at fES T ' ,1‘ o 1995 ABSTRACT AGGREGATION BIAS IN THE DEMAND FOR MONEY By Edward Elliot Veazey The primary purpose of this research is to determine whether a single macroequation should be relied upon as an accurate description of the demand for money in the United States. Many authors have as- sumed that a single equation does adequately describe total U.S. money demand, and they have proceeded on that basis with empirical analysis involving a few arbitrary macrovariables. The rate of return on four to six month commercial paper, for example, is one of the most fre- quently used interest rate variables and it is usually treated as "the" rate of interest with the implication that it adequately repre- sents all of the various interest rates and thus the opportunity cost of holding money. Likewise, some measure of national income is usually treated as adequately representing all budget constraints. In spite of the fact that the problems associated with using such aggregates in regression analysis have been known for some time, they have been ig- nored or assumed away by monetary empiricists. In this research, separate monetary demand equations are esti- mated for each of the forty-eight continental United States and the District of Columbia using interest rate and income series applicable to each particular state. Then weighted averages of the variables Edward Elliot Veazey are calculated for use in a single macroequation. These macrovariables are constructed with fixed weights so that the analysis will fall within the scope of linear aggregation. The principal conclusions of this research are as follows: (i) Estimation of demand for demand deposits at the state level yields parameter estimates which conform generally with prior expectations based on economic theory. (ii) The system of state demand equations is not consistent with a single macroequation which describes aggre- gate demand in terms of linearly aggregated macrovariables. (iii) Es— timates based on such a misspecified macroequation cannot be assumed to be unbiased; therefore, conclusions based on these estimates are suspect. Calculations based on state estimates indicate a bias in one of the macroparameter estimates equal to 86% of the estimate. AGGREGATION BIAS IN THE DEMAND FOR MONEY By Edward Elliot Veazey A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1973 (fl ACKNOWIEDGMENTS I wish to express my sincere appreciation to Professor Jan Kmenta, my dissertation chairman, who advised, encouraged, and inspired me throughout my entire course of study at Michigan State University. I wish also to thank Professor Warren Samuels for his advice and criticism, and especially for his sincere personal interest. Professor Robert Gustafson also gave generously of his time and advice whenever it was needed. Many other people were instrumental in the completion of this research. I would like to acknowledge the help received and thank Carl Gambs, Alan Shelly, and Professors Maurice Weinrdbe and Mark Ladenson for their advice and comments. Appreciation is also extended to the National Science Foundation and the Graduate Council for their financial support. Finally, I wish to thank my wife, Jennie, for her many hours of typing and editing, and especially for her support and encouragement during the long duration of my graduate studies. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . LIST OF TABLES Chapter I. II. III. INTRODUCTION Purpose . . . . . . . Summary of Chapters . THEORETICAL DERIVATION OF THE DEMAND FOR MONEY Demand Theory . Portfolio Theory . Efficient Portfolios Empirical Use Alternative Specifications of Demand Functions Friedman . Latane . Chow . Teigen . Laidler THE AGGREGATION PROBLEM . Consistent Aggregation . . Aggregation over Individuals . Convenient Macrovariables Macroequation . . Aggregation of Assets . Aggregating Interest Rate Variables The General Case . . . . Macrovariables . . General and Specific Inconsistency . Aggregation and Estimation Aggregation and R2 Variance of Disturbances Correlation Coefficients . Specification Error in Microrelations Disaggregation iii Page ii Chapter Page IV. ESTIMATION I . . . . . . . . . . . . . . . . . . . . 6h Definition of State Variables . . . . . . . . . . . . 65 Demand and Time Deposits . . . . . . . . . . . . 65 Interest Rates . . . . . . . . . . . . . . . . . 66 Income . . . . . . . . . . . . . . . . 68 Adjustment Variables . . . . . . . . . . . . . . 72 Macrovariables . . . . . . . . . . . . . . . . . 72 Equation Estimates . . . . . . . . . . . . . . . . 7h Macroequation Estimates . . . . . . . . . . . 7H Macrovariables in Microequations . . . . . . . . 82 V. ESTINmTION II . . . . . . . . . . . . . . . . . . . . 95 State Demand Equations . . . . . . . . . . . . . . . 95 Tests for General Consistency . . . . . . . . . . . . 97 Transformation of Variables . . . . . . . . . . 109 Aggregation of Demand and Time Deposits . . . . . . . llh General Inconsistency . . . . . . . . . . . . . . . . 115 Specific Inconsistency . . . . . . . . . . . . . . . ll8 Aggregation Bias . . . . . . . . . . . . . . . . . . 118 VI. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 120 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 128 iv LIST OF TABLES Table 1. Comparison of Macroequation Estimates . 2. Estimates and Restricted Estimates: D = f (BO, RT, RS, YP, RTS*,YP ) . . . . . 3. Estimates and Restricted Estimates; (D+T)=f(BO, RT, RS, YP, RTS*, YP ) . . . . h. ZA Estimates By State thBo+BlRTt+B2RSt+B3Yt+BuYPt+B5Dt_1+Et 5. ZA Estimates By State (D+T)=BO+B1RTt+B2RSt+B3Yt+BuYPt+B5(D+T)t_l+E 6. Tests of Consistent Aggregation for 6 New England States . . . 7. Tests of Consistent Aggregation of Di for Unrelated States . . . 8. Tests of Consistent Aggregation of D1 + T1 for 9 Unrelated States 9. Macroparameter and Aggregation Bias Estimates t 102 108 CHAPTER I INTRODUCTION The era of econometric forecasting is currently being dominated by large models which rely on nationally aggregated data. Thousands of man and computer hours are spent each year in attempts to predict the sum total of goods and services which will be produced in the next year. Likewise, major effort is directed toward determining national indices of unemployment, prices, corporate profit, interest rates, etc. In prior years, lack of convenient disaggregated data and a need to limit the scope of early efforts probably dictated the heavy concentra— tion on national aggregates. But now it seems that the marginal gains will be larger if additional effort is applied to collection and anal- ysis of disaggregated data. The value of a national economic index to most people depends on how well it describes the economic environment relevant for a partic- ular set of individuals, and an estimate of gross national product-- even a positively known, correct estimate--would be of no more use to most individual businessmen than a forecast of gross national rainfall would be to an individual farmer. It is possible that some of the national indices describe most of their local counterparts very well and thus are a useful summary of information. It is just as likely, however, that some of the national indices are no more relevant for Particular geographic and economic segments of the country than the national forecast of rainfall. Pur ose The purpose of this research is to determine whether the demand for money in the United States is adequately described by a single macroequation based on national data. Demand equations are estimated for each of the states and the estimates are used to investigate the effects of aggregating to a single marcoequation. Previous authors have estimated the demand for money using every convenient functional form and a wide variety of variables. In each case, however, the esti- mates have been based on macroequations, and little or no attention has been given to the problems which might be caused by aggregation. This is a serious oversight. If a system of microequations does not meet the conditions for consistent aggregation, then least squares estimates of the macroparameters cannot be assumed to be unbiased and conclusions based on their values are in doubt. Not only are the esti- mated parameters apt to be biased estimates of the average micropara— meter, but in the event of large scale inconsistency even unbiased estimates would be of limited value. The greater the inconsistency the less the macroequation tells about the demand in any particular microunit and the less useful it is as a summary of information. An important indication that aggregation bias might be a problem in demand for money equations comes to light with a close look at the Portfolio theory approach to the demand for liquid assets. Reference is Often made to portfolio theory as a rationale for regressing quan- tities of particular assets on a vector of interest rates and other eXOSerious variables. The portfolio theory approach is developed in Chapter II and, under certain assumptions, it does yield equations which describe quantities of assets as linear functions of interest rates. However, when the equations are expressed in this linear form,_ the coefficients of the interest rate terms are implicitly defined as non-linear functions of, among other things, the investor's antici- pated second moments of the probability distributions of the future returns to each asset. That is, the investor is assumed to antici- pate some probability distribution for the return to each asset, and the coefficients of the interest rates in the linear equations are functions of the second moments of these probability distributions. One of the standard assumptions of regression analysis is that the coefficients in the regression equation are constant for all ob- servations. This assumption would be met in the portfolio theory equations if, among other conditions, the investor's anticipations re- garding the probability distributions of the returns were the same for all observations. Since anticipated distributions are not observable, a direct test of their stability is not possible. However, we might reasonably assume that investors formulate their subjective antici- pations in some systematic way according to actual values of past observations. It might be argued, for example, that an investor pre- dicts the variance of the return to time deposits by calculating a sample variance from past observations. Then if all investors calcu- late the same values for the sample moments, there might be some justification for the assumption that anticipated distribution moments, and thus coefficients in the portfolio equations, are constant for all individuals. The common usage, in economic literature, of terminology which im- plies that interest rates are adequately described by a single rate called "the rate of interest" makes it easy to erroneously conclude that each investor faces the same rate of return on, say, time deposits. If this were true, then of course all investors would have the same data available for use in calculating sample moments. However, inves- tors do not receive the same return to such assets as time deposits and savings and loan shares. Even granting the assumption that return to a particular asset is the same throughout each state, there are wide differences in the levels and changes in the return to liquid assets in different states. Likewise, there is large disparity in sample moments calculated from past data in different states. Recognition of the disparity in interest rates in different states leads naturally into doubts about the appropriateness of aggregating asset and interest rate variables over all states for use in a macro- equation regression. With different investors facing different current and past interest rates, the agrument that the coefficients in the portfolio equations are equal for all investors becomes exceedingly tenuous. It seems unlikely that investors with disparate interest rate experience would, nevertheless, have equal anticipations for the prob- ability distributions of future returns. Thus, it seems unlikely that the coefficients in portfolio equations would be equal for all indi- viduals in every state. Aggregation bias appears a clear possibility. Even without the specific example afforded by portfolio theory, there are important considerations which suggest that the information gained from a disaggregated approach would justify the extra effort. States are economically less heterogeneous units than is the nation as a whole. Banking laws differ among various states as do the laws gov- erning savings and loan associations, mutual savings banks, and other financial intermediaries. Usury laws vary among states; the variety of charges applied by national retailers to their credit customers in different states illustrates the variety of state legal environments for financial transactions. The degree and nature of industrialization also varies from state to state. Different states tax and control their corporations differently and, of course, different geographic regions have different comparative advantages. It seems unlikely that the aggregate demand for money in North Dakota responds to the interest paid on short-term commercial paper in the same proportion as in New York. And if the response is not the same in these two places, then combining their demand deposits in a single macroequation for use in regression analysis may be a serious error. One important purpose of most regressions is to estimate the values of coefficients and draw conclusions or make predictions based on the estimates. When aggregate equations are used in place of a larger number of disaggregated relationships, estimates of the re- gression coefficients are apt to be biased and conclusions drawn from the estimates are apt to be erroneous. Summary of Chapters Chapter II presents several models of demand for liquid assets ‘Which have been used in recent empirical research. Two of the models are based on microtheory--one on the theory of demand, and the other On portfolio theory--and the other models are best classified as macro- models in spite of the appeals of the various authors to microtheory in defending various characteristics of their models. This chapter both summarizes the important empirical work in monetary economics and introduces the various model specifications which are used in Chapter IV. Chapter III deals in a general way with the problem of aggregation associated with using macromodels in place of micromodels. The first part develops the conditions under which the non-stochatic part of the macrorelation is always consistent with the system of microrelations. Since these strigent conditions seem unlikely to hold in general, this leads directly to the questions of what importance the inconsis- tencies play when estimates are made of the parameters of marcoequa- tions. The second part of this chapter develops answers to this question by investigating the impact of aggregation on the parameter estimates, on the goodness of fit, and on prediction. Chapter IV contains detailed descriptions of the data and the procedure by which macrovariables are formulated to fall within the scope of linear aggregation analysis. Each of the specifications of Chapter II is estimated using linearly aggregated macrovariables, and the general results are compared with the results obtained by the original author. In Chapter V, statewide data are used, first to estimate the Iparameters of the entire system of state demand equations and then to 'test the hypothesis that the equations may be consistently aggregated 't0 a single macroequation. Equations for both demand deposits and the sum of demand and time deposits are subjected to this estimation and testing. The final chapter contains a summary of results and the important conclusions and implications. CHAPTER II THEORETICAL DERIVATION OF THE DEMAND FOR MONEY The demand equations for liquid assets are developed in completely different fashion depending on whether the model is purely theoretical or meant to be used in empirical work. Virtually all of the recent empirical studies of demand for money are studies of macrorelations. Typically they involve regressions of total money supply on net national product and some interest rate or index of interest rates. In recent literature, by far the most common approach in estimating monetary relationships is to use aggregate data for the nation as a whole with the underlying implicit assumption that the macrovariables used in the study are related in a stable way according to the proposed macrorelation. On the other hand, the purely theoretical development of monetary theory has shifted quite heavily to microeconomic analysis. The macro- relations of the classical quantity theory have given way to micro- relations based on an individual's maximization of utility. In the Classical approach, money was singled out as a unique and separate entity conveniently described and limited by quantity theory equations MV = PT OI' Keynes personalized the demand for money somewhat by focusing on motives for holding money, in particular the speculative and transactions motive. Then Hicks brought monetary theory definitely within the range of micro- " . standard commodity selected from economics by referring to it as the rest to serve as standard of value."2 He emphasized that money, like other commodities, has close substitutes. "The fact that money and secu- rities are close substitutes is absolutely fundamental to dynamic economics."3 Hicks also made an important contribution to the early development of portfolio theory.)4 He introduced into monetary theory the concept of the rate of return to a portfolio as having a range of possible values and the idea that the width of this range was indication of the risk incurred by the individual. Markowitz took these notions of expected rather than certain return and dispersion as a measure of risk and developed a comprehensive formulation of portfolio theory.5 Then TObin 6 grounded the analysis in utility theory. 