m -s WWW“, «-7 ".mm—uuu- ._.. .3...~« -: “guy-x ‘ . . n. ‘ - 2—w- w—w—“wm—‘— ““m‘ sEHb L131” THE DEMAND FOR MAJOR Hou VAN ‘ c NOMETRI ECO. ALY; 3'57 AN APPLIANCES NIVERSITY “:MI'CHIGANQSTATQEU :Thééis‘tor‘the>0e§réefibf£h’ . .2 infra. .3. 5.. ms; 4 ,. ”v _. a, .3 .,.. fl .4 4 N... . ... y . . w . ... .. ind. . A/ x .4 . .hana.." .a . r/fimeanv. a: 4, a? ”5,. 5, . ”r wwwifiwdémw fl ,_..,.,._,_4:M,y_§ £395... .A r . anew} é'Jmt . Mr... 9/3.,“ .321? ..,M.q.¢;.vv r. I; 14 5 5.4 1' t l 3.“...wa , f ., a, (5&1?) I IV - 1 g .Y/‘JIV'Jgff.’ . J t; e. 4 $.11}; . ; #4.. Z: 1515.... . .I. 5.7:: 7 rrrni}.. ..#-.. 5..., x! )4}... if? .I' r»: , . .. (it! 1. 2. its: :vlr..p»rpl;rz L 9.3.4711, . .x .2... rial? ... ,. fozvrllv31 ‘92:: I} 1.. 4.. #2,“? This is to certify that the thesis entitled The Demand for Major Household Appliances: An Econometric Analysis presented by William Richard Cron has been accepted towards fulfillment of the requirements for Ph.D. degreein Economics MM # v / Major professor Date December 2|I I972 0—7639 ‘— amomc av " HUAE & 80H? Bllllll BlllllElll Illl‘.. Liam“! amnzns snmamyl. mums! ABSTRACT THE DEMAND FOR MAJOR HOUSEHOLD APPLIANCES: AN ECONOMETRIC ANALYSIS By William R. Cron This study attempts to examine alternative approaches to specifying a demand function for several major household appliances. 0f the existing approaches, a stock adjustment and a habit persistence model are examined in detail. Demand analysis in the area of durable goods seems to have developed independently of the great amount of work performed on the estimation of demand functions based upon maxi— mization of a utility function subject to a budget constraint. In this paper an analysis of the existing approaches to the study of durable demand are examined to determine if they have a tie to the classical utility maximization framework. Two alternative approaches to the development of demand function are then attempted. The first approach accepts the constant elasticity of demand model as a good first order approximation to the demand func— tions in question and proceeds to impose demand restrictions (notably symmetry) derived from the utility maximization framework as a set of Lagrangean constraints. The second approach specifies a particular utility function and proceeds to derive the demand function by maxi— mizing this function subject to an assumed budget constraint. The stochastic specifications for each of the models are examined and estimation methods which are consistent with the various William R. Cron stochastic formulations are presented. Each model is then estimated using aggregate U.S. data for the period 1950 to 1970. An approach to expanding the data base upon which the demand functions are esti— mated is presented in the form of a method for combining time series and cross—sectional observations. The pooling method, which is applied to the existing "ad hoc” models, allows for observations which are heteroskedastic and cross—sectionally correlated, as well as time— wise autoregressive. All of the models performed very well if judged by their resultant R 's. In addition, the use of a two-stage generalized linear regres— sion method for pooling time series and cross—section data caused sig~ nificant improvements in the efficiency of the regression estimates. An examination of the linear expenditure system of demand equations revealed that this functional form is compatible with the habit per— sistence type of model, provided the income variable is interpreted as a type of "supernumerary income." The constant elasticity of de— mand model is shown to offer a plausible alternative to specification of a functional form for the demand equations, which the imposition of demand restrictions as side conditions is sh0wn to be relevant and to offer an improvement in the efficiency of the estimates obtained. THE DEMAND FOR MAJOR HOUSEHOLD APPLIANCES: AN ECONOMETRIC ANALYSIS By William Richard Cron A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1972 To: My wife and family ii ACKNOWLEDGEMENTS I am forever in the debt of Dr. Jan Kmenta for the invaluable assistance he has rendered to me during my course of study at Michigan State University. He suggested the topic of this dissertation to me, guided me in each stage of its prepara— tion, and, when the future appeared bleak, he never ceased to be a beacon of encouragement. I would like to thank Dr. Anthony C. K00 and Dr. Robert L. Guftason for agreeing to serve on my committee, and Dr. James Ramsey for his helpful comments on an earlier draft. Acknowledgement must be made to Wayne State University for providing the necessary hours of computer time on the University's IBM 360 computer. A special thanks is also extended to Mr. David Johnstone for his invaluable programming assistance and his incisive comments which caused me to rethink many areas. TITIE DEDICATION ACKNOWIEDGEMENTS TABLE OF CONTENTS TABLE OF CONTENTS LIST OF TABLES LIST OF APPENDICES CHAPTER 1: CHAPTER 2: CHAPTER 3: INTRODUCTION A REVIEW OF DURABLE DEMAND MODELS l\)l\)f\)i\)[\)l\)l\)l\) OOQOm-P’me R003 and von Szeliski Farrell Chow Stone and Rowe - Nerlove Houthakker and Taylor Suits Household Appliance Studies Summary DURABIE DEMAND MODELS wont» .l .2 .3 3.h 3.5 Introduction Review of Consumer Demand Theory "Ad Hoc" Models 3.3.1 A Stock Adjustment Model 3.3.2 Modifications of the Standard Model 3.3.3 Habit Persistence: An Alternative Model An Intertemporal Utility Model 3.h.l Constant Elasticity of Demand System 3.A.2 Linear Expenditure System Summary iv l2 l5 l7 17 22 CHAPTER 1+: CHAPTER 5: BIBLIOGRAPHY APPENDIX A: APPENDIX B: STOCHASTIC SPECIFICATION AND ESTIMATION METHODS h.l Introduction 11.2 "Ad Hoc" Models h.2.l Standard Specifications h.2.2 Alternative Stochastic Specifications h.2.3 A Procedure for Combining Cross-Section and Time Series Observations h.3 Constant Elasticity of Demand Models h.h Linear EXpenditure System EMPIRICAL TESTS OF THE MODEIS 5.1 Introduction 5.2 The Data Base 5.3 "Ad Hoc" Model Results 5.3.1 Two Stage Generalized Linear Regression Method Results Constant Elasticity of Demand Model Results Summary and Conclusion \J'lU1 \nF‘ DATA USED IN THE EMPIRICAL TESTS STATES INCIUDED IN REGIONS USED FOR COMBINING TIME SERIES AND CROSS-SECTION DATA 68 68 7o 71 77 85 91 91+ 91+ 95 105 111 116 120 127 1A3 TABLE 5-1 5-2 5-3 5-h 5-5 5-7 5-8 A-l LIST OF TABLES Results of Ols Estimates of Stone-Rowe—Nerlove Model OLS Estimates of Houthakker and Taylor Model OLS Estimates of Stone-RoweANerlove Model Using Pooled Data by Regions Two Stage Generalized Linear Regression Estimates of Stone-RoweANerlove Model Using Pooled Data by Regions OLS Estimates of Houthakker and Taylor Model Using Pooled Data by Region Two Stage Generalized Linear Regression Estimates of Houthakker and Taylor Model Using Pooled Data by Region Unrestricted OLS Estimates of CED Model Restricted Aitken Estimates of CED Model Retail Value of Manufacturers' Sales (Aggregate U.S. Data) 1950-1970 Independent Variables Used in Regressions (Aggre— gate U.S. Data) 1950-1970 Estimated Dollar Value of Retail Sales by Region 1959-1969 Independent Variables Used in Regressions by Region 1959-1969 vi PAGE 100 103 106 107 108 109 112 113 128 129 130 136 LIST OF APPENDICES PAGE APPENDIX A: Data Used in the Empirical Tests 127 APPENDIX B: States Included in Regions Used for Combining Time Series and Cross-Section Data 1113 Chapter 1 Introduction Expenditures on durable commodities represent a significant propor— tion of annual national expenditure. In light of this significance, it would be generally felt that the literature on the demand for these prod— ucts would be extensively documented. The expectation would also be that the amount of research performed would be distributed over the various types of durables roughly in relation to the importance the item assumes in the consumer budget. A review of the literature does not confirm our expectations. Instead, we find a relatively small number of studies, with those that do exist being concentrated on either aggregate durable expendi- ture, or, if applied to a specific commodity, being applied primarily to automobiles. The durable commodity presents a unique problem. Durables by their nature provide services over a period of time, thus implying that the im— pact of the existing stock on the current purchase decision must be con— sidered. The classical theory of demand contained as arguments consumption of commodies, which for a non—durable good coincided with its purchase. However, for a durable commodity consumption does not occur simultaneously with purchase, and its availability, or the stock of a durable good, affects the amount consumed in a given period. Since durables were not conformable to the standard utility framework, the recourse was to neglect the idea of utility maximization in favor of models based on generalizations as to how an individual might be expected to formulate implicitly his purchase plans. 2 The lack of readily available data contributed to making studies of household appliance demand scarce. Available figures on retail sales of durables other than automobiles are subject to a considerable amount of measurement error and are not extensively published. Data on the stocks of these durables are practically non—existent. In contrast, the auto- mobile industry has large amounts of reliable data available to the re— searcher. Automobile companies have developed their reporting practices to the extent that franchised dealers report sales summaries by ten-day periods, with some daily information developed by means of warranty card counts. This study is concerned with the demand for a few of the major house- hold appliances. Specifically, three appliance groupings representing the functional purposes of cold storage, food heating, and laundry service were selected for analysis. To place the analysis in historical perspective, some of the past contributions to our knowledge of durable demand are reviewed in Chapter 2. Chapter 3 then proceeds to develop some of the existing models and addi— tional models based on a utility maximization concept for adaptation to household appliance demand. The objective is to develop models which are plausible in terms of observable behavior and to discover if the classical utility maximization framework provides a plausible background against which new models may be developed or existing models connected. In Chapter 4 alternative stochastic formulations of the models presented in the pre- vious chapter are examined and estimation methods consistent with these specifications are developed. Additionally, a procedure for increasing the amount of available data by an appropriate combination or regional data for a number of years is developed. 3 Before we can proceed to turn our attention to the estimation of demand functions, we must be certain the relationship being estimated is truly a demand function.1 By virtue of the Cournot-Walras theory of eco— nomic equilibrium there is a tendency for the market totals of demand to quual the market totals of supply, so that the observed quantity variables might determine a sort of hybrid between the true demand and supply func— tions. A general approach to the problem is to specify both a supply and demand model, and to estimate both functions simultaneously. An alterna- tive is to specify assumptions for the functions, which, if correct, will allow a demand function to be estimated by single equation methods. In this paper we will proceed with the later approach, and assume the supply function is of the recursive type. A simplified example of a supply and demand model, where the supply function is assumed to be of the recursive type, is given by dt = D(pt) st = S(pt_1) Pt = Pt-l + b(dt—l " St...]_) a S , and s where dt’ dt—l’ t t—l refer to demand and supply at periods t and t-l, and pt and pt—l refer to the prices at periods t and t—l. This model explains the price during period t as an adjustment in the price from period t-l. If b is specified to be positive, the price will rise with excess de- mand and fall with excess supply, as would be expected by the classical 1An elementary discussion of the problem can be found in Lawrence R. Klein, An Introduction to Econometrics, New York: Prentice—Hall (1962), pp. 10-19. A more complete discussion of the problem is contained in H. Wold and L. Jureen, Demand Analysis, New York: Richard D. Irwin (1953). theory of economic markets. Starting from some initial values of d, s, and p, the above model allows a series for each of the three variables to be recursively calculated. Single equation estimates of the demand equa— tion in this system will produce estimates which can be identified as per- taining to the demand function, and not a combined market equilibrium function. The household appliance industry is felt to be typical of a market where the quantity supplied is not instantaneously adjusted to changing market conditions, but instead one in which the quantities are adjusted as the result of a deliberate policy consideration with some lag in im- plementation. Given this condition the recursive system appears to be a reasonable working hypothesis for this industry. Chapter 2 A Review of Durable Demand Models Our intent in this chapter is to present various models which could be adapted to household appliance demand. Accordingly, the models selected have often been developed in connection with durables other than appliances. Studies that attempted to measure the impact of various explanatory variables through existing models are not dis— cussed. Following the review of the models, a brief section to illustrate the application of demand models to household appliances is included. The studies specifically treated are those of R005 and von Szeliski, Farrell, Chow, Stone, Rowe and Nerlove, Houthakker and Taylor, and Suits. In the section dealing with household appliance studies the contributions of Burnstein, Miller and Wu are presented. The symbols used in the discussion will be those of the original authors. 2.1 R008 and von Szeliski One of the earliest attempts to develop a consistent theoretical framework for household durables was Supplied by C. F. R005 and Victor von Szeliskil in a study of U.S. automobile demand. In this study demand, taken in the sense of total retail passenger sales of automobiles, is considered to consist of two basic components, new owner sales and replace— ment sales. The factors determining the individual components are then explored. lRoos, C. F. and von Szeliski, Victor, "Factors Governing Changes in Domestic Automobile Demand", in The Dynamics of Automobile Demand, General Motors Corporation, New York, N.Y., (1939). 