36

Piyu 3/ne

Piyu Yue, a research associate at the 1C2 Institute, Universityof Texas as Austin, was a visiting scholar at the FederalReserve Bank of St. Louis when this article was written. LynnDietrich provided research assistance. The author wishes tothank Professor Leon Lasdon for the GRG2 Fortran code andProfessor Douglas Fisher for providing the data. Estimationsthat appear in this article were calculated at the University ofTexas at Austin and are the responsibility of the author

A Microeconometric Approachto Estimating Money Demand:The Asymptotically IdealModel

HE DEMAND FOR MONEY plays a criticalrole in macroeconomics. In conventional moneydemand analysis, the demand for real moneybalances is typically expressed as a function ofsuch variables as real income, the expected rateof inflation and the nominal interest rate.’ Em-pirical investigations using these variables havenot been particularly useful in predicting thedemand for money or in formulating and evalu-ating monetary policy.2

More recently, a number of researchers haveattempted to estimate money demand in a man-ner consistent with mictoeconomic foundations.

Even in these cases, however, the empiricalresults have been largely discouraging.’

This paper reviews the general micro-econo-metric approach to estimating the demand for

money, culminating with an advanced micro-econometric model, called the AsymptoticallyIdeal Model (AIM). AIM is applied to U.S. time-

series data and the results are compar’ed brieflywith those from previous studies. AIM resultsare consistent with microeconomic theory andprovide insight into the behavior of money de-mand in the 1970s and 1980s.

‘See Friedman (1956) for one of the most comprehensivediscussions of the money demand function.

2These functions have been subject to several unexplaina-ble shifts and often imply a larger liquidity effect than istypically experienced. Perhaps the most dramatic exampleof this phenomenon occurred in the early l9BOs with theyet unexplained break in the income velocity of Ml. Forthis and other examples, see Goldfeld (1976), Friedman(1984), Lucas (19BB) and Rasche (1990).

3Frequently, the estimated own price elasticities of demandfor monetary assets are positive, implying that their de-mand curves slope upward. For example, see Serletis(1988), Fisher (1989) and Moore, Porter and Small (1990).

FFflFRAI PFSFRVF RANK OF SiT LOUIS

1 37

MICROECONOMIC MODELING

As a result of developments in macroeconomictheory over the past two decades, “almost allmacroeconomists agree that basing macroeco-nomics on firm microeconomic principles shouldbe higher on the research agenda than it hasbeen in the past.”4 Problems arise, however,when aggregate, macroeconomic data are usedto estimate microeconomic-based models ofmoney demand. Some of these problems are il-lustrated by a simple example that uses two ap-proaches to microeconomic modeling: thedemand function approach and the utility func-tion approach.’

The Demand Function Approach

Consider an economy where the representa-tive consumer allocates income between a com-posite consumer good, A, and a monetary asset,M, that yields monetary services. The consumer’sobjective is to maximize the utility function(subject to a budget constraint), given the priceof the composite commodity and the user costof the monetary asset. Let P, u and E denotethe price level (the price of A), the nominal usercost of holding one real unit of M and total ex-penditures (or income), respectively.’ The con-sumer’s decision problem is expressed by

Max f(A, M) = AVMI-r

subject to PA + uM = E.’

For simplicity, the utility function, f, is Cobb-Douglas, where r, an unknown parameter,characterizes the consumer’s taste or prefer-ence. The optimal solution to the consumer’s de-

‘Mankiw (1990), page 1658.‘Some economists argue that aggregate data cannot be ap-plied to microeconomic models without considering theproblems of aggregation. Aggregation problems are notdiscussed in this paper, although the aggregation errormight be one source of the unsatisfactory performance ofconventional money-demand functions.

‘The user cost of holding a unit of a real monetary asset iscomputed by the formula,u p’(t) [R(t) — i(t)Jlll + R(t)], where p’(t) is the “true” costof living index defined as the geometric average of theconsumer price index and the consumption goods deflator,R(t) is the benchmark interest rate or the maximum rate inthe economy at each period and i(t) is the interest rate onthe monetary asset, The formula is derived from a widelyapplicable consumer decision model.

‘Distinct views about money have resulted in two ap-proaches to analyzing consumer demand for money. In thefirst approach, money is viewed as a commodity whichprovides a monetary service flow to holders. Thus, realbalances of the monetary assets directly enter the con-

cision problem yields ordinary demand functionsfor A and M. In this case, the demand functionsare:

(1) A = rE/P = rI(P/E) = G,(uIE, PIE, r), and

(2) NI = (1— r)E/u = (1— r)I(uIE) = G2(uIE, PIE, r).

The demands for A and M are functions, G, andG,, respectively, of E, P, u and the unknownparameter r. Because the budget constraint islinear in P and u, the normalized price, PIE, anduser cost, u/E, can replace P, u and E. In gener-al, demand functions can be expressed by nor-malized prices (including the user cost) and theunknown parameter. This parameter can be es-timated by simultaneously fitting equations (1)and (2) using data on real quantities of A and Mand the normalized price and user cost.

This approach is called the “demand function”approach because estimation begins after de-mand functions are specified. For this approachto yield meaningful estimates, however, thespecified system of demand functions must cor-respond to the neoclassical utility function fromwhich they were derived. Consequently, theconditions for estimating the system of demandfunctions are fairly restrictive. For instance, theRotterdam model (a well-known demand systemused in empirical studies) requires specific formsfor demand functions and specific constraintson parameters during estimation.’

Even if these conditions are satisfied, however,the Rotterdam model is still highly restrictivebecause the assumed underlying utility function(either Cobb-Douglas or CES) is a member of anarrow class of utility functions with constantelasticities of substitution.’

IIIIIItI,III1IIIII

sumer’s utility function along with real consumption. In theother approach, money is viewed as intrinsically worthless;consumers hold it only to finance current and future con-sumption. As a result, real money balances do not enterthe consumer’s utility function per Se. Instead, the liquiditycost of holding real money balances is taken into accountin the budget constraint. Feenstra (1986) shows that thesetwo approaches are equivalent.

‘For an application of the Rotterdam model to the money-demand system, see Fayyad (1986). For the theory of theRotterdam model, see Barnett (1981).

~ is easy to encounter difficulties using the demand func-tion approach. The failure to specify functions correctly orimpose relevant restrictions can result in biased or ineff i-cient parameter estimates.

. NOVEMBERIDECEMBER 1991

38

The Utility Function Approach

The utility function approach to demand esti-mation also has been used in empirical studies.To understand this approach, reconsider theconsumer’s decision problem and the demandfunctions shown in equations 1 and 2. in theutility function approach, demand functions forA and M are substituted into the utility func-tion, f(A, ML to obtain the indirect utilityfunction,

h(v,, s’,, r) = f[A(v,, v,, r), M(v,, x’,, r)],

where v, = P/E, v, = u/E. Because the indirectutility function has properties that are the in-verse of those for the utility function, it is moreconvenient to use the reciprocal of the indirectutility function,

F(v,, ~‘,, r) = 1/h(v,, v,, r)’°

By definition, demand functions can be ex-pressed in terms of their expenditure shares,5, = AP/E and s, = Mu/E. That is,

A = s,/v, and M = s,/v,.”