1J. M. Keynes, The General Theory of Employment, Interest, and , New York: Harcourt Brace and Co., 1936, pp. 195-200. Mone 2 J. R. Hicks, Value and Capital, 2nd ed., Oxford: Oxford LThiversity Press, 19h6, p. 170. 3 Ibid. J. R. Hicks, "A Suggestion for Simplifying the Theory of Money," EsxDriomica, New Series, Vol. 2 (1935), pp, 1-19, 5Harry Markowitz, "Portfolio Selection," The Journal of Finance, -V01.. 7 (March, 1952), pp. 77-91. James Tobin, "Liquidity Preference as Behavior Toward Risk," Bfiflfliew of Economic Studies, Vol. 25 (February, 1958), pp. 65-86, IO Both portfolio theory and demand theory have gone largely untested and even ignored in empirical work except for an occasional casual reference.7 They are nevertheless important since they led, through analogy, to the development of estimable aggregate demand equations. The next two sections of this chapter will present essential ele- ments of the demand.theory and portfolio theory approaches to the demand for money. The third section will review the specification of demand for money functions encountered in current empirical research. Demand Theory The traditional theory of consumer behavior is stated in terms of flows rather than stocks and so, in order to utilize it in the demand for liquid assets, there must by an assumed relationship between the amount held of a particular asset and the flow of services which it generates. The exact nature of the service flows and the motivation for holding assets received a great deal of attention in the early transition from.the quantity theory approach to demand theory. Demand fknr:money, for example, was described in terms of "transactions" and "ereculative" demand with the names indicating different motives for lhOining and different service flows provided by money balances. How- eVWBI‘, just as in demand theory no attention is paid to the exact nature Of' trtility, it was a natural development in monetary theory to deempha- SjJZEE the exact nature of the service flows and treat them the same as any‘ ther consumer good. \ 7An important exception to this is Edgar L. Feige, Demand for Liquid ffififizpsz A.Temporal Cross-Section Analysis, Englewood Cliffs, N.J.: Prentice-Hall, Inc. 196A. Feige very carefully develops the microtheory on which his model is based. His work is discussed below. 11 The variables included in theoretical discussions of demand functions are income, tastes, and the prices of the commodity demanded and related commodities. The application of this to demand for financial assets is fairly straight forward except that the concept of price needs to be developed. Edgar Feige suggests the following approach.8 The price of holding a particular asset and enjoying its service flows is basically an opportunity cost. It is the interest foregone in holding that asset in place of an asset whose return is entirely pecuniary. Feige lets RO represent the rate of return on the hypothetical asset whose return is entirely pecuniary. This is a troublesome concept because there is no asset which is completely described by its rate of return with no other characteristics to provide satisfaction of "service flows" to its owner. For practical purposes, however, RO could also be considered as the highest return available among the competing assets. Then if Ri repre- sents the return to the ith asset, the price or opportunity cost Pi can be defined as the difference between R0 and Ri' Witkl prices thus defined and with the usual qualifying assumption that tastxes and preferences are constant, demand.equations for money and Oth£21~ liquid assets can be defined in terms of income and a vector of iTlte'rest rates on closely related assets. 1me list of assets that should be included among "closely related" monetary assets is necessarily arbitrary. One possibility in limiting 8Feige, pp. 16-18. ~xk P\ A N .mv‘ A. A F“ «a a._ 12 the list is to define specific necessary characteristics that the assets must have. For example, it might be reasonable to assume that in order for an owner to consider an asset a close substitute for money it must be stated in a fixed nominal amount and be readily convertible at close to that fixed amount into another of the closely related assets. Friedman and Schwartz suggest that this limited set includes cur- rency, demand deposits, time deposits, deposits at mutual savings banks, savings and loan shares, cash surrender value of life insurance, and series E government savings bonds.9 In their empirical tests, however, they considered only four combinations of these seven assets in order to reduce the scope of their investigation. For several practical reasons most empiricists reduce the list even further. The most frequently cited, closely related monetary assets are currency, demand deposits, and time deposits. Portfolio Theory The distinguishing characteristic of the portfolio approach is that it exqflicitly includes risk as an influencing factor in the selection of asseflss. The investor is assumed to regard the future returns to the asserts in his portfolio as random variables and to adjust his portfolio on 'tlie basis of their joint probability distribution. Risk is quantified as 'tlie variance of the return to the total portfolio and the investor is aSSIInned to minimize risk for the given level of expected return. Marlicnvitz stated this assumption more formally: ". . . the investor . S’Milton Friedman and Anna J. Schwartz, Monetary Statistics of the )hxrted.States, New York: National Bureau of Economics Research, 1970. ll‘ 91. n, H w; 1,- 13 would want to select one of those portfolios . . . with minimum variance for given expected return or more and maximum expected return for given . n10 variance or less. In looking for a rationale for the investor to regard expected value and variance of a portfolio as the parameters relevant to his investment decision, TObin discovered the clever device of parametric restrictions on the investors utility of return function. Specifically, Tobin shows that focus on the mean and variance can be justified by the assumption that the utility function is quadratic and the investor acts to maximize the expected value of utility. A full development of this approach leads to the same set of efficient portfolios as does Markowitz's mean variance rule but in more convenient form--a form more apparently akin to the linear regression models in recent empirical studies. Efficient Portfolios Let the following variables denote the investors anticipations regarding the returns to alternative assets: q = the proportion of the total portfolio held in the ith i asset i = l, 2, ... N. Ri = random variable representing the investort;anticipated return to the ith asset. Ui = E(Ri) = the mean or expected value, of the return to asset i. Mij = E(RiRj) = the second moment of the joint prObability distribution of assets 1 and j. From the above parameters associated with the individual assets it is iEMossible to derive the parameters for the total portfolio. Dropping \ loMarkowitz, "Portfolio Selection," p. 81. 1A the subscript to denote total portfolio rather than a particular asset within it, we have R = E qiRi U = E(R) = Z q. U. l 2 M = E(R ) = I 2 gig M i j j 13 1 Then if utility of return is represented as the quadratic equation: U(R) 2 a0 + alR + a2R a0 + aquiRi + a2zzqiquiRj the expected value of that utility is E[U(R)] = a0 + align u + a Ziiqiqm qu 13 Maximizing the expected value of utility subject to the constraint that the total portfolio equal the sum of the individual assets is a straight- forward problem readily suited to the Lagrange technique. If A repre- sents the Langrangean multiplier, the objective function can be written as L = a0 + a lzqi Ui + a “Ziq qMiJ qui ' ij J Thenn 3L _ 531— alUi + 2a2§quij + A 3L _ 'é‘X‘Eqi'l \ 11 1This is equivalent to the assumption that the utility function 355 euiequately approximated by the terms in the Taylor series expansion “firlcfln involve only the first two moments of the distribution. 15 When the first partials are set to zero, they can be represented in matrix notation as L1 = 159,, where _IT M_— 2a2Mll 2a2M’l2 . . . 2a2MlN 2a2M21 2a2M22 . . . 2a2M2N l H] 2a2MNl 2a2MN2 . . . 2a2MNN l l l l O ._ N+lxN+l " '.N+lxl Q = (- -alU]j) -alU2 -alUN Ll- N+le BY Sholving for the efficient set qi (i = l, 2, ... N), we SEt qi = Det(Mé)/Det(M) 16 where Mi indicates the matrix formed by replacing the ith column of _M_ wi th Li and where Det(_Mi) and Cof(_Mij) indicate the determinant of Mi and the cofactor of the i, j element of M respectively. Expanding Det(Mi) on the ith column (which is U) we get .) Det(yf) = alUlCof(Mii) + alUQCof(M21) + ... + alUNCOf(MNl + Cof(MN+l,i) By using the last expression, the efficient portfolio can be represented in such a way that the optimum quantity of the ith asset is a linear Mction of the means of the anticipated returns to all of the assets: (1) qi = Bi,N+l + BilUl + BiZU2 + + BiNUN whe re Bij = alCof(Mji)/Det(M) i = (1 ... N+l) Erical Use In order to make use of this model in any quantitative way, additional assumptions are required to reduce the number of unknown Parameters. Donald Farrarl2 used historical data on rates of return to compute sample moments for large groups of securities and used that infOI‘mation to compare actual with efficient portfolios held by mutual f.111'1618 under a large number of possible specifications of the utility fuflction parameters. In the empirical work done on demand for money, the typical use of equation (1) is to support by analogy a macromodel which includes among the regressors one or more interest rates. The most convenient £1 . l2Donald Farrar, Investment Decision Under Uncertainty, Englewood Cllffsa New Jersey: Prentice Hall, 1962. l7 assumptions to limit the unknown parameters are the following: (i) the investor feels the interest rate is no more likely to rise than to fall and thus that and (ii) the parameters other than Ui in the expression are constant ove r all observations. Also an assumption must be made which relates the actual amount held to the desired amount and makes a provision for a stochastic disturbance.l3 Alternative Specifications of Demand Functions In contrast to the models derived from assumptions regarding con- S‘umer behavior and designed to describe the demand function of an indivi- dual , the models of this section have been formulated by the authors as aggregate functions. For the most part, the authors make no reference to any specific micromodel. They formulate their models from the start as macromodels and reiterate the common rationale for inclusion of pa. rt :1 cular variables . x 13The partial adjustment model is well suited for this purpose. Zr:‘etting qi* denote the desired quantity of the ith asset, the change 1!} actual quantity from one period to the next is assumed to be propor- tlonal to the difference between actual and desired: (lit ' qi,t-l = ki(qi*t - qi,t-l) See for example G. Chow, "On the Long-Run and Short-Tun Demand for “Omar," Journal of Political Economy, Vol. 71+ (April, 1966) pp. 111—131; 1- Friend, "The Effects of Monetary Policies on Nonmonetary Financial InStitutions and Capital Markets," in Commission on Money and Credit, M138 Capital Markets, Englewood Cliffs, New Jersey: Prentice Hall, 1963, PP. 165-268; M. Hamburger, "Household Demand for Financial Assets," W, Vol. 36, No. 1 (January, 1968) pp. 97—118. T. . ~ . . .5. 18 The following descriptions are not exhaustive of the models proposed for the demand for money function, but rather represent an arbitrary selection from those models that have attracted considerable attention in economic literature. Their presentation here serves two purposes: In the first place the rationale used in developing these models is typical of that used in almost all empirical monetary economics. Thus a discussion of these models constitutes a general review of a larger body of work than the four or five titles suggest. Secondly, each of the equations discussed will be used in Chapter III as a basis for com- pari son of results derived by using similar formulations but differently constructed data. The purpose of this comparison will be to show that the macrovariables defined in this study for the purpose of bringing the analysis with the framework of linear aggregation are comparable with the variables commonly used. Eriedman Friedman does not present the exact specifications of his empirical Work and thus it is difficult to find in his voluminous work a specific, conCrete formulation of an empirically estimated demand for money funCtion. His contributions to monetary history and the data required for regression analysis are referenced in nearly every important empiri- cal work in the field, yet he has made scant use of regression analysis. ThuS we do not attempt in the next chapter to reproduce a particular equation of Friedman but it seems worthwhile to include in this review W ...—‘ ' v—‘e ~ in ~.\~ ..n‘ \r . 19 one of his early important articles, "The Demand for Money: Some Theoretical and Empirical Results."l)+ Friedman produced this article in an attempt to explain the seem- i ngly contradictory evidence on income velocity from secular and cyclical data. Secularly, changes in the real stock of money per capita are posi- tively correlated with changes in real per capita income. Using measure- ments at the bottoms of troughs in twenty cycles between 1870 and 1951+ Friedman found the following result: "A 1 percent increase in real income per capita has . . been associated with a 1.8 percent increase in real cash balances per capita and hence with a 0.8 percent decrease in income velocity."ls This decrease in velocity over the long run is in direct contradiction to the pattern it follows within cycles. Even though the real stock of money expands and contracts in conformance with the short-run cycles, the changes are not enough to leave velocity con- stant. While income increases 1 percent, real money increases only about a. fifth of 1 percent so that velocity tends to rise during cyclical e3:138-nsions and fall during cyclical contractions. Friedman's explanation of this phenomenon represents one of the early treatments of money as a durable consumer good yielding a flow of Services proportional to the stock. However, his preoccupation with the macrovariable velocity results in the development of a macromodel rather than a micromodel, even though Friedman borrows heavily from the language of microeconomics. The first step is an application of the X 11L . Milton Friedman, "The Demand for Money: Some Theoretical and EmPlfr‘ical Results," The Journal of Political Economy, Vol. 67 (August, 1959) , pp. 327-351. 15l‘bid., pp. 328-329. 20 permanent income hypothesis which implies that the quantity of money demanded, like the quantity of other goods, is adapted to permanent rather than current income. Under this hypothesis replacing income wi th permanent income in the velocity formula would provide a more accurate reflection of demand for money, and since permanent income is by construction more stable than current income, the new permanent income velocity is a more stable series than the straight income velocity Calculations. In fact Friedman finds that although this permanent income velocity seems to conform positively to the cycles, it would take only small changes in the price index used in reducing income to real terms to convert this positive conformance to the negative conformance implied by the secular results. Thus Friedman proposes that not only income but also the price level has a corresponding "permanent" series, and if velocity were calculated by using permanent income adjusted by this per- marlent price index, it would yield the same results for both cyclical and secular data. Friedman's test of this theory is extremely roundabout. First he measures the secular velocity using permanent income and prices. Then he Computes cyclical velocities for each cycle in his series by working backwards from secular or permanent velocity to the standard unadjusted velOCity formula. He compares these calculated values of velocity with ObServed values and finds a high correspondence.l6 x . 