5 6 New owner sales are assumed to be a product of the number of potential new owners and the probability that an individual selected at random from the potential new owner group will purchase a car. The potential new owners are assumed to be given by the difference between an upper limit of the car maintaining ability of the country and the number of cars in operation. This upper limit is a product of f(t), the number of families at time t, and m(t), which is a measure of the maximum ownership level per family. At this point the authors introduce an innovation by assuming the maximum ownership level m(t) is continuously changing in response to other factors. Functionally this was given by m(t) = B(t) p“5 I8 d6, where p is the price of cars, I is per capita income, d is average car life, B(t) represents all neglected factors, and 8 and 6 are the elasticities of m(t) with respect to I and d/p respectively. The probability a potential owner will purchase a new car is assumed to vary as the product becomes known and wanted and facilities for use built up. It was felt that this influence could be measured by the cars in operation (C). Other factors affecting the probability of purchase are thought to be income (1), price of new cars (p), and the trade in ratio of used car allowance to new car price (T). This is given by A0 = A3(t)p_u TB IY c, where A0 is the probability of purchase, p, I, T and C are as defined above, A3(t) represents neglected factors,2 and a, B, and Y are parameters to be determined. 2Although A3(t) and B(t) are specified to include neglected factors, they are not necessarily the same since they measure the effect of these factors on different variables. 7 The complete specification for new owner sales is given by sN = A0 [f(t) . m(t) - c] = A3(t)p‘°‘ TB IY c [f(t)°B(t)'(p/d)-6 IE — c] , where all of the terms are as previously defined. Replacement sales are assumed to be proportional to the theoretical scrapping based on survival tables for cars in operation. Theoretical scrapping was recognized as only one of the many factors which contribute to replacement sales. In keeping with an earlier paper by de Wolff3 the factor of proportionality is assumed to be a function of the price of a new unit, the trade—in price ratio, and per capita income. Using A4(t) to measure neglected factors, retaining the other symbols from the new owner sales, and using X for theoretical scrapping produces an equation for replacement sales as follows: SR = A4(t) p‘o‘ TB I'Y x. Combining new car sales and replacement sales, and adding an assumption that new car sales should be reduced by some fraction of replacement sales to allow for interdependent effects produces an equation for estimation of retail sales, 3 = A3(t) p'O‘TBIY :V‘C(f-B-(p/d)_(51b — C) + A6 x + G], where G is a measure of the interdependent effect of replacement sales, A6 has now replaced A4(t) because the exponents for p, T, and I have been combined in the equation for retail sales, and all other factors are as previously defined. However, the equation fitted by the authors 3P. de Wolff, "The Demand for Passenger Cars in the United States", Econometrica, 6, (1938), pp. 113—129. represents a slight modification of this form. The result obtained 3 =jt1'20 pt"65 [.0254 Ct(M3 — ct) + .65 x2 ] , t t where jt represents supernumerary money incomes, M3 is the maximum ownership level at time t, th is a measure of replZcement pressure6 and other variables are as previously defined. The price and income elasticites are -.65 and +1.20 respectively. The remainder of the study deals with the implications of other variables that might be significant. In addition, the authors experi- mented with different concepts of the variables included and fitted the equations to different time periods. 2.2 Farrell A study of the demand for automobiles in the United States which attempted to incorporate a utility maximization concept was made by M. J. Farrell.7 The automobile market is viewed as being made up of a series of interrelated markets corresponding to model age groups. For each market a demand function of the form (1) Xi = fi (y.P,t) is specified, where Xi represents the demand for ownership in the i'th 4Roos and von Szeliski, op. cit., equation 16, p. 60. 5The concept of income used is that of supernumerary money income which is defined as the excess of disposable income over subsistence expenditures. The symbol (jt) has been used to represent this variable as opposed to (I) which was utilized previously in the derivation of the final form. 6The variable X2 is a measure of replacement preSSure at time t obtained by applying 5 shifting mortality table to the age distribution of passenger cars. 7M. J. Farrell,"The Demand for Motor Cars in the United States,' Royal Statistical Society Journal, 87, (Part II, 1954), pp. 171—200. 9 group, y represents income, P represents a vector of prices of the current and prior model year groups, and t is assumed to represent individual taste for various age group cars. An assumption that the supply function for each group, other than new cars, is perfectly inelastic with reSpect to prices and income allowed Farrell to express the demand for new cars as a function of its own price (P), income (y), tastes (t), and the known quantities (X2. . ., Xn). Several additional assumptions were made in order to specify the form of the demand function. Using U to represent the highest price ik the k'th individual would pay for an i year old car if no other car were available to him, the assumptions are given by: (1) No individual owns more than one car; (2) For all k, U (i = l,....,n—l), or > ik Ui+l,k that an individual will place a higher value on a newer car than an older one; (3) For all i, k, Ui is a function of nk, where nk = k ukyk with yk being the k'th individual's income and “k is a constant repre— senting his tastes; (4) BUi/3n>0 (i = l,....,n); (S) B(Ui - Uj)/8n>0 (i = 1, ...... ,n—l; j = i+l, ...... ,n); (6) The distribution of individual incomes is uniquely determined by the national or community income (Y), and may be represented by the frequency function F(Y,y); (7) u is a random variable, distributed independently of v, with frequency function f(u) and may be interpreted as representing the taste variable, including many factors, such as family size or geographical location, which might not normally be implied by the word "tastes". Based on the above assumptions Farrell derived an aggregate demand 10 function of the form ” (l/Y)G. (P. ) (2) X1 = [F(Y,y) dy f 1‘1 1‘1f(u) du (i = l,.....,n), 0 (l/Y)Gi(P1) where X1 is the demand for 1 year old automobiles, F(Y,y) and f(u) are the income distribution and taste functions mentioned above and Gi(Pi) and G. 1—1(Pi—l) are functions defining the upper and lower limit of incomes respectively between which an i year old car would be purchased. For estimation it was more convenient to express the above function as follows: (3) xi = [ F(Y,y) dy f f(u) an, 0 (l/y)Gi(Pi) ti where xi = X,. 2321 3 To apply equation (3) the form of the functions F(Y,y), f(u) and Gi(Pi) must be specified. As a general method cross—section data were used to estimate F(Y,y) and f(u). These estimates were then regarded as known and a time series regression was used to determine Gi(Pi)' The second integral in equation (3) may be represented by Qi and r-wi q. = Q. 1 j=l j distributed which gives a function for qi as (i = 1,....,n). It is assumed that log(u) is normally (4) q. = N dx (i = 1, ...... ,n), 1 I, where 2 N = (um ) e‘(" /2) and (5) A. = (1/0) log(Gi) - (l/G) log (Y) (i = 1, ..... ,n). 11 The proportion of families owning a car not more than 1 years old (qi), and average group income (y) was calculated from budget studies. Values of Xi corresponding to the qi obtained from the budget data were obtained by reference to a table of the normal integral. OLS method was then applied to obtain estimates of Gi (the highest price at which an i year old car would be purchased) and 1/0 for each age group. In performing the estimation, equation (5) was fitted to observations which were weighted by an estimate of their sampling variance. For new cars this procedure produced an estimate of Gi of 7,530 and of U of .387. Estimates were also obtained for older age groups. To obtain estimates of F(Y,y) information as to income distribution in 1941 and the aSSumption8 AF(AY, Ay) = F(Y,y) were used. Utilizing this assumption allowed equation (3) to be written as oo (6) xi = I FO(Y,y)dy I N du 0 (1/0)(log Gi(Pi) — log y — log Yt), where Yt is the ratio of national income in year t to that in 1941, FO(Y) is the distribution of income given by the 1941 data and the other symbols are as previously defined. It is then possible to calculate a value of kit corresponding to each observed value of xit in the time 8Farrell points out that this condition is satisfied if the incomes of all families vary proportionately. 12 series, where kt has been substituted for Yt such that equation (6) is satisfied. Farrell ends up with a demand equation m (1/0) log (H _ /Y) (7) X = I F (y)dy f i 1 N du, 0 ° (l/o) log (Hi/y) where r G i Hi= [b Y (Pit - Pi+l,t ‘Pot (3+C1t)) with the G1 and 0 having been obtained from cross section data as i explained, and Fo(y) from the 1941 income distribution study. The a1, bi’ and c are parameters that have been estimated from calculated 1 values for kit and an assumed linear relationship for G1 (Pi)’ The following values of a1, bi’ and c1 for new cars were estimated: ai = .7257; b1 = 1.02; c1 = —.045. An additional facet explored in the study was the complication introduced to the cross section data because of difference in habits between urban, rural non—farm and farm families. 2.3 9223 Gregory C. Chow9 has provided a study of the U.S. automobile market in which he attempts to develop demand functions for new automobile ownership consistent with the utility maximization framework. Using data for the years 1921 to 1953, he presents two models to explain automobile ownership. The first model, referred to as the "existing theory", explains the per capita stock of automobiles in the United States as a 9Gregory C. Chow, Demand for Automobiles in the United States, Amsterdam : North Holland Publishing Company, 1957. 13 function of its relative price and income. The specific function is of the form X = JpaIbu, where X is the per capita stock of automobiles, p is the relative price per unit of the durable, I is the income variable in per capita terms, u is the random disturbance term, a, b, and J are parameters to be estimated. For estimation the equation is transformed to log linear terms and the logarithm of price regressed against the other variables. Using expected per capita income for the income variable and including a trend term produced the best fit, ln(p) = —5.854 - 1.048 1nX + 2.007 lnIe + .00238t R2 = .949. (.065) (.256) (.00433) The second model of automobile ownership, referred to as the "alternative theory", explains automobile ownership by introducing a hypothesis on the desired asset structure of the individual into the existing theory. Specifically, it is assumed the individual desires to maintain a constant ratio between his durables and other assets, which includes his stock of money and securities. Following this approach, Chow proposes a function of the form X = JpaIb Mgu, where X, p, and I, are as previously defined and Ma is a monetary variable.10 The estimated equation is found to be ln(p) = -5.420 — .9751n(X) + 1.701 ln(I ) + .237 ln(Ma) (.087) 2 (.356) (.184) R = .951. 10This variable is defined as the per capita stock of currency, demand deposits and time deposits in commercial banks, in 1937 dollars held by all sectors of the American economy, except the banking sector. 14 In testing both the existing theory and the alternative theory, Chow experimented with two income variables, per capita disposable income and a longer run expected per capita income. In the context of the existing theory the inclusion of the expected income term outperformed the disposable income concept. Including expected income as a variable in a demand equation, which excluded the trend variable, improved the R2 from .898 with the disposable income concept to .948, while the coefficient of the income term was significant for both concepts. Employing the expected income variable in the alternative theory improved the R2 from .920 with the disposable income concept to .951, while causing the monetary variable to become insignificantly different from zero. Based on these reSults, the alternative theory is shown to exhibit a slightly larger R2 than the existing theory, no matter which concept of income is utilized. However, two possible arguments which affect the interpretation of the results should be considered. The alternative theory may have given better results than the existing theory when disposable personal income was used only because the stock of money is a better approximation to an appropriate concept of income than is disposable personal income. A second argument advanced is that the existing theory has come out as well as it has relative to the alternative theory, when ”expected income" is used in both, simply because ”expected income" is a better measure of the equilibrium stock of money than the empirical definitions employed in the regressions. After applying the models to explain automobile ownership, Chow proceeds to examine three purchase models. The first is derived by noting new purchases in period t equals the difference between the total 15 stock at the end of t and the depreciated stock left over from t-l. The second and third models respectively are derived by interpreting the arguments in the utility function as purchases and then including a variable representing the relative price of old cars which are substitutes for new cars. The best result is obtained from the third model when the price variable used in the regression is the price index of the total car stock. Choosing this price variable as the dependent variable the result is p = —12.492 + .4685 I - 17.072 X' — 12.420 X R2 = .924, (.0294) 6 (3.909) (1.412) ‘ where p is the price variable, Ie is expected income, X' is the purchases of period t, and Xt— is the stock of passenger cars at the end of period 1 t-l. 2.4 Stone and Rowe—Nerlove Stone and Rowe have proposed a model of durable demand that has made a considerable contribution to empirical research.11 Their model, which is based on the concept of a stock adjustment mechanism, has the desirable feature that no data on stocks of appliances are required. However, some estimation problems were originally encountered, but these were later handled with some modifications suggested by Nerlove.12 The model with the Nerlove modifications is one of the models to be tested in the current paper and so only a brief heuristic discussion of its methodology and a presentation of its results in empirical application will be given here. 11Richard Stone and D.A. Rowe, "The Market Demand for Durable Goods," Econometrica, 25, (July, 1957), pp. 423—43. 12Marc Nerlove, "The Market Demand for Durable Goods: A Comment," Econometrica, 28, (February, 1960), pp. 132—142. 