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A microeconometric approach to estimating money demand: the asymptotically ideal model.

The demand for money plays a critical role in macroeconomics. In conventional money demand analysis, the demand for real money balances is typically expressed as a function of such variables as real income, the expected rate of inflation and the nominal interest rate.(1) Empirical investigations using these variables have not been particularly useful in predicting the demand for money or in formulating and evaluating monetary policy.(2)

More recently, a number of researchers have attempted to estimate money demand in a manner consistent with microeconomic foundations. Even in these cases, however, the empirical results have been largely discouraging.(3)

This paper reviews the general micro-econometric approach to estimating the demand for money, culminating with an advanced micro-econometric model, called the Asymptotically Ideal Model (AIM). AIM is applied to U.S. timeseries data and the results are compared briefly with those from previous studies. AIM results are consistent with microeconomic theory and provide insight into the behavior of money demand in the 1970s and 1980s.

MICROECONOMIC MODELING

As a result of developments in macroeconomic theory over the past two decades, "almost all macroeconomists agree that basing macroeconomics on firm microeconomic principles should be higher on the research agenda than it has been in the past."(4) Problems arise, however, when aggregate, macroeconomic data are used to estimate microeconomic-based models of money demand. Some of these problems are illustrated by a simple example that uses two approaches to microeconomic modeling: the demand function approach and the utility function approach.(5)

The Demand Function Approach

Consider an economy where the representative consumer allocates income between a composite consumer good, A, and a monetary asset, M, that yields monetary services. The consumer's objective is to maximize the utility function (subject to a budget constraint), given the price of the composite commodity and the user cost of the monetary asset. Let P, u and E denote the price level (the price of A), the nominal user cost of holding one real unit of M and total expenditures (or income), respectively.(6) The consumer's decision problem is expressed by

Max f(A, M) = [A.sup.r][M.sup.1-r],

subject to PA + uM = E.(7)

For simplicity, the utility function, f, is Cobb-Douglas, where r, an unknown parameter, characterizes the consumer's taste or preference. The optimal solution to the consumer's decision problem yields ordinary demand functions for A and M. In this case, the demand functions are:

(1) A = rE/P = r/(P/E) = [G.sub.1](u/E, P/E, r), and (2) M = (1-r)E/u = (1-r)/(u/E) = [G.sub.2](u/E, P/E, r).

The demands for A and M are functions, [G.sub.1] and [G.sub.2], respectively, of E, P, u and the unknown parameter r. Because the budget constraint is linear in P and u, the normalized price, P/E, and user cost, u/E, can replace P, u and E. In general, demand functions can be expressed by normalized prices (including the user cost) and the unknown parameter. This parameter can be estimated by simultaneously fitting equations (1) and (2) using data on real quantities of A and M and the normalized price and user cost.

This approach is called the "demand function" approach because estimation begins after demand functions are specified. For this approach to yield meaningful estimates, however, the specified system of demand functions must correspond to the neoclassical utility function from which they were derived. Consequently, the conditions for estimating the system of demand functions are fairly restrictive. For instance, the Rotterdam model (a well-known demand system used in empirical studies) requires specific forms for demand functions and specific constraints on parameters during estimation.(8)

Even if these conditions are satisfied, however, the Rotterdam model is still highly restrictive because the assumed underlying utility function (either Cobb-Douglas or CES) is a member of a narrow class of utility functions with constant elasticities of substitution.(9)

The Utility Function Approach

The utility function approach to demand estimation also has been used in empirical studies. To understand this approach, reconsider the consumer's decision problem and the demand functions shown in equations 1 and 2. In the utility function approach, demand functions for A and M are substituted into the utility function, f(A, M), to obtain the indirect utility function.

h([v.sub.1], [v.sub.2], r) = f[A([v.sub.1], [v.sub.2], r), M([v.sub.1], [v.sub.2], r)],

where [v.sub.1] = P/E, [v.sub.2] = u/E. Because the indirect utility function has properties that are the inverse of those for the utility function, it is more convenient to use the reciprocal of the indirect utility function.

F([v.sub.1], [v.sub.2], r) = 1/h([v.sub.1], [v.sub.2], r).(10)

By definition, demand functions can be expressed in terms of their expenditure shares, [s.sub.1] = AP/E and [s.sub.2] = Mu/E. That is,

A = [S.sub.1]/[V.sub.1] and M = [S.sub.2]/[V.sub.2].(11)

In this way, demand functions can be obtained without solving first-order conditions. Consequently, no matter how complicated the utility function might be, the derivation of share equations and demand functions is straight-forward.

Of course, if the utility function is relatively simple and well-behaved (for example, when the Cobb-Douglas function is used), there is no need to use the utility function approach. However, if the utility function includes more than two goods or is sufficiently complicated, the Lagrange multiplier procedure cannot be used to derive demand functions.

THE SEMI-NONPARAMETRIC

METHOD FOR ESTIMATING THE

DEMAND SYSTEM

The critical step in applying the utility function approach is the specification of the proper reciprocal indirect utility function, F. To simplify the terminology, the term "utility function" will indicate "the reciprocal of the indirect utility function" in the following discussion.

