Are 'risky assets' substitutes for 'monetary assets?'.

Drake, Leigh ; Fleissig, Adrian R. ; Mullineux, Andy 等

I. INTRODUCTION

Financial innovation over the past couple of decades has enhanced the liquidity of many financial assets, particularly noncapital certain ("risky") assets. Further, financial assets previously not regarded as monetary assets, such as mutual funds and particularly money market mutual funds, now provide monetary services such as liquidity and transactions services,(l) Given the increased liquidity of such funds, the question arises as to whether risky assets are substitutes for capital certain monetary assets? If noncapital certain assets are indeed substitutes for capital certain assets and are also being used for transactions purposes, then agents may be treating "risky" assets as money. Although this issue has been addressed in the United States in the context of the addition of bond funds to M2 by for example, Duca [1995], no such research has yet been conducted in the United Kingdom.

Including risky assets in the analysis contradicts the work of Ando and Shell [1975], Spencer [1986, Appendix I, pp. 184-93] and Spencer [1994]. Broadly, they argue that since some nonrisky assets provide monetary services, and probability also liquidity, individuals would not hold risky assets for these purposes.2 In contrast, Hicks [1935] states that falling transaction costs and rising wealth encourages individuals to increase the proportion of risky assets in portfolio holdings. Most studies generally assume that "risky" assets fail to provide significant monetary services and, for example, have been excluded from the asset demand analysis of Barr and Cuthbertson [1991a], Drake, Fleissig, and Swofford [1997] and money demand studies such as Taylor [1987], Belongia and Chrystal [1991], Drake and Chrystal [1994], and Drake [1996]. Alternatively, Barr and Cuthbertson [1991b] use only risky assets in their asset demand system analysis.

The degree of substitution between risky assets and capital certain assets is an empirical issue. While noncapital certain (risky) assets, mutual funds, and other financial assets may well be becoming closer substitutes for what people are using as money, they are in no way perfect substitutes for the highly liquid "cash assets." To evaluate the role of noncapital certain assets we construct a noncapital certain (risky) Divisia aggregate consisting of equities, government bonds, and unit trusts. The Divisia index allows for less than perfect substitution between assets and weights component assets according to their differing degrees of "moneyness," which may vary over time due to many factors, for example, financial innovation; see Barnett [1980]. This methodology has been widely applied to monetary aggregation and money demand studies in a number of countries, as in Belongia [1996], Drake [1996], Drake and Chrystal [1997], and Anderson, Jones, and Nesmith [1997].

To determine whether risky assets are substitutes for capital certain assets, we use two other Divisia aggregates constructed from traditional measures of money. The first of these is an aggregate of highly liquid zero yield cash assets. Given that the component assets all have zero own rates of return, however, Divisia aggregation is equivalent in this case to simple summation. The second aggregate is a Divisia aggregate of interest bearing capital certain assets. Including risky assets complicates the analysis because the degree of risk aversion will affect the estimates of substitution between capital-certain and noncapital certain assets. Thus, we also analyze how different degrees of risk aversion affect the results. The substitution relationship between assets is estimated from an asymptotically ideal model (AIM).

The choice of an AIM is in line with the recommendation of Barnett, Fisher, and Serletis [1992] that flexible systems of demand equations should be used that allow for nonlinear optimizing behavior by economic agents. Furthermore, this paper can be seen as an extension of the recent application of this model to U.K. data by Drake, Fleissig, and Swofford [1997].(3) The main contribution of this paper is to add risky financial assets to the more traditional set of U.K. "monetary assets" considered by Drake, Fleissig, and Swofford. Other related work using U.K. data is that of Barr and Cuthbertson [1991a], using the almost ideal demand system (AIDS), which is only flexible at a single point of approximation, and Drake [1992], which uses the Translog model, which has a similarly restricted degree of flexibility.(4) In both papers a fairly narrow range of traditional, capital certain monetary assets is considered.

The main result of this paper is the evidence that risky assets are substitutes for traditional monetary assets. Moreover, risky assets are often stronger substitutes for the highly liquid cash assets than are interest bearing deposit assets. In addition, as the degree of relative risk aversion increases, substitution between risky and capital certain assets generally falls.

The remainder of the paper is organized as follows: section Il outlines the incorporation of risky assets in monetary aggregation, section III discusses the AIM specification; section IV examines the various measures of elasticity of substitution; section V provides details on the data; section VI discusses the estimation and results, and section VII concludes.

II. INCORPORATING RISKY ASSETS

Constructing monetary aggregates using the Divisia index number technique requires data for both the prices and quantities of the relevant assets. Quantities are generally measured by the real per capita holdings of the assets. Measuring the opportunity cost of an asset, however, depends on the degree of risk in the return on the asset.

Capital Certain Assets

The rental price or user cost ([[Pi].sub.it]) for capital certain assets, derived by Barnett [1978] is:

(1) [[Pi].sub.it] = ([R.sub.t] - [r.sub.it]) / (1 + [R.sub.t]),

where [R.sub.t] is the benchmark rate of return in period t and [r.sub.it] is the own rate of return on the ith asset at time t. The benchmark rate is often taken as the return on an asset that is held as a store of value having no monetary services, for example, government bonds. A problem arises when the return on the benchmark asset is less than any capital certain asset. Following Patterson [1991], we use the envelope approach in which the benchmark rate is set to the highest return of all the assets. Capital certain monetary aggregates are constructed from the rental prices and quantities of these monetary assets using a Divisia index.

The CAPM Extended Divisia Monetary Aggregate under Risk

Recently, Barnett and Liu [1995] and Barnett, Jensen, and Liu [1997] extended the rental price formula expressed in equation (1) to allow for the "riskiness" of noncapital certain assets. They use the consumption capital asset pricing model (CCAPM) to show that the risk adjustment amounts to subtracting a risk premium from the individual asset returns. For notational convenience, they convert the nominal rates of return [r.sub.it