16 A basic premise is that net investment (i.g,, the difference between beginning of period and end of period stocks) occurs in fixed proportion to the difference between desired and actual stocks. New purchases are simply equal to the sum of depreciation, which is aSSumed to be determined by declining balance method, and net investment. By a series of successive substitutions, a reduced form for the model which contains only observable magnitudes is obtained, but some of the variables in the reduced form depend on an arbitrary selection of the asset's life. In this context one of the Nerlove modifications was helpful for he pointed out that estimating the reduced form based on various values for the asset's life and selecting the life which gives the highest R2 will produce maximum likelihood estimates. The modified model was applied by Stone and Rowe in estimating the demand for two groups of durables, furniture and hardware.13 Using annual data for the period 1922—38, the estimate of the reduced form for the furniture group produced the best results as follows: Aq = —4.54 + .0139 9(u/p) + 1.92 0(n/p) - .20 E'lq R2 = .781, (1.21) (.0074) (2.50) (.06) where n = 2 was the life used, Aq is the change in the expenditure on furniture, E-lq is expenditure on furniture lagged one period, u/p is income deflated by its own price, U/p is an index of all other prices divided by its own price, and 9 is an operator dependent on n. A second application of the model was made to quarterly data for a comparable (but slightly narrower) commodity grouping in the postwar period 1953—58. l3Richard Stone and D. A. Rowe, "The Durability of Consumers' Durable Goods," Econometrica, 28, (April, 1960), pp. 407—416. 17 2.5 Houthakker and Taylor Houthakker and Taylor have used a habit persistence type model in studying the demand for several types of products.14 The details are given in the next chapter and therefore will not be presented here. In essence, the authors have assumed that purchases of a durable good occur in response to a "state" or stock variable. This method, as in the stock adjustment model of Stone and Rowe—Nerlove, does not require data on the stock, but rather eliminates the stock through a series of algebraic sub- stitutions and manipulations. The findings relevant to the current study are those obtained when the model was applied to kitchen and other household appliance expenditures. The results are qt = —29.11 — .1715 qt_l + .0411 Axt + .0418 Xt-1 + .6830 zt R2 = .988, (4.126)(.1504) (.0059) (.0057) (.2208) where qt is a measure of purchases at time t, Xt is an income measure equal to total per capita personal consumption expenditure in year t, Axt = Xt — Xt 1 and Zt is a variable introduced on the third pass in the _ s three—pass least squares15 method of estimation utilized. 2.6 Suits An analysis of the U.S. automobile market incorporating a complete 14H. Houthakker and L. Taylor, Consumer Demand in the U.S., 1929-1970, Cambridge, Mass.: Harvard University Press, (1966). 15 The three—pass least squares method of estimation was utilized because the equation estimated contains a lagged dependent variable and an autocorrelated error term. The composite error term in this model is assumed to be of the form u = Au + E . The Z is intended to be a consistent estimate of u . Details of the three— —pass least squares method of estimation can be found in L. Taylor and T. Wilson, "Three— Pass Least Squares: A Method for Estimating Models with a Lagged Dependent Variable," Review of Economics and Statistics, XLVI, (November, 1964), pp. 329—346. 1 18 supply and demand model was formulated by Daniel B. Suits.16 The analysis covered the period 1929 to 1956. The model contained four basic equations for a) the demand for new cars by the public (Rd); b) the supply of new cars by retail dealers (RS); c) the supply of used cars by retail dealers (Rg); and d) the demand for used cars by the public (R3). They are given by (P-U) a) Rd = a1 __—fi——_ + a2 Y + a3AY + a0 + ul, 1)) Rs=blP+b2W+b3T+bO+u2, ' C) RS-C1R+C0+u39 l d) Rd — dl (U/M) + d2 Y d3AY + d4 3 + d0 + u4, where the symbols (other than the R's) may be interpreted as: P, the price of a new car, Y, disposable income, U, the average real price of used cars, M, the number of months the average automobile installment contract runs, S, the stock of used cars, W, the real wholesale price of new cars, T, retailer operating costs, u 4, random disturbances 1’ u2, u3, u to measure the impact of omitted factors, and the a's, b's, c's, and d's are coefficients to be determined. These equations are then subjected to a series of substitutions to eliminate several of the variables for which data were unavailable. The reduced form for the model is given by R = cl (P/M) + c2 Y + C3AY + C4 8 + C0 + u5, 16D. B. Suits, "The Demand for New Automobiles in the United States, 1929 to 1956," The Review of Economics and Statistics, 40, (August, 1958), pp. 273, 280. 19 where u is now a linear combination of ul, uz, u3, and u The 5 4' regression coefficients were estimated by applying OLS to the first differences of the equation. Differencing the equation was done in an attempt to minimize the autocorrelation effects of time series data.17 The result when this equation was fitted to annual data for the period 1929 to 1956 was AR = .106AY — .234 (P/M) - .507AS - .827AX + .115 R2 = .93 (.011) (.088) (.086) (.261) where the A represents a "change", AX is a dummy shift variable, (AX = O for all years except 1941 when AX = +1 and 1952 when AX = -1), and all other terms are as previously defined. The demand equations for new and used cars include a variable to account for the influence of credit conditions. In the demand equation for new cars this variable is given by (P-U)/M while in the demand equation for used cars it is U/M. The symbols are: P, the price of new cars, U, the price of used cars, and M, the number of months an average installment contract runs. The interpretation of these two credit variables is that they are a measure of the average monthly payment. In testing his model without the inclusion of a term to measure credit conditions, the R2 was only .80, while the coefficient for price had the wrong sign and was no longer significant. 17The use of first differences will remove the autocorrelation only if the residuals were autocorrelated according to a first order autoregres- sive scheme and the autocorrelation parameter was equal to +1, an unlikely circumstance. For a proof and discussion of this statement, see J. Kmenta, Elements of Econometrics, New York: Macmillan Company, (1971), pp. 289—292. 20 A later article by Suits extended this model in several reapects.l8 The income concept was reformulated to be supernumerary income which was defined as disposable income less a subsistence level of income. To select the subsistence income level, various levels were tried and the one which gave the highest R2 selected. This turned out to be $1500. Upon incorporation of the new income concept the R2 was improved from .782 to .851. Account was also taken of the possible influence of the age composition of the stock of used cars has on the new car market. This was accomplished by introducing lagged sales into the regression, but it was shown that, although the coefficient was of the right sign, it was not statistically significant. A third experiment was to isolate separate price and credit responses. Upon separation it is shown that the demand elasticities with respect to wholesale price was higher than elasticity with respect to retail price. Originally the model was fitted to both a prewar and a postwar period, and a test whether the relations were the same in both periods was conducted. In this test it was discovered that the relations held for both periods but the impact of the stock of cars on the road was significantly different. 2.7 Household Appliance Studies . 1 . . . M. L. Burstein 9 has attempted to focus 1nformat1on on the computation of price and income elasticities for household refrigeration in the 18Daniel B. Suits, "Exploring Alternative Formulations of Automobile Demand," Review of Economics and Statistics, 43, (Feb. 1961), pp. 66-69. 19M. L. Burstein, "The Demand for Household Refrigeration in the United States," in A.C. Harberger, The Demand for Durable Goods, University of Chicago Press, Chicago (1960), pp. 99-145. 21 United States. The model used for estimation is log 8* = a + B 1 log (P*) + B2 (log Y) + B3 T + u, where 8* is a measure of per capita consumption of services,20 P* is a measure of the real price of refrigerators, Y is per capita real income, T is a trend variable, and u is a random disturbance term. The author experimented with two alternative income concepts — per capita di3posable income and per capita expected income. Using the disposable income concept and a depreciation rate of .10, the best results obtained were as follows: 3* = a - 1.172 P*l + .825 Ye + 1.246 T + u R = .997. (0.195) (.212) (0.380) Some interesting aSpects of this study include the calculation of a price index adjusted for quality changes and the utilization of the concept of a unit of refrigeration service for the aggregation of appliance stocks. H. Laurence Miller21 has provided another empirical study of refri— gerator demand. The R003 and von Szeliski model forms the basis for his study, but it is never applied with the completeness contained in the authors' original study of automobile expenditures. Miller's study attempts to estimate several linear regressions in which the stock of refrigerators per household at the end of year t is the dependent variable. The independ— ent variables (examined individually) include average personal income, the proportion of households having electricity available, and the proportion of owner-occupied households at the end of year t. The regressions are applied to cross sectional State data for a number of years, but no attempt 0The measure of consumption must be based on knowledge of the stock and on assumptions regarding the depreciation rate. Since the stock is unknown, it must be estimated by using sales figures and mortality tables. It is this requirement of the knowledge of the stock which makes this model unsuitable for our present investigation. 2111. Laurence Miller, Jr., "The Demand for Refrigerators: A Statis— tical Study,” Review of Economics and Statistics, 42, (May, 1960), pp. 196—202. .. .-¢‘.. ' .4 22 to pool the State data for a number of years is attempted. Estimates of new refrigerator sales are obtained by combining information as to the demand for the stock, the existing stock and expected scrapping. This approach is not useful for the present study in that the data on stocks are not directly available and must be estimated based on sales and mortality tables. A third study for an aggregate of household durable goods is provided by De—Min Wu22 who examines the purchases of household durable goods in response to various input and status variables. Specific features of this study include attempts to: 1) disaggregate the income variable into different observable components that are heterogeneous in their effect on the durable purchase; 2) study the effects of lagged input and lagged status variables; 3) consider the effect of inter— dependence of decisions and different family characteristics on the purchase decision. The model used is a combination of a two stage decision model in which the individual first decides whether or not to purchase and then, given the decision to purchase, decides on the amount of purchase. Although the model is very fruitful for the study of cross sectional family budget data it does not appear to be applicable to the aggregate time series data which are the concern of the present study. 2.8 Summary Of the studies presented only the Suits and the Farrell studies attempted to construct complete supply and demand models. However, as pointed out in the first chapter, the circumstances in a particular industry could justify a procedure in which only demand equations are 22De-Min Wu, An Empirical Analysis of Household Durable Goods Expenditure, Unpublished Ph.D. dissertation, University of Wisconsin, (1963). 23 Specified. Both the Chow and Farrell studies attempted to provide a utility basis for their model. In this regard the Farrell model appeared to be very promising but its application in our present study is limited. This is because the model utilizes cross sectional data rather than time series observations, and it assumes the existence of a fairly well developed second hand market which does not exist for household appliances. None of the previous studies attempted to incorporate an intertemporal utility basis. The Stone and Rowe—Nerlove and the Houthakker and Taylor studies were selected for inclusion in the present investigation. These studies were selected because they offered the dual advantage of not requiring data on stocks and at the same time were in concert with generally accepted notions as to how consumers formulate their plans. Chapter 3 Durable Demand Mbdels 3.1 Introduction Durable demand studies, as demonstrated by our review of past work, have historically been of the "ad hoc" variety. That is, they have pro— ceeded by utilizing a peculiar mixture of armchair theorizing and economic theory to explain the quantities of durable goods demanded. The two pri- mary models of this type are the "stock adjustment" model and the "habit persistence" type model. In comparison with the armchair theorizing of the durable demands studies, the estimation of complete demand systems based on the concept of utility maximization has received much attention.1 The primary objective has been an explanation of the allocation of budget re— sources on competing commodities by a representative typical consumer. Part of the contribution of classical demand theory has been the development of theoretical restriction which should be exhibited by demand functions. However, in a large portion of the work on the demand for specific come modities, no attempt has been made to incorporate these restrictions. This is particularly in evidence in the area of durable demand studies where there is a noticeable void in the development of demand equations based on utility theory. One reason for this neglect has been that classical theory is not well-developed in the area of intertemporal problems which the de— mand for durables would entail. In the current chapter we will attempt to develop demand models that give attention to the inherent durable stock 1For a summary of the work that has been done on the specification and estimation of a complete set of demand equations, see A. Zaman, Formulation and Estimation of a Complete System of Demand Equations, (Unpublished Ph.D. Thesis, Michigan State University, 1970). 24 25 problems and incorporate the classical restrictions. As part of the in— vestigation, it will be shown that, subject to a slight reinterpretation of the variables, the habit persistence model is consistent with utility max- imization. In particular, it can be shown to be derivable from the linear expenditure system of section 3.4.2. Before proceeding to consider durable demand models of both the "ad hoc" and utility based types, a brief review of consumer demand theory will be undertaken to provide a background against which our models will be developed. 