Flexible Functional Form Modeling

Cobb-Douglas and CES functions have been used extensively in theoretical and applied work because of their relative simplicity. Despite their apparent successes, however, such use has been criticized. For example, if there are more than two goods, the CES utility function can only generate demand systems when each pair of goods has the same constant elasticity of substitution.(12) Unless there is prior information to the contrary, however, the elasticities of substitution should be determined by the data rather than restricted by the specification of the utility function. This limitation has motivated researchers to look for utility functions that are more flexible and allow for data-determined elasticities of substitution.

Flexible functional form models have attracted considerable attention in economics literature since the early 1970s, when it was proposed that the translog and generalized Leontief functions should replace neoclassical utility functions. It was recognized that the values of the elasticities of substitution are determined by the value of the utility function and the values of its first- and second-order derivatives are evaluated at its extreme point. Consequently, if the values of the utility function and these derivatives can be estimated, so too can the elasticities of substitution. This idea forms the basis for the flexible functional form approach.

A functional form is said to be flexible if its level and the first- and second-order derivatives at a point in its domain are allowed to reach the respective values of the "true" utility function at that point. The true utility function is assumed consistent with the properties of the data, so that, in principle, elasticities of substitution consistent with the data can be estimated.

One flexible functional form is derived from a Taylor series expansion where all terms greater than second-order are eliminated, that is,

(3) F = [a.sub.omicron] + [EPSILON.sub.i a.sub.1 x.sub.i] + [EPSILON.sub.i EPSILON.sub.j a.sub.ij] [x.sub.i x.sub.j.sup.13]

This approximation is flexible because it has enough free coefficients, [a.sub.o], [a.sub.i], [a.sub.ij], to allow for any desired value of the first- and second-order derivatives 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 = In(f(x)) and [x.sub.i] = In([q.sub.i]),

where f denotes the utility function and [q.sub.i] represents the quantity of good i. The generalized Leontief function is attained by letting

F = (f(x))[sup.1/2] and [x.sub.i] = ([q.sub.i])[sup.1/2].

The coefficients in these functional forms can be estimated and, in turn, the demand system and the elasticities of substitution can be derived.

Caveats For Flexible Functional

Forms

Theoretically, the second-order Taylor approximation can attain flexibility only at a single point or in an infinitesimally small region. Hence, estimates of the elasticities of substitution are valid only for the range of observations covered by the sample data. Therefore, the second-order Taylor series approximation should be viewed as "locally flexible."

Such models are also subject to another, potentially more serious, problem. Experience has demonstrated that regularity conditions are frequently violated! Therefore, the restrictions that microeconomic theory imposes on consumer behavior are not embedded in these flexible functional forms. This point is illustrated later in the empirical section of this paper.

In an attempt to solve these problems, microeconomists have developed a variety of flexible functional forms that maintain their flexibility and have larger regularity regions.(14) A family of such flexible functional form models has been proposed (for example, Barnett's (1981) minflex Laurent model).(15) To gain global regularity, however, additional constraints are imposed on the parameters which result in a loss of local flexibility. This tradeoff between flexibility and regularity is characteristic of flexible functional form modeling. None of these models is both globally regular and globally flexible.(16)

Semi-nonparametric Method

Gallant (1981) created the "semi-nonparametric method" to remove the local flexibility limitation. His method specifies a series of models that approximate the underlying utility function at every point in the function's domain. Hence, the models are globally flexible.

The "semi-nonparametric method" is built upon a well-known result in mathematics: a Fourier series expansion can converge to any continuous function.(17) In contrast to the local convergence of the flexible functional forms, the Fourier series can approximate a continuous function in the entire domain. Gallant proposed to use the Fourier series expansion to specify a series of utility functions that can converge to any neoclassical utility function. Because neoclassical functions are a subset of continuous functions, the property of the Fourier series expansion will guarantee asymptotic convergence to an underlying neoclassical utility function.

Fourier series modeling consists of a series of expansions of models, with succeeding models nested in the preceding one. When the sample size increases, higher-order models can be specified by simply adding more terms of the component functions. For instance, the first-order model is defined by the utility function,

[Mathematical Expression Omitted]

The [j.sup.th]-order model is defined by the utility function,

[Mathematical Expression Omitted]

Asymptotically, the model contains an infinite number of terms and unknown parameters. Therefore, asymptotic inference based upon the Fourier series expansion models is free from functional-form specification error. This is its principal advantage.

In empirical work, however, the number of terms must be finite. Consequently, the properties of lower-order Fourier models become decisive. The harmonic component functions, such as sines and cosines which are frequently used in engineering and physics, are not suitable in economic applications because they do not satisfy the usual regularity conditions, such as monotonically increasing and strictly quasi-concave. This means that lower-order Fourier series models can violate regularity conditions.(18) Nevertheless, Gallant's approach permitted micro-econometric models to achieve both global regularity and global flexibility.