16It comes as no surprise but should be pointed out that the money VBfriable which Friedman used throughout the entire analysis includes t11119 deposits as well as demand deposits and cash. 21 Latane Like Friedman, H. A. Latane uses some informal empirical procedures in his analysis but he also includes a regression equation which is used in our Chapter IV. In "Cash Balances and the Interest Rate-~A Pragmatic Approach,"l7 Latane proposes four hypothetical aggregative equations involving two variables, the money supply as a proportion of income, m/Y, and the interest rate, R. He rejects two of these specifications wi thout the use of regression analysis and retains two closely related specifications because he was unwilling to prespecify one of the wrariables as independent and the other dependent. The first equation Latane considers is the crude Cambridge version of the quantity theory of money: NVY = K He rejects this outright, as do most economists now, as an inaccurate description of reality. "It is apparent from the data for the past 33 years . . . that M/Y is much more stable than either M or Y, but 18 that, even so, it is subject to wide variations." Latane similarly rejects a second specification 2 + + 14 BO 31(1/R) BZY as implying relationships which are readily refuted by the data. x 17H. A. Latane, "Cash Balances and the Interest Rate--A Pragmatic APProach," Review of Economic and Statistics, (November, 195%), PP- h56-h6o. l8rbid., p. #58. E Iv t b ..I. 22 Then he derives two regression lines M/Y = .oo7u/R + .11 1/R = 95.11 M/Y - 2.1+l+ by using the ordinary least squares method on the two variables M/Y and l/R treating first one and then the other as the dependent variable. 93111 S technique is not very sophisticated by current standards but by using a line which fell between his two estimates, Latane was able to predict other observations to his satisfaction. "The fit seems to indicate that the structural relations established from the 1919-1952 data. had some significance both over the longer period and currently."19 C how The most distinguishing feature of Gregory Chow's "On the Long-Run and Short-Run Demand for Money"20 is the distinction he makes between long-run or equilibrium demand for money and the short-run or current demand. Much of his analysis of the long-run is based on earlier work 1n the area of consumer durables. "The prices of services from durable gGods depend on the prices of the goods and the interest rate; the price of Services from money depends on the rate of interest. The relevant lnCome variable will be some measure of permanent income provided that 19mm. , p. 159. Gregory C. Chow, "On the Long-Run and Short-Run Demand for Money," Mnal of Political Economy, Vol. 7L1 (April, 1966), pp. 111—131. 23 the economic unit has a fairly long horizon in making its decisions."2l Thus Chow's first empirical work is with the equation (1) Mt = a0 + 3'1th + agac + et where YPt is permanent income at time t and cat is a stochastic distur- bance. He estimates the parameters of this regression with both untrans- formed data and the natural logarithms of all variables and finds long-run income and interest elasticities of approximately 1 and -.75 respectively. Then, in order to show that permanent income is a better eggplanatory variable than either wealth or current income, he reestimates the equation twice including first wealth and then income with the other variables previously included. His conclusion is that the permanent income is better than either wealth or current income.2.2 The adjustment mechanism which he uses has two components: one is equivalent to the stock adjustment model and it is this part which Chow re fer-s to as the long-run or equilibrium component; the other, short-run component is simply a constant proportion of the change in the relevant COnStraintuin this instance permanent income. To test only the rele- V‘a‘nCe of the first component Chow estimates (2) Mt=€o+£lYPt+€2Rt+€M +6 3 t-1 t \ 211‘bid., p. 113. 22An important consideration at this point which Chow ignores is that if his adjustment specification in his short-run analysis is corI‘ect then equation (1) has a specification error. 2h The equation is derived as an application of the partial adjustment model by substituting M-M =kM*-M t t-l (t t—l) into the equation: M* - a + a YP + a R + e t o l t 2 t t k in this expression is the partial adjustment coefficient, and M: is desired money balance. All of the estimates are in accordance with Chow's expectations and he takes assurance from the fact that when the logarithms of observations are used, the estimate of the coefficient of adjustment, k, is about .5 and this is consistent with his estimates for £1 and g2 of approximately half the magnitude estimated for al and 8.2 in equation (1). If equation (1) is an accurate estimate of desired money, then the partial adjustment relation yields .X. 14 = kM + 1-k M t t ( ) t-l 1 + (1+k)e M=ka+aYP+aR +1-kM ( l t 2 t) ( ) t_ t t 0 Thus , if k were equal to .5, g1 and :2 should be half of a:L and a2 since they are related as: 51 l = ka E‘22 2 The complete specification of Chow's model includes current income among the regre sso rs : = + + + M + + (3) in B0 BlYPt B2Bt B3 t_l BuYt et gm in 25 Y enters as part of the expression Y—kYP which represents saving if con- sumption is assumed to be proportional to permanent income. On this assumption and with the further assumption that a constant portion of new saving is held as money, Chow arrives at specification (3). Clearly, the coefficients are not the same in (3) as in (2) or (1). For one thing, the coefficient of permanent income must now be interpreted as having a component with -k as a factor and we thus expect B1 in this specification to be less than in the other two specifications. Again Chow's estimates conform to his expectations. Teigen The model proposed by Ronald Teigen in "Demand and Supply Functions for Money in the United States: Some Structural Estimates"23 is a three equation structural model with three endogenous variables--the money stock, the short-term interest rate, and income. Unfortunately a thorough analysis of Teigen's complete model is beyond the scope of the present research and estimation of only Teigen's demand for money equation leaves Open.the possibility of simultaneous equations bias. HOwever, since the sanme criticism could be levelled at all of the other demand equations invenstigated in this work when doubt is cast on the exogeneity of money suppiky,'we include here a description of Teigen’s demand equation and estilnate it in Chapter III for comparison with other results. Teigen's ‘Uxxiretical development of the demand for money is very similar to 23Ronald Teigen, "Demand and Supply Function for Money in the United States: Some Structural Estimates," Econometrica, Vol. 32, No. A (October, 1961*) 3 Pp. 1476-509 . 26 Tobin's transactions demand model.2h Both make use of the argument that the availability of savings deposits and other equally liquid assets makes money an unlikely choice as a means of holding wealth. Savings deposits are free of risk of capital loss due to interest rate changes and therefore dominate money as a store of wealth. Teigen concludes that " . . . under present institutional arrangements, there should exist only a transactions demand for money."25 Relying heavily on Tobin's development of the by now well known square root inventory formula for transactions demand, Teigen first derives the equation for the ith individual (1) Mi = kYi/2R and then generalizes it to the form (2) M = BOYBlRB2 Rather than follow the usual procedure of estimating this function by ‘mflcing the logarithms of all the observations, Teigen jumps to the fig tune formulation M = B + B Y + B R Y + B M (3) t o l t 2 t t 3 t-1 This is the equation Teigen uses in his structural estimates but fortu- namely he was also interested in the question of simultaneous equation bias and estimated (3) by single equation least squares. Although he 2)‘lJames Tobin, "The Interest Elasticity of Transactions Demand for Cash," Egyiew of Economics and Statistics, Vol. 38 (August, 1956), pp 0 2hl-2h7 . 25Ibid., p. h83. 27 does find evidence of serious simultaneous equation bias in estimating the supply equation, his single equation estimation of demand does not differ very much from the structural estimation. The ratios of struc- tural to single equation elasticity estimates are all just slightly greater than one. Laidler "The Rate of Interest and the Demand for Money--Some Empirical Evidence"25 contains eight variations of the basic equation M =13 +BY +BR t 0 1t 2t For each of his two definitions of M, one including time deposits and the other including only demand deposits and currency, Laidler estimates the above relationship for both a long (the yield on twenty-year bonds) and a short (the yield on h-6 month commercial paper) rate of interest. His income variable is real per capita permanent income generously pro- vided by Friedman. All variables are transformed to natural logarithms ami regressions are run on these logarithms and on their first differ- encens. Thus Laidler provides a wide variety of the possible combinations for iregressions of money on income and interest rate. In this article Laidler makes no pretense of deriving his model. He sinqfly'estimates a regression equation in which money demand is a :finurtion of income and an interest rate, and assumes that permanent in- come wcrks better than other income variables and that the choice of 25David Laidler, "The Rate of Interest and the Demand for Money-- Some Empirical Evidence," Journal of Political Economy, Vol. 7h (Decem- ber, 1966), pp. 5h3-555- 28 particular short rate and long rate indices is arbitrary and probably of little importance. Laidler's major conclusions are that there is a stable relationship between the demand for money and the rate of interest and that the short rate performs better than the long one. CHAPTER III THE AGGREGATION PROBLEM The overwhelming majority of empirical studies of demand for money (including all of those reported in the last chapter) use aggregate data for the nation as a whole. There may be several reasons for this but undoubtedly one of the most important is the ready availability of suit- able data, especially since the publication of the monumental A Monetary 1 History 9f the United States by Friedman and Schwartz. Another factor which may be of equal importance is the nature of the historical develop- ment of monetary theory. At the time when utility theory and other microeconomic concepts were developing, the monetary sector was still being represented by equations such as MV = PT. The emphasis of this equation is entirely on aggregates to the complete exclusion of concepts such as a single individual's demand for a particular asset. This con- ception of the money function as a relationship between aggregates has persisted in current empirical work despite the change in emphasis that monetary theory assumed beginning with Hicks and Keynes. In defense of the aggregation approach, it must be pointed out that it is still possible to set up monetary equations which right from the lMilton Friedman and Anna J. Schwartz, a Monetary History 9: fire United States 1867-1960. National Bureau of Economic Research, Studies in Business Cycles, No. 12., Princeton, N.J.: Princeton University Press, 1963. 29 30 start are specified as macrorelations. This simply requires an assump- tion that aggregate variables are directly related. However, even in this case it is reasonable to want to go behind the macrorelations to investigate their implications for the corresponding microrelation. Grunfeld and Griliches have argued that aggregation may actually be beneficial. They argue " . that in practice we do not know enough about microbehavior to be able to specify microequations perfectly. Hence, empirically estimated microrelations, whether those of individual consumers or producers, should not be assumed to be perfectly specified either in an economic sense or in a statistical sense. Aggregation of economic variables can, in fact frequently does, reduce these specifica- tion errors."2 Thus, there may be an "aggregation gain" in addition to whatever aggregation error may be present. The question of whether the overall result of the aggregation is a gain or a loss must be answered in terms of the goal of the research being attempted. If the purpose of the research were to determine in- formation regarding specific microparameters, it would take more than a sweeping generalization about reduced specification error to draw appro- priate conclusions from estimates drawn from aggregate data. On the other hand, if the primary research goal were a set of estimates which best predicted the value of a particular aggregate, then it is entirely possible that estimates based on the aggregates would perform better than those based on the disaggregated data. 2Yehuda Grunfeld and Zvi Griliches, "Is Aggregation Necessarily Bad?," The Review 9: Economics and Statistics, Vol. M2 (February, 1960), p. l. 31 There are alternatives, of course, to the two positions discussed. In deciding against the use of observations on single individuals, whether for theoretical reasons or because of data restraints, we do not necessarily have to accept complete aggregation as the alternative. A great deal of data is available for much smaller cross-sectional units than the total United States and, in the particular case of the monetary variables required in this study, data are potentially available by individual bank and publicly available both by states and by Federal Reserve districts. The question then arises as to what level of aggregation is best. Of course, the answer depends as before on the criteria used for judg- ing but in any event the problems caused by aggregation and the proper- ties of estimates based on aggregated data should be explored. The literature on this subject is meager. There are some early discussions in a series of articles in Econometrics3 but the first and still the most important systematic treatment is H. Theil's Linear Aggregation 2f h Economic Relations. Theil's principal contribution is in defining the links between micro and macrorelations. He makes explicit the sources of aggregation bias under several sets of assumptions but his analysis is limited almost entirely to the case of linear aggregation of linear microrelations. 3L. R. Klein, "Macroeconomics and the Theory of Rational Behavior," Econometrica, Vol. 1h (19u6), pp. 93-108; K. May, "The Aggregation Prob- lem for a One Industry Model," ibid., pp. 285-298; Shou Shan Pu, "A Note on Macroeconomics," ibid., pp. 299-302; L. R. Klein, "Remarks on the Theory of Aggregation," ibid., pp. 303-312. hHenri Theil, Linear Aggregation 9£_Economic Relations, Amsterdam: North Holland Publishing Company, 195k. 32 That is, the analysis is based on microrelations which are assumed to be linear. This may be regarded either as an important special case or as an adequate approximation for small changes in the variables. The macro- relations are also assumed to be linear and the variables used in the macrorelations are assumed to be linear aggregates of those used in the microrelations. These conditions are clearly restrictive but Theil's analysis is important both for the special cases which he does cover and for the theoretical framework he provides as point of departure in further analysis. Consistent Aggregation Actually there are two distinct problems which are often lumped to- gether under the term "aggregation bias." The first has to do with the consistency of the macrorelation with the microrelations and the second concerns the estimation of the parameters of the macrorelation. In this section we will examine the possibilities of consistency both for aggre- 5 gation over individuals and for aggregation of various assets. Aggregation over Individuals Suppose we assume that the ith individual's demand for demand deposits is a function of income and interest rates and is adequately 5The aggregation discussion is stated in terms of "individuals to conform with existing literature, but "microunit" could be substituted for individual to create the obvious generalization. The empirical work in following chapters deals with states rather than individuals as the basic microunit. 33 6 represented by the following equation: i i i i i i i i (1) AD _ BO + BYY + BTRT + BSRS The quantity of an asset is represented by A. Subscripts D, T, and S indicate demand deposits, time deposits, and savings and loan shares, respectively. Superscript i indicates ith individual or microunit. In addition, suppose that in place of this microequation an analagous macro- equation is proposed which, it is hoped, will not contradict the under- lying microequations. Theil labels this "the problem of good aggrega- tion" and states the "Rule of Perfection for a Macroequation." There is no contradiction between the macroequation and the microequation corresponding to it, for whatever values and changes assumed by the microvariables and at whatever point or period of time.7 Aggregation which satisfies this rule is what we have referred to as consistent aggregation. Convenient Macrovariables Quantity variables such as the number of dollars held as demand deposits and the amount of income have convenient macrovariables defined as the simple summations of their corresponding microvariables. Dropping the superscript from a microvariable to denote the corresponding 6The stochastic disturbance is dropped from this expression as an alternative to stating all of the arguments which follow in terms of the expected value. 