3.2 Review of Consumer Demand Theogy The typical approach to demand theory is based on the concept of a consumer with given income and tastes maximizing his utility subject to his limited resources. Starting with an assumed utility function and budget constraint, the derivation of demand functions, utilizing classical calculus thechniques, had been extensively discussed by many 3 has investigated the axiomatic foundation of demand authors.2 Uzawa theory and a brief partial summary of his findings will be given here. The basic axiom, which may be referred to as the axiom of choice, provides the explanation as to how the consumer selects his particular consumption bundle among competing alternatives. The axiom states that an individual with given prices and a fixed expenditure level, which may not be exceeded, will select that combination of affordable goods 2For example, see P. A. Samuelson, Foundations of Ecgggmic Analysis, Cambridge, Mass.: Harvard University Press (1947); J. R. Hicks, Value and Capital, Oxford: Clarendon Press (1939); J. R. Hicks and R. G. D. Allen, "A Reconsideration of the Theory of Value," Economica, 1, (1934), pp. 52-75 and 196—219. 3 H. Uzawa, "Preference and Rational Choice in the Theory of Con— sumption," in Proceedings of a Symposium on Mathematical Methods in the Social Sciences, 1959, (K. J. Arrow, S. Karlin, and P. Suppes, eds.). 26 which is highest on his preference scale. This axiom in turn implies the existence of a preference relation over the set of all conceivable commodity bundles. A commodity bundle will be assumed to consist of an n dimensional vector x or y whose elements x or yi are assumed to 1 represent the quantity of the i th commodity in the bundle. In addition, the collection of all positive bundles of which x and y are elements will be denoted by 9. A complete statement of the existence of a preference relation is: "There exists a dichotomous binary preference relation P defined on Q." The assumption of a dichotomous, binary relation implies it is possible to make pairwise comparisons over all bundles x E 0, while a "preference" relation may be defined as a relation P on the set of all conceivable commodity bundles possessing the prop— erties of irreflexivity, transitivity, monotonicity, convexity, and continuity. The properties of a preference relation may then be expanded as follows: I. Irreflexivity: For any x e 9, x PIX where P means the negation of P. This implies that each bundle is as good as itself. 11. Transitivity: For any x, y, z E 0, the relation xPy and sz xPz. III. Monotonicity: For any x, y E 9 such that xiZy,, i = 1, ...n, and x >y , for some i, then xPy. The implication of o t is statement is that given two bundles with the first containing more than the second for some of the goods and at least the same amount of the other goods in the bundle then the first bundle will be preferred. IV. Convexity: For any x, y 6 D such that x¥y and xPy, then 81—A)x+Ay) Px for all 0 < A < 1. The implication of this property is that on any budget hypersurface there will exist a unique point preferred to all others on the surface. Stanford, California: Stanford University Press, (1959), Chapter 9. 27 V. Continuity: For any x06 {2, the set {x: x € $2, x0 P x} is an open set in 0. Based on the existence of a preference relation with the properties as stated, Uzawa4 has shown by applying Debreu's5 Theorem there exists a continuous function U(x) defined on 9 such that For any x, y 60, xPy iff U(x) > U(y). In addition, properties III and IV imply that U(x) is monotone and strictly quasi-concave. In addition to possessing a utility function, the consumer is faced with the existence of a budget constraint which separates the commodity space 0 into attainable and unattainable regions. Formally the attain- able space is given by {xn‘p'x.§Y, x > 9} where x and Q are as pre— viously defined, p' is nxl vector of prices, and Y is a scalar repre— senting the predetermined amount of expenditure that must not be exceeded. Samuelson6 has then stated the general problem as "an individual confronted with given prices and confined to a given expenditure selects that combination of goods which is highest on his preference scale." Utilizing a utility function derived previously, which preserves a preference ordering, the problem may be stated mathematically as maximize U(x) subject to Y=p'x . 41818., p. 135. 5G. Debreu, "Representation of a Preference Ordering by a Numerical Function," in Decision Processes, (R. M. Thrall, C. H. Coombs, and R. L. Davis, eds.), New York: John Wiley & Sons, (1954), pp. 159~165. 6P. A. Samuelson, op. cit. p. 97. 28 A further assumption, not provided for previously by the existence of a continuous function, is that U(x) is at least twice differentiable. This addition provides for the utilization of the classical calculus techniques in the derivation of demand functions and their restrictions. Maximization of a function subject to a constraint is achieved by utilizing the Lagrangean technique. In this case, the Lagrangean func- tion is 1) L = U(x) + A (Y — p'x), where A is the undetermined Lagrangean multiplier. Differentiating this function with respect to the n x's and A and setting each deriva— tive equal to 0 gives the n+1 equations which are the first order con- ditions necessary for maximization. 2) U1 - A pi = O (i = l, 2, ..., n) and Y - p'x = 0, where U1 is the derivative of u with respect to the i th quantity and p1 is the i th price. The n+1 conditions involve 2(n+l) variables (-A, X1, X2, ..., Xn’ p1, p2, ..., pn Y). Demand equations are derived by solving n+1 of the variable in terms of the remaining n+1 as follows: 3) X1 = h1 (pl, ..., Pn’ Y) (i = 1, ..., n) -A = f (pl, ..., pn, Y). The hi represents the demand functions which are the object of our investigation. It is legitimate to solve for n+1 unknowns in terms of the other n+1 variables by virtue of the negative definiteness of the Hessian matrix U, defined as follows: Un 1 C : C I O O I Unn where the elements of U(Uij) represent the second order derivatives of the utility function with respect to x1 and xj respectively. A single subscripted U (i.e., U1) will similarly indicate the first derivative of U with respect to xi. A property which follows from the assumption of a twice differentiable utility function is Uij = Uji' This property will be of considerable importance in the development of one of the restrictions on the demand functions generated. The demand functions generated are subject to various restrictions. This is confirmed by Samuelson's frequently quoted statement7 "... utility analysis is meaningful only to the extent that it places hypo- thetical restrictions upon these demand functions." Before proceeding with their derivation, however, several additional terms must be defined and explained, as many of the restrictions are framed in elasticity terms. The price elasticity of the 1 good with respect to the j th price, defined as the percentage change in quantity of the i'th good in response to a percentage change in the j th price is 5) e = 8(log Xi) ij B(log pj) = (Pj/xi) (BXi/Bpj) (i, j = 1, ..., n) while the price elasticity of the marginal utility of income is = 3(10 A) = ~ _ 6) eAj ——-&———3(10g pj) (pj/A/ (BA/BPJ). 7Ibid., p. 97. 30 The income elasticities are 7) E1 - 3<1°3 X1) =(Y/x1) (3X1/3Y) 3(1og Y) and _ 3(log A) = 8) EA 3(108 Y) (Y/A) (BA/AY) respectively,where the E1 and EA are defined as the percentage change in X and A per percentage change in income. 1 The change in quantity resulting from a given price change may be decomposed into substitution and income effects. This may be seen by examining the total differential of the n+1 first order conditions. The n+1 differentials are Ulldx1 + Ulzdx2 . . . . . . . . . .+ Ulndxn + pldA = (—A)dp 021d}:1 + Uzzdx2 ... . . . . . . . .+ 02ndxn + pzd - (~A)dp2 9) . Unldxl + Unzdxz o o o o o o o o a 0+ Unndxn + pnd A = (-X)dpn ‘Dldxl - p2dx2 o o o o o o o o o 0—6. pndxn = dY-depl-depz . . . . . . .-xndpn These may be regarded as n+1 equations in n+1 unknowns (n dxi's and d(-A) ). Solving utilizing Cramer's rule gives n (4A) D dp + (dY — S n Xk dp ) D .7 10) dxj 3: 51:1 ij i Jkgl k “+19 :1 (j=l9 °° '9 n) 1 D .J and n n 11) d(-A) = E :1=1 ('A) D1, n+1dp1 + (dY ‘;,k=1 Xkdpk Dn+1, n+1 D 9 Where D is the determinant U11 U12 . . . . . . . Uln p1; U21 U22 0 o o o . u o U213. P2 ‘- . . . . i D = I . . . 1 . . . . ! iUnl Un2 . . . . . . . Unn pn! : i I . gpl p2 . . . . . . . pn 0 a, while Dij is the cofactor of the element in the i th row and j th column. Dividing both sides of equation (10) by dpi (1 = l, ..., n) while holding the remaining n-l prices and Y constant gives 12) axj/api = ("D D11 "xi Dn+l.j (1 = 1, n), D while dividing both sides by dY with n prices constant gives 0 D . 13) axj/ay = —Ei%LL— These results are brought together in Slutsky's relation (Kji) which is a measure of the income compensated effect of a price change as follows: 14) K.. = (8X 31 j/api) + x1 (axj/ay) = (—x) (Dij/D). An alternative price elasticity measure, which incorporates the Slutsky term above, will be referred to as the "Slutsky" price elasticity of demand (Sij)' It is given as follows: 15) s.. = (pi/Xi)Ki ij j ' Finally, the budget share of the i th good (W1) may be written as 16) Wi = (piX1)/Y' 32 The first set of restrictions come from a consideration of the n+1 equation of the first order conditions. A simple examination of this "budget constraint" reveals n 17) I § pix1 = Y (Adding up restriction). If the budget equation is differentiated and then both sides multiplied and then in turn divided by Y/X, the second condition becomes n 18) II é WiEi = l (Engel Aggregation). i=1 The third set of restrictions derived from the budget equation is obtained by differentiating the budget equation with respect to price (pj), mul- tiplying both sides by pj/Y, and converting the derivatives to elasticity measures. The restriction is then "‘1n 19) III 2 wieij = —Wj (j = 1, ..., n) (Cournot Aggregation). i=1 Samuelson8 has downgraded the importance of these restrictions since they are direct conSequences of the definition of the budget equation. He states, "At best, they could but reveal that we have not applied our de- fined operations with numerical accuracy." Inspection of the Determinant D shows that it is symmetrical since U-. = U..- 1] Jl From this it follows that —A D . _) D.- 20) K = (i__i__ii_ = i__i__1i. = K.., ji D D 11 31b1d., p. 106. 33 or in more familiar terms our fourth restriction is j + Xj(3X1/8Y) (i,j=1,...,n) (Symmetry Relation). 21) IV SXj/api + Xi(3Xj/3Y) = 3Xi/8p An alternative expression of the same restriction, utilizing the defini- tion of Slutsky's price elasticity, may be written as 22) IV(a) wisij = szji (1, j=l, ..., n). Finally, it is noted the demand functions are homogeneous of order zero. The validity of this statement is demonstrated by an examination of the first order conditions which are seen to be unaffected by a pro- portional increase in all prices and income. By virtue of Euler's theorem, we have 23) V (BXi/api) pl + (BXi/sz) p2 + . . . + (BXi/BY) Y=0 (i=1,...,n) (Homogeneity Restriction), or in elasticity terms 24) V(a) eil+e12+...+ein+Ei=0. The five conditions stated form the background upon which meaningful de— mand functions should be estimated. In particular,conditions IV and V provide meaningful restrictions which should be maintained. 3.3 "Ad Hoc" Models As a prelude to an examination of "ad hoc" models of demand for durable goods, it is necessary to specify the units in which demand is measured. Two possibilities are: (l) a simple aggregate of units, where distinctions between various sizes and models are neglected in the aggrega— tion; and (2) the total dollar value of units, where each unit is weighted 31+ by its unit value, whose representation is assumed to be its price. The specification of measurement units is necessary because the theoreti— cal underpinnings of various models are often more conformable to a particular mode of measurement. In this section it is assumed that consumers desire a stock level sufficient to cover their expected use levels. This assumption can be justified by the indivisible nature of appliances. An appliance may be considered as a store of potential services which, in the normal case, are released over some time span often referred to as the expected life. In an individual case the life may be lengthened or shortened depending on the intensity of use, but, in the aggregate, it is reasonable to assume that the level of services attainable is a monotonically increasing func— tion of the level of stocks. Conversely, this assumption implies that it is necessary for an in— dividual to have a stock of appliances if he wishes to utilize their ser— vices. Caution must be taken in drawing direct conclusions from observa~ tion of individual use of appliances, for the stock itself can be owned by the consuming unit or leased from an owner unit. At the present time there is no well developed rental market for appliances in which transactions, and hence prices, can be observed. The leasing arrangements that do take place occur primarily in connection with the rental of a living unit, such as an apartment, where the only price that can be ascribed to the rental service is the differential between an equipped and unequipped unit. Washers and dryers do have one additional type rental outlet with coin—op laundries where a direct rental price can be observed. Rental units must be considered for they present the possibility that the demand model should incorporate one explanation for owner units and a separate explanatJon for rental units. 35 3.3.1 A Stock Adjustment Mbdel The basic model used in this study9 is attributable to Stone and Rowe.lo Implicit in this formulation is the concept of a desired stock level which varies over time as a result of changing conditions. The desired stock level is assumed initially to be a function of per capita income and price. An individual utility function might exist, but it is never formally stated. One might envision that each consumer possesses a utility function, constant over time, in which stocks of appliances are one of the arguments. Prices and income go to make up the budget con- straint, and, hence, desired stock levels would shift with these para~ meters. The desired stock equation is written as I-l S*= a+bp+c (P/TT). where 3* represents a desired stock, p represents the income variable, P/w represents relative price, and a, b, and c are coefficients to be determined. New purchases, q, are defined as the Sum of depreciation, u, and net investment, v, I—2 q = u + v, where v is simply the increment in stock levels from one period to the next, I-3 v = ES — S with S the stock at the beginning of the period and where E is an operator such that, I-4 Ee x(t) = x(t + 9% 9The model will be presented in its non-stochastic form only in this section. A discussion of its stochastic form is found in Chapter 4. 