In this way, demand functions can be ob-tained without solving first-order conditions.Consequently, no matter how complicated theutility function might be, the derivation of shareequations and demand functions is straight-forward.

Of course, if the utility function is relativelysimple and well-behaved (for example, when theCobb-Douglas function is used), there is no needto use the utility function approach. Howevei’, ifthe utility function includes more than two goodsor is sufficiently complicated, the Lagrange mul-tiplier procedure cannot be used to derive de-mand functions.

THEMETHOD FOR ESTIMATING THEDEMAND SYSTEM

The critical step in applying the utility func-tion approach is the specification of the proper

reciprocal indirect utility function, F. To simpli-fy the terminology, the term “utility function”will indicate “the reciprocal of the indirect utili-ty function” in the following discussion.

Flexible Functional Form Modeling

Cobb-Douglas and CES functions have beenused extensively in theoretical and applied workbecause of their relative simplicity. Despite theirapparent successes, however, such use has beencritirized. For example, if there are more thantwo goods, the CES utility function can onlygenerate demand systems when each pair ofgoods has the same constant elasticity of substi-tution” Unless there is prior information to thecontrary, however, the elasticities of substitu-tion should be determined by the data ratherthan restricted by the specification of the utilityfunction. This limitation has motivated research-ers to look for utility functions that are moreflexible and allow for data-determined elastici-ties of substitution.

Flexible functional form models have attractedconsiderable attention in economics literaturesince the early 1970s, when it was proposedthat the translog and generalized Leontief func-tions should replace neoclassical utility func-tions. It was recognized that the values of theelasticities of substitution are determined by thevalue of the utility function and the values of itsfirst- and second-order derivatives are evaluatedat its extreme point. Consequently, if the valuesof the utility function and these derivatives canbe estimated, so too can the elasticities of sub-stitution. ‘I’his idea forms the basis for the flexi-ble functional form approach.

A functional form is said to be flexible if itslevel and the first- and second-order derivativesat a point in its domain are allowed to reachthe respective values of the “true” utility func-tion at that point. l’he true utility function is as-sumed consistent with the properties of thedata, so that, in principle, elasticities of substitu-tion consistent with the data can be estimated.

‘°Theduality theory states that if the reciprocal of the in-direct utility function, F, is nondecreasing and quasi-concave with respect to normalized prices, the respectiveutility function, f, must be nondecreasing and quasi-concave with respect to quantity variables. In this sense,the utility function is equivalent to its reciprocal indirectutility function.

“Expenditures shares can be obtained by using the modi-

fied Roy’s identity from duality theory. That is,5, = (v, dF/dvJlV’VF(V,r), i = 1, 2,

where the vector V’ = (v,, v2) and the gradient vectorV F(V,r) = (dF/dv,, dFldv2)’. See Diewert (1974).

‘2See Uzawa (1962).

IFEDERAL RESERVE BANK OF ST. LOUIS S

One flexible functional form is derived fi’om aTaylor series expansion where all terms greaterthan second-order are eliminated, that is,

(3) F = a0 + I,a,x, + ~

This approximation is flexible because it hasenough free coefficients, a

0, a,, a,~, to allow for

any desired value of the first- and second-orderderivatives of function F.

Two frequently used flexible functional forms,the translog and generalized Leontief functions,are given by simple substitution into equation 3.For the translog function,

F = ln(f(x)) and x, = ln(q),

where f denotes the utility function and q,represents the quantity of good i. The general-ized Leontief function is attained by letting

F = (f(x))” and x, = (q1)½.

‘The coefficients in these functional forms canbe estimated and, in turn, the demand systemand the elasticities of substitution can bederived.

Caveats For Flexible FunctionalForms

Theoretically, the second-order Taylor approx-imation can attain flexibility only at a singlepoint or in an infinitesimally small region.Hence, estimates of the elasticities of substitu-tion are valid only for the range of observationscovered by the sample data. ‘rherefore, thesecond-order Taylor series approximationshould be viewed as “locally flexible.”

Such models are also subject to another,potentially more serious, problem. Experiencehas demonstrated that regularity conditions arefrequently violated! Therefore, the restrictionsthat microeconomic theory imposes on con-

sumner behavior are not embedded in these flex-ible functional forms. ‘this point is illustratedlater in the empirical section of this paper.

In an attempt to solve these problems, micro-economists have developed a variety of flexiblefunctional forms that maintain their flexibilityand have larger regularity regions.” A family of

such flexible functional form models has beenproposed (for example, Barnett’s (1981) minflexLaurent modefl15 To gain global regularity,however, additional constraints are imposed onthe parameters which result in a loss of localflexibility. This tradeoff between flexibility andregularity is characteristic of flexible functionalform modeling. None of these models is bothglobally regular and globally flexible.”

Semi-nonparametric Method

Gallant (1981) created the “semi-nonpam-ametricmethod” to remove the local flexibility limita-tion. His method specifies a series of modelsthat approximate the underlying utility functionat every point in the function’s domain. Hence,the models are globally flexible.

The “semi-nonparametric method” is built upona well-known result in mathematics: a Fourierseries expansion can converge to any continu-ous function.’~In contrast to the local conver-gence of the flexible functional forms, theFourier series can approximate a continuousfunction in the entire domain. Gallant proposedto use the Fourier series expansion to specify a

series of utility functions that can converge toany neoclassical utility function. Because neoclas-sical functions are a subset of continuous func-tions, the property of the Fourier series expan-sion will guarantee asymptotic convergence toan underlying neoclassical utility function.

Fourier series modeling consists of a series of

expansions of models, with succeeding modelsnested in the preceding one. When the samplesize increases, higher-order models can he speci-

‘5lnstead of the Taylor series expansion, Barnett used theLaurent expansion to enlarge the regularity region andmaintain enough parametric freedom to satisfy require-ments for flexibility. See Barnett (1983).

“See Diewert and Wales (1987).~ function must be integrable or, more generally, it must

lie in a 1-lilbert space. See Telser and Graves (1972).

39II

III

I‘IIIIIII‘III

“This equation is written in general form, where ; denotesall the constants (that is, the function evaluated at thepoint of interest and the partial derivatives evaluated at thesame point). The use of this general form to estimatethese equations is one of the procedure’s limitations be-cause the point about which the expansion is made is esti-mated by the data, rather than being specified by theresearcher. Hence, there is no guarantee that this pointwill necessarily correspond with the maximum value of thefunction itself.

‘4The regularity region is the subset of the domain of theutility function in which all regularity conditions aresatisfied.