The AIM Demand System

To solve the problems of Fourier series models, another infinite function series, called the Muntz-Szatz series, is adopted. A typical form of the series is expressed as:

[Mathematical Expression Omitted]

The Muntz-Szatz series expansion converges to a continuous function, and any continuous function can be approximated by the Muntz-Szatz series.(19) Consequently, this series can be used to approximate a neoclassical utility function asymptotically.(20)

The Muntz-Szatz series is a linear combination of a set of special power functions. In contrast to the Fourier series, the component functions of the Muntz-Szatz series, [q.sub.i.sup.1/2], [q.sub.i.sup.1/4],..., [q.sub.i.sup.1/2], [q.sub.j.sup.1/4], ..., are neoclassical functions. In other words, they are monotonically increasing and quasi-concave with respect to variables [q.sub.i] and [q.sub.j]. The Muntz-Szatz series is necessarily neoclassical, however, only if all of the coefficients, [a.sub.i], [a.sub.ij.sup.k], [b.sub.ij.sup.k], ..., are non-negative, because only positive linear combinations of the neoclassical component functions are necessarily neoclassical. As a result, the coefficient-restricted Muntz-Szatz series can approach a neoclassical function but may not approach any continuous function. Imposing these restrictions guarantees that the estimated function will not violate regularity conditions.

The Muntz-Szatz series is used in place of the Fourier series in Gallant's semi-nonparametric method. A series of models can be defined by increasing degrees of the Muntz-Szatz approximations. Under the parameter constraint, these models are globally regular; the respective utility functions are neoclassical everywhere in their domain. When the sample size increases, higher-order models can be specified with more free parameters to best fit the data and derive the elasticities of substitution that the data suggest. Hence, the Muntz-Szatz series gives rise to a model that is asymptotically globally flexible. Even a low-order approximation requires a fairly large number of parameters to be estimated, however. Hence, while the model is asymptotically globally flexible, finite samples will limit the researcher's ability to fully utilize this property.

The model has two additional features that make it particularly attractive for applied work. First, although there are a relatively large number of free parameters to be estimated, it is impossible to overfit the noise in the data. Because movements due to measurement errors are irregular and cannot be expressed by the neoclassical component functions, the model simply

(1) See Friedman (1956) for one of the most comprehensive discussions of the money demand function. (2) These functions have been subject to several unexplainable shifts and often imply a larger liquidity effect than is typically experienced. Perhaps the most dramatic example of this phenomenon occurred in the early 1980s with the yet unexplained break in the income velocity of M1. For this and other examples, see Goldfeld (1976), Friedman (1984), Lucas (1988) and Rasche (1990). (3) Frequently, the estimated own price elasticities of demand for monetary assets are positive, implying that their demand curves slope upward. For example, see Serletis (1988), Fisher (1989) and Moore, Porter and Small (1990). (4) Mankiw (1990), page 1658. (5) Some economists argue that aggregate data cannot be applied to microeconomic models without considering the problems of aggregation. Aggregation problems are not discussed in this paper, although the aggregation error might be one source of the unsatisfactory performance of conventional money-demand functions. (6) The user cost of holding a unit of a real monetary asset is computed by the formula, u = [p.sup.*](t) [R(t) - i(t)]/[1 + R(t)], where [p.sup.*](t) is the "true" cost of living index defined as the geometric average of the consumer price index and the consumption goods deflator, R(t) is the benchmark interest rate or the maximum rate in the economy at each period and i(t) is the interest rate on the monetary asset. The formula is derived from a widely applicable consumer decision model. (7) Distinct views about money have resulted in two approaches to analyzing consumer demand for money. In the first approach, money is viewed as a commodity which provides a monetary service flow to holders. Thus, real balances of the monetary assets directly enter the consumer's utility function along with real consumption. In the other approach, money is viewed as intrinsically worthless; consumers hold it only to finance current and future consumption. As a result, real money balances do not enter the consumer's utility function per se. Instead, the liquidity cost of holding real money balances is taken into account in the budget constraint. Feenstra (1986) shows that these two approaches are equivalent. (8) For an application of the Rotterdam model to the money-demand system, see Fayyad (1986). For the theory of the Rotterdam model, see Barnett (1981). (9) It is easy to encounter difficulties using the demand function approach. The failure to specify functions correctly or impose relevant restrictions can result in biased or inefficient parameter estimates. (10) The duality theory states that if the reciprocal of the indirect utility function, F, is nondecreasing and quasi-concave with respect to normalized prices, the respective utility function, f, must be nondecreasing and quasi-concave with respect to quantity variables. In this sense, the utility function is equivalent to its reciprocal indirect utility function. (11) Expenditures shares can be obtained by using the modified Roy's identity from duality theory. That is,

[Mathematical Expression Omitted]

where the vector [V.sup.'] = ([v.sub.1], [v.sub.2]) and the gradient vector

[Mathematical Expression Omitted]