71bid., p. lho. 31+ macrovariable we define the aggregates wig-1% ._ Zri l (2) K: I Wealth is another quantity variable which is frequently aggregated in the same way as A: and Y. Interest rates are ratios rather than quantities and their treat- ment requires special attention. The macrovariable for interest rate used in empirical work is certainly not a simple summation of the rates applicable to individuals. A simple average of individual rates might realistically be proposed but the more common approach is to define the macrovariable as a weighted average of the microvariables with weights equal to the proportion invested. (3) Ba =§ QR: W3. = Ai/ EA; Since w: is defined as the proportion of asset Aa, earning R: this for- mula is obviously equivalent to dividing total interest paid by total Cluaarltity invested. The convenience of this ratio in constructing data L11'1<101:L‘l:>tedly explains its wide usage. M30 1‘0 equation In order to investigate the problem of consistency we now postulate 3- macroequation based on the above macrovariables = + + + R (1*) AD Bo BYY BTRT BS s Consistency, or the rule of perfection, requires that AD, as defined by macroequation 1+, be equal to the summation of All) in the microequations 35 for all values and changes of microvariables. This problem has received considerable attention but from a slightly different point of view. Whereas we have begun with the con- venient definitions of macrovariables most often used in empirical work and asked what consistency implies for the parameters, the usual approach is to begin with microvariables and sometimes a relationship between micro and macroparameters and then attempt to develop macrovariables with some set of desirable properties. This latter approach is more often called an index number problem than an aggregation problem and some aspects of the problem have received considerable attention. How- ever, since monetary empiricists have for the most part chosen convenient macrovariables rather than theoretically elegant index numbers it seems appropriate to approach the problem as one of consistency between micro and macrorelationships starting with the variables and relationships actually in use in empirical studies. With the macrovariables defined by equations (2) and (3) and rela- tionships defined in (l) and (h) it is easy to show that consistency implies the following relationships among parameters: (5) B0 = EB: l 2 N (6) BY:BY=BY="':BY (7) Ba = Bi/Wi = Bi/wi = ... = BE/wg a = T,S In a later section we will develop and prove a general case but the above results are illustrative. They follow from the requirement in the rule of perfection that any change in a microvariable has equal effect 36 on D whether through the microequation (l) or the macroequation (h). Assuming, for example, a change in microvariable R; applicable to individual i's demand for A5, we equate the changes resulting in (l) and (A) i__ii_ __ _ ii (8) AAD _ BTART _ AAD _ BRTART _ BRTWRTART Since asset T and individual i are simply illustrative examples, it is clear that equation (7) is a straightforward generalisation of equation (9). It should also be clear that equation (6) could be explained in the same way by substituting Y for R in equation (8) and letting wi equal 1 for all individuals since each income receives equal weight in the macrovariable Y. Before generalizing these results we will look at two other special cases, aggregation of assets and aggregation of interest rates. Aggregation of Assets We rewrite here the equations proposed in the last section which describe the demand for two aggregates, AD and AT: (10) A D BoD + BYDY + BTDRT + BSDRS AT Suppose now that further aggregation is desired and, as is typical inrnonetary analysis, a macroequation is desired which describes the demand.for a combination of the already aggregated variables, for 37 example, A.D + AT. We write this proposed aggregated equation as linear in the set of macrovariables already used and let * stand for D + T: (ll) A* = 30* + By*Y + BT*RT + BngS The rule of perfection for equation (ll) requires that it be consistent for changes in variables with the summation of AD and AT described in equation (10). This summation is written (12) Z A = E (BOa + BYaY + BTaRT + BSaRs) a=D,T a It is very easy to show that the rule of perfection imposes the follow- ing constraints on the parameters: (13) 2 Bja = B. J* j = O,Y,T,S a=D,T That is, each parameter in the aggregate equation must equal the summation of the corresponding parameters in the microequations. Aggregatingilnterest Rate Variables Except for the two different interest rate terms, expression (10) is very similar to those actually used in formulating aggregate demand for money functions. The one further step required is a reduction of the number of interest rate variables. Suppose, for example, that the following macroequation is proposed: (1h) AD = 130 + BYY + BRR The two interest rates in (10) are replaced by a single new interest rate variable resulting in additional constraints on the parameters if 38 the rule of perfection is to be met. Two definitions will be considered for the new macrovariable R. The first is a simple extension of the case already described in equa- tion (3). There the aggregation involved individuals whereas here it involves other large aggregates but in both cases the new variable is a weighted average with weights equal to quantities invested. Let R in (1%) be defined: (15) R = (E AaRa)/2 Aa a=T,S a=T,S or equivalently where With this definition of r in (1A) and consistency assumed between (1A) and (10), the following constraints must be met by the parameters (16) B0 = BOD BY=BYD BR = BED/Va a = T,S These results are perfectly analagous to those discussed on page 35. Alternatively, suppose R in (12) is not a weighted average of rates included in (10) but rather some completely unrelated series such as the 39 rate on four- to six-month prime commercial paper or the rate on long- term government bonds. Each of these has been used in demand for money macroequations and been accepted as an important explanatory variable. However, there is no set of constraints on parameters which would allow a specification with either of these among the macrovariables to meet the requirement of the rule of perfection. That is, there is no pos- sibility for perfectly consistent aggregation of the microrelation (l) with macrorelation (lb) if R is not related in a definite way to the microvariables in (l). The General Case All of the above examples are special cases of the generalization which we develop and prove in this section.8 We have assumed that there are a number of microrelations which describe for each individual the quantity which he demands of each asset as a linear function of several microparameters. Generalizing our notation we let Xi be the l x K1 row vector of l...N) in each of exogenous parameters facing the ith individual (i his demand equations. Let B; be the K x 1 column vector of parameters applicable in the demand for asset Aa. Subscript a in this study will be limited to identifying the assets D, T, and S but in general we can 8The generalization of this section could be derived as an exten- sion of results provided by Theil, ibid., pp. luo—luz. Theil did not bother to generalize his results to include weighted rather than simple aggregation of microvariables. Also, the simplification which results from.introduction of matrix notation warrants developing the entire Proof. A0 let a = l,...H, where H is the number of assets under consideration. Then,corresponding to microequation (h), we write i i i (17) A8 _ X B3 X1 is written without subscript to indicate that the same variables appear in the demand equations for each asset. The system of micro- equations which describes the demand for the ath asset by all individuals may now be written. _ l ... _ (18) Al F- x1 o . . . o A B1 8 8. 2 2 2 A8 0 X . Ba . o N N A8 0 . . . o xN Ba u... c .l J- _ le Nxzkl zlel The off-diagonal 0's in the matrix whose diagonal elements are the vectors of exogenous microvariables X1 (i = l,...N) represent appro- priately dimensioned row vectors of zeros. For example, all off-diagonal elements in the ith column are l x Kivectors of zeros to conform with the l x Ki dimension of X1. 1+1 — fi — — LetA= Al , X: Xlo...oT,B=i—Bl a a a A: orL B2 a .o N N Aa o ...oxN Ba Now expression (18) may be written A8 = XBa Again, this expression represents the system of equations which describe each individual's demand for the ath asset. We can expand the system one more step to include each individual's demand for each asset by writing ll (19) A XB A=[AlA2...AH],B=[BlB2...BH] NxH KixH Expression (19) contains equations of N individuals for H assets or NH total equations. Macrovariables Let A*, X*, and Ba denote macrovariables which are assumed to be related according to (20) (20) A* = X*B* Eadh.macrovariable is a linear combination of some set of microvariables. In the case of A we assume specifically that this macrovariable is a A2 simple summation over all i and some subset of the possible values of a. Without loss of generality, we let the included assets be in the first H* columns of matrix A. Then by using a l x N row vector of 1's and a le column vector with 1's in the first H* elements and zeros in the remaining H-H* elements, (21) i = [l l l ... l ] N l x N i = [1 ... o ... ] n* 1H* lfli we can write macrovariable A* as (22) A* = 1NA1H* The K* elements of the vector of exogenous macrovariables X* are more generally defined as linear combinations of their corresponding micro- variables. Note that, as in the case when a single interest rate, R*, is the macrovariable under consideration, we include the possibility of more than one ”corresponding" microvariable. A macrovariable may be a weighted sum over individuals of more than one of the microvariables in each individual's equation. That is, the jth element of X* need not necessarily be the weighted sum over N individuals of only the jth element in each individual's vector of microvariables. It might also include, for example, element j + l or some other element from.each individual's vector. The diagonal elements of X can be put in row vector form by pre- multiplying X by the vector iN defined above . _ l‘ 2 , (23) lNX _ [x x ... xNJlszl l A3 Then by suitably defining a matrix of weights w* the macrovariables X* may be derived as (2”) X* = iwa* Each column of the matrix W* contains the weights used in forming a single macrovariable. To form the macrovariable for income Y* as a simple sum of all Yi, W* would contain column vector W*y with elements wjy where wjy 1 if the jth element of iNX is an income variable 0 otherwise In general, each column of W* may be thought of as a column of column vectors say V‘ 1‘1 W*z = WZ W2 z mil ZKixl where W; has Ki elements corresponding to the elements in Xi. The non- zero elements are the weights attached to the corresponding micro- variables in forming the macrovariable X*za the zth element of X. With the notation established it is now exceedingly simple to discover the conditions under which the aggregate of the dependent microvariables is consistent with the macroequation (20), that is, under Ah what conditions (22) is consistent with (20). Rewriting these expres- sions we have Ag. = X*B* and A* = iNAiHX- Substituting for X* and A using (2M) and (19), we have (25) iwagag = iNXBiH* Then by differentiating both sides with respect to the vector of micro- variables iNX we can find the necessary and sufficient condition for consistency: (26) W*B* = BiH* In most instances, each microvariable is used in computing only one macrovariable. When this is the case, each row of W* has only one non- zero element. It may also be that each macrovariable is a simple sum- mation over individuals of a single corresponding microvariable. Then W* may be written as a column of identity matrices. w* = I KzKl i=l,2, ... N #5 If in addition A* is a summation over individuals of only a single asset, the ath (i.e., if iH* has a single non-zero element as the ath element) then we have the well known condition9 It is clear that the consistency conditions stated in (26) are much too stringent to be expected to hold in general for equations describing the demand for liquid assets. Even in the simplest case in which the macro and microrelations have the same number of variables and each macrovariable is a weighted average of corresponding microvariables the consistency condition is unlikely to be fulfilled. As we saw in equa- tions (6) and (7), consistency even in this simple case requires that the partial derivative with respect to any change in interest rate in a microequation be exactly proportional to the weight that interest rate has in the corresponding macrovariable. This same condition holds when the macroequation has but a single interest rate variable which is a weighted sum of all the interest rate variables in each macrorelation. In the simple case it is at least conceivable that the condition might hold. If, for example, the weighting was based on quantity held, if each macrorelation was homogeneous of degree one, and if all interest rates changed in the same proportion, then the condition would be ful- filled. In the second case, however, with a single macrovariable 9R. G. D. Allen, Mathematical Economics, 2nd ed., New York: St. Martin's Press, 1966, pp. 69h-72M. All presents these results for a restricted model in the course of his concise and slightly simplified presentation of the principal contributions of Theil's Linear Aggregation, 9p, cit. A6 corresponding to several interest rates in each microequation the con- sistency condition contradicts basic economic precepts. Letting w§a be the weight attached to the jth microvariable in the ith individual's demand for asset a, we write out the rows of (26) which correspond to individual i as: i wya By* — Bya i : BRTa i WRTaBR* i i WRS 8BR-X- : BRS a Since each of the weights is a positive number and the ratio of each interest rate parameter to its corresponding weight is equal to BR*’ these conditions imply that all partials have the same sign whether they are own first partials or cross partials. This means that the direction of change is the same for every asset when any interest rate changes. Few economists would be willing to accept this g priori restriction. General and Specific Inconsistengy Before looking at the effects of using a macrorelation which is not consistent with the system of microrelations it will be useful to dis- tinguish between general and specific consistency and to define corre- sponding measures of inconsistency. So far, we have dealt only with general consistency which is a property of a system of equations when all of the microvariables are free to take on any values. In this use, consistency is a property of a model and does not depend on particular ‘values of microvariables. A? In later sections, however, our primary interest will be in the use of a model with actual observed values of the microvariables. Thus we are not as concerned about general consistency with maximum.degrees of freedom for the variables as with the consistency in particular in- stances when the microvariables are given and thus in a sense have zero degrees of freedom. It is clearly possible for a system of equations to be consistent for a specific set of values and not be consistent in general, so the distinction is useful. To develop measures of inconsistency corresponding to these two uses we will consider the conditions of equations (26) in the special case when A* is a simple sum over individuals of only one type of asset, say asset a. Then iH* contains a single non-zero element and (26) may be written (27) W*B* = Ba The most obvious measure of the inconsistency of a system of equations would seem to be the difference between W*B* and Ba' However, since these are vectors rather than scalars perhaps a more convenient measure would be the inner product of the difference (28) d'd measure of general inconsistency where d = W*B* - Ba This measure would be zero if and only if all of the consistency con- ditions of (27) were met and it would, of course, increase in value the larger the absolute value of any deviation from the consistency #8 conditions. Of greater practical importance to the purpose of this study is a measure of inconsistency for a specific set of values of the micro— variables X and the macrovariables iNXW*. To establish this measure we will assume that the vector d defined in (28) has at least one non-zero element and ask what effect this has on consistency for a particular set of values. We can rewrite (25) substituting