NOVEMBER/DECEMBER 1991

40

fled by simply adding more terms of the compo-nent functions. For instance, the first-ordermodel is defined by the utility function,

F = a0 + I~aq + ~ + I~bcos(Qq~)

Asymptotically, the model contains an infinitenumber of terms and unknown parameters.Therefore, asymptotic inference based upon theFourier series expansion models is free fromfunctional-form specification error. This is itsprincipal advantage.

In empirical work, however, the number ofterms must be finite, Consequently, the proper-ties of lower.order Fourier models become dcci-sive. The harmonic component functions, suchas sines and cosines which are frequently usedin engineering and physics, are not suitable ineconomic applications because they do not satis-fy the usual regularity conditions, such asmonotonically increasing and strictly quasi-concave. This means that lower-order Fourierseries models can violate regularity conditions.”Nevertheless, Gallant’s approach permittedmicro-econometric models to achieve both globalregularity and global fleidbility.

The AIM Demand System

To solve the problems of Fourier seriesmodels, another infinite function series, calledthe Muntz-Szatz series, is adopted. A typicalform of the series is expressed as:

‘l’he Muntz-Szatz series expansion convergesto a continuous function, and any continuousfunction can be approximated by the Muntz-

Szatz series’5 Consequently, this series can beused to approximate a neoclassical utility func-tion asymptotically’°

The Muntz-Szatz series is a linear combinationof a set of special power functions. In contrastto the Fourier series, the component functionsof the Muntz-Szatz series, q~,q~ q~q~are neoclassical functions. In other words, theyare monotonically increasing and quasi-concavewith respect to variables q~and q~.The Muntz-Szatz series is necessarily neoclassical, however,only if all of the coefficients, a~,a1~,b~,.., arenon-negative, because only positive linear combi-nations of the neoclassical component functionsare necessarily neoclassical. As a result, the co-efficient-restricted Muntz-Szatz series can ap-proach a neoclassical function but may notapproach any continuous function. Imposingthese restrictions guarantees that the estimatedfunction will not violate regularity conditions.

The Muntz-Szatz series is used in place of theFourier series in Gallant’s semi-nonparametricmethod. A series of models can be defined byincreasing degrees of the Muntz-Szatz approxi-mations. Under the parameter constraint, thesemodels are globally regular; the respective utili-ty functions are neoclassical everywhere in theirdomain. When the sample size increases, higher-order models can be specified with more freeparameters to best fit the data and derive theelasticities of substitution that the data suggest.Hence, the Muntz-Szatz series gives rise to amodel that is asymptotically globally flexible.Even a low-order approximation requires a fairlylarge number of parameters to be estimated,however, Hence, while the model is asymptoti-cally globally flexible) finite samples will limitthe researcher’s ability to fully utilize thispi-operty.

The model has two additional features thatmake it particularly attractive for applied work.First, although there are a relatively large num-ber of free parameters to be estimated, it is im-possible to overfit the noise in the data. Becausemovements due to measurement errors are ir-regular and cannot be expressed by the neoclas-sical component functions, the model simply

“Moreover, the Fourier series models can easily overf it thenoise of the data. Usually, the measurement errors of eco-nomic variables can be decomposed into a pure whitenoise plus some high-frequency periodic functions. Theselatter functions might be mistaken for useful information iftheir frequencies are close to that of the sine and cosinefunctions in the Fourier series models.

“Once again, the function must be integrable or, moregenerally, lie in a Hilberf space. See Telser and Graves(1972).

2cSee Barnett and Jonas (1983).

+ I~d1sin(Qq~)

The jthorder model is defined by the utilityfunction,

F = a0 + I~a~q~+ ~X~a~qq~+ ~

+ I1I~d~~sin(jQq,)

f = a0 + ~~afq~’ +

+ II~a~q~q~

+ II~b~tlj’~q~

I~a~2q~+ 11a? q~ +

+ I~I~a~ciJ’ q~+ -

+ E1X~fq~q~+-..

FEDERAL RESERVE BANK OF ST. LOUIS

ignores them. Also, because the componentfunctions are not periodic, the high-frequency,periodic movements in the data are likewiseignored.

Consequently, models based on the Muntz-Szatz series expansion are globally regular andare asymptotically globally flexible. This is whythey are called Asymptotically Ideal Models(AIMS).”

ESTIMATION OF THE AIM MONEYDEMAND SYSTEM

Four subaggregated goods are included in theempirical work presented here: a consumergood, A4, and three monetary assets, A,, A, andA,. A4 is an aggregate good of consumer dura-bles, nondurables and services and its respectiveaggregate price is denoted by p~A1 consists ofcurrency, demand deposits of households andother checkable deposits; A, is composed of sav-ings deposits at commercial banks and thrifts,super NOW accounts and money market depositaccounts; and A, is small time deposits at com-mercial banks and thrifts.” For each subset, anaggregate quantity is defined as a sum of percapita real balances of the component monetaryassets.” The opportunity cost of holding a unitof a real monetary asset is measured by its usercost, a quantity-share-weighted sum of the in-dividual user costs that compose it. The usercost of A is denoted by uj.

The representative consumer solves an op-timal allocation problem by selecting real con-sumption, A4, and real balances of the monetaryassets, A,, A, and A,, to maximize the utilityfunction f(A,, A,, A,, A4), subject to the budgetconstraint, given p~,u,, u,, u, and the total ex-penditure, E. Following the utility function ap-proach and using the first-order AIM model, wespecify the reciprocal indirect utility functionfor the four goods case as:

“See Barnett and Yue (1988) and Barnett, Geweke and Yue(1991).

F(v,a) = a,v,” + a,v,½ + a,v,’<’ + a4v4~

’

+ a,v,~v,” + a6v, ½v½+ a7v,vv,~

+ a,v,~’v,~’+ a9v,~’v,~.

+ a,0

v,~’v4~

’+ ~

+ a,,v,’~’v,~’v,’~’+ a,,v,½v,½v4½

+ a,4v,v.v,’/v4a + ~

where v,, v,, v, and v, are the normalized prices,and a,, a,, ..., and a,, are the parameters of theindirect utility function.

The share equation for each good is derivedfrom equations 3 using the modified Roy’s iden-tity. These are

5, = S,IS, s, = S,/S, s, = S,!S, and 54 = (1 — 5,

s,), where

+ a,,v,vv,½v,½+ ~

+ a,4v,’~v,~’v

4’.+ a,,v,’/.v,½v,½v

4v~and

S = a,v,½ + a,v,” + a,v,” + a,v4½+ 2a,v,”v,”

+ 2a~v,~~v,½+ 2a7v,~’v

4~’+ 2a,v,”v,’~’

+ 2a9v,”v4” + 2a,0v,~v

4½+ 3a,,v,~’v./’v,”

+ 3a12

v,’~v,~’v4~

’+ 3a,,v,’1~

v,’v,½

+ 3a,4

v,½v,~.v4~

)+ ~

Only the first three share equations are in-dependent and can be written generally as:

(4) s = S/S = g(v,a) for i = 1, 2, 3.