10R. Stone and D. A. Rowe, op.cit, pp. 423~43. ’77 .... -— ......i. «~- ~ - 36 The portion of the stock consumed as depreciation in each whole period is assumed to be a proportion l/n of the stock at the beginning of the period, where n is a measure of the asset's life in years. New purchases WLH have a smaller depreciation rate than an identical asset held from the beginning of the period, simply because there has been less time to utilize its services. For this reason, depreciation in the year of purchase is assumed to be l/m of the purchases during the period where 1/n>l/m and 1—5 1/m = 1 — E'IBEEIIE7YEZIST"- 11 The combined depreciation is then I—6 u = S/n + q/m . This equation may also be expressed in terms of the opening stock and net investment as follows: 3 _ = _L S L . I 7 u in(m—l)] + [111.1] v \— ~— During a given period there may exist a discrepancy between the de— sired and actual stock which the consumer tries to rectify, but it is not assumed that the entire discrepancy will be closed within one period. Rather, it is assumed that only a fixed proportion r of this difference will be made up in any given period. This implies that net investment v is an increasing function of the size of the gap, as follows: I-8 v = r (8* — S). When this assumption is applied to total durable purchases of an individual family, the behavioral implications appear reasonable, since durable acquisi- tion frequently requires significant expenditures. A typical family is W 111b1d., equation 32, p. 430. 37 constrained by their income in relation to the cost when it comes to acquiring durables, so a choice as to which durables should be acquired in a given period must inevitably be made. When applied to individual appliances, the rationale is not as clear cut. For many families the purchase decision is an "all or nothing" decision. They will either tend to purchase the unit this period or post- pone the decision until a future period, such that r will have a value equal to 1 or 0, depending on whether or not the item in question was pur— chased. In an aggregate setting, r would tend to have a value between 0 and 1 depending on the frequency of purchase in relation to the total dis- crepancy between desired and actual stock. The value of r is assumed to be constant from period to period and location to location in our formulations. HoweVer, a more realistic assumption would make r a variable which assumes different values for each year and location in the analysis. This variable (r) may then be considered to be influenced by causal factors such as family income and prices of the product. In developing their model, Stone and Rowe encountered some diffi— culties which were later solved by Nerlove.12 In particular, it was necessary to assume a value for n a—priori and, secondly, it was necessary to use up degrees of freedom in computing values of S and u for use in the model. The solution for the second problem just raised was contained in a substitution procedure which eliminated the need for direct computation of a series for S and u. The first problem of an arbitrary choice for n and consequently m was resolved by computing the regression under various 12M. Nerlove, op. cit. 38 1 specifications for n and m, where m was computed from an approximation based on n, and then selecting that value which yielded the highest R2. This procedure yielded results which approximated maximum likelihood estimates.13 In deference to these problems the model developed here follows the Nerlove adaptation closely. In fact, the portion of the model developed to this point incorporates some of the Nerlove influence, for the desired stock equation (I—l) is linear in contrast to a loga- rithmic formulation used by Stone and Rowe. An equation for the end of period stock, ES, may be obtained from the definition of net investment (1—3) I—9 ES = S + v, which may be rewritten in terms of beginning of period stock and pur— chases by incorporating the equation for total purchases (I-2), and de— preciation (I—6), ES = S + q — u I—lO = S + q — S/n - q/m HH The net investment equation may be written as ES — s = r (3* — S) I—ll ES = r S* + (l—r)S, and this may be further simplified by incorporating the definition of ending stock (I-lO) above. n4} 1.4 ———, s + L———- q = rS* + (1—r)S n J m '1 I—l2 rm r ”m(l - r n)7 . = ———- 8* + !————-——-——— S q km-lfl L n(m—l) j 13Ibid. 39 This equation could not be applied to empirical data as S* is not an observable magnitude, and data on stocks of appliances, which is currently unattainable, are required. If knowledge of purchases for a sufficiently long period of time were available, and the life (n) were known, a series for stocks could be computed from the approximation for m and the ending stock equation (I-lO), which is a first order difference equation, expres- sible for the beginning stock after successive substitution, as follows: (In-1)] E (n-1)E- I—13 S {rm-1) 9= 1 An easier alternative to direct computation of the stocks from this equa— tion is substitution of this equation in the purchase equation (I—l3) giving I—14 _ m r S* + 1-r n ' n—l'e E—e q _ (m-l) (n—l) 0:1 n q. This yields after applying a Koyck transformation to simplify. _ m r _ 3:1 -1 _ —l I-15 q —[%E:Ii}[s* n ) E SE] + (l r) E q. The second problem of a non—observable S* is solved by substitution of our desired stock equation (I—l). This yields the final non-stochastic form for the model. I-16 q = a' + b'[ p _GE?§E_IO]+C '[(P/n) — (n; W) (P/n)] + r' E_1q where ' _ 3113 b _ [m-l ] b 40 3.3.2 Modifications of the Standard Model A basic framework for a model has been developed, which must now be explored for possible modifications to improve its results when applied to the demand for household appliances. These modifications expand the model in two basic ways: (1) by introducing additional explanatory vari- ables; and (2) by changing certain formulations within the model. Of particular relevance for household appliances are the influences of liquid assets and other monetary variables, such as interest rates, on household appliance demand. Suits and Sparks,14 in studying the influence of liquid assets on five categories (automobiles, other durables, food, other non-durables, and services), found that this variable15 was a sig— nificant factor in explaining expenditures on the "other durable" cate— gory. An exact rationale as to why it was significant was not given, but it was strongly suspected that liquid assets were an indicator of the longer run status of households. This is in contrast to current income which is felt to exert its influence on the short run spending decision. Evidence offered in support of this rationale includes Friedman's "permanent income" hypothesis16 and Modigliani, Brumberg and Ando's "life cycle" hypothesis.17 Although these theories were applied to explaining total consumption expenditures, their implications are felt to be relevant to appliance expenditures. 14D. Suits and G. Sparks, "Consumption Regressions with Quarterly Data," in The Brookings Quarterly Econometric Model of the United States, (J. Duesenberry, G. Fromm, L. Klein, and E. Kuh, eds.), Chicago: Rand McNally Company (1965). 15Liquid assets were defined as the sum of currency, demand deposits, and fixed value redeemable claims as estimated in the Federal Reserve Board's "Flow of Funds." 16M. Friedman, The Theopy of the Consumption Function, Princeton; Princeton University Press (1967). 111 Further support for liquid assets as a determinant of the demand for household appliances was provided in a study by Klein.18 Assets in liquid form were felt to be available for the purchase of durables or at least to provide the means for a down payment. Both this rationale and the previous one are felt to be sufficiently strong so as to make liquid assets a candidate to be tested in the models. A further study of the influence of monetary variables on consumer durable expenditures was made by M. J. Hamberger.19 As part of his study, the author attempted to estimate the role of liquid assets, which were separated into two categories: those reflecting actions taken by the monetary authorities, and those measuring consumer liquidity. 0f particu— lar interest is the latter category which included: Mel: the consumer stock of demand deposits plus currency : M plus consumer holdings of time and savings accounts at commercial banks, mutual savings banks, and savings and loan associations. Mc3 Mc3 was found to be significant and accordingly will be the variable to be used in the current study. There are two basic mechanisms through which liquid assets can act: 1) Directly, by influencing net investment, 2) Indirectly, by influencing the desired stock level. 17A. Ando and F. Modigliani, "The Life Cycle Hypothesis of Savings," American Economic Review, 53 (March, 1953), pp. 50-84. 18L. Klein, "Major Consumer Expenditures and Ownership of Durable Goods," Bulletin Oxford University Institute of Statistics, 15, (November, 1955), pp. 387-414. The first rationale was also used by Klein, "A Postwar Quarterly Model: Description and Applications," in Models of Income Deter- mination, Princeton University Press, 1964, where liquid assets were con- sidered as a proxy for total wealth. 19M. J. Hamberger, "Interest Rates and the Demand for Consumer Durable Goods, American Economic Review, 57, (December, 1967), p. 1144. 112 If L is assumed to represent the liquid asset in question at time t, the assumptions are expressable as I-l7(a) v = r(S* - S) + dL and I—17(b) S* = a + b + c (P/n) + dL respectively. After incorporation of I-l7(a) in Equation I—16 of the model, the reduced form for estimation under this assumption becomes 1—18 q = a' + b' p — (Eh—1 p + c'
0,
where the factors tending to cause an increase in the state variable are
assumedtx>bezafixed proportion f of current purchases.
1+9
The depreciation term represents the using up of the stock at time
t, and is assumed to follow a declining balance method of depreciation.
That is, depreciation for the t th time period is assumed to be a fixed
proportion l/n of the stock at time t, where n represents the asset's
life in years.
III-3 u(t) = (l/n) S(t)
This expression makes some concessions to reduce the complexity of the
problem in that depreciation on current purchases is ignored. Theoreti—
cally, this would be justified only if all purchases were concentrated
at the end of period. However, its omission will have only a minor in-
fluence on the empirical results obtained. It is noted that this assump-
tion contrasts with the Stone-Rowe-Nerlove depreciation formulation of
section 3.3.1 equation I-6 and their assumed uniform distribution of
purchases.
The objective of the remaining calculations is to obtain a reduced
form.which will contain only measurable magnitudes. Equation (III—3) for
depreciation is substituted in equation III-2(b) and S(t) is eliminated
by using III—1 giving
”1’4 '3“) = M“) 7%,") [q(t) — a — c acoj.
Equation III-l is assumed to be continuous and differentiable with re—
spect to "t", so that upon differentiating we obtain
111-5 q(t) = b S(t) + c 0(t),
where the dot above each symbol is to be interpreted as the derivative
with respect to time. This equation is solved for S in explicit form
and then used in equation III-4 yielding the following expression after
simplification,
50
III-6 q(t) = a/n + (b f -(1/n))q(t) +1c b (t) + c/n p(t),
which is a first order differential equation involving only observable
magnitudes.
The preceding derivation has been carried out by assuming the
equations were continuous in nature. Since empirical data is in dis-
crete form a suitable approximation must be made. Exact magnitudes for
a given period r (which may be later defined as a year) are given by the
following integrals.
to+r
Jf q(t) dt.
q =
to
to
t
_ o+r
0 t
o
t
.. o+r
St = S(t) dt.
0 t
0
Since S(t) refers to a stock level which is not cumulative over time,
Sto must be interpreted as an average stock level over the n th period.
Equation III-6 is a continuous function, so a discrete approximation
for it must be obtained. The conversion is accomplished by integrating
first for period to to to+r and then for period t to to+2r' Upon sub-
o+r
traction, the difference in equation III—l between discrete periods is
given by
III-8 - - 7 = b 8 — S + c " — 7 .
qto+r qto ( to+r to) (pto+r oto)
The remaining portion of the calculations parallels the continuous
case with the added assumptions that the between period change in the
average stock level may be approximated by
III—9 s - 5 = r 2 A*S + A *S
tm °to (/)< 1:01., to),
51
* *
where Sto+r and Sto are the changes in the stock during the t0 and
to+r periods respectively. This may be recognized as a linear approximation
to the change in the average stock level between periods to and to+r' If
the average stock (St) is changing according to some exact linear function,
the approximation would produce a perfect fit. If this approximation is
accepted and utilized in equation III-8, the result can be written as
111-10 ‘ - “ = (r/2) b A*S + A*S ) + c(5 — 5 )
9 1t ( t to to+r to ,
o+r o o+r
which after substitution for A*S and A*S becomes
to+r t0
III—11 ' — ‘ = r _ r _ _ -
9to+r qto (r/Z) b[f qto+r (l/(nb))(qto+r at cote”)
+ffi -(1/(nb))(fi -ar-CE )]+c(5 ~5 ).
to to to to+r to
If r is specified to represent 1 on a time scale, equation III—l
becomes after simplification
r l + (l/2)[b f + (l/n)1]_
_ _ = (a/n)
111 12 qto+r [1 _ (1,2)[b f _ (1/n)]J + 1 _ (1/2)[b f - (l/n)]_tho
c[1 + (1/2n)] 1- +[ c[1 - (l/2n)]
+ 1 — (172)“) f - (umfl pto+r +L1 — (1/2) [b f — (l/n)]J
Finally, a variable with a subscript t is specified as pertaining to the
o+r
t th period, and with subscript t to the t—l period. If A0, A1, A2, and
0
A3 are written for the coefficients above, the form for estimation becomes
III—13 qt = A + A qt_1 + A2 pt + A
0 1 p
3 t—l'
In the original formulation Houthakker and Taylor wrote pt as equal to
(0t “ pt-l) + 0 pt + pt—l to reduce the incidence of multi—colli-
t-l =
nearity. If this algebraic substitution is utilized, the form for
S2
estimation becomes
III-14 qt 3 A0 '1' A1 qt_1 + AZApt + A3pt_1,
where now A3 is specified to be
cln
1 - (1/2) (b f - (l/n)) .
3.4 An Intertemporal Utility Model22
Historically durable demand studies based on a concept of utility
maximization have received little attention. Instead, application of
demand systems have been concentrated primarily in the area of non-
durables commodities. A motivating force behind this avoidance has been
a desire to avoid the intertemporal problems inherent in analyzing com—
modities which are not fully consumed in the period of acquisition. How-
ever, this reasoning had some merit, for it allowed the author to bring
into clearer focus the particular model under scrutiny.
In the investigation of the demand for household appliances a
plausible rationale for the inherent intertemporal problems must be pro—
vided. Specifically, the existing stock, the life span of the durable,
and expected future incomes and prices should be considered as well as
the conventional variables of current income and prices. The general
approach will be to impose restrictions which will reduce the demand
equations thereby generated to estimatible form.