(12) See Uzawa (1962). (13) This equation is written in general form, where [a.sub.o] denotes all the constants (that is, the function evaluated at the point of interest and the partial derivatives evaluated at the same point). The use of this general form to estimate these equations is one of the procedure's limitations because the point about which the expansion is made is estimated by the data, rather than being specified by the researcher. Hence, there is no guarantee that this point will necessarily correspond with the maximum value of the function itself. (14) The regularity region is the subset of the domain of the utility function in which all regularity conditions are satisfied. (15) Instead of the Taylor series expansion, Barnett used the Laurent expansion to enlarge the regularity region and maintain enough parametric freedom to satisfy requirements for flexibility. See Barnett (1983). (16) See Diewert and Wales (1987). (17) The function must be integrable or, more generally, it must lie in a Hilbert space. See Telser and Graves (1972). (18) Moreover, the Fourier series models can easily overfit the noise of the data. Usually, the measurement errors of economic variables can be decomposed into a pure white noise plus some high-frequency periodic functions. These latter functions might be mistaken for useful information if their frequencies are close to that of the sine and cosine functions in the Fourier series models. (19) Once again, the function must be integrable or, more generally, lie in a Hilbert space. See Telser and Graves (1972). (20) See Barnett and Jonas (1983). ignores them. Also, because the component functions are not periodic, the high-frequency, periodic movements in the date are likewise ignored.

Consequently, models based on the Muntz-Szatz series expansion are globally regular and are asymptotically globally flexible. This is why they are called Asymptotically Ideal Models (AIMs). (21)

ESTIMATION OF THE AIM MONEY

DEMAND SYSTEM

Four subaggregated goods are included in the empirical work presented here: a consumer good, [A.sub.4], and three monetary assets, [A.sub.1], [A.sub.2] and [A.sub.3]. [A.sub.4] is an aggregate good of consumer durables, nondurables and services and its respective aggregate price is denoted by [p.sup.*]. [A.sub.1] consists of currency, demand deposits of households and other checkable deposits; [A.sub.2] is composed of savings deposits at commercial banks and thrifts, super NOW accounts and money market deposit accounts; and [A.sub.3] is small time deposits at commercial banks and thrifts. (22) For each subset, an aggregate quantity is defined as a sum of per capita real balances of the component monetary assets. (23) The opportunity cost of holding a unit of a real monetary asset is measured by its user cost, a quantity-share-weighted sum of the individual user costs that compose it. The user cost of [A.sub.i] is denoted by [u.sub.i].

The representative consumer solves an optimal allocation problem by selecting real consumption, [A.sub.4], and real balances of the monetary assets, [A.sub.1], [A.sub.2] and [A.sub.3], to maximize the utility function f([A.sub.1], [A.sub.2], [A.sub.3], [A.sub.4]), subject to the budget constraint, given [p.sup.*], [u.sub.1], [u.sub.2], [u.sub.3] and the total expenditure, E. Following the utility function approach and using the first-order AIM model, we specify the reciprocal indirect utility function for the four goods case as:

[Mathematical Expression Omitted]

where [v.sub.1], [v.sub.2], [v.sub.3] and [v.sub.4] are the normalized prices, and [a.sub.1], [a.sub.2], ..., and [a.sub.15] are the parameters of the indirect utility function.

The share equation for each good is derived from equations 3 using the modified Roy's identity. These are

[Mathematical Expression Omitted]

Only the first three share equations are independent and can be written generally as:

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

When the additional parameter normalization [a.sub.1] + [a.sub.2] + [a.sub.3] + [a.sub.4] = 1 is imposed, one parameter, for example [a.sub.4], can be eliminated by substitution. Hence, in the case of four goods, the first-order AIM system contains 14 free parameters. (24)

The share equations are nonlinear with respect to the normalized prices and hence, to income and prices as well. By the definition of expenditure shares, demand functions can be expressed as [A.sub.i] = [s.sub.i/v.sub.i]. The complicated nonlinearity of the share equations, however, makes it impossible to derive a closed-form expression for the demand functions, such as the conventional linear or log-linear functions of income, prices and interest rates. Fortunately, the estimated parameters and share equations can be used to compute the income and price elasticities for consumer goods and monetary assets.

Estimation of the AIM Demand

System

The AIM model is estimated by a maximum likelihood procedure under the assumption that each share equation in (4) has an additive error term, [epsilon.sub.it]. That is,

[Mathematical Expression Omitted]

The disturbances are assumed to be independent and identically distributed multivariate normal random variables with zero mean and covariance matrix, [summation]. The sample disturbance covariance matrix, [summation], is defined as

[Mathematical Expression Omitted]

where N is the sample size, and the sample disturbance, [epsilon.sub.t], is computed by

[Mathematical Expression Omitted]

Maximizing the likelihood function for the system is equivalent to minimizing the generalized variance, [bar summation]. (25)

The estimation was accomplished using a nonlinear program (GRG2). To find a global optima, an extensive search over a large range of initial conditions was conducted. Because of the complex nonlinearity of the AIM demand system, the true maximum likelihood estimates are difficult to obtain. The possibility of missing the global optima was reduced, however, by an extensive search of the parameter space. (26)

All parameters are subject to non-negativity constraints to guarantee that global regularity conditions are satisfied. Because inequality constraints limit the applicability of the existing theoretical sampling distribution theory, the usual methods for testing hypotheses cannot be used. (27)

Income Elasticities and Price

Elasticities

Because the share equations are so complex, AIM does not yield explicit functional forms for demand functions. This is the consequence for correctly embedding utility maximization into an econometrically estimable demand system that can be used to compute economically meaningful income and price elasticities.