When the additional parameter normalization a,+ a, + a, + a4 = 1 is imposed, one parameter,for example a,, can be eliminated by substitu-

41IIIIIII’I$II1II$‘lII

5, = a,v,v~ + a,v,½v,’)

+ a,,v,”v,”v,~’ +

+ a5v,”v,” + a,v,½v4’~

a,,v,½v,½v4~~

+ a,,v,½v,’/~v4~

)+ ~

5, = a,v,~’ + a,v,½v,’1 + a,v,”v,” + a9v,½v,~~

+ a,,v/’v,’~’v,’~’+ ~

+ a,4v,~’v,~.v4’.+

5, = a,v,’~’ + a6v,”v,”

a, ,v, ‘7

’v, ‘~‘v,‘~‘v4½,

+ ~ + a,0v,½v

4v~

22This division of monetary assets is proposed in Fisher(1989), who has performed separability tests over a varietyof deposit categories included in M2. He found that this di-vision (A,, A,, A,, A4) is weakly separable in terms of theGeneral Axiom of Revealed Preference (GARP) test. Theweak separability condition offers theoretical assurancethat A,, A,, A, and A, are meaningful aggregate goods bythe definition of economic aggregation theory. However,the partition, ~(A,,A,, A,), A4 did not pass the GARP testfor weak separability. This raises the question of the exis-tence of the monetary aggregate, M2 (M2~A,+A,+A,). If

this is the case, the demand system of consumption andthe subaggregate M2 monetary assets are the appropriateway to study demand for money in terms of the economicaggregation theory.

“The reader is cautioned that these results are not directlycomparable with those using conventional money demandspecifications because the latter employ conventionalmonetary aggregate data that differ in several respectsfrom those used here. First, business demand deposits areexcluded from the data used here, so that Ml is not equalto A1 and M2 is not equal to A, + A, + A,. Moreover, un-like the conventional monetary aggregates, each of thesub-aggregates is seasonally adjusted separately.

a NOVEMBER/DECEMBER 1991

tion. Hence, in the case of four goods, the first-order AIM system contains 14 free parameters.’4

The share equations are nonlinear with respectto the normalized prices and hence, to incomeand prices as well. By the definition of expendi-ture shares, demand functions can be expressedas A, = s,/v,. The complicated nonlinearity of theshare equations, however, makes it impossibleto derive a closed-form expression for the de-mand functions, such as the conventional linearor log-linear functions of income, prices and in-terest rates. Fortunately, the estimatedparameters and share equations can be used tocompute the income and price elasticities forconsumer goods and monetary assets.

Estimation of the AIM DemandSystem

The AIM model is estimated by a maximumlikelihood procedure under the assumption thateach share equation in (4) has an additive errorterm, s,,. That is,

(5) s = g(v,a) + E,,, i = 1,2,3.

The disturbances are assumed to be indepen-dent and identically distributed multivariate nor-mal random variables with zero mean andcovariance matrix 1. The sample disturbancecovariance matrix, X, is defined as

= (,, i,~’,)/N,

where N is the sample size, and the sample dis-turbance, i,, is computed by

= s — g, (v,a), for i = 1, 2, 3.

Maximizing the likelihood function for the sys-tem is equivalent to minimizing the generalizedvariance, ]~.“

The estimation was accomplished using a non-linear program (GRG2). To find a global optima,an extensive search over a large range of initialconditions was conducted. Because of the com-plex nonlinearity of the AIM demand system,the true maximum likelihood estimates aredifficult to obtain. The possibility of missing the

global optima was reduced, however, by an ex-tensive search of the parameter space.26

All parameters are subject to non-negativityconstraints to guarantee that global regularityconditions are satisfied. Because inequality con-straints limit the applicability of the existing the-oretical sampling distribution theory, the usualmethods for testing hypotheses cannot beused.”

Income Elasticities and PriceElasticities

Because the share equations are so complex,AIM does not yield explicit functional forms fordemand functions. This is the consequence forcorrectly embedding utility maximization into aneconometrically estimable demand system thatcan be used to compute economically meaning-ful income and price elasticities.

The Allen Partial elasticities of substitutionand income elasticities are defined and expressed by the following formulas:28

for i # j,

for i =

SAIp, I Ss, 5 S 1 Ss, S Ev~= = — —— — + — (— +—) —3p,A~ v, Sp, vp, v, v, SE p, s’,

where p, are the prices (and user costs), Aldenotes the income-compensated demand functions for the ith asset, s, denotes the expenditureshares and E denotes total expenditures. Theelements, a,,, constitute a symmetrical matrixcalled the Allen Partial matrix.

The income elasticities are defined by

SA,E Ss, E

SEA, SE s,

and the uncompensated price elasticities aredenoted by

‘4The higher-order AIM system contains many moreunknown parameters; see Barnett and Vue (1988).

“See Barnett (1981).2STo estimate parameters, a, i = 1, 2, 3, 5, . . , 15, initial

values of the unknown parameters were assigned. A num-ber of different initial values of a,, 0.01, 0.03, 0.05, 0.08,0.1, 0.2,..., 1.0, 2.0,..., 10.0 were used.

“See Dufour (1989), Kodde and Palm (1986), Wolak (1989)and Barnett, Geweke and Yue (1991).

“See Diewert (1974) and Barnett and Yue (1988).

42 II‘IIII

a = ~ = I Ss, s, 1 Ss, 5, Ev,v~

v, Sp~ v, v, SE p, 5,5,

IIIII

.1’II

FEDERAL RESERVE BANK OF St LOUIS I

S A,p,= ________ = sia’i —

Sp,A,

where A, are the ordinary or uncompensateddemand functions. The connection between com-pensated and uncompensated demand functionsis stated by the usual Slutsky equation.

Gross substitutability and complementarity isprovided by the off-diagonal terms of the un-compensated price elasticity matrix. If v~, is posi-tive, good i and good j are substitutes; in otherwords, when the price of good i rises, demandfor good j increases to replace a cutback in de-mand for good i. If it is negative, they are com-plements—an increase in the price of good i (or jicauses the demand to fall for both goods.

Similarly, the pure substitution effects are de-fined by the Allen Partial matrix. If the utilityfunction obeys regularity conditions, the owncompensated price elasticities (s,o,,) and (a,,),must be negative. Hence, the compensated priceelasticity matrix represents potential movementsalong the consumer’s indifference curves andcan he used to examine whether the estimatedunderlying utility function satisfies regularityconditions.