22The intertemporal model considered here will not attempt to des-
cribe the allocation of consumption over time, but rather it will try to
incorporate the implication of stocks on current behavior. For a dis-
cussion of the broader problem see R. H. Strotz, "Myopia and Inconsistency
in Dynamic Utility Maximization," Review of Economic Studies, 23, (1956),
pp. 165-180. (This article attempts to describe the allocation process
53
The typical utility function may be written as
1) V u = u(xl, x2, ..., xn),
where x1 is defined as the consumption of the i th commodity. For
durables, consumption has been defined as the fraction of the existing
stock used, typically referred to as depreciation of the stock. The
object of the demand equations then became consumption as opposed to the
purchase of the durable good. As part of this investigation, a reformu—
lation of the problem to that of durable purchases will be made. The
first step will be to modify the utility function by redefining its
arguments in terms of the "expected utility from purchases." As is
readily seen, this approach ignores the utility derivable from the
existing stock, while the budget constraint must also be reinterpreted
to be consistent with this modification.
The conventional budget constraint, which may be written as
n .
2) Y = :;7 pixi (lFl, ..., n),
L_Ji=l
deals with a consumer allocating a fixed money income (Y) among competing
alternatives (xi), subject to a fixed price (pi). For a durable good the
budget constraint as written implies that its entire price is paid for out
of current income. This implication makes the assumption of a budget con-
straint in the form of equation 2 questionable. An alternative approach
recognizes that only a portion of the price of the durable good must be
outlayed in any given period. Using this alternative assumption, the
revised budget constraint may be written as
through the use of the calculus of variations.) A summary of the classical
Lagrangean analysis of the problem is contained in J. M. Henderson and
R. E. Quandt, Microeconomic Theory, New York: McGraw—Hill Company, (1971),
Chapter 8, pp. 293—333.
511
m n
3) Y I Z I] pixi + Z pj-Ixj s
1=1 j=m+l
where x1 and xj represent the quantities purchased of durable and non—
durable goods respectively, pi and pj represent their prices and d repre-
sents the proportion of the total price of durables purchased that is
paid out during the current period.23 It is assumed h is a constant in
our revised budget constraint (equation 3), but a broader interpretation,
that will not be pursued here, could consider d as a variable dependent
on other factors. The rate of interest should be a prime consideration
as a factor which might have significant effect on d, as it is a measure
of the cost of borrowing against future income.
The term "expected utility from purchase" requires a more concrete
interpretation. For this, it is assumed the consumer under consideration
has in mind a definite consumption pattern or use plan, consistent with
the physical composition of the asset which will generate services in
each year of asset ownership. The consumer then evaluates this service
stream in terms of its utility worth today which will be referred to as
E1. It is further assumed that R1 is proportional to the amount of cur-
rent period purchases so that the utility function may be written as
4) u = u (1,, xj) (1=1, ..., m), (j=m+1, ..., n),
or upon substitution of the proportionality assumption as
U = u (pxi, Xj) (i=1, ..., m)’ (j=m+1, ..., n),
23This approach is an adaptation of an approach developed by Vernon
Smith in an analysis of investment expenditures. In his approach he at-
tempts to isolate the cost on current account for both indestructible and
fixed life capital goods. See V. L. Smith, Investment and Production,
Cambridge, Mass.: Harvard University Press, (1961), pp. 68-70 and 109-111.
55
where U is the redefined utility which we suppose the consumer to be
maximizing, x1 and xj are quantities purchased of durable and non-
durable goods respectively, and p is the factor of proportionality.
The proportionality assumption can be supported in two alternative
ways. On one hand, it can be supported if it is assumed the discount rate
is zero and the asset services are released according to a declining
balance method of depreciation. This may be demonstrated as follows:
a) It is assumed n is the depreciation rate and
xi and Xi are as previously defined.
b) With zero discount rate, we may write the "ex—
pected utility from purchase" as the sum of the
service generated in each period the durable is
expected to serve as follows:
/‘
_ _ 2 _ n-l 1
5) Xi _ 7] [xi + (l - n)Xi + (1 T1) xi ...+ (1 n) xi .3
5
which becomes after collecting terms and simplifying
A
X1 = [1 - awn] X1-
c) It is readily seen that (l — (l—n)n) is a constant,
and hence the proportionality assumption is maintained.
A second approach would be to assume some positive rate of dis—
count, but with services being released in uniform amounts each period.
Using the symbols as defined above, but using r for the discount rate
and 0 for the straight line depreciation rate, we can write
A 1 1 1
6 .= —— ——
) x1 “”10 [(1n) + (1+r)2 + (1+r)n],
Again, the above expression can be factored into a constant times xi to
obtain the desired result.
The assumption of a constant depreciation rate (n or ¢) for all assets
regardless of life, whether applied in a declining balance or straight line
56
manner, is a little hard to accept. A more palatable assumption is that
the depreciation rate is a decreasing function of the life of the asset.
Using pi for the depreciation rate of the i th asset, this assumption is
given by
7) Pi = C(13):
where
api/Bt < 0,
and where t is considered the life of the asset in years. A further
assumption is that future income and prices will be regarded as known
with certainty.
With these amendments to the standard utility maximization model,
the problem may be stated as
maximize U = u(Dixi, xj) (i=1, ..., m; j=m+l, ..., n)
m n
subject to Y = E hpixi + E ijj'
i=1 j=m+l
Forming the Lagrangean expression L and maximizing with respect to xi,
xj and the Lagrangean multiplier A gives the first order conditions
BL/axi = piSu/Sxi -Ahpixi = 0 (i=1, ..., m),
BL/axj = Bulaxj — Apjxj = 0 (j=m+1, ..., n),
m .n _
BL/BA = Y —r S h P1x1 + \77 13ij = 0-
L __)i=1 pm“, j=m+1
It is readily seen that the properties of the revised demand equa—
tions will not be affected by the above changes. Specifically, the addi-
tivity restrictions can be assured as they were simply derived from a
budget constraint which was unchanged under the proportionality assumption.
57
Further, the utility matrix
will retain its symmetric nature under the modified version, so its
inverse will also be symmetrical which in turn implies the symmetry
relations (restriction IV - Section 3.2) will hold.
3.4.1 Constant Elasticity of Demand Systems
A choice must now be made as to the functional form for the demand
equations. The simplest system to be considered falls under the heading
of a constant elasticity of demand (CED) system. This system has in the
past been one of the more commonly utilized forms, deriving its popularity
from its ease of application and straightforward interpretation of the
parameters as elasticity coefficients. The theoretical justification for
the system lies in application of Taylor's Theorem, for this theorem
states it is possible to approximate an elementary function by a Taylor
series expansion. The constant elasticity formulation in essence assumes
a first order approximation with the remainder terms being subsumed under
the random error term in estimation. The system may be represented as
Ei ei- . .
1) Xi = aiy (Hj pj J) (1. J = 1. ---, n)
in its non-linear form, or upon transformation by logarithms,
2) log xi = log a1 + Ei (log y)-F§;"jeij (108 Pj),
58
where x1 is the quantity of the i'th good, pj, the price of the j'th
good, y, total expenditure and E1 and eij are the Engel and Cournot
elasticities respectively, which are assumed constant.
It is precisely this constancy assumption which has, on one hand,
built the simplicity in the model and, on the other, rendered the form
incompatible with traditional demand restrictions. Wold and Jureen24
have stated in connection with demand relations that if the system if of
the constant elasticity type then "unless all elasticities are equal to
unity, such a function cannot satisfy the balance relation in the whole
range of the variables involved." This was given in more explicit form
by Zaman25 as follows:
The budget share of the i'th commodity is defined as
P x.
_ i 1
3) i — -—-—— ,
Y
or in logs
4) log wi = log pi + log xi — log y.
If prices are assumed constant and we consider the total differential of
the function, we get
5) d (log wi) = d (log xi) - d (log y)
= (Ei - l) d(log y),
d(log xi)
where the Ei is the income elasticity and its definition as d(log y)
is utilized. Carrying out the calculations we have
6) dwi/wi = (E1 — 1) dy/y.
24H. Wold and L. Jureen, op. citL, p. 106.
25A. Zaman, op. cit., pp. 98-99.
59
0
Now starting from an initial position of income y , prices p0, and the
i'th good budget share wg, we can state that the budget share of any
commodity must be bounded by 0 and 1, thus for the i'th commodity we
have
O<(wg_+dw1) <1.
or or
—wi < dwi < l — wg,
(1 -wg)
Iff [{Ei — 1) — 1],.0 <(1:i — 1) (y0 + dy) < [(E — 1) +___O_] yo
w .
1
The sign of E1 then determines the limits within which the income, (y - dy),
must remain if its budget share is to be a positive finite magnitude.
Only if E1 = 1 will income lie between plus and minus infinity. Thus, if
income elasticities are constant and are not all equal to unity, the budget
constraint will be violated.
In this study we will try to salvage this system by considering
only a subsystem of equations and not a complete system. This possibility
was recognized by Wold and Jureen26 who pointed out that for part of the
field it would be perfectly possible to yield a demand function of the
constant elasticity type. Actually, there are two alternatives which
can be recognized here. On one level we can think of the elasticities
and their values as a particular solution to the set of differential
equations defining the demand functions. This approach was exploited
by Zaman27 in deriving two alternative systems which were consistent
with utility maximization. The second approach which will be utilized
26H. Wold and L. Jureen, op. cit.
27A. Zaman, op. cit.
60
here is attributable to Court.28 In this paper the author recognizes
that additivity restrictions which are relevant in complete systems may
be effectively ignored in subsystems. In a complete system this implies
7) Zpixi = y,
where Pi’ xi and y are as previously defined, or alternatively, upon
replacing xi by its demand function di’ we have the more general specifi-
cation
8) Zpidi = y.
i
The interpretation of this equation is that the true demand functions in
a complete system must "add up" to fulfill the budget equation. This pos—
sibility under a constant elasticity specification was seen to prevail
only under very unlikely circumstances. There is no reason, however, to
assume that all demand equations are of the same form so that in studying
a subset of the complete system it is plausible to assume that the other
equations are of such a form so as to satisfy the additivity conditions
without explicitly considering their particular formulation.
Additionally, since the CED system is an approximation to the true
form, the system must be constructed so as to exhibit the other properties
of homogeneity and symmetry. The homogeneity property is handled in tra—
ditional fashion by selecting a price from the n possibilities and deflating
all other prices and income in the sybsystem by dividing through by the
selected price. Symmetry in the system is then imposed by incorporating
a set of exact linear restrictions.
28R. H. Court, "Utility Maximization and the Demand for New Zealand
Meats," Econometrica, 35 (July—October, 1967), p. 424—446.
61
The elasticity of substitution between the i'th and j'th good is
defined as
S .
9) 013' = 41-1,
W1
where sij is the Slutsky income compensated elasticity of demand and wj
is the j'th good budget share. The symmetry conditions require
10) Cij = Oji o
By utilizing the Slutsky relation
11) s-- = e , + ij
and the definitions of oij the symmetry conditions can be expressed in
Cournot elasticity terms, which has the advantage of being directly cal-
culable. This gives
. .+ = .+,
12) eiJ/wJ Ei eji/w1 E3.
The complete statement of the symmetry condition can then be made by
forming a matrix
lwl
II
0
N
N
13)
Lofil . . . . . ofin
,
which is symmetrical to satisfy restriction IV of Section 3.3, and nega-
tive semi-definite to insure maximization and homogeneity respectively.29
In application the symmetry conditions are not exact. Alternatives
are then to use an exact formulation which would cause the restrictions to
be of non-linear form or force a linear form by calculating the elasticities
of substitution at given values of the budget proportions. The most
29P. A. Samuelson, 0p. cit., p. 113.
62
appropriate value being the mean of the budget proportion. The latter
alternative appears to be the most promising and it was thus selected
by Court and will be utilized here. The symmetry conditions may then
be written as
14 k +E =1. +
) jeij 1 1811 E1’
where the ki and kj are now calculated constants. The estimation problem
is then to estimate n equations of the form
n
15) log xi = ei0 + % j_1eij 108(Pj/P) + E1 108(Y/P) + 6i
(i = 1, ..., n),
where 51 is the random error term, subject to the (n/2) (n~1) restrictions
of the form of equation(1®. Details of these computations are given in
the next chapter.
3.4.2 Linear Expenditure System
A popular form for demand functions has been the linear expenditure
system, which is derivable from a particular form for the underlying
utility function. It is considered here because it exemplifies the approach
of deriving demand equations from specific utility functions, and the be—
havior implied by its form appears reasonable in the context of household
appliances. In addition, this system aggregates perfectly over both in—
dividuals and commodities due to the linearity of the Engel curves. The
system itself was proposed by Klein and Rubin,30 while its inherent
2
utility basis was proveded by Samuelson31 and Geary.3
30L. R. Klein and H. Rubin, "A Constant—Utility Index of the Cost
of Living," Review of Economic Studies, 15, (1947), p. 84—87.
63
Specifically, the demand equations of the linear expenditure
system may be represented in their non—stochastic form as
n
l) x:L = C1 -[a1/pi] Zk=lpkck+ [ai/Pi] Y9
while the resulting expenditure equations are
2) pixi = pici + a1 (Y - E kpkck) (i=1, ..., n),
where p1 represents prices, Y is income, x the quantity of the i'th good
i
demanded, while the Ci represent what is referred to as the subsistence
bundle of the i'th good. The consumer may be thought of as first allo-
cating his income to the subsistence bundle and then determining the
"additional" expenditure based on his remaining income.