The Allen Partial elasticities of substitution and income elasticities are defined and expressed by the following formulas: (28) for i [is not equal to] j,

[Mathematical Expression Omitted]

where [p.sub.i] are the prices (and user costs), [A.sub.i.sup.c] denotes the income-compensated demand functions for the [i.sup.th] asset, [s.sub.i] denotes the expenditure shares and E denotes total expenditures. The elements, [sigma.sub.ij], constitute a symmetrical matrix called the Allen Partial matrix.

The income elasticities are defined by

[Mathematical Expression Omitted]

and the uncompensated price elasticities are denoted by

[Mathematical Expression Omitted]

where [A.sub.i] are the ordinary or uncompensated demand functions. The connection between compensated and uncompensated demand functions is stated by the usual Slutsky equation.

Gross substitutability and complementarity is provided by the off-diagonal terms of the uncompensated price elasticity matrix. If [eta.sub.ij] is positive, good i and good j are substitutes; in other words, when the price of good i rises, demand for good j increases to replace a cutback in demand for good i. If it is negative, they are complements - an increase in the price of good i (or j) causes the demand to fall for both goods.

Similarly, the pure substitution effects are defined by the Allen Partial matrix. If the utility function obeys regularity conditions, the own compensated price elasticities ([S.sub.ii sigma.sub.ii]P)and([sigma.sub.ii)], must be negative. Hence, the compensated price elasticity matrix represents potential movements along the consumer's indifference curves and can be used to examine whether the estimated underlying utility function satisfies regularity conditions.

The computed elasticities of the AIM demand system are compared with other money demand systems in the next section. Because of the complexity of share equations, a numerical method is used to compute the partial derivatives of the expenditure shares with respect to prices and income that occur in the elasticity formula. The computation of elasticities is calculated using the estimated share equations. Time series of the elasticities are produced by substituting time series of normalized prices and respective partial derivatives into the elasticity formula.

EMPIRICAL RESULTS OF THE AIM

MONEY DEMAND SYSTEM

In this section, the AIM demand system is estimated and the income and substitution elasticities are compared with those for the translog and Fourier demand systems. In addition, characteristics of monetary assets relative to consumer goods are analyzed.

Estimates of Parameters and Income

and Price Elasticities

Table 1 displays the coefficient estimates from the AIM demand system derived by U.S. quarterly data from 1970.1 through 1985.2. (29) These parameters represent the consumer's taste or preference and determine the utility function that underlies the estimated AIM demand system. Because the taste parameters are assumed to be constant, the consumer's utility function and preference did not change over time. The estimates of [a.sub.1] and [a.sub.2] were zero due to the non-negativity constraint. (30)

[Tabular Data Omitted]

The estimated Allen Partial elasticities of substitution and income elasticities are reported in table 2. The numbers represent the averages and their standard deviations (in parentheses) over the sample period. Table 3 displays the estimated substitution and income elasticities from the translog and the Fourier series models previously reported by Fisher (1989, page 103).

[Tabular Data Omitted]

Table 4 presents the average uncompensated price elasticities and their standard deviations over the sample period for AIM. The corresponding elasticities for the other two models are not available.
Table 4
Uncompensated Price Elasticities
 [A.sub.1] [A.sub.2] [A.sub.3] [A.sub.4]
[A.sub.1] -.534 .075 .084 -.126
 (.004) (.010) (.011) (.009)
[A.sub.2] .069 -.537 .080 -.123
 (.005) (.007) (.010) (.007)
[A.sub.3] .157 .162 -518 -.322
 (.005) (.009) (.004) (.004)
[A.sub.4] -.022 -.023 -0.15 -.985
 (.003) (.005) (.003) (.003)
NOTE: Standard deviations in parentheses.


What's Wrong with the Translog

and Fourier Demand Systems?

In the translog demand system results (shown in table 3), the positive sign of [theta.sub.11] indicates that the regularity condition is violated. This result suggests that the higher the opportunity cost of holding currency and demand deposits, the greater their demand. Given this violation of the "law of demand," the results from the translog demand system must be considered suspicious at best and, at worst, unreliable.

[Tabular Data Omitted]

Problems in the Fourier series demand system cannot be seen in table 3 because the numbers reported there are the average values of these coefficients. According to Fisher, however, except for [eta.sub.40], the income elasticities and Allen Partial elasticities of substitution changed signs frequently over the period. (31) For example, in 1970, [sigma.sub.12] was significantly negative, implying complementarity; in 1971-1972, it was significantly positive, implying substitutability; and then in 1974-1975, it became negative again. Figure 1 displays a [theta.sub.12]-comparison of the Fourier and AIM money demand systems. It is inexplicable that currency and demand deposits, [A.sub.1], and savings deposits and money market deposit accounts, [A.sub.2], should be complements during some periods and substitutes during others.