‘rhe computed elasticities of the AIM demandsystem are compared with other money demandsystems in the next section. Because of the com-plexity of share equations, a numerical methodis used to compute the partial derivatives of theexpenditure shares with respect to prices andincome that occur in the elasticity formula. Thecomputation of elasticities is calculated usingthe estimated share equations. ‘I’ime series ofthe elasticities are produced by substituting timeseries of normalized prices and respective par-tial derivatives into the elasticity formula.

EMPIRICAL RESULTS OF THE AIMMONEY DEMAND SYSTEM

In this section, the AIM demand system is esti-mated and the income and substitution elasticitiesare compared with those for the translog andFourier demand systems. In addition, charac-teristics of monetary assets relative to consumergoods are analyzed.

‘9Douglas Fisher provided all of the data used to estimatethe AIM demand system. He had previously used thesedata to estimate the translog and the Fourier demand sys-tems. Hence, the empirical results presented here can becompared directly with his. See Fisher (1989).

Estimates of Parameters and In-come and Price Elasticities

Table 1 displays the coefficient estimates fromthe AIM demand system derived by U.S. quar-terly data from 1970.1 through 1985.2.29 Theseparameters represent the consumer’s taste orpreference and determine the utility functionthat underlies the estimated AIM demand sys-tem. Because the taste parameters are assumedto be constant, the consumer’s utility functionand preference did not change over time. Theestimates of a, and a, were zero due to thenon-negativity constraint.’°

The estimated Allen Partial elasticities of sub-stitution and income elasticities are reported intable 2. The numbers represent the averages

‘°Tothis extent, the AIM model is at odds with the data be-cause the estimated parameters would have been negativeif unconstrained.

43

~‘/ ~‘~> ~~

/

,1’ /~/~/

IIIIIIIIIIIII$III

Mtiit4M~4z\~

SPfl~4S~~~fr/,,4~<; 7//~\/\//// ~/ /osawlrsattha~ ~ ~ /.

F4/ rc/4 //// ~ ~~)*&//2 // 4/c/ //

ari~’ea/~ ~S~!4fl~ /

NOVEMBER/DECEMBER 1991

44

and their standard deviations (in parentheses)over the sample period. Table 3 displays the es-timated substitution and income elasticities fromthe translog and the Fourier series modelspreviously reported by Fisher (1989, page 103).

Table 4 presents the average uncompensatedprice elasticities and their standard deviationsover the sample period for AIM. The cor-responding elasticities for the other two modelsare not available.

What’s Wrong with the Translogand Fourier Demand Systems?

In the translog demand system results (shownin table 3), the positive sign of ~ indicates thatthe regularity condition is violated. This resultsuggests that the higher the opportunity cost ofholding currency and demand deposits, thegreater their demand. Given this violation of the“law of demand,” the results from the translogdemand system must be considered suspiciousat best and, at worst, unreliable.

Problems in the Fourier series demand systemcannot be seen in table 3 because the numbersreported there are the average values of thesecoefficients. According to Fisher, however, ex-cept for v~,,,,,the income elasticities and AllenPartial elasticities of substitution changed signsfrequently over the period.” For example, in1970, a,, was significantly negative, implyingcomplementarity; in 1971-1972, it was signifi-cantly positive, implying substitutability; andthen in 1974-1975, it became negative again.Figure I displays a o,,-comparison of the Fourierand AIM money demand systems. It is inexplica-

/ /~ // / / /‘:~~ :~ ~ ~Zf-

~

/ ~:_ ~

4S~~ <Fat4~:~ /~ /// 4/4/ /

~ 4~/; ~

~ 4~~/~< / /4~

/4 / ////4/44/ //c//

24//~

ble that currency and demand deposits, A,, andsavings deposits and money market deposit ac-counts, A,, should be complements during someperiods and substitutes during others.

Empirical Inference ofCharacteristics of Monetary Assetsby the AIM Demand System

The anomalies observed with the translog andthe Fourier series demand systems do not occurin the AIM demand system. The own-priceelasticities are negative and all estimated elastici-ties maintain their signs over the entire sampleperiod. Moreover, their smaller standard devia-tions indicate that they are more stable; this canalso he seen in figure 1. In the Allen Partialmatrix, the diagonal elements are all negative

‘1See Fisher (1989), pp. 105-06.

FEDERAL RESERVE BANK OF ST. LOUIS

Figure 1The Allen Partial Elasticity of A1 and A,

while the off-diagonal elements are positive.This implies that the three monetary aggregatesand aggregate consumption are substitutes foreach other in the presence of income compensa-tion. Moreover, the pure substitution effect be-tween each pair of the three aggregatedmonetary assets is much greater than betweenthe consumption good and monetary assets.

The income elasticities in table 2 are all Posi-tive, with the income elasticity of the consump-tion good about unity, and the incomeelasticities of the three monetary assets roughlyequal to 0.5.” These results suggest that con-sumption goods and monetary assets are normalgoods.

Table 4 shows that the uncompensated cross-price elasticities of (A,, A,), (A,, A,) and (A,, A,)are positive, implying that these monetary assetsare gross substitutes. The uncompensated cross-price elasticities of (A,, A,), (A,, A,) and (A,, A4)are negative, indicating that consumption goodsand monetary assets are gross complements.

These results show that, if the user costs ofsavings deposits (or money market deposit ac-counts, or small time deposits) rise, consumerswill shift their funds to, demand deposits (or to

S2This may not be justified by the unity income elasticity inthe reduced form of the aggregate money-demand func-tion. As pointed out in the text, no reduced form demandfunctions are estimated in our study and the incomeelasticity is defined by the microeconomic approach.

Because their income elasticities are significantly lessthan one, the monetary assets in our study are not “luxurygoods,” as has been claimed in some previous research.See Serletis (1988).

30

25

20

10

45

checkable deposits or currency). An oppositeshift of funds will take place if the user costs ofcurrency, demand deposits and checkabledeposits should rise. If these user-cost changesare sufficiently large, ignoring the cross-priceeffects among monetary assets will producelarge errors in their demand functions.

Monetary services and consumer goods are

10 consumed jointly. If a consumer increases the

consumption of commodities for some reason,o the demand for monetary assets is also in-

creased. This is consistent with the idea thatconsumers hold monetary assets to finance cur-rent and future consumption. Also, the negative

-is ~ indicates that price inflation will reduce de-

mand for monetary assets.Not surprisingly, the own price elasticities of

monetary assets are greater than their crossprice elasticities. Similarly, it is not surprising tosee that the cross-price effects of a change inthe price of consumption goods on the demandfor monetary assets are greater than the cross-price effects of a change in the price of mone-tary assets on consumption. Consequently, em-pirical results from the AIM demand systemprovide a reasonable quantitative analysis of thecharacteristics of monetary assets—characteris-tics that are broadly consistent with conventionalviews of demand for money.

A DYNAMIC ANALYSIS BY THEAIM DEMAND SYSTEM

Constructing a dynamic model of demand formoney has been difficult because the currentstate of economic knowledge about dynamic be-havior is incomplete; the dynamics of money de-mand are still very much a “black box” mystery.”