Ba + d for W*B* to allow for general inconsistency. This gives (again assuming ng has a single unit element): (29) iNX(Ba+d) = iNXBa Then the amount by which quality (29) fails to be satisfied is a measure of the specific inconsistency. Denoting this measure as L we have (30) L = iNXd L is a scalar, the dot product of iNX and d, and it is obvious that gen- eral consistency is not a necessary condition for specific consistency. Aggregation and Estimation Most of the implications of estimating a macroequation which is not consistent with a system of microequations have been developed in the 10 literature and some of them are well known. However, with a slight loSee especially Theil's Theorem 7 in Linear Aggregation, 2p. cit., pp. 120-121; and H. A. J. Green, Aggregation 32 Economic Analysis, Princeton: Princeton University Press, 1965, pp. 99-106. A9 modification of the matrix notation established above it is possible to simplify considerably the presentation of these results. Suppose we have T observations on each of the N microequations which we now assume contain a stochastic element e%: i _ i i i (31) Aat _ XtBa + et Now if we generalize our notation to let Aa’ X and ea represent matrices of all Observations on the variables r‘w -- .r; - (32) Aa = Ail , ea = 9:1 , x1 = X1 , x = xlo ... o .Aig eé2 x; o x? O O .O AiT e:T X1 o oxN A2 e2 "ikK _- NTxNK al al AN eN aT L_aT ‘7 NTxl NTxl We can write the entire system of microequations for all Observationsll (33) Aa = XBa + ea llWe confine our attention at this point to the case in which the dependent macrovariable is the sum.of dependent microvariables describing a single asset—-for example, all demand deposits or all time deposits but not both. 50 Now corresponding to the vector iN which facilitated the formation of the macrovariables we define the matrix iT to use in creating the T observations on each macrovariable, (31+) iT=[Im,I,I, ] .L T T ITTXNT where IT is a T dimensional identity matrix. Then letting A* be the T x 1 matrix of observations on the dependent macrovariable which we continue to define for each time period as the summation of all the dependent microvariables, A1 a? we have (35) A* = iTAa = lTXBa + iTea Similarly we define the matrix of observations on the macrovariables Z* as (36) 2* = iwa* and introduce the T x 1 vector of macrodisturbances (37) a. = iTea Thenall observations on the proposed macroequation under condition of general consistency may be written as (38) A* = iTXW*B* + 8* 51 We now consider the OLS estimates of B*.12 Theil's approach is to take the expectation of 3* (39) E[D*] E[(z;z*)'lz;A*] or, using (35) (No) Em] (Z~£z*)’lZ.LiT>CBa Since (ZAZ*)-lZAiTX describes the matrix of coefficients we would find by regressing each vector of iTX on 2*, expression (MO) duplicates results achieved by Theil "The parameters estimated are sums of weighted averages of microparameters . . .--the weights being equal to the co- efficients of those regression equations which are Obtained when the statistical method used for the estimation of the microequation is applied to the linear equations that de- scribe the exogenous microvariables as functions of the exogenous macrovariables . . ."13 Another way to approach this result is to rewrite (38) making use of d defined in (28) and allowing for general inconsistency (A1) A* = iTXW*B* + iTXd + e* 12As Theil points out, the following discussion could very easily be generalized to any estimation procedure which is linear in the de- pendent variable and unbiased under the condition that the expected value of E* is zero for all observations. Theil, Linear Aggregation, p. 119. Also see Theil's presentation in "Specification Errors and the Estimation of Economic Relationships," Revue Institute Inter- nationale g3 Statistique, Vol. 25 (1957), pp. hl-5l. l3Theil, Linear Aggregation, p. 121. 52 The vector d in this expression may be thought of as another group of unknown regression parameters and their omission from a regression equation is a common type of specification error. It is well known and very easy to show that application of OLS to only part of an equation leads to parameter estimates which are under general conditions both biased and inconsistent.lu Looking again at the expected value of the OLS estimate of B* based only on the macrovariables we can rewrite (39) as (A2) E[ 3*] = E[(Ziz*)‘1ziA*] Then, using (Al), we can write (#3) E[é*] = B* + (ZlZ*)‘lZ*z%Xd This expression clearly separates out that part of B* which is appro- priately defined as aggregation bias. The direction and magnitude of the bias quite clearly depends on both the covariances of the macro and microvariables over the sample period and on the elements of d. One other approach to the relationship between micro and macro- parameters will provide useful information. In our last approach we postulated some true set of macroparameters and then analyzed the prop- erties of estimates of these parameters based on our fig ggfig definition of the "true" model and "true” parameters. In comparison, we now adopt 11+See Jan Kmenta, Elements 9£_Econometrics, New York: Macmillan, 1971, PP- 392-395. 53 ' a definition which is an fig post definition of the "macroparameters,' based on our estimates, and then examine the properties of the "para- meters" given the nature of the estimates.15 Specifically, we define the "macroparameters" to be the expected value of the OLS estimates B*. Then we write the macrorelation. (AA) A* = Z*E(E*) + u* A comparison with (35) yields (A5) 2*E(é.) + u. = i XBa + i e T T a Then, using (MO), we obtain "-1'. _. . (M6) 2*(Z*Z*) Z*1TXBa + u* — lTXBa + lTea and since (242*)"1ZiiTX is a matrix of coefficients obtained in the OLS regression of I X on Z we can define T . A , -l .- (A?) lTX = 2*(z*z*) Z*lTX so that u* in (MM) can be written as (M8) u* = (iTX-iTX)Ba + lTea and (MM) may be rewritten as (’49) A* '2 Z*E<:é*) + (iTX-1T£)Ba + iT ea 15"Parameters" is used with quotation marks because this use con- tradicts the generally accepted notion of a parameter. 5M Under this specification, it is clear that if each vector in iTX is an exact linear function of 2* so that iTX = iTX, the macrorelation may be A written without the (iTX — iTX)Ba term and the error term is the simple sum of errors in the microrelations. Another implication of specification (A9) is that if 3* in (M1) is set equal to E(B*) then (50) iTXd = (iTX _ iTX)Ba 2 Aggreggtion and In most of the demand for liquid asset studies, R2 is at least as important to the researcher as are the parameter estimates. Unfortun- ately, Theil and most other students of the aggregation problem have ignored the goodness of fit aspect of the aggregation problem. Grunfeld and Griliches are important exceptions and their article, "Is Aggregation Necessarily Bad?", was important to many of the results in this sec- tion.16 One of the drawbacks of their presentation is lack of distinc— tion between "residuals" and "disturbances."l7 The usual practice, of course, is to let "disturbance" denote the unknown (and usually random) element in a regression equation and let "residual" be the calculated different between the dependent variable and its predicted value with the prediction based on estimates of parameters. 16Grunfeld and Griliches, 22. cit. 17For example, ibid., p. 6, they explain a quotation from Theil as meaning "...the residual variance of the macroequation must be larger than the variance of the sum of residuals from the microequations." Theil's statement actually applies to disturbances and sums of dis- turbances. 55 Variance of u* Let us assume that the microdisturbances, ea, all have finite variances. Since the macrodisturbances u* are seen in (M8) to equal the sum of microdisturbances plus the non-stochastic term (iTX-iTX)Ba, the variances of the elements of u* equal the variances of the elements of e IT a. A (51) Var [u*] = Var [(iTX—iTx)Ba + i ea] = Var [iTeg T This means that in terms of the variance of the disturbance we do not have an.g priori reason for choosing the aggregate model over the sum of the micromodels. Perhaps a better indication of the predictive power of the models than the variance, or second moment about the mean, is the second moment of the disturbance about zero. For the sum of the micromodels this measure will be the same as the variance if the elements of ea are as- sumed to have zero mean. Letting it be the tth row of iT we can write the sum.of microdisturbances at period t as itea. Then (52) E[i e —E[i e 12 = Var [i e ] t a t t a For the macrodisturbance, however, the measure is different. Using (51) we have 2 (53) E[u§t] E[(itX-iti)Ba + itea] (5M) Var [itea] + [(itX-itX)Ba]2 2 Var [itea] 56 It is clear from this expression that E[u§t] is greater than or equal to Var [itea] so that in general the second moment about zero of the macro- disturbance is larger than the variance of the sum of the microdistur- bances. Two conditions are immediately obvious which change the weak in- equality in (5M) to an equality: (1). If itX is a linear function of th so that i X equals itX: (2). If W*B* equals Ba so that d is a vec- t tor of zeros by (50) we have (55) ith = (itx-itx‘)sa = 0 While it is true that these are only sufficient and not necessary con- ditions for the equality, it is also true that they are not apt to be met in the demand for liquid asset specifications under consideration. Thus on the basis of second moments about zero the macrorelation suffers some disadvantage. Correlation Coefficients R2 can be represented as 1 minus the ratio of the sample second moment about zero of the residuals to the sample second moment about the mean of the dependent variable. Thus for the macroequation we have (56) R: = l-(SS*/SA*) where Sd* and S represent sample second moments about the mean of the Ax- calculated residuals G* and dependent variables A*. Then we follow Grunfeld and Griliches in defining a "composite R2” which measures the percentage of the aggregate dependent variable which is explained by summing the predicted values of the dependent variables from each 57 microrelation. We do this by finding the residuals for each micro— equation and adding together all the residuals for a given year to form a composite residual for that year. The vector of these composite residuals may be written 2; = iTe and we let Sé+ be the second moment of these residuals about zero. Corresponding to these composite dis~ turbances we have a composite dependent variable which at each time period is calculated as the sum of all the dependent microvariables for that period. Since this is exactly the same as the definition of the dependent macrovariable we can write second moments about the mean for each of them as S Now we define a composite RE as A*‘ 2 (57) R+ = 1 - (Sé+/SA*) 2 A comparison of R* and RE shows that the relationship between these two measures of goodness of fit depends on the relationship between S£* and A 88+. In light of the result implied by (5M) that Su*ZSe+ it is tempting to think that perhaps REzRE. Of course, this is not the case. The moments in R2 are moments of residuals not of disturbances and when that distinction is kept in mind it is not surprising to find that RE £22.9E greater than RE. Nevertheless, it is well known that under certain assumptions commonly made in model specifications Sa and Sé+ when * adjusted for degrees of freedom are unbiased estimates of Su* and Se+ so that E[Sfi*] Z E[Sé+] and on this basis we might expect the composite correlation coefficient R3 to be greater than RE. In two separate cases, however, Grunfeld and Griliches found that the aggregate correlation coefficient was at least as large as the composite correlation 58 coefficient which led them to the conclusion that "...aggregation is not necessarily bad if one is interested in the aggregates."18 In view of the importance attached to the R2 statistic in the literature, this conclusion of Grunfeld and Griliches deserves further attention. In the empirical studies of demand for liquid assets the co- efficient of determination has probably been used to justify more con- clusions than any other single statistic. Almost every researcher in the monetary field has used R2 in his defense of selecting one reported equation specification over another, to say nothing of the perhaps thousands more specifications which are never reported. Friedman 2 to define money! His carries it one step further. He has used R wording is a little less direct but nonetheless he uses R2 to select the correct dependent variable in the demand for money equation. "The cri- terion was that the correlation between the total called money and national income be higher than between each of the individual compon- "19 ents of the total and national income. Specification Error 22 Microrelations Before calculating and comparing in the next chapter the actual values of R5 for the macroequation and the analagous microequation com- 2 posite coefficient of determination R+ , we will generalize a rationale given by Grunfeld and Griliches to explain their experience in con- 2 sistently finding R* > RE. We have assumed throughout the previous 181mm, p. 10. 19Friedman and Schwartz, Monetary History, p. 177. 59 analysis that the microrelations were exactly and correctly specified. This assumption is useful under many circumstances but it clearly is impossible to expect it to always be true in practice. Now if the microrelations are not specified correctly then their residuals con- tain a term which is a result of the specification error, and the second moments which we calculate from the residuals will be larger on the average than the actual second moment of the disturbances in the true model. Under these conditions, it is impossible to specify §_priori that E[S;*] Z E[Sé+] because the relationship depends not only on the effect of aggregation but on the effect of the specification error in both the microrelations and in the macrorelations. Suppose that instead of estimating with the true model A = XB + e we use the incorrect model A 2 VB + disturbance. It will be important to the later analysis that V is defined similarly to X as a block diagonal matrix of exogenous variables. Now if we estimate B by OLS we have: (58) a = (V'V)‘lv'A Then if we use these estimates of B to calculate residuals e we get A (59)é A-VB e + XB - VB (I - V(V'V)'lV')(XB + e) and as long as V is non-stochastic and all of the true disturbances have zero mean we can write the expected value of the sum of squared WV 6O residuals: (60) was) E(e'MVe) + B'X'MVXB where (I - V(V'v)-lv') Mv Similarly we find the expected value of the sum of squared composite residuals (61) E(é'i.£,iTE) = E(e'MVi,]'jiT'ufe) + B'X'Mvir'riTNR/XB Equation (61) is the expression we wish to compare with the sum of squared residuals in the macroequation A* = V*B* + u*. Here V* is assumed to be a T x K matrix of observations on a set of exogenous macrovariables. If we estimate B* by OLS we have a situation very similar to one just presented and we can write (62) 51* iTA — wig T . + . MV*(1TXB 1T8) MV* is defined similarly to MV with the substitution of V* for V and appropriate changes made in the dimension of the identity matrix. NCw we can write the expected value of the sum of squared residuals of the macroequation: (63) E[u'u] = e'igPMWMWiTe + B'X'in1V*MV*iT)CB In comparing the expressions in (61) and (63) it will simplify the analySi£3 a great deal if we make a restrictive assumption regarding 61 the distribution of the microdisturbance e. Specifically, we assume that (6M) E[ee'] =0 21 where 02 is a positive real number and I is an appropriately dimen- sioned identity matrix. While this assumption is clearly restrictive there are many circumstances, especially with cross-section and time series data combined, when observations can be transformed so that the new disturbance term conforms at least asymptotically to the above speci- fication. The first term on the right hand side of (61) is E[eMVifiTMVe]. Now because V can be partitioned as a block diagonal matrix, MV is also block diagonal. Mé, the diagonal component corresponding to the ith microequation is IT - Vi(Vi'Vi)‘lVi' where IT is a T dimensional identity matrix. Now since the matrix iTiT may be partitioned into NN sub- matrices all equal to IT, the matrix product MViTiTMV can be partitioned with submatrices on the diagonal equal to MéMé. Now since MV is idempotent the product MViTiTMV has diagonal elements identical to MV’ (65) E [e'MviqiTMVe] =0 2 Trace [MvifiTM] :0 2 Trace MV = (NT — NK)J2 where K is the number of exogenous variables in each microequation. We assume that the observations are linearly independent so that V'(V'V)'lV has rank and trace equal to NK. . . . . 