Empirical Inference of

Characteristics of Monetary Assets

by the AIM Demand System

The anomalies observed with the translog and the Fourier series demand systems do not occur in the AIM demand system. The own-price elasticities are negative and all estimated elasticities maintain their signs over the entire sample period. Moreover, their smaller standard deviations indicate that they are more stable; this can also be seen in figure 1. In the Allen Partial matrix, the diagonal elements are all negative while the off-diagonal elements are positive. This implies that the three monetary aggregates and aggregate consumption are substitutes for each other in the presence of income compensation. Moreover, the pure sub in exchange for the service of predictable immediacy. The market-maker also confronts traders who have inside information, however, and who can therefore speculate profitably at the expense of the market-maker.(39) The market-maker must charge everyone a wider spread to compensate for losses to the information-motivated traders.

Because of the relatively abstract nature of currencies as commodities, it is difficult to construct examples of "inside" information on foreign exchange rates. One exception is money supply announcements, which, if known before publicly distributed, might provide a basis for profitable speculation. Another form of information that can be construed as inside information is knowledge of an arbitrage opportunity. Consider a hypothetical market in which there are numerous decentralized market-makers who do not quote spreads, but single prices at which they are willing both to buy and sell. Unless there were a perfect consensus among the market-makers on the value of the foreign currency, all of them would be vulnerable to arbitrage. A decentralized market makes a perfect consensus difficult to achieve. Without centralizing price information, it is impossible to know if no arbitrage opportunities exist. A bid-ask spread, in contrast, allows a market-maker to include an error tolerance in her prices, thus facilitating a price consensus: it is easier to get bid-ask spreads to

overlap than to get scalar prices to coincide. The spread also provides the market-maker with some degree of protection from adverse selection in the form of arbitrage.

The bid-ask spread is also affected by inventory considerations. This idea dates back at least as far as Barnea and Logue (1975).(4) The notion of a desired inventory level for the market-maker underlies all of these models. In the simplest case, the desired level is set at zero, and a constant spread is shifted up and down on a price scale to equalize the probability of receiving a purchase order with that of receiving a sale order. The result is that the expected change in inventory is always equal to zero, and (with all trades for one round lot) the inventory level follows a simple random walk.

An undesirable implication of random-walk models of inventory is the inevitable bankruptcy of the market-maker. Finite capitalization levels for market-makers impose upper and lower bounds on allowable inventories. Because inventory follows a random walk, with probability one it will reach either its upper or lower bound in a finite number of trades.(41) The dynamic optimization models of Bradfield (1979), Amihud and Mendelson (1980) and Ho and Stoll (1981) resolve this problem. They conclude that a market-maker, optimizing his bid and ask prices over time in the face of a stochastic order flow, will shift both bid and ask rates downward (upward) and increase the width of the spread when a positive (negative) inventory has accumulated.(42)

We should expect two of these three rationales for the spread to apply to market-makers' bid-ask spreads in the foreign exchange market. Because there are numerous market-makers, competition should eliminate their ability to earn monopoly rents by charging a premium for predictable immediacy per se. The adverse selection argument does apply in the foreign exchange market, however, since the spread allows market-makers some protection against arbitrage opportunities. Arbitrage opportunities can be construed as a form of inside information in a market where price information is not centralized. In accordance with the dynamic optimization models, a market-maker's inventory level should affect the spread, widening and shifting it as inventories accumulate.

Brokers' Spreads

So far, the discussion of the bid-ask spread has focused on models in which bid and ask prices are set by individual market-makers. The dual role of the stock exchange specialist suggests that this is only part of the story. Spreads are produced in two fundamentally different ways. It is only when limit orders are sparse that a NYSE specialist must step in as a market-maker to provide an "orderly market."(43) When limit order volume is sufficient, the specialist acts as a broker, accounting for incoming limit orders on the limit order book, and pairing market orders against them. Cohen, Maier, Schwartz and Whitcomb (1979) note that inadequate attention has been given to the fact that not all prices are market-maker spreads. The market often makes itself without specialist assistance, through the aggregation of limit orders on the book.

The foreign exchange market differs from the NYSE in that the market-making and brokerage roles are separated: market-makers do not act as brokers, and brokers do not make markets. Therefore, it is even more appropriate to model brokered spreads as determined in a fundamentally different way from market-maker spreads. The separation of roles also has other implications for modeling foreign exchange brokerage.

A brokered spread is the combination of the best bid and best ask, received by the broker as separate limit orders. This arrangement might be modeled as a pair of extreme order statistics from independent distributions of purchase and sale limit orders. The distribution of these statistics would have to be conditional on limit order volume and on the fact that the best ask must always exceed the best bid, since crossing orders transact immediately and are removed from the book.(44) Perhaps because of its complexity, such a derivation has not been attempted.

Cohen, Maier, Schwartz and Whitcomb (1979) model limit orders as generated by "yawl" distributions. These distributions satisfy three heuristics for the incentives of investors placing limit orders.(45) The heuristics are motivated by a notion of the centralized exchange as a market for immediacy; placers of limit orders produce immediacy, and placers of market orders consume it. This relationship between limit and market orders is formalized in Cohen, Maier, Schwartz and Whitcomb (1981), where each half of the brokered spread is assumed to be generated by a compound Poisson process. A minimum brokered spread results: if the limit order's bid (ask) price is sufficiently close to the specialist's ask (bid), the benefit to the investor of being able to specify the price of a limit order is overwhelmed by the cost of foregone immediacy.