The utility function underlying this form is of the type
n
3) U(x) = E ai log (xi — Ci) ai > 0;
i = 1
.n
g ai = 1, x1 — C1 > 0.
i=1
Since the demand functions are directly derivable from the above utility
function we may be sure that these demand functions exhibit the desirable
properties of homogeneity, additivity, and symmetry. The only parametric
restriction required is
n
4) 231:1
i=1
to satisfy the Engel aggregation condition, and this is applicable to com-
plete systems only.
31P. A. Samuelson, "Some Implications of Linearity," Review of
Economic Studies, 15, (1947), pp. 88—90.
32R. C. Geary, "A Note on 'A Constant-Utility Index of the Cost of
Living'," Review of Economic Studies, 18, (1949), pp. 65—66.
64
The utility function as stated is of the directly additive variety,
thus limiting its scope of application. This is pointed out by Houthakker
who states that additivity reduces the scope of substitution and compli-
mentarity to the barest minimum.33 Under direct additivity the substitu-
tion effect between the i'th and k'th commodity may be written as
5) Rik = ('Y) (axi/BY) (Bxk/BY),
where K1 is the substitution effect, and Y is the money flexibility
k
parameter, while its own—price substitution effect is
6) piKii = y(3xi/8Y) (l - pi ° axi/BY).
In the first case, the cross price substitution effect is seen to be pro-
portional to the income derivatives 05 both commodities and to the income
flexibility, while in the second case its own-price substitution effect
is proportional to the money flexibility parameter, income derivative,
and marginal propensity to spend on commodities other than the i'th.
From these conditions Houthakker has noted that inferior goods and com—
plements are ruled out while the substitution effect is relegated to the
more general type in the sense of competing for a place in the consumer's
budget.
It has been felt that by choosing broad aggregate classifications
the above conditions could be more readily met. Appliances, however,
can be seen to possess many of the desirable characteristics. No previous
studies have found appliances to be an inferior type good, nor would we
a—priori expect inferiority to be the case. In considering appliances
in terms of three categories, refrigerators-freezers, ranges and ovens,
and laundry products, which reflect their service functions, the desired
33H. S. Houthakker, "Additive Preferences," Econometrica, 27,
(1960), pp. 244-257.
65
characteristics of substitution and complementarity may be met. The
groups are not complements in the technical sense of function per-
formed nor are they directly substitutable in terms of their natural
function. In a given year we would expect each group to compete for a
place in the consumer's budget as required by the general substitution
idea.
The original empirical implementation of the linear expenditure
3" suffered from some deficiencies which were later
system by Stone
- 35 36
p01nted out by Parks and Pollak and Wales. These concerned the
stochastic properties of the system which will be considered in the next
chapter, and the lack of a satisfactory explanation for the "subsistence
bundle" of purchases. Stone recognized the problem and attempted to in-
corporate a time trend in explanation,
_ *
7) cit _ Ci + cit,
Pollak and Wales have suggested an alternative which appears more prom-
ising. The effect of habit formation can be incorporated by basing cur-
rent period subsistence purchases on last period's purchases,
8) Cit = c; + cixi,t-l,
or as a modification of this form
= +
9) cit ci cizi,t-1’
where 2. represents a variable such as average consumption of the
1,t-l
i'th commodity over a number of years or the highest attained level of
34 .
J. N. R. Stone, "Linear Expenditure Systems and Demand Analy51s,"
Economic Journal, 64, (1954), pp. 511-527.
35R. W. Parks, "Systems of Demand Equations: An Empirical Comparison
of Alternative Functional Forms," Econometrica, 37, (1969), pp. 629-650.
36R. A. Pollak and T. J. Wales, "Estimation of the Linear Expendi-
ture System," Econometrica, 37, (1969, pp. 611—628.
66
consumption during an appropriately selected lag period.
The expenditure equation for appliances provides the opportunity
to incorporate a habit persistence mechanism of the Houthakker and Taylor
type considered in Section 3.3.3. The subsistence bundle, which could
be interpreted as a type of replacement sales, can be written as a linear
function
*
10) Cit = Ci + Ci 31(15):
*
where C1 is a constant, while Si(t) is a state or stock variable as be—
fore with coefficient ci. Upon substituting this equation in the basic
demand equation of the linear expenditure system we have
11) x1 = c: + ci Si(t) + bi[(Yt -2:Jk(pktckt)/pit].
This equation corresponds very closely to equation III—l of section
3.3.3 presented earlier, with the exception of income which is now in—
II
terpreted as a type of "supernumerary income deflated by current price.
The corresponding expenditure equation becomes
.»_W ’1
* .
12) pitxit - ci + C1 pit Si(t) + bi LYt — Ziok pkt Ckfij.
Utilizing the definitions and eliminations presented earlier and defining
Y — /' .
t (...;k pkt ckt , , _ , ,
0- r ——““—E;"“—““— , the equation for estimation in non-stochastic
it
form becomes
13) x- = A + Ai xi + A pt + A
, t-l 2 3 pt—l’
which is identical in form to equation III—13 of Section 3.3.3.
67
3. 5 Summary
In summary, it is observed that the "ad hoc" and utility maximiza—
tion models have some relationship. A clear example was given by the
"ad hoc" models of the H & T habit persistence type which were found to
have their implicit roots in utility maximization, as was evidenced by
their link—up with the linear expenditure system. However, it must be
pointed out that those models which rely on an explicit formulation of
the utility function have a built—in drawback in the arbitrariness that
marks any choice of utility function. As is often the case, the empirical
nature of the hypothesis restricts testing to proving only whether the
implied behavior is plausible. The CED system, as opposed to the linear
expenditure system, presents no claim to justification from an explicit
utility function. Its only claim is that of an approximation which is
consistent with utility maximization. The final choice as to which is
the "true" demand function must remain unanswered as no one can know the
truth with certainty. All that can be done in this paper is to present
the theoretical merits of each and their empirical results so an appro—
Priate choice can be made.
Chapter 4
Stochatic Specification and Estimation Methods
4 . 1 Introduction
Each of the models considered to this point in time has been pre-
sented in only a non-stochastic form. Of these models we see that both
"ad hoc" models have attempted to incorporate a "stock" effect for
durables into their computations, while the utility based models have
more or less accepted the stock as given and attempted to explain the
consumer expenditure allocation in terms of utility maximization be-
havior. The "ad hoc" models have been extensively utilized in many
demand studies for particular goods, while of the utility based models,
only the CED model has had considerable application for this purpose.
In addition, the CED applications have been of limited extent, for demand
restrictions, notably symmetry, have not been generally included.1 The
linear expenditure system model has been applied primarily to the esti-
mation of complete demand systems.
The original Stone and Rowe model was applied to the determination
of the demand for clothing and household durable goods in the United
Kingdom. Fairly good results were obtained in that the regressions
accounted for around 90% of the observed variance in each case. In a
later study Utilizing the amendments as suggested by Nerlove, Stone
1A significant exception to this was the application to a "non—
durable" (meats) in New Zealand by Robin Court, op. cit. It is this
article which has suggested the possibility of its extension to durable
commodities.
68
69
and Rowe again found a good fit when applying their model to British
data for various categories of household durables. Houthakker and
Taylor likewise have found reasonably good results in application
of their model to demand projections for the U.S.. One of the many con-
stant elasticity formulations, which is mentioned here due to its
popularity, is attributable to G. C. Chow, who studied the demand for
automobiles in the U.S. and also found a "good fit" for his model.
In View of the seemingly good results which each of the models have
enjoyed, the choice as to which is the "best" demand model must obviously
be made on grounds other than simple comparisons of st or other sin-
gular statistic derived from the data. All that can be expected from
an application is affirmation that the model is a logical candidate for
consideration. The final choice must then rest upon a comparative con-
sideration of the theoretical underpinnings of the models, the plausibility
of their stochastic specifications, and their ability to make accurate
forecasts. The theoretical considerations have been the topic of the pre—
vious chapter. This leaves the task of formulating the various stochastic
specifications to which the model will be subjected, and the development
of estimation methods compatible with these specifications, to the present
chapter.
The "ad hoc" models will be estimated using both aggregate U.S. data
for a twenty—one year period, and data by region for an eleven year period,
while the utility based models will be estimated using only the aggregate U.S.
data. Our first consideration will be to specify stochastic assumptions
for the "ad hoc" models, which are applicable to the aggregate data, and
to suggest some estimation methods which consider these specifications.
70
Following this summary, the additional stochastic specifications involved,
when the combined time-series and cross-sectional data are utilized, are
presented, and a procedure for incorporating these specifications into the
estimation method is developed.
In regard to the "utility based" models, we will attempt to specify
stochastic specifications and estimation methods compatible with the
particular model and data under consideration. Specifically, stochastic
specifications for the CED model, when it is to be applied to aggregate
U.S. data, are examined. An estimation procedure which incorporates
these specifications,and the symmetry restrictions given in Chapter 3, is
then developed. Some additional problems in the estimation of the linear
expenditure system are also explored.
4.2 "Ad Hoc" Models
4.2.1 Standard Specifications
The "ad hoc" models derived in equations I—16 and III-13 in Chapter 3
are repeated here in their reduced form as follows:
1a) Stone and Rowe — Nerlove
. 7 ‘ -
q = A' + B' [o — 1'“—'1>E-1p] + (2' [(11/11) — {—n‘lh—Hp/oJ + r'E‘lq,
\ n n
lb) Houthakker and Taylor
1 (2)
l —1 _
o + A3(p/11) + A4E (p/n) + ASE q,
= + _
q A0 + Alp AZE
where q represents the quantity purchased, 0 represents income, and p/n
represents relative price.
2The Houthakker and Taylor model shown here includes relative price
(P/U) as an explanatory variable.
71
If the simplest stochastic specification concerning these forms is
made, the assumption of a random error term measuring the cumulative effect
of all remaining influences is attached additively to each reduced form
giving
2a) Stone and Rowe -— Nerlove
q = A' + B' [p — (Ell) E—lo] + C' [(p/fl) — CEZlAE-l(p/n{]+ r'E'lq + at,
n n
I
2b) Houthakker and Taylor
= —l -l —l
q A0 + Alp + AZE p + A3(p/n) + A4E (P/fl) + ASE q + 8t.
The standard assumptions concerning 5t is that it is a normally dis—
tributed random variable with the following specifications:
1. E(st) = 0 Zero mean
2. E(etes) = 0 (t¥s) No autocorrelation
3. E(Ei) = 02 Homoskedasticity
Estimation can be carried out utilizing ordinary least squares (OLS),
or, if desired, the lack of autocorrelation specification may be dropped
implying that the estimation procedure should be amended to include an
adjustment for removal of the time effect before application of OLS.
4.2.2 Alternative Stochastic Specifications
The annexing of a random error term to the reduced form renders these
equations suitable for estimation, but their interpretation from a
behavioralistic point of view is somewhat questionable. To alleviate
this situation an error structure must be incorporated to recognize that
72
the final reduced structure is a combination of behavioralistic and
purely definitional equations. The consequences for estimation of the
reduced form must then be investigated for some of the standard stochastic
specifications (e.g., homoskedasticity, etc.) may be lost by the manipu—
lations.
As an initial step, consider the desired stock equation III—l of
3.3.1
3) S*t = a + bpt + c(p/n)t,
This equation is intended to define some desired level to which the
individual is striving. An error term ult’ appended additively, may be
interpreted as the cumulative effect of factors other than those explicitly
considered. This gives
4 S* = a + b + c /n + u .
) t at (p )t It
If the rest of the equation of the S—R—N model is considered to hold
without modification, the reduced form associated with this specification
may be written as
F .
5) qt = av + b'Lpt — Bil-pt_l] + cl [(p/n)t — (Bil) (p/fl)t_1]
_ l .
- .. j 1
+ ' + _E£. _ E:lj
r qt'l [m—1:]Lult < n j u.1,t—1 -
Since m, r, and n are considered as fixed parameters of the model,
the composite error term
g = mr' f n—l' 1
1t m—l Eult _ n u11t‘1i
k ‘4
may be looked upon as a linear combination of normally distributed
random variables which is also a normally distributed random variable.
73
A second alternative in the S—R—N model is to assume the adjustment
of actual to desired stock as given by equation I-8 of 3.3.1 includes an
error term u2t which is intended to measure the random influence in the
adjustment process.
6) v = r(s* — s) + u
2t
The reduced form associated with this specification is
n
+ r! + m __ n-l
qt-l [~m_1j E1“ (T, )“2,t—1],
where the composite error term is given by
7) qt = a' + b' [at — (Ell) pt_:] + c' {(P/fl)t - €EiL' (P/n)t_1£
l 1
Combining both specification simultaneously will produce a combined
error term
8) s = ._EL_? u + ru — 2:1 u + ru )i
t ym-l L< 2t 1t) ( n ( 2,t—1 1,t—1 }.
The primary error specification in the H&T habit persistence model
involves attaching the error to equation III—1 of 3.3.3 which describes
the current level of demand in terms of its explanatory variables. The
error term can be considered as encompassing other influences not
explicitly included in the formulation. This equation, including the
error term at, may be written as
9) q(t) = a + b S(t) + c p(t) + €(t).