1., . The first term on the right hand Side of (63) IS elTMV*MV*lTe' 62 We assume that V* has full column rank equal to K* and it is easy to show (66) E[ei'MV*MV*iTe] = N(T-K*)02 A comparison of (65) and (66) shows that if each expression were cor- rected for degrees of freedom, as it would be if R2 were used in place of R2 as goodness of fit criterion, the results would be equal. The relationship between (61) and (63) thus depends on the other terms B'X'MviéiTMVXB and B'X'ifMV*MV*iTXB. These are easily reduced to B'X'MVXB and B'X'ifMV*iTXB and then B'X'B — B'X'V(V'V')'1V'XB and B'X'igiTXB - B'X'iiV*(V*V*)'lV*iTXB. Without more specific information on the nature of V and V* not much more can be said. It is readily apparent. however, that the relationships depend on how closely related the variables used in the regression are to the true variables. The first expression depends on the correlation of X and V and the second on iTX with V*. Disaggregation The arguments in the previous sections have been based on the assumption that each individuals demand for a particular asset could be written as a linear function of a particular set of variables. This is the microtheory approach to the demand for money and there are clearly defined problems when aggregation of the microequation is attempted. However, the majority of the specifications described in Chapter I have to be classified as macroequations and none of them pretends to be a linear aggregation of all microequations. It is tempting to think that in these cases there are no aggregation problems. However, unless the 63 author of a particular macromodel is willing to place absurd restric- tions on the applicability of his model, the problems of aggregation or, perhaps more appropriately, the problems of disaggregation still require investigation. To take an extreme example, it seems most unlikely that an author would deny the applicability of his micromodel if a single individual whose balances had been included in the specification happened to emmigrate. A little more realistically it is doubtful that any of the authors in Chapter I would admit that total demand could no longer be written as a linear function of macrovariables if data for Hawaii and Alaska were excluded. Since the exact boundary between macro and micro- economics is not defined we are left wondering at what point in the process of excluding individuals one by one from the macromodel does the model cease to apply. In this research we investigate the problems of aggregation when the macromodel is assumed to apply to units as small as single states. When this is the case we can write an equation for each state which is linear in some set of variables and the problems associated with aggre~ gating over all states are exactly analagous to those associated with aggregating over individuals. Actually, even the question of aggregating over individuals will be explored in this paper by using statewide data. Essentially, we ask in both micro and macro tests whether summing only over individuals within a single state is better than aggregating all individuals in all states. CHAPTER IV ESTIMATION I The purpose of this chapter is to explore the practical effects of aggregation on the estimation of demand for money equations. In order to do this, we first estimate with aggregate data the equations devel- oped by the several authors discussed in Chapter I. It would have been desirable to repeat estimation techniques of those authors with their data and thus hopefully reaffirm exactly their results, but several circumstances dictate against this technique. Probably most important is the fact that virtually none of the data series used by the original authors are available on a statewide basis, so that even if the aggre- gate data could be reproduced (or borrowed), the state data would still be lacking. Further, in order for the estimation procedures and data to conform to the theoretical framework established in the last chapter, the macrovariables must be fixed weight aggregates of microvariables, and the data used by the original authors were not constructed in this way. Convenience often dictates choice of variables and with all of the published series of corporate, government, and bank interest rates it is hardly surprising that the previous authors selected readily available series rather than constructing weighted averages. Thus the first step in investigating the effects of aggregation on empirical work is the formulation of variables which conform to the theoretical framework established in the last chapter. 6M 65 Definitions 2: State Variables There has been a great deal of controversy in the monetary field as to the appropriate definition of money but the area of disagreement has been over what, if anything, should be included in money besides currency and demand deposits. Yet this basic accepted quantity, the sum of currency and demand deposits, is an aggregate of quite dissimilar quantities and it is entirely possible that these quantities are most accurately described with separate demand equations.l Unfortunately, figures for currency in circulation are not available at the state level so that this aspect of aggregation cannot be explored. Demand and Time Deposits The finest available breakdown of deposits by ownership is the sum of deposits held by individuals, partnerships and corporations. It could easily be argued that these IPC demand and time deposits are also heterogeneous aggregates and ought to be broken down further. In this study, however, IPC demand and time deposits at insured commercial banks by state are the finest breakdown attempted of the asset vari- ables. There is more justification for this than the availability of data. The purpose of the empirical work in this chapter is to discover the effects of aggregation to national totals on estimating demand functions for money. The deposits used in those aggregate equations by previous investigators have been IPC deposits. Any division of assets into quantities accurately defined by a single equation is arbitrary lPhilip Cagan has explored this topic with national data in "The Demand for Currency Relative to the Total Money Supply," Journal 2: Political Economy, Vol. 66 (August, 1958), pp. 303-328. 66 and the IPC figures deserve attention if for no other reason than because they have been used and were useful in the past. The specific figures used in this study are IPC demand and time deposits at all insured commercial banks, by state, taken from the Reports of Call For 19u9-68. Interest Rates The following quotation by Laidler typifies the rationale for selecting an interest rate to use in empirical work. Laidler explained his selection of the rate on M-6 month commercial paper as ". . . prompted by the fact that a short rate seemed a more appropriate proxy for the opportunity cost of holding money than a long rate, and partly because the variable performed slightly better than the yield to matu- rity on 20-year bonds in a series of preliminary tests."2 Friedman uses this same sort of argument based on opportunity cost to explain his use of yield on corporate bonds.3 The fact is that almost any return could be rationalized in one way or another and the high corre- lation among rates would probably yield unimportant differences when one was substituted for another. In this reasarch, the rates of return on time deposits and on savings and loan shares are assumed to be the relevant Opportunity costs of holding demand deposits or, alternatively, they are assumed to be directly related to the prices of the service flows from the closely related assets. All rates are calculated in the same way, namely by dividing total interest payments to each asset in a 2Laidler, "Some Evidence," p. 55. 3Friedman, "The Demand for Money," p. 3M5. 67 given state by the average quantity of the asset held for that period. Then for each state a third interest rate is calculated which is a weighted average of the other two rates. Although a number of reason- able alternatives exist for the selection of weights there is little practical difference in the resulting numbers and with no precedent in the empirical work the choice is in any event completely arbitrary. The central purpose of exploring the effects of linear aggregation eliminates from consideration the immediately apparent index based on weights which change each year in proportion to quantity invested. If the analysis is to conform to the framework of linear aggregation the weights must remain constant for the entire period under consideration. Roy Gilbert proposes that an appropriate index could be constructed with the use of principal component analysis and he includes the follow- ing as a desirable property of such an index: "In the case of both the Paasche and Lespeyres indices the selection of the weights depends upon the arbitrary choice of the base year . . . . Thus the choice of base year affects the characteristics of the index. With the principal component method the estimates of the weights are objectively determined by the data."u Gilbert also describes other desirable properties of the principal component method but for the present purpose the extra computational effort of the procedure does not seem.merited. Of course the choice of base year in the Lespeyres index is arbitrary but the choice of data to use in the component method is also arbitrary and the weights depend entirely on the data selected. uRoy F. Gilbert, "The Demand for Money: An Analysis of Specifica- tion Error," Unpublished Ph.D. dissertation, Michigan State University, 1969, pp. M6-M7. 68 A preliminary investigation of the changes in a Lespeyres type index resulting from different base year selection reveals very small differences. The rates of return to time and savings deposits for the United States as a whole in 19M9, calculated as total interest paid divided by the average asset value, were .913% and 2.518% respectively. This is the year of both the largest absolute and the largest percentage difference in the two rates for the twenty years covered in this study. Yet the choice of base year weights has a very small effect on the resulting average. The 1968 base weights of .528M for time deposits and .M7l6 for savings and loan shares yield a 19M9 index rate of 1.669%. The 1959 weights yield 1.613%. The largest difference that choice of base years can possibly make is a little over .3% between the unusual year of 19M9 when time deposits heavily outweighed savings and loan shares (2.7 to 1) and 1961 when the two quantities of assets were almost equal. As the interest rates move closer together over time, it is clear that choice of weights becomes even less important so that in 1968 (when the rates are nearly equal) the choice of base year weights can change the index by no more than .O5%. The base year 1968 was selected to construct the index. It is the latest year for which data were collected and the weights are between the extremes of 19M9 and 1961. Income Total personal income and per capita personal income are available on a state basis. The particular series used in this research is a Department of Commerce series uniformly constructed for the years 69 l9M8-l968 inclusive.5 The permanent income data had to be constructed. In Spite of the fact that Friedman's permanent income work has come under severe attack ever since its initial publication,6 many researchers continue to use figures supplied by Friedman in their calculations which involve perma- nent income. This practice has several shortcomings. In 1969 Colin Wright tried unsuccessfully to reproduce Friedman's figures even though he used essentially the same data and original formulation of permanent income's definition. He states: "My estimates of the weight current income had in determining permanent income differed from those obtained by Friedman and did so in a systematic and interesting manner . . . . If my results are correct, then not only are the original calculations made by Friedman suSpect but the use to which they have been put by Friedman and others needs modification."7 Gilbert also sharply criticizes Friedman. After pointing out an obvious case of omitted variables, Gilbert subjected Friedman's con- sumption function to four tests for specification error which ". . . resulted in rejection at better than the 5% confidence level by all 8 four tests." By estimating much more inclusive consumption functions, Gilbert was able to find one which passed the specification error tests 5United States Department of Commerce, Office of Business Econom- ics, Survey gf Current Business, Vol. M9, No. 8 (August, 1969), pp. 1 -15. 6Milton Friedman, 5 Theory 2: the Consumption Function, Princeton: Princeton University Press, 1957. 7Colin wright, "Estimating Permanent Income: A Note," Journal pf Political Economy, Vol. 77 (September/October, 1969), p. 8M6. 8Gilbert, "Demand," p. 50. 70 and gave reasonable estimates of other parameters.9 From his estimates he calculated that the weight which current income receives in deter- mining permanent income is .52. This figure is right in between the .3 to .M range of Friedman and the .7 to .8 range which Wright calcu- lates so it was considered a reasonable choice in creating the state permanent income data. Since actual estimation of the series for each state was not possible, it was necessary to make an assumption about the initial values of the permanent income series. Real per capita personal income in 191+7 and 19M8 are equal and approximately 1.5% above the 1919 figure. In 1950, income jumps by almost 7%. The years prior to 19M7 are dis- torted by the war. In our calculations, we use l9M8, the middle year of the three year period of almost constant income, as a base in calculating the permanent income series. We assume that in this year permanent income is equal to the actual measured value of current income and although this choice is arbitrary it is certainly no more arbitrary than Friedman's creation of past income data by extrapolating backward using an assumed 2 per cent growth rate. Friedman creates data both to begin his series and to replace war year data. While 2% may be a good overall index of growth rate it is more than twice the actual rate for the war years of the 19MO's. The growth in per capita real income in 19M8 prices from $1,301). in 19112 to $1, 365 in 19A? implies an exponential growth rate of a little less than 1 per cent per year. Thus using a 2% rate to span these war years or create the additional data would 9Gilbert, "Demand," p. 62. 71 overstate considerably the actual growth. It is hard to imagine that people's expectations for these early post war years were 5% higher than the acutal incomes when the actual income levels were at an all time high. Fortunately, as with the interest rate variables, the selection of a particular weighting scheme and an initial value for the series seems to have little practical effect on the results. In a preliminary check on the data a simple linear regression was run on data for the state of Michigan. The series based on a weight of .52 for current income with an assumed initial value equal to 19M8 current income was regressed on the values developed by Feige for use in his estimation of the demand for liquid assets. Feige bases his weighting scheme on Friedman's work but makes no mention of how he develops initial values for his series. The correlation between Feige's figures and those used in this research is .9958 and it is well known that if the correlation were 1 the effect of substituting one for the other in another regression would be exactly the same as scaling the original variable with a linear transformation. Sudh a scaling has no effect on the usual t tests of significance of the variable. Friedman and Wright both use real per capita income in constructing their permanent income series. The original data used in this study are similarly scaled so that with Yt interpreted as per capita real personal income the formula used in constructing permanent income for each state was * * Yt 2' .52Yt + .M8Yt _ 1 As pointed out above, the series was begun in 19M8 with the assumption 72 that current real per capita income for that year was equal to permanent or expected income. Adjustment Variables The population data were calculated as the value implied by the per capita and total income figures described above. However, the figures obtained in that way were subjected to a preliminary comparison with an independently formulated set of population estimates10 and all twenty of the figures checked fell with .05% of each other. Asset values and income series were adjusted to real terms by dividing by the consumer price index for all items with 1969 equal to l. The original series from United States Department of Labor, Bureau of Labor Statistics, Handbook 2: Labor Statistics 1971 was transformed to make 1969 the base year. Macrovariables The macrovariables used in this study are fixed weight linear ag- gregates of the microvariables just described. The unsealed quantity variables, demand and time deposits and current nominal income, are simple summations over all states of the correSponding microvariables. The real variables corresponding to these nominal ones are similarly constructed. Since the consumer price index is used to adjust the nominal data in all states, the real macrovariables are computed equiv- alently as either the simple summation of real microvariables or the 10United States Department of Commerce, "Population Estimates and Projections," Current ngulation Reports, Series P-25, No. M36 (January, 1970), p. 13. 