Because models of the informational content of brokered spreads are few, the literature offers little guidance in modeling brokered quotes in the foreign exchange market. The yawl distribution is the only explicit distributional form for brokered spreads in the literature. Unfortunately, its heuristic basis cannot be transferred directly to the foreign exchange market, because market-makers there differ from stock market investors. Indeed, this may be an instance in which the foreign exchange market informs microstructure theory rather than the other way around. The extant approaches to brokerage treat it as a service facilitating predictable immediacy. This aspect of brokerage is redundant in the foreign exchange market, because of the multitude of market-makers, each providing immediacy. This redundancy suggests instead that foreign exchange brokerage serves some other function.

One motive for trading through a foreign exchange broker is to maintain anonymity - the name of the bank placing a limit order is not revealed unless a deal is consummated and then only to the counterparty.(46) Anonymity is valuable, because revealing a need to buy or sell a currency puts a market-maker at a bargaining disadvantage. In addition, anonymity can help pair market-makers who ordinarily would not contact each owas less sensitive to changes in the user cost of [A.sub.3]. Hence, the shift of funds from [A.sub.3] to [A.sub.1] should have been moderate despite a substantial increase in the user cost of [A.sub.3].

These results are roughly consistent with developments during the period. In November 1978, commercial banks were authorized to offer automatic transfer service (ATS) from savings accounts to checking accounts. Other interest ceiling-free accounts were also introduced in the early 1980s. In January 1981, NOW accounts were introduced nationwide. These financial innovations should have encouraged consumers to shift funds from savings accounts and money market deposit accounts in [A.sub.2] into NOW accounts in [A.sub.1]. This may have increased the interest sensitivity of demand for [A.sub.1].(35)

Simulating the Growth Rates of

Monetary Aggregates

A further investigation of the behavior of monetary aggregates can be made by a dynamic simulation. For example, suppose that demand for [A.sub.i] has been derived by utility maximization and expressed by the ordinary demand functions of price and user costs and total expenditure

(6) [A.sub.i] = [G.sub.i] ([u.sub.1], [u.sub.2], [u.sub.3], [u.sub.4], E).

The total differentiation of (6) results in [Mathematical Expression Omitted]

Dividing both sides of (7) by (6) and using definitions of the uncompensated price elasticities and the income elasticity gives [Mathematical Expression Omitted]

Using time series of the elasticities and the growth rates of price and user costs and total expenditure, the right-hand sides of the equations in (8) are computed. In this way, the growth rates of demand for [A.sub.i] can be "simulated."

The actual and simulated growth rates of demand for monetary aggregates and consumption are displayed in figures 5 through 8. The simulations match the actual growth rates fairly well, especially for consumption. The simulation rates of monetary assets had large fluctuations around the actual growth rates in the periods 1972.4-1975.1 and 1978.2-1982.2. Fluctuations in interest rates and inflation rates were substantial during each period, causing corresponding fluctuations in growth rates of user costs.(36) These changes are reflected directly in the simulation rates.

Because the AIM model is static, sharp changes in user costs are necessarily reflected in correspondingly sharp changes in the simulated growth rates of aggregates. Hence, it is not surprising that the simulation errors are large during periods when there are sharp changes in user costs. Nevertheless, figures 5 through 8 suggest that the AIM demand system has captured many of the characteristics of the U.S. monetary system during the sample period.

Can the AIM Demand System Explain

the Case of the Missing

Money?

Although the simulation of the growth rate of [A.sub.1] indicates that the AIM model produced relatively large errors during the period of "missing money" (1973.4-1976.2), an analysis of the AIM results might provide a clue.(37) During this period, there was a sharp decline in demand for M1; conventional money demand equations consistently overpredicted demand.

A potential explanation can be obtained by considering figures 9 and 10. Figure 9 shows the partial derivative of demand for [A.sub.1] with respect to its own user cost. Figure 10 shows the partial derivative of demand for [A.sub.1] with respect to the user costs of [A.sub.2], [A.sub.3], [A.sub.4] and E. Figure 9 shows a sharp rise in the rate of change in demand for [A.sub.1] with respect to a change in its user cost. Indeed, figure 11 shows that the user cost of [A.sub.1] increased relative to the price level. Hence, the demand for [A.sub.1] should have declined by a proportionately larger amount than the rise in its user cost. Conventional linear money demand equations with fixed regression coefficients could not accommodate this nonlinearity because, in such linear demand systems, the coefficients (derivatives) are assumed to be constant. However, figures 9 and 10 clearly suggest that this is not the case. Moreover, ordinary least squares is relatively sensitive to "outliers." Consequently, there will be substantial changes in the estimated regression coefficients when the equations are estimated over periods when these derivatives change significantly.