. W—J
74
Defining the discrete analog of S(t) for the rph_period as
t
_ o+n
10) 8to 7 “f" e(t)dt
t0
and utilizing this definition in the derivation of the reduced form
produces
11) qt = A0 + Alqt-l + AzApt + A3pt—1 + zit,
where th is the composite error term given as
uh.
_ ; -n
th - g- — l/(2n) 8t — [l + 1/(2nljet_l,
L-
An alternative specification would identify the random error term as
occurring in equation III-2b of 3.3.3 which describes the change in the
state variable during period t. This modified version can be written as
12) S(t) = fq(t) - u(t) + at,
where at is the disturbance term which was annexed. The interpretation of
this equation is that adjustment in the state variable occurs in relation to
the net purchases of the period, while the st measures the random influence
in the adjustment from period to period. The reduced form will be the
same as equation III—14 except for the disturbance term which becomes
13) Z2t = (1/2)(et + €t_1).
If both specifications hold simultaneously, the composite error
would then become
14) Z = Z
_ ' " ‘1,
3: 1t + 22, - [:1 + 1/2 — 1/(2n)_]et - L1 + 1/2 + 1/(2n)..
J’t~1'
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII . j
75
All of the previous specifications have resulted in reduced forms
of the general type
15) Y = a + BXt - AYt_1 + ut — Au
t t—l’
where Yt is the endogenous variable to be explained and Xt and Yt—l
are the explanatory variables. The equation may be recognized as the
reduced form resulting from the application of a Koyck transformation
to a geometrically distributed lag equation. Three assumptions re—
garding the composite disturbance
16) at = ut — Aut_1
have been recognized in the literature.3 The simplest assumption would
assume the u follow a first order autoregressive process
t
17) Ht = put_1 + Zt
with p = A, where the Zt are assumed to be independent, normally dis—
tributed random variables with zero mean and constant variance.
zt m N(0,ozz)
The composite error term would then become
18) s = Z
making OLS the appropriate method of estimation.
3See A. Zellner and M. Geisel, "Analysis of Distributed Lag Models
with Applications to Consumption Function Estimation," Econometrica, 38
(Nov., 1970), pp. 865-889, for a discussion of these assumptions and their
implications for estimation. This section will rely heavily on their
work. An excellent summary of these specifications and estimation of
distributed lag equations is contained in J. Kmenta, Elements of Econome-
trics, Macmillan, 1971.
76
The second assumption regarding the error term is that the ut's
are normally and independently distributed variables with zero mean, constant
variance, and no correlation over time. This implies certain desirable
properties of ut carry over to at as follows:
1. Normality: Since 51 is a linear combination of normally
distributed random variables, it too is normally distributed.
2. Unbiasedness: E(5t) = E(ut)— AE(ut_1) = 0.
3. Homoskedasticity:
E(ut — Aut_l) = E(u%) + 12E(u%_1)— 2AE(utut_l)
= 02 + A202 = a constant.
However, it is no longer appropriate to claim a complete lack of auto—
correlation as demonstrated by the relationship between 5t and Et-l'
19) E(etet_l) = E(utut_1) + A2E(ut_lut_2) — AE(utut~2) — AE(uE_1)
= -Aoz.
In addition, a is no longer independent of the regressors as may be shown by
t
the relationship between at and Yt—l'
_ _ = _ 2
20) E(eth_l) — E(ut — Aut_l)(o + BXt-l + AYt_2 + ut—l Aut_2) Ao ,
The lack of independence renders the OLS estimates inconsistent. For this
reason alternative methods of estimation must be sought.
The last specification to be considered follows closely the lead of
the first in that it is assumed the ut's are mN(O,g2), and follow a first
order autoregressive process. However, it is not assumed the autoregression
parameter p is equal to A. As with the second case, normalcy, unbiasedness,
77
and homoskedasticity carries over to st, while the stare now autocorrelated
over time and not independent of the regressors.
In this study we will assume that the first specification presented
(p=A) is applicable. This choice will avoid the problem of having to
specify the elements of the variance—covariance matrix of disturbances
and make it possible to use OLS to estimate the "ad hoc" models from the
aggregate data. In addition, previous demand studies, utilizing the "ad
hoc" models, have proceeded on the basis of this implicit assumption.4
4.2.3 A Procedure for Combining Cross-Section and Time Series Observations
As part of our investigation the possibility of extending the model
by expanding the data base must be explored. If the first part of this
paper is thought of as building a model which is theoretically consistent,
then this part must be considered as an attempt to refine the estimates by
supplying additional data to which the model may be applied. Initial attempts
at estimating regression parameters from cross-section and time series
observations proceeded by using cross-section data, such as those for
states, firms, or households, to derive some estimates, and then following
up by holding these estimates constant at the computed values while esti—
mating the remaining parameters from time series observations.5 An
alternative procedure which we shall follow involves simultaneous esti—
mation of all regression coefficients from pooled time series and cross—
section data. Specifically it is our intention to utilize the data for
4See for example R. Stone and D. A. Rowe, op. cit. and M. Hamberger,
op. cit..
5See for example H. Staehle, "Relative Prices and Postwar Markets for
Aalimal Food Prices," Quarterly Journal of Economics, 59 (1944-45),
pp. 237-279. A good summary of the procedure can be found in L. Klein,
A 'Iextbook of Econometrics, (Evanston, Illinois: Rowe Peterson & Co., 1953).
\ I
78
individual selected regions for a number of years. To this end we will
develop a procedure that leads to the application of the generalized
linear regression method. In the remainder of this paper this method
will be referred to as the "Two Stage Generalized Linear Regression"
method (TSGLR).
The earliest attempt at providing estimates from pooled data used
an "analysis of covariance" technique.6 The essence of this technique
is to provide dummy variables for the firm and time effects. This is
shown as
k
21) Yij = Z1 + Wj + > Flsrxr’ij + Uij ,
where Zi is intended to be a dummy variable representing the firm effects,
Wj a dummy representing the time effect and Xr ij a variable which varies
,
over both firm and time dimensions. This technique will produce estimates
which are both unbiased and efficient. However, there are some drawbacks
of the method, as pointed out by Maddala.7 The use of an extremely
large number of "dummy" variables may tend to eliminate large portions of
the variation between the dependent and explanatory variables. This
would be especially true if the between firm or time period variation is
large. In addition, there is a loss of degrees of freedom due to the
large number of independent variables, and the interpretation of the
dummy variables is awkward.
6See for example I. Hoch, "Estimation of Production Function Parameters
Combining Time—Series and Cross—Section Data", Econometrica, 30 (January, 1962),
pp. 34-53; F. R. Johnson, "Some Aspects of Estimating Statistical Cost
Functions", Journal of Farm Economics, 46 (February, 1964), pp. 179-187;
Y. Mundlak, "Estimation of Production and Behavioral Functions From A Combi-
nation of Cross-Section and Time-Series Data", in Measurement and Economics,
(C. F. Christ, Ed.), Palo Alto, California: Stanford University Press,
(1963), pp. 138—166.
7G. S. Maddala, ”The Use of Variance Components Models in Pooling
Cross—Section and Time—Series Data", Econometrica, 39 (March, 1971),
pp. 341-359.
79
An alternative approach is known as the "error component method."8
This method assumes that the regression equation can be written as
k
22) Yit = °‘ +E Brxr,it + sit ’
r=l
where a and Br (r = l, ..., k) are the intercept and slope parameters
respectively, Y represents the independent variable, Xr it represents
’
it
the explanatory variables, and the error term Eit is now composed of
three parts as follows:
23) sit = U1+Vt +wit (1= 1, nj),(t = 1, 2, T),
where the parts are assumed to represent random components with U1
representing the firm effects, Vt representing the time effect, and
Wit representing a component varying over both dimensions. Since the
components are random variables, we make the further assumption that
each component is normally distributed with 0 mean and constant variance
2, andqifl2 respectively. In addition, the following inter—
o 2
u ’ O
V
relationship between components is specified:
24) E(Uin) = E(init) = E(Vtwit) = 0,
E(Uin) = 0 for 1 ¥ j,
E(VtVs) = 0 for t ¥ s,
B(witwis) = E(witwjt) = E(witsz) = 0.
If these assumptions are investigated for their implications it may
be shown that the variance of the composite error term Eit is a constant.
8T. D. Wallace and A. Hussain, "The Use of Error Components Models
in Combining Cross Section with Time Series Data", Econometrica, 34
(July, 1966), pp. 585-612.
This is given as follows:
2 _ 2
25) E(eit) — ECUi + Vt + Wit)
= 02 + 02 + 02.
u v w
The covariance between cross sectional units is
26) E( ) = E(Ui + Vt + wit)(Uj + Vt + Wit)
Eitejt
for (i#j),
ll
Q
and the covariance over time is given by
27) E( = E(Ui + Vt + wit)(ui + VS + Wis)
eiteis)
= ou for (t#s).
According to (27), the covariance over time of the error term (cit)
is the same for any two time periods. The implication of constant auto—
correlation effect holds no matter how far the periods were separated in
time. This is in contrast to the general consensus that regards the
autocorrelation effect as diminishing as the distance in time of the
errors increases, as would be exemplified by a first—order autoregressive
scheme. The second objection involves the assumptiOn of homoskedasticity.
Error component model assumptions were seen to generate a variance of
2 2 2 . .
o + 0V + ow. The assumption of the same variance of the errors for two
u
distinct regions again is hard to accept, especially when utilizing data
by regions with extreme geographic and economic differences. A third
problem arises from the constant covariance between cross sectional
units. In the 'error com onent' a roach this was shown to be 02 for
p pp V
two distinct cross sectional units. A much more flexible assumption
81
would allow the interdependence between cross sectional units to vary
depending on the two units under consideration.
A method, suggested by Kmenta,9 will be utilized in this study.
The advantage of this approach is that it avoids the three unacceptable
implications referred to above. This method uses the generalized
linear regression method, where the estimates of the variance—covariance
matrix of disturbances are based on residuals from OLS estimated equations.
The estimates of the regression coefficients given by this method will be
both consistent and asymptotically efficient.
The variance-covariance matrix of residual error terms (9) can be
written as
— 2 '7
E(ell) E(ellelz). . . E(ellslT) E(811€21) E(ellezz). . E(€11€NT)‘
2
E(512511)“612) ° E(amen) E(E12821) E(E12822)° ' E<€lZENT)
' 3
_ 2 v
Q - E(elTell)E(echlz). . . E(elT) B(EIT 21) E(e1T 22). . B(EITENT):
2
E(521511)E(s21s12). . E(62151T) E(521) E(621e22). . E(621€NT)i
i
2 l
E(822611)“622612)‘ ' E(ezleu) E(E22821) E(€22) E(522%qu
i
i
LF(€NT€11)E(€NT612)' E<€NT612) E(amen) E(EN'rezz)' B(ENT) _J,
where the e is a NT x 1 vector of residuals with element 5
it’ where the
first subscript 1 refers to the cross sectional unit and the second t,
9J. Kmenta, Elements of Econometrics, Macmillan, 1970, Chapter 12,
pp. 508-517.
82
to the time period. If the elements of this matrix were known a-priori
then we could utilize this information in the Aitken formulas,
29) §= (x' Q'IX)‘l (X' fly)
with variance-covariance matrix
30) Var-Cov (E) = (X' Q—1X)_l.
Since this information is not known, we must provide consistent estimates
of the elements of Q. In providing these estimates, stochastic specifica-
tions will be imposed which are more realistic than those of the preceding
models.
The most complete specification assumes the disturbances are "cross—
sectionally correlated and time—wise autoregressive." Since the data
represent observations drawn from nine regions, we assume that there
is heteroskedasticity between regions, and that the disturbances are
not mutually independent over geographically defined boundaries. Indepen—
dence would in fact be more a condition of the economic configuration of
the regions than their geographical boundaries. The specification of
this model is given by
B(e. ) = a. (heteroskedasticity),
) = (mutual correlation),
E<€it€jt Oij
= re io
Eit piei,t—l + uit (autoreg 33 n)
with
“it m N(0’¢ji)’
Em mmam02344
“no
¢.oo o.mo mm.¢ HNmNo
¢oeo o.m0 mm.¢ oomoa
¢.eo o.m0 mm.¢ Newmm
¢uow o.mo mmI¢ ommmm
¢Ioo o.mo mm.¢ hmmoH
¢.oo Onmo mm.¢ ochNo
«.00 o.mo nm.¢ oo¢¢m
¢Ioo ovmo mm.¢ ammoo
¢Ioo o.mo mm.¢ ommwN
m600 m6mo mm.¢ ¢oomm
m.o¢ nomo mm.¢ OHHmH
0.00 m.mo mm.¢ mmflmm
mooa Mono mm.¢ .wson
m.oo m.m@ mm.¢ memmu
w.oo mpmw mm.¢ maomw
mooo Momo mmo¢ Nooom
Q.oo m.mo mm.¢ Nooaa
w.00 momo mm.¢ hmobm
IIIIIIIw wwmmmm .onuwm mCZCm «<4 .mzownnuzv
«cu . mod 20 msqa wzoozw
“do Hdu smwmmszu 442Cmawa
oooa I omoH
zouwwd >m mzoumwmewd Z~ mea mmqmq~m<> FZwQZwamozu
.otz~sZOUV
«I4 m4m4»
NooH
mama
d