P2* N1* N2* 0* 1955 1 97.5 96.6 96.0 96.3 4.6 2 96.7 97.1 96.7 96.5 4.3 3 98.1 96.8 96.1 96.7 3.5 4 97.4 97.1 96.3 96.4 2.7 5 98.1 98.2 96.3 96.2 2.1 6 98.6 97.8 96.4 96.7 2.6 7 99.1 98.3 96.3 96.6 3.4 8 97.6 97.7 96.9 96.9 3.6 9 98.2 97.2 96.2 96. 3.8 10 97.6 97.3 97.0 97.2 3.7 11 97.9 96.4 96.4 97.0 2.7 12 98.7 95.8 96.5 97.6 2.1 13 98. 95.3 96.6 97.6 4.3 1956 1 96.7 94.4 96.7 97.6 3.2 2 97.2 94.2 96.6 97.8 4.5 3 96.1 94.6 97.6 98.3 4.8 4 97.0 95.8 97.8 98.3 5.5 5 98.5 97.5 97.9 98.3 6.9 6 100.6 100.1 97.9 98.1 7.6 7 101.1 101.3 98.6 98.5 7.8 8 100.1 102.6 99.8 98. 8.0 9 98.6 99.1 99.7 99.5 8.2 10 98.2 99.2 100.0 99.6 8.0 11 101.7 98.5 99.0 100.3 6.6 12 100.7 99.0 100.0 100.7 4.1 13 100.1 97.7 99.6 100.8 3.9 1957 1 97.5 97.8 101.2 101.2 4.5 2 99.2 99.0 101.2 101.2 4.7 3 99.1 98.1 101.2 101.6 4.7 4 102.3 99.0 100.3 101.7 5.0 5 100.5 99.7 101.6 101.9 5.2 6 101.1 101.7 102.1 101.8 6.1 7 102.3 102.3 102.0 102.0 6.5 8 102.4 102.8 102.3 102.1 7.5 9 101.4 102.2 102.5 102.2 6.8 10 99.5 101.5 103.2 102.3 8.7 11 99.8 101.0 102.9 102.4 6.7 12 101.0 99.7 103.4 103.9 5.5 13 102.0 99.9 102.7 103.6 6.1 Table G-3 continued 124 Year P eriod P1* P2* N1... N2* C" 1958 1 101.8 102.1 103.2 102.7 8.5 2 102.0 102.5 103.1 102.9 9.9 3 102.4 104.6 103.5 102.6 11.5 4 104.4 105.5 103.0 102.6 13.2 5 103.9 106.2 103.1 102.1 13.7 6 105.1 105.6 102.4 102.2 13.6 7 104.5 105.4 102.8 102.5 13.6 8 102.4 105.2 103.7 102.5 14.4 9 101.1 102.9 103.5 102.8 17.4 10 100.1 108.9 104.0 100.5 12.9 11 100.4 108.7 103.5 100.0 10.0 12 101.9 101.9 102.9 102.9 9.1 13 101.1 100.9 103.4 102.6 8.1 *Pl denotes the consumer panel food price index; 1955-58 ' 100. P2 denotes the Detroit BLS price index of food; 1955-58 = 100. N1 denotes the nonfood price index based on the BLS Consumer Price Index and P1; 1955-58 = 100. N2 denotes the nonfood price index based on the BLS Consumer Price Index and P2; 1955-58 2 100. C denotes the percent of the workforce unemployed-~expectation variable. ROOM USE ONLY MICHIGAN STATE UNIV. LIBRARIES VIHIINNI“WWW”IHI‘NIWIIHHVllWlWlVHIWI 31293011036062