73 summation of the nominal microvariables with the summation then adjusted by the consumer price index. The ratio variables require a little more attention. It was necessary to construct fixed weight indices for both real income per capita and the interest rates on time deposits and savings and loan shares. As with the index of the two interest rates within each state, these also were constructed with weights from the base year 1968. The real income per capita macrovariable is the weighted average of the corresponding figures of each state with the weights proportional to the 1968 population for each state. Each macro-interest rate index is similarly a weighted average of the state data with weights proportional to the 1968 quantities of the assets held. Perhaps a more common procedure than the one just described for constructing indices is to use a weighting scheme which changes each year in proportion to the change in the selected weighting variable, i.e., population or asset value. Although this procedure could not be used and still keep the research within the scope of fixed weight linear aggregation, the indices produced by the two methods are not signifi- cantly different. This is not a surprising result. There are two circumstances in which a variable weight index correSponds exactly to the fixed weight index, and both of the conditions are very nearly met by many economic variables. The two indices are the same if all of the variables used for weighting change in the same proportion in each state eadh year. Either of these conditions is sufficient and both are often approximated by economic data. 7M Equation Estimates In this section we take up the problems of estimating the param- eters of the regression equations. In general, we will want to estimate each specification using data both from total United States and from each individual state. The regressions using total U.S. data are fairly straightforward. They are based on assumptions which the original author made either explicitly or implicitly in his original specifica- tions of the model. In general, these are classical assumptions and are so well known they need almost no explanation. The regressions using state data, on the other hand, lend them- selves to an estimation procedure developed by Zellner.ll In this procedure, information about the error terms is used to provide esti- mators asymptotically more efficient than single equation estimators. Macroequation Estimates Table 1 presents the results of ordinary least squares estimation of all of the demand for money specifications developed in Chapter I using in each case the macrovariables just defined. It is obvious that all of these equations cannot be correctly specified at the same time, even with respect to which variables are included, let alone with respect to functional form or the scaling and transforming of these variables. A more subtle inconsistency of these equations appears in the nature of the disturbance terms. If, for example, the demand for money is written in one instance as linear in income and interest rate llArnold Zellner, "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal pf American Statistical Association, Vol. 57 (June, 1962), p . 3M8-368. 75 with the assumption that e is a normally distributed random disturbance which has zero mean and constant variance, then the same assumption cannot consistently be made for the disturbance term When the original equation is rewritten in the logs of the original variables. That is, if and e2 = ln(A) - ln(X)B then el and e2 cannot both be distributed the same. Unfortunately this argument, although correct, is not very useful. Economic theory does not pretend to specify the distribution of the stochastic disturbance in demand equations. As a matter of fact it is only recently that a stochastic element has figured in economic analysis at all. Thus the nature of the disturbance must be assumed and the most convenient assumptions are those which, if they were true, would yield estimates with desirable properties. In this reSpect economists have, by default, allowed their tools to dictate their assumptions. All of the results in Table l are derived from least squares esti- mation. This is the technique which the original authors applied to each of the equations with the implicit or explicit assumptions that it would yield unbiased estimates of both the coefficients and their variances. When the results in Table l are interpreted in conformance with the same assumptions most of the conclusions are in general agree- ment with those previously drawn. Certainly the results support the 76 .mEHop spammo aom Hoop me was meofipmowo omonp :H moaomflam> hpflpcoow o£B* .moOflpmfi>oe pampompm was mononpeouom ea monomflm .mpflmomoo asap mosaoefi pew oooaoxo op >Ho>flp noommoa oooflmop homes mam m2 pom H2 .homom Hofloaoeeoo Speoe on: Go oaofih u om .mopmaomaoo eaopnwooa so mama» N am .qm poo mzw pooaado moms oedema .om one mac peoaaoo mom: domfloe .qm pom om neon one «oeooofl pooGsanm m.omeooflam mom: aoaoflmq .qm one oeooefl pooomeaom m_osEdoHsm “mamaaoe poossso ea mzw was moans :flnm> m.30£o .moaoofiam>oaome popmmoawws adamoefia wean: modem wasp cw podwppno mopwefipmo one Boaop.zapooaflo Go>flw one «moaomflam> we: moaws “Hogpoo Hocflwwao one an poeflmpoo mopmeflpmo one acoflpmsoo zoom mom ”wopoz Eco; A88 em: omma.- Foam- amo.a amamm m2 some Aowaa.v Amoamv Aomma.v Ammomv mam. mmma- sewn- mmam. oasmm m m a m as om Amwmv Ammao.v mama- some. swede as sage Aommmv Ammo.v Amemav Ham. mFOm- oma. madam a a m ma om mm moanoflam> pooooomopoH .no> ao£p5< .oz .mom Hoeflmfiao magazHemm onaapomomoaz mo zomHm pemoQUMoocH .aw> nonpo< .oz .mom Hocflwfiao A.o.wsoov a moose 78 Samoa ammo; Como; $23 a mam. asmm. mama. mmmm.- mace. smom. 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Aa+mvsa< *ma Evade Sign om Asmo.v Awmm.v mmm. maa.- mam. amo.- Amzvsa woaoaaa mam. amo.- 6mm. amo.- Aazvaa woaoaao Amoa.v Abom.v Aoao.v awn. mmm.- Ho.a moo.- Amveae *ma Amine Earns on m moapmflaw> pcooeomooeH .am> asgpae .02 m .mom Hmeflmfiao Ao.osoov a mom(w*B* 048.13.. Each of the first six rows of the 6N x 5 matrix Q_lw* has a l as the only non-zero element. The last 6N—6 rows consist entirely of zeros and we have the following null hypotheses: N 1 2 3 *° = = = : H01 . £1 E 1 El . . . El 0 2 3 N H 2*: E; = 6 = . . . g = O 0 ET 1;]? 11:1, *0 : : : H03 . g Y gY o o o EY O 2 _ 3 _ N ._ 2 *’ : 3 : N : H05 0 g Dt_l th-l o o th-l O lkich of these hypotheses conforms to the standard format of the usual 'F test for the relevance of a group of regression variables. In the last three hypotheses a; is equivalent to (B: - Bi). Similarly 112 53% = (1/wfiT)B%T - (l/wlii'l‘mli‘T so that H02 through H05 are exactly equivalent to HO2* through H05*. The nature of 8% and the presence of 6% in HOl* permit testing one final hypothesis which is implied by the consistency condition. If RT and RS are to be aggregated to the vari- able RTS in a macroequation, consistency implies that (l/WéT)B§T must equal (l/W;3)B§S for all states. 8% is equal to the difference between these weighted parameters in the ith state and thus Hol* is the hy- pothesis that this consistency condition is met in all states. Together Hol* and H02* imply H01 and H02 but the converse is not true. Results in Table 7 confirm earlier results for the six New England tates. Again the null hypothesis of consistent aggregation is rejected at a Significance level less than 1 percent. Hol* stands out by virtue of its low F statistic. If there is a difference in the weighted parameters of the interest rates RT and RS, the sample contains too little information to uncover it. Thus, this particular test cannot be used in argument against the use of RTS as an adequate index of interest rates. This result is not very surprising. Aside from the fact that RT and RS may well affect demand deposits in proportion to the quantity held of T and S, RT is highly correlated with RS and high correlation between two independent variables in a regression makes it difficult to sort out the contribution of either variable alone. A test similar to that in Hol* was performed on each of the nine States independently to test whether the index (WéT)R% + (WES)R; was a COIisistent aggregate of R% and R; in the demand deposit equation for eamfli state. The weights WéT and WES were obtained analagously to those Ilsed.in.the national index with each weight equal to the proportion held ill the corresponding asset in 1968. When the null hypothesis 113 TABLE 7 TESTS OF CONSISTENT AGGREGATION 0F D1 FOR 9 UNRELATED STATES Null Hypothesis F Significance 01°F 2 N H 1*: 61 = E = E3 = = a = 0 .8057 .612 O_ l l 1 l 2 3 N H 2*: 6 = 6 = = a = 0 3.0523 .008 0 RT RT RT 2 3 H 3*: E = 6 = = E = 0 8.8688 <.0005 0 Y Y Y 8* £2 63 EN 0 2 002 HO . YP — YP - - YP — 3. 997 . 62 3 N 88 H 5*: = a = = a = 0 1. 72 .175 O Dt-l D81 D81 H 6*: H 1* H 2* ... H * 8.6 1 .000 O (0.0. 05) 57 < 5 llh (l/leiT)B%T = (l/W%5)B%S‘was tested in 9 different states, the highest F statistic obtained was 1.56 with a significance level of only .233. The average F value was much lower and thus even less significant. Our conclusion from these tests is that inconsistent aggregation cannot be demonstrated to result from the use of the index RTS in place of RT and RS in either the aggregate demand equation or those of the individual states. However, other null hypotheses implied by the necessary condition for consistent aggregation seem extremely unlikely to hold so that, taken as a group, they point very strongly to the rejection of the consistent aggregation hypothesis. Demand and Time Deposits When the aggregation to a single macroequation involves two depen- dent microequations which have the same independent microvariables, an additional degree of freedom is created for the microparameters. Thus suppose we assume that the demand in the ith state for demand deposits plus time deposits can be written: YP+ (D+T):_ i = i i 01 i+ i (D+T)t fiRTW fit hYt YP t “D+T General consistency of the macroequation (written without superscripts), . = + (D+T)t ao+aRTSRTS,t+aYYt+aYPYPt+aD+T(D+T)t-l e *with.the system of microequations for all states implies a set of con- ditions in a: equivalent to those implied for B; in the aggregation of i . . i i . D , 148., Hol* through HO6*. However, if D and T are each linear :functions of the same variables given in the demand for their sum, Ifi--+ Tl, then a: (azT, S, YP, Y, At-l), may be interpreted as the sum 115 of the corresponding parameters in the equations for D1 and Ti alone. Then the null hypotheses tested for consistency in aggregating over states pertain to these sums rather than to individual parameters in the equation for either Di or Ti alone. Again a transformation of variables simplifies the analysis. Table 8 gives the results of testing Hol*+ through H06*+ with exactly the same procedure used in testing Hol* through H06* in Table 7 for the aggre- L“ gation of D1 alone. Each of the five null hypotheses necessary for con- sistent aggregation over states is rejected at a level of significance 5 less than 10 percent and the hypothesis that all conditions are simul- taneously met is again rejected at a significance level of less than 1 percent. Including time deposits in the definition of money does not seem to reduce the inconsistency of aggregating over states. The tests of the preceeding chapter establish with very high probability that state demand equations cannot be consistently aggregated to a single macroequation whether the dependent variable is demand deposits or the sum of demand and time deposits. Conditions for general consistency imply restrictions on the parameters of every state and yet null hypotheses which postulate the existence of the conditions for as few as six states are very strongly rejected. In the following sections we present estimates of several measure- ments which describe the extent of the inconsistency and its importance in estimation. General Inconsistency The vector d was defined in Chapter III as the difference between W*B* and B and its inner product d’d was suggested as one measure of the 116 TABLE 8 TESTS OF CONSISTEN‘I‘ AGGREGATION OF D1+T1 FOR 9 UNRELATED STATES Null Hypothesis H Significance ofF l 2 N H l*+: a = a = a = = a = 3.9657 <.0005 o l l l 1 2 N H 2*+: E = E3 = = E ‘ 1.8075 .082 0 RT RT RT + 2 3 N H 3* : a = a = . = a — 2.7190 .009 0 Y Y 2 3 N H u*+; = = . = = . l .000 O EYP EYP EYP h 75 5 < 5 3 N H 5i+z 5D = a = = a 6.0678 <.0005 O t-l Dt-l Dt-l H 6*+ n.9788 <.0005 H 1* H 2* ... H * ( O , O 3 05 ) 117 amount by which the system of equations fails to meet the conditions for general consistency It is clear that without knowing B* and B we can- not calculate d. However, by using the vector of unbiased ZA estimates, ~ B, we can calculate an estimate of B* which, ifWB were equal to B, would give us a lower bound on d'd. We simply utilize the well known property of the least squares estimator, write B=W*B*+d and calculate B4 as the vector which minimizes the sum of squared terms in d. 5.. = (w*'w)'lw*'l§ In the case at hand, (WiW)—l is a diagonal matrix and such that i — l ~i me = NE Ba (a = Y, YB, Dt_l) and "H *“"l‘_—{ (w1 Bi new?L Hi ) * = i2 i2 RTs 2(WRT+WRS) RT RT RS Rs ~ 1 If B were equal to B, then d'd could not be less than d'd. Using d'd in a manner analagous to the sum of squared errors and then treat- ing the inner product of the vector of state estimates, B'B, as total sum of squares, we can separate B'B into the proportion which is explained by W¥B* and that which is due to general inconsistency. The proportion of B'B explained by W£B for the estimates of B, by state, given in Table h, is .208. Since d'd is a lower bound under the con- ditions stated, inconsistency accounts for at least 80% of the total sum of squares. 118 Specific Inconsistency General consistency permits maximum degrees of freedom for the microvariables. For specific consistency, all that is required is that B* and B satisfy the following relationship for the particular values of the variables observed at a specific time: iXW*B* = iXB When this equality fails to hold, the vector of differences from all observations can be written as iXd. Again using d to approximate the true vector, we calculate iXd and then proceed, as with the general inconsistency measurement, to find the proportion of B'X'i'iXB which is due to specific inconsistency. The nature of the microvariables in X is such that specific incon- sistency seems to be considerably less than general inconsistency. d'X'i'iXd is slightly more than 5% of total sum of squares. Aggregation Bias Aggregation bias was defined in Chapter II as the amount by which the expected value of the least squares estimate of B* differed from the actual value, i.e., E(B*) — B*. From (M3) in Chapter III this difference equals (242*)‘124 iTXd where 2* is the T x K* matrix of observations on the macrovariables. Again using d as an estimate of d we can get an idea of the size of aggregation bias by performing least squares regression of the vector of estimated values of specific ll9 inconsistency, iTXd, on the matrix of macrovariables.2 The estimated amounts of aggregation bias are given in Table 9. TABLE 9 MACROPARAMETER AND AGGREGATION BIAS ESTIMATES sfiTS a, s,, fibt-l Estimates —5203.7 .092 .020 .535 Std. Error 2125.h .095 .120 .126 Est. Agg. Bias hh92.6 -.Oll .007 .077 Bias/Estimate .86 .12 .35 .1t Aggregation bias in the interest rate parameter stands out immediately, but all of the calculated biases are greater than 10% of the parameter estimates. 