Conventional money demand equations may be misspecified for another reason, as well. Usually they include a single short-term interest rate intended to reflect the opportunity cost of holding money. AIM analysis indicates that the demand for moeny does not depend on a single "representative" interest rate, but on its user cost and the user costs of "close" substitutes (recall that the demand for [A.sub.1] was sensitive to changes in the user costs of [A.sub.2]). Hence, conventional money demand equations may produce misleading results when interest rates change relative to the user costs of M1 or relative to the user costs of close substitutes for M1.

Therefore, the case of "missing money" and "unexplained" parameter shifts in conventional money demand functions may result from the fact that they are essentially linear approximations to nonlinear demand functions. If so, they intuitively will provide much poorer approximations during periods when there are dramatic changes in user costs.

Is The Money Demand Function

Stable?

The erratic behavior of conventional money demand functions and, more recently, the income velocity of M1, have led many researchers to assert that the demand for money is "unstable".(38) Others have asserted that money demand is stable based on the observed stability of the consumption function.(39)

The AIM demand system integrates demand for both consumption and money and then estimates them simultaneously. These estimates suggest that, while the own price and cross price elasticities show considerable variation due to changes in the price level and user costs, they change little on average over the period (see figures 3 and 4).(4) Moreover, the estimated income elasticities for all three monetary aggregates are nearly constant (see figure 2). Of course, these results are obtained from a model where the estimated parameters are time-invariant, that is, the preference function is constant. Because of this, it is necessarily true that demand functions are "stable." Nevertheless, the relatively good performance of AIM provides some promise that, like consumption, the demand for money will ultimately be shown to be a stable function of a relatively few economic variables - in this case, income and user costs.

CONCLUSION

Two distinctly different micro-econometric demand system approaches to the demand for money were presented and discussed. An advanced AIM demand system was presented and estimated using U.S. time-series data. Unlike other utility function-based approaches, AIM estimates are consistent with microeconomic theory. Dynamic simulations of the growth rates of various monetary aggregates and consumption suggest that the estimated AIM model performed well; nevertheless, the largest simulation errors occurred in periods when there were relatively sharp swings in user costs or inflation. This is perhaps not too surprising given the static nature of the AIM analysis.

An analysis of changes in income and cross price elasticities are suggestive of portfolio shifts among monetary aggregates in the 1970s and 1980s consistent with the observed behavior of these aggregates. The results of AIM suggest that the reported failure of conventional linear (or log-linear) money demand equations may result from trying to fit fundamentally nonlinear functions with linear ones. The results shown here suggest that this problem will be particularly acute whenever there are sharp changes in user costs. Unfortunately, these are precisely the times when AIM performance was also poor. The key to solving this problem in AIM, however, is to find a way to make AIM explicitly dynamic. It may not be necessary to assume that consumer preferences are unstable.

The sampling distribution theory for AIM has not been worked out at this time, so relevant hypothesis tests cannot be conducted yet. Also, because the time series on the relevant user costs of monetary aggregates is limited, the available data cover a relatively short sample period. These factors, coupled with the fact that even low-order (first-order) AIM systems require a relatively large number of estimated parameters, place severe limits on attempts to evaluate the performance of AIM using out-of-sample forecasts. Despite these problems, the estimated AIM system appears to have captured many of the characteristics of monetary assets and offers some useful explanations to puzzling empirical issues. Hence, these results are encouraging to those who believe that microeconomic principles, such as utility maximization, can be applied usefully to macroeconomic problems.

(35) See Thornton and Stone (1991) for a discussion of this possibility. (36) Some terms are essentially zero and can be ignored. The following growth rate equations are accurate enough to produce the simulation: [dA.sub.1/A.sub.1] = [eta.sub.11 du.sub.1/u.sub.1] + [eta.sub.12 du.sub.2/u.sub.2] + [eta.sub.13 du.sub.3/u.sub.3] + [eta.sub.44 du.sub.4/u.sub.4] + [eta.sub.10]dE/E [dA.sub.2/A.sub.2] + [eta.sub.22 du.sub.2/u.sub.2] + [eta.sub.23 du.sub.3/u.sub.3] [dA.sub.3/A.sub.3] = [eta.sub.33 du.sub.3/u.sub.3] [dA.sub.4/A.sub.4] = [eta.sub.42 du.sub.2/u.sub.2] + [eta.sub.43 du.sub.3/u.sub.3] + [eta.sub.44 du.sub.4/u.sub.4] + [eta.sub.10]dE/E.

In the equation for [A.sub.1] there are more affecting elements; the own and cross-price effects of [A.sub.2], [A.sub.3] and [A.sub.4] are important in simulating the growth rate of [A.sub.1]. The growth rates of demand for the other two monetary aggregates, however, are determined mainly by their own price effects and cross-price effect, [n.sub.23]. This suggests that ignoring the substitution effects of non-M1 components of M2 might be one of the factors that discredit reliability of the conventional M1 demand function. (37) See Goldfeld (1976). (38) For a discussion of the velocity of M1 and an analysis of some of the explanations, see Stone and Thornton (1987). (39) For example, see Friedman (1956) and Lucas (1988). (40) In the parlance of modern time-series analysis, these elasticities are said to be stationary, that is, mean reverting. However, no formal tests of stationarity were performed in this study.

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Date:Nov 1, 1991
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