Unlike most multivariate time series models,the AIM model is static. It does not considerspecific dynamic effects among monetary assetsand consumption goods. The utility function isnot intertemporal and its parameters are time-invariant—the consumer’s preference is not per-mitted to change over time.

~ situation occurs in a recent debate in economic litera-ture; see Hendry and Ericsson (1991) and Friedman andSchwartz (1991).

1970 70 72 70 74 70 70 77 70 70 00 01 02 03 04 1900

IIIIIIIIIIII

IIIII

NnvpMRrwnprnnncD 1001

Figure 2Income Elasticities E1, E2, E3 and E4 for A,, A2,A3 and A4

46

Figure 3Uncompensated Own Price Elasticities

Nevertheless a simple dynamic analysis canbe used to examine the AIM demand system.Time series of income and price elasticities canbe computed using estimated share equations.Movements in these elasticities reflect bothchanges in user costs and the consumer’s reac-tions to such changes. Both of these are reflectedin the shares, 5. In this way, the dynamics ofthe AIM demand system can he investigated eventhough demand for money is stable byassumption.

This dynamic analysis is displayed in figures2, 3 and 4. Figure 2 shows that the incomeelasticities of the three monetary assets are rela-tively constant over the entire sample period. Incontrast, the price elasticities (shown in figures3 and 4), exhibit sizable fluctuations. Major shiftsin price elasticities occurred during 1973.1-1974.4and 1978.2-1982.2. During these periods, a num-ber of studies have reported that demand formonetary aggregate Ml was “erratic.”~

Price and user cost elasticities moved drasti-cally during these periods. In the second period(1978.2-1982.2), the cross price elasticity, n~,rose by 50 percent of its 1977 level, implyingthat demand for A, (currency plus demand de-posits, plus checkable deposits) became muchmore sensitive than it was previously to changesin the opportunity cost of holding A2 (savingsdeposits and money market demand accounts).Meanwhile, there was a sharp rise in the user

Figure 4Uncompensated Cross Price Elasticities

cost of A2. These factors would appear to ac-count for the major shift of funds from A2 to A,during the period.

The opposite price elasticity, ~23’ also rose by20 percent. Nevertheless, it was less than 80percent of the value of rj~~and the rise in theuser cost of A, was more modest than that ofA3- It was observed that the opportunity cost ofA2 increased much faster than that of A,. Hence,the actual flow of funds from A, to A, might notbe significant.

The cross price elasticity, vj,,, dropped 30 per-cent in 1979, implying that the demand for A1

345ee Goldfeld (1976) and Friedman (1984).

I

<4~4~‘~H

/ /\ ~~/M1~ K<,.

o //c\/ ~

7 ~%~ ~ c1~1

III

3970 71 73 73 75 76 77 78 75 85 II 62 03 84 1581

IIIIII

1571 VI 72 73747176777678 65 83 82 83 64 555

IIIIII~1I

F~flFRM RFSFRVF RANK OF St IOUIS

was less sensitive to changes in the user cost ofA,. Hence, the shift of funds from A3 to A,should have been moderate despite a substantialincrease in the user cost of A~.

These results are roughly consistent with de-velopments during the period. In November 1978,commercial banks were authorized to offer au-tori atic transfer service (ATS) from savings ac-counts to checking accounts. Other interestceiling-free accounts were also introduced inthe early 1980s. In January 1981, NOW ac-counts were introduced nationwide. These finan-cial innovations should have encouragedconsumers to shift funds from savings accountsand money market deposit accounts in A2 intoNOW accounts in A5. This may have increasedthe interest sensitivity of demand for A,.33

Simulating the Growth Rates ofMonetary Aggregates

A further investigation of the behavior of mon-etary aggregates can he made by a dynamicsimulation. For example, suppose that demandfor A, has been derived by utility maximizationand expressed by the ordinary demand func-tions of price and user costs and total expen-diture

(6) A, = C1 (u,, u,, u,, u4, E).

The total differentiation of (6) results in

(7) dA~= I dG,/du, du1 + dG1/dE dE.

Dividing both sides of (7) by (6) and using defi-nitions of the uncompensated price elasticitiesand the income elasticity gives

(8) dA1/A, = I ~ (du1/u~) + ~,, (dE/E).

Using time series of the elasticities and thegrowth r-ates of price and user costs and totalexpenditure, the right-hand sides of the equa-tions In (8) are computed. In this way, thegrowth rates of demand for A, can be“simulated.”

The act ual and simulated growth rates of de-mand for monetary aggregates and consumptionare displayed in figures 5 through 8. The simu-lations match the actual growth rates fairlywell, especially for consumption. The simulationrates of monetary assets had large fluctuations

35See Thornton and Stone (1991) for a discussion of thispossibility.

47

Figure 5Growth Rates of Aggregate A, Actual vs. Simulation

0.10

0.55

0.00

—0.05

—5.001

—0.10

0.30

0.40

0.00

—0.55

—5.10

—0-05

Figure 6Growth Rates of Aggregate A2 Actual vs. Simulation

07 07

‘7-7

050 / 00

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,,lr: e “~ I ,~C ~ -

0 ~ ‘~sK -- ;—-* ~ ‘‘7 0

‘7 -000 4/7,7/ -< -“ // 000

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-—-‘77/ /77,//’7,/~ // <‘7I’--7-02 ~‘ /7 7/7’ ‘711 *,,,,~ ,, , -425

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~7/

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17071 7 73 74 7 7 77 70 75 80 1 3 03 41

0-5

0.0

—00

—1.0

—0.0

070 7173737470767770 70 00 00 02 03 04 01=

IIIIIII’I$I1I1IIIpI

Figure 7Growth Rates of Aggregate A3 Actual vs. Simulation

0070 717273747570777070 00 01 02 83 04 1100

a Mn%FtRRnrD,ncrni.nco I OOI

Figure 8Growth Rates of Aggregate A4 Actual vs. Simulation

around the actual growth rates in the periods1972.4-1975.1 and 1978.2-1982.2. Fluctuations ininterest rates and inflation rates were substan-tial during each period, causing correspondingfluctuations in growth rates of user costs.3°These changes are reflected directly in the sim-ulation rates.

Because the AIM model is static, sharp changesin user costs are necessarily reflected in cor-respondingly sharp changes in the simulatedgrowth rates of aggregates. Hence, it is not sur-prising that the simulation errors are large dur-ing periods when there are sharp changes inuser costs. Nevertheless, figures 5 through 8suggest that the AIM demand system has cap-tured mahy of the characteristics of the U.S.monetary system during the sample period.

Can the AIM Demand System Ex—plain the Case of the MissingMoney?

Although the simulation of the growth rate ofA2 indicates that the AIM model produced rela-tively large errors during the period of “missingmoney” (1973.4-1976.2), an analysis of the A1Mresults might provide a clue.~~During this peri-od, there was a sharp decline in demand for

‘6Some terms are essentially zero and can be ignored. Thefollowing growth rate equations are accurate enough toproduce the simulation:dA1/A, = 110du,/u, + qo

2du

2/u, + rj,,du,/u

3+

+ rncdEfEdA,/A, = rj

22du

2/u

2+ ~

23du

3/u,

dA3/A, = tj33

du3

Iu3dAdA4 = r

142du

21u

2+ ~

43du

3/u

3+

144duJu

4+ 1,0dE/E.

In the equation for A0 there are more affecting elements;the own and cross-price effects of A2, A3 and A4 are im-

48

Ml; conventional money demand equations con-sistently overpredicted demand.

A potential explanation can be obtained byconsidering figures 9 and 10. Figure 9 showsthe partial derivative of demand for A1 withrespect to its own user cost. Figure 10 showsthe partial derivative of demand for A1 withrespect to the user costs of A,, A3, A4 and E.Figure 9 shows a sharp rise in the rate of changein demand for A, with respect to a change inits user cost. Indeed, figure 11 shows that theuser cost of A5 increased relative to the pricelevel. Hence, the demand for A1 should havedeclined by a proportionately larger amountthan the rise in its user cost. Conventionallinear money demand equations with fixedregression coefficients could not accommodatethis nonlinearity because, in such linear demandsystems, the coefficients (derivatives) are assumedto be constant. However, figures 9 and 10 clear-ly suggest that this is not the case. Moreover,ordinary least squares is relatively sensitive to“outliers.” Consequently, there will be substan-tial changes in the estimated regression coeffi-cients when the equations are estimated overperiods when these derivatives change signifi-cantly.

Conventional money demand equations maybe misspecified for another reason, as well.Usually they include a single short-term interestrate intended to reflect the opportunity cost ofholding money. AIM analysis indicates that thedemand for money does not depend on a single“representative” interest rate, but on its usercost and the user costs of “close” substitutes(recall that the demand for A, was sensitive tochanges in the user costs of A,). Hence, conven-tional money demand equations may producemisleading results when interest rates changerelative to the user costs of Ml or relative tothe user costs of close substitutes for Ml.

Therefore, the case of “missing money” and“unexplained” parameter shifts in conventionalmoney demand functions may result from thefact that they are essentially linear approxima-

portant in simulating the growth rate of A,. The growthrates of demand for the other two monetary aggregates,however, are determined mainly by their own price effectsand cross-price effect, n23. This suggests that ignoring thesubstitution effects of non-Mi components of M2 might beone of the factors that discredit reliability of the conven-tional Mi demand function.

375ee Goldfeld (1976).

IIIIIIIIIIIIIIIIII

FEDERAL RESERVE BANK OF St LOUIS a

Figure 9The First Derivative of A1 with Respect to Own User Cost

Respect to User Costs,

tions to nonlinear demand functions. If so, theyintuitively will provide much poorer approxima-tions during periods when there are dramaticchanges in user costs.

.Is The Money Demand FunctionStable?

The erratic behavior of conventional moneydemand functions and, more recently, the in-come velocity of Ml, have led many researchersto assert that the demand for money is “unsta-38For a discussion of the velocity of Mi and an analysis of

some of the explanations, see Stone and Thornton (1987).30For example, see Friedman (1956) and Lucas (1988).40ln the parlance of modern time-series analysis, these

elasticities are said to be stationary, that is, mean revert-

49

Figure iiUser Cost of A, vs. Price of A4

0-’SO

0-120

0-000140

0-0701

0-050 -

0-0251

0-0 00

0-0 25

0-00 0

0-075

- 0-050

0030

ble.”~~Others have asserted that money demandis stable based on the observed stability of theconsumption function.~°

The AIM demand system integrates demandfor both consumption and money and then esti-mates them simultaneously. These estimates sug-gest that, while the own price and cross priceelasticities show considerable variation due tochanges in the price level and user costs, theychange little on average over the period (seefigures 3 and 4)40 Moreover, the estimated in-come elasticities for- all three monetary aggre-gates are nearly constant (see figure 2). Of course,these results are obtained from a model wherethe estimated parameters are time-invariant,that is, the preference function is constant.Because of this, it is necessarily true that de-mand functions are “stable.” Nevertheless, therelatively good performance of AIM providessome promise that, like consumption, the de-mand for money will ultimately be shown to bea stable function of a relatively few econonlicvariables—in this case, income and user costs.

CONCLUSION

Two distinctly different micro-econometric de-mand system approaches to the demand formoney were presented and discussed. An ad-vanced AIM demand system was presented andestimated using U.S. time-series data. Unlike

ing. However, no formal tests of stationarity were per-formed in this study.

1070 7072717475 70 77 70 70 00 00 82 03 04 ‘005

Figure 10The First Derivatives of A0 withPrice and Income

UA0

IIIIII1I11

IIIIIIIIIa

0.1W~ ___-

0070 70 77737470 76 77 70 70 00 00 02 03 00 0005

Mfl17c00otOin~r00lO0rr5 Inn-c

REFERENCES

Barnett, William A. Consumer Demand and Labor Supply(Amsterdam: North-Holland, 1981).

_______- “New Indices of Money Supply and the FlexibleLaurent Demand System;’ Journal of Business and Eco-nomics Statistics (January 1983), pp. 7-23.

Barnett, William A., and Andrew B. Jonas. “The Muntz-Szatz Demand System: An Application of a Globally WellBehaved Series Expansion’ Economics Letters Vol. 11,No.4 (1983), pp. 337-42.

Barnett, William A., John Geweke, and Piyu Vue. “Seminon-parametric Bayesian Estimation of the Asymptotically IdealModel: the AIM Consumer Demand System;’ in WilliamBarnett, James Powell, and George Tauchen, eds., Non-parametric and Seminonparametric Methods in Economet-rics and Statistics, Proceedings of the Fifth InternationalSymposium in Economic Theory and Econometrics (Cam-bridge University Press, 1991).

Barnett, William A., and Piyu Vue. “Seminonparametric Esti-mation of the Asymptotically Ideal Model: the AIM DemandSystem’ in George Rhodes and Thomas Fomby, eds.,Nonparametric and Robust Inference, Advances in Econo-metrics, Volume 7 (1988), pp. 229-51.

Diewert, W. E. “Applications of Duality Theory;’ in MichaelD. Intriligator and David A. Kendrick, eds., Frontiers ofQuantitative Economics, Vol. II (Amsterdam: North-Holland,1974), pp. 106-71.

Diewert, W. E., and t J. Wales. “Flexible Functional Formsand Global Curvature Conditions[ Econometrica (January1987), pp. 43-68.

Dufour, Jean-Marie. “Nonlinear Hypotheses, Inequality Res-trictions, and Non-Nested Hypotheses: Exact SimultaneousTests in Linear Regressions’ Econometrica (March 1989)pp. 335-56.

Fayyad, Salam K. “A Microeconomic System-Wide Approachto the Estimation of the Demand for Money;’ this Rewew(August/September 1986), pp. 22-fl

Feenstra, Robert C. “Functional Equivalence Between Li-quidity Costs and the Utility of MOney’ Journal of MonetaryEconomics (March 1986), pp. 271-91.

Fisher, Douglas. Money Demand and Monetary Policy (TheUniversity of Michigan Press, 1989).

Friedman, Benjamin. “Lessons From the 1979-82 MonetaryPolicy ExperimenL” American Economic Review (May1984), pp. 382-87.

Friedman, Milton. “The Quantity Theory of Money—A Restate-ment;’ in Milton Friedman, ed., Studies in the Quantity The-ory of Money (University of Chicago Press, 1956).

Friedman, Milton, and Anna J. Schwartz. “Alternative Ap-proaches to Analyzing Economic Data’ The American Eco-nomic Review (March 1991), pp. 39-49.

Gallant, A. Ronald. “On the Bias in Flexible FunctionalForms and an Essentially Unbiased Form: The FourierFlexible Form,” Journal of Econometrics (February 1981),pp. 211-45.

- “The Fourier Flexible Form2 American Journal ofAgricultural Economics (May 1984), pp. 204-08.

Gallant, A. Ronald, and Douglas W. Nychka. “Semi-Nonparametric Maximum Likelihood Estimation;’Econometrica (March 1987), pp. 363-90.

Goldfeld, Stephen M. “The Case of the Missing Money;’Brookings Papers on Economic Activity (3:1976), pp.683-739.

Hendry, David F., and Neil FL Ericsson. “An EconometricAnalysis of U.K. Money Demand in Monetary Trends in theUnited States and the United Kingdom by Milton Friedmanand Anna J. Schwartz[ American Economic Review (March1991), pp. 8-38.

Judd, John R, and John L. Scadding. “The Search for a Sta-ble Money Demand Function: A Survey of the Post-1973Literature[ Journal of Economic Literature (September1982), pp. 993-1023.

50 IIITiII

other utility function-based approaches, AIM es-timates are consistent with microeconomic the-ory. Dynamic simulations of the growth rates ofvarious monetary aggregates and consumptionsuggest that the estimated AIM model performedwell; nevertheless, the largest simulation errorsoccurred in periods when there were relativelysharp swings in user costs or inflation. This isperhaps not too surprising given the static na-ture of the AIM analysis.

An analysis of changes in income and crossprice elasticities are suggestive of portfolio shiftsamong monetary aggregates in the 1970s and1980s consistent with the observed behavior ofthese aggregates. The results of AIM suggestthat the reported failure of conventional linear(or log-linear) money demand equations mayresult from trying to fit fundamentally non-linear functions with linear ones. The resultsshown here suggest that this problem will beparticularly acute whenever there are sharpchanges in user costs. Unfortunately, these areprecisely the times when AIM performance wasalso poor. The key to solving this problem inAIM, however, is to find a way to make AIM ex-plicitly dynamic. It may not be necessary toassume that consumer preferences are unstable.

The sampling distribution theory for AIM hasnot been worked out at this time, so relevanthypothesis tests cannot be conducted yet. Also,because the time series on the relevant usercosts of monetary aggregates is limited, theavailable data cover a relatively short sampleperiod. These factors, coupled with the fact thateven low-order (first-order) AIM systems requirea relatively large number of estimated parame-ters, place severe limits on attempts to evaluatethe performance of AIM using out-of-sampleforecasts. Despite these problems, the estimatedAIM system appears to have captured many ofthe characteristics of monetary assets and offerssome useful explanations to puzzling empiricalissues. Hence, these results are encouraging tothose who believe that microeconomic princi-ples, such as utility maximization, can be ap-plied usefully to macroeconomic problems.

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I4!

II

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FEDERAL RESERVE BANK OF St LOUIS

51

Kodde, David A., and Franz C. Palm. “Wald Criteria for Joint-ly Testing Equality and Inequality Restrictions’ Econo-metrica (September 1986), pp. 1243-48.

Lasdon, Leon S., and A. D. Warren. “GRG2 User’s Guide”University of Texas at Austin, 1989.

Lucas, Robert E., Jr. “Money Demand in the United States: AQuantitative Review[ Carnegie-Rochester Conference Serieson Public Policy No. 29, (Autumn 1988), pp. 137-68.

Mankiw, N. Gregory. “A Quick Refresher Course in Macro-economics,” Journal of Economic Literature (December1990), pp. 1645-60.

Moore, George R., Richard D. Porter, and David H. Small.“Modeling the Disaggregated Demands for M2 and Ml:The U.S. Experience in the 1980s,” in Peter Hooper andothers, eds., Financial Sectors in Open Economies: Empiri-cal Analysis and Policy Issues (Board of Governors of theFederal Reserve System, 1990), pp. 21-97.

Pierce, James L. “Did Financial Innovation Hurt the GreatMonetarist Experiment?” American Economic Review (May1984), pp. 392-96.

Rasche, Robert H. “Demand Functions for Measures of U.S.Money and Debt,” in Peter Hooper and others, eds., Finan-cial Sectors in Open Economies: Empirical Analysis andPolicy Issues (Board of Governors of the Federal ReserveSystem, 1990), pp. 113-61.

Serletis, Apostolos. “Translog Flexible Functional Forms andSubstitutability of Monetary Assets’ Journal of Businessand Economic Statistics (January 1988), pp. 59-67.

Stone, Courtenay C., and Daniel L. Thornton. “Solving the1980s Velocity Puzzle: A Progress Report;’ this Review(AugustlSeptember 1987), pp. 5-fl

Swofford, James L., and Gerald A. Whitney. “Flexible Func-tional Forms and the Utility Approach to the Demand forMoney: a Nonparametric Analysis,” Journal of Money,Credit and Banking (August 1986), pp. 383-89.

Telser, Lester G., and Robert L. Graves. Functional Analysisin Mathematical Economics (The University of ChicagoPress, 1972).

Thornton, Daniel L., and Courtenay C. Stone. “Financial Inno-vation: Causes and Consequences;’ in Kevin Dowd andMervyn K. Lewis, eds., Current Issues in Monetary Analysisand Policy (MacMillan Publishers, 1991).

Uzawa, Hirofumi. “Production Functions with ConstantElasticities of Substitution[ Review of Economic Studies(October 1962), pp. 291-99.

Wolak, Frank A. “Local and Global Testing of Linear andNonlinear Inequality Constraints in Nonlinear EconometricModels’ Econometric Theory (April 1989), pp. 1-35.

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NOVEMBER/DECEMBER 1991