2Theil calculates an estimate of aggregation bias by the formally equivalent procedure of first calculating a matrix G = (zlz*)-lzii X and then multiplying to get the vector of biases, Gd. Since we already have the vector of specific inconsistency measures, L¥Xd, Theil's approach would be an unnecessary duplication of effor in this instance. See H. Theil,“Principals of Econometrics, New York: John Wiley and Sons, 1971. pp. 5627566. ‘_— CHAPTER VI CONCLUSIONS The principal conclusions of this research are as follows. i) Esti- mation of demand for demand deposits at the state level yields parameter estimates which conform generally with prior expectations based on economic theory. ii) The system of state demand equations is not con- sistent with a single macroequation which attempts to describe aggre- gate demand in terms of linearly aggregated macrovariables. iii) Esti- mates based on such a misspecified macroequation cannot be assumed to be unbiased; therefore, conclusions based on these estimates are suspect. Of course these conclusions have only been firmly established for the variables and functional form used in this study. However, the reestimation of macroequations, which were estimated by other authors using different variables, establishes that the variables of this study are not widely divergent from those used previously, and in fact, pro- duce comparable results in estimation. This result makes it seem likely that problems of inconsistency and aggregation bias occur in other specifications and with other variables. One of the implications of inconsistency is that the parameter estimates of the macroequation provide no information about the demand for money in any particular location. This arouses some curiosity as to why aggregate demand is of any interest at all. Of course state 120 121 demand is also aggregate demand and perhaps the next step in disaggre- gating should be at the SMSA level. However, states are less hetero- geneous economic units than is the country as a whole and it may be that inconsistency and aggregation bias are insignificant problems within a state and thus that state demand equations provide a useful summary of information. This would be a fruitful area for further re- search. The primary purpose of this research was to determine whether a single macroequation should be relied upon as an accurate description of the demand for money in the United States. Many authors have as- sumed that a single equation does adequately describe total U.S. money demand and they have proceeded on that basis with empirical analysis involving a few arbitrary macrovariables. The rate of return on four to six month commercial paper, for example, is one of the most fre- quently used interest rate variables and it is usually treated as "the" rate of interest with the implication that it adequately repre- sents all of the various interest rates. However, in spite of the use of macrovariables in the equations and in the related empirical anal- ysis, most studies nevertheless appeal to microeconomic concepts in developing a theoretical framework. The interest rate in most studies is described as an opportunity cost, and national income or wealth is used as a budget constraint. It is this appeal to the language of microtheory which suggested that consideration of state demand for money equations might prove ‘worthwhile. It would be possible to propose a linear relationship be- tween macrovariables which did not rest on microtheory. However, once 122 an author formulates the tempting rationale that interest rate is anal- agous to opportunity cost and income serves as budget constraint, it is difficult to argue at the same time that the demand relationship is not applicable to large subgroups of the population. The rates of re- turn to time deposits and savings and loan shares within each state are, for most people, much more realistic indications of the oppor- tunity cost of holding demand deposits than the return on four to six month commercial paper which is so frequently used in empirical work. Justification for the disaggregation of demand for money was also derived from portfolio theory. Interest rates in different locations exhibit considerable variation both in their absolute levels and in their patterns of change, and in view of the disparity of past histor- ical data in the various states, it seems highly unlikely that inves- tors within each state would hold similar expectations regarding the probability distribution of future returns. Thus there is considerable doubt whether a single macrovariable can adequately represent the ex- pected value of the future return on financial assets in all states. Moreover, since the coefficients of interest rates in the linear equa- tion derived from portfolio theory also depend on the nature of the investors' anticipated prdbability distribution, it seems unlikely, on theoretical grounds, that the coefficients would be the same in all states. This anticipation of disparity in state coefficients and the superiority of state versus national variables in explaining variation in demand for money was overwhelmingly supported by the empirical work in this study. State interest rate and income variables were calcu- lated and demand equations were estimated for each of the states. 123 The coefficients of interest rate and income variables differed widely among the states and the national indices of interest rate and income could not be shown to be significant explanatory variables in most of the states. In every state, the hypothesis that the state income and interest variables were not important in explaining variation in the demand for money was rejected at a 1% level of significance. On the other hand, when national income and interest rate variables were used as regressors in the state demand equations, in most states they did not contribute significantly to the variation in quantity of money demanded. These results implied that the problems of aggregation in demand for money could not be assumed away on the basis that the na- tional variables adequately represented the opportunity cost and bud- get constraint for each of the states separately. If the converse had been true, if the state interest and income variables had proven in- significant and the national macrovariables had explained the varia- tion in demand for money in each state, then many of the problems associated with aggregation would have disappeared. The macropara- meters could have been interpreted as the sum of the corresponding parameters in all the states. Estimates of macroparameters likewise would have been equal to the sum.of corresponding estimates in all the states and there would have been no possibility of aggregation bias. Of course the macroparameter estimates would give no indication of the disparity in parameters between states, but all summary statistics re- sult in some information loss--a loss which is offset by convenience or some other consideration. The inadequacy of the national variables to explain state demand for money, and the significance, at the same time, of the state interest 121+ rate and income variables indicated that there might be serious prob- lems associated with estimating the demand for money as a function of national macrovariables. Of course the same sort of problem might exist with the state demand functions, and it might prove worthwhile in further research to reestimate for, say, the standard metropolitan statistical areas to determine whether variables calculated for the state as a whole adequately explain the demand for money in the in- dividual SMSA's. But for the present study, equations for each state were a convenient level of disaggregation and, as regression models, they were at least as acceptable, under such standard criteria as R and the F test for significance of variables, as macromodels for the total United States have been. Thus, if there is a single national macroequation, there are also fifty, equally valid state demand equa- tions; there do not seem to be any geod reasons for rejecting the state equations which would not be equally valid in rejecting the na- tional equation. Thus the problems of aggregation had to be confronted. The con- ditions under which a single national equation is consistent with a set of underlying state equations are highly restrictive. In order to bring the analysis within the scope of linear aggregation, macrovari- ables had to be constructed as fixed weight linear aggregates of the corresponding variables from each state. Although a number of rea- sonable alternatives existed for the selection of weights, there was little practical difference in the resulting indices. The quantity variables, money and income, were simply added over all states to give their corresponding macrovariables. National interest rate indices were constructed as weighted averages of the state interest rates 125 with the weights proportional to the amount invested in each asset in the base year, 1968. For further verification of the reasonableness of these variables they were used to reestimate (by least squares) the demand for money specifications previously estimated by other authors. The reestimation confirmed that the new variables produced results in regression analysis which the original authors probably would have accepted as comparable to their own results. Thus, it seems likely that the results of tests for consistent aggregation and aggregation bias are more generally applicable than to just the variables created in this research. The tests for consistent aggregation left little doubt that the system of state demand for money equations is inconsistent with a single macroequation for the nation as a whole. Essentially what this inconsistency means is that the quantity of money implied by demand equations in each of the states does not necessarily add up to the quantity implied by the demand equation for the nation as a whole. This need not have been the case. It is perfectly feasible for a system of microequations to be consistent with a macroequation and, in fact, several authors have assumed that this condition was met. But in this study, when the necessary conditions for consistent aggrega- ‘tion were stated as a null hypothesis, the hypothesis was strongly re- jected in each of two tests. The conditions were not met, even in the two small groups of states selected for the tests. In order to determine the extent of the inconsistency, the vector of estimates from all of the states was split into two parts--a vector of state parameters which would be consistent with a single macroequa- tion, and a residual vector whose non-zero elements indicate inconsis- 126 tency. In case of general consistency of the state parameter estimates with a single macroequation, this residual vector would consist en- tirely of zeros and, of course, would have an inner product equal to zero. But in this study, the inner product of the vector of residuals was 80% of the total inner product of the vector of state parameter estimates. This result overwhelmingly reaffirmed the results of the earlier tests which indicated that the system of state demand equations was not consistent with a single macroequation. With inconsistency thus firmly established, two sets of calcula- tions were made to estimate the effect of this inconsistency when re- gressions are run under the assumption of a single macroequation. It is possible for a set of state demand equations to be consistent with a single national equation for a specific set of exogenous variables even though the conditions for general consistency do not hold. If, for example, all of the state income variables moved proportionally then a single aggregate variable could be consistent with all of the state income variables. While it is true that the state variables used in this study are highly correlated with one another they never- theless do not move in direct proportion and we have a positive value for the measure of specific inconsistency for the twenty year period of this study. This measure was constructed by forming two vectors of estimates of total money demanded for each of the years in the study. One set of estimates was obtained by adding together the esti- mates from each of the states; the other set was calculated by using the estimated parameters of the macroequation and the national exo- genous variables. The difference between these vectors would be a zero vector in the absence of specific inconsistency. Instead, in 127 this study that vector of differences had an inner product equal to 5% of the inner product of the vector of estimates based on state data. This indicates considerable disparity between state and nationally based estimates. Probably the most significant effect of this specific inconsis- tency is the bias created in the estimates of the national parameters when total money is regressed on national interest rate and income variables. Based on the results of this study, the estimate of the interest rate parameter derived from national variables might be biased by as much as 86%, and all of the other parameter estimates have biases of at least 10%. Due to the widespread practice of dropping insig- nificant variables from regression equations, aggregation bias may have already resulted in some specifications being eliminated which, in the absence of aggregation bias, might describe money demand quite well. In the specifications which are reported, the biases can lead to erroneous conclusions about the importance of particular variables. BIBLIOGRAPHY r . . BIBLIOGRAPHY Allen, R. G. D., Mathematical Economics, 2nd ed. New York: St. Martin's Press (1966). Cagan, Philip, "The Demand for Currency Relative to the Total MOney Supply," Journal of Political Economy, Vol. 66 (August, 1958), PP 303- 32 28 Chow, G., "On the Long-Run and Short- Run Demand for MOney," Journal of Political Economy, Vol. 6% (April, 1966),p pp 111- 131. Cramer, J. 8., "Efficient Grouping, Regression and Correlation in Engle Curve Analysis*," American Statistical Association Journal, Vol. 59 (March, 196M), pp. 233-250. Farrar, Donald, The Investment Decision Under Uncertainty, Englewood Cliffs, New Jersey: Prentice Hall, 1962. Feige, Edgar L., Demand for Liquid Assets: A Temporal Cross- Section Anal sis, Englewood Cliffs, New Jersey: Prentice Hall, Inc. , 196E. Friedman, Milton, A Theory of the Consumption.Function, Princeton: Princeton University Press, 1957. Friedman, Milton, "The Demand for Money: Some Theoretical and Empirical Results," The Journal of Political Economy, Vol. 67 (August, 1959), PP- 327-351- Friedman, Milton, and Schwartz, Anna J., A MOnetary History of the United States 1867-1960, National Bureau of Economic Research, Studies in Business Cycles, No. 12, Princeton, New Jersey: Princeton University Press (1963). Friedman, Milton, and Schwartz, Anna J., Monetary Statistics of the United States, New York: National Bureau of Economic Research, 1970. Friend, I., "The Effects of Monetary Policies on.Nonmonetary Financial Institutions and Capital Markets," Commission on.Money and Credit, Private Capital Markets, Englewood Cliffs, New Jersey: Prentice Hall, 1963, pp. 16512I8. 128 129 Gilbert, Roy F., "The Demand for Money: An Analysis of Specification Error," Unpublished Ph.D. dissertation, Michigan State University, 1969. Goldberger, Arthur S., Econometric Theory, New York: John Wiley & Sons, Inc., 196M. Green, H. A. John, .Aggregation in Economic Analysis - An Introductory Survey, Princeton, New Jersey: Princeton University Press (1963). Grunfeld, Yehuda, and Griliches, Zvi, "Is Aggregation Necessarily Bad?," The Review of Economics and Statistics, Vol. #2 (February, 1960), p. l. "'11 Hamburger, M., "Household Demand for Financial Assets," Econometrica, Vol. 36, No. 1 (January, 1968), pp. 97-118. 2‘1)..- Lemma“ Hicks, J. R., "A Suggestion for Simplifying the Theory of Money," Economica, New Series, Vol. 2 (1935), pp. 1-19. Hicks, J. R., Value and Capital, 2nd. ed., Oxford: Oxford University Press, 19h6. Johnston, J., Econometric Methods, New York: McGraw—Hill Book Co., 1963. Keynes, J. M., The General Theory of Employment, Interest, and Money, New York: Harcourt Brace and— Co., 1936. Klein, L. R., "Remarks on the Theory of Aggregation," Econometrica, Vol.1h (19u6), pp. 303— 312. Klein, L. R., "Macroeconomics and the Theory of Rational Behavior," Econometrica, Vol.1h (l9h6), pp. 93- 108. Kmenta, Jan, Elements 2: Econometrics, New York: Macmillan (1971). Laidler, David, "The Rate of Interest and the Demand for Money--Some Empirical Evidence," Journal of Political Economy, Vol. 7h (December, 1966), pp. 5E3- 55 5. Laidler, D., "Some Evidence on the Demand for Money," The Journal of Political Economy, Vol. 76 (February, 1966), pp. 55- 68. Latane, H. A., "Cash Balances and the Interest Rate-2A Pragmatic Approach," Review of Economic and Statistics, (November, 195M), pp. h56- A60— Markowitz, Harry, "Portfolio Selection," The Journal of Finance, Vol. 7 (March, 1952), pp. 77-91. May, K., "The Aggregation Problem for a One Industry Model," Econometrica, Vol.1h (19u6), pp. 285- 298. 130 Pu, Shou Shan, "A Note on Macroeconomics," Econometric, Vol. 1h (19h6), PP- 299-302. Teigen, Ronald, "Demand and Supply Functions for MOney in the United States: Some Structural Estimates," Econometrica, Vol. 32, No. A (October, 196A), pp. M76-509. Theil, Henri, Linear Aggregation of Economic Relations, Amsterdam: North Holland Publishing Company (1955). Theil, Henri, Principles 9: Econometrics, New York: John Wiley & Sons, Inc. (1971). Theil, Henri, "Specification Errors and the Estimation of Economic Relationships," Revue Institute Internationale d3 Statistique, Vol. 25 (1957), pp. u1-51. Tobin, James, "The Interest Elasticity of Transactions Demand for Cash," Review of Economics and Statistics, Vol. 38 (August, 1956), pp. 2El-2h7. Tobin, James, "Liquidity Preference as Behavior Toward Risk," Review 2: Economic Studies, Vol. 25 (February, 1958), pp. 65-86. United States Department of Commerce, Office of Business Economics, Survey'g£ Current Business, Vol. M9, N0. 8 (August, 1969), pp- lH-lS- United States Department of Commerce, "Population Estimates and Projections, Current Population Reports," Series P-25, No. M36 (January, 1970), p. 13. United States Department of Labor, Bureau of Labor Statistics, Handbook 23 Labor Statistics 1971. Wright, Colin, "Estimating Permanent Income: A Note," Journal of Political Economy, Vol. 77 (September/October, 1969), pp. 8H5-850. Zellner, Arnold, "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of American Statistical Association, Vol. 57 (June, 1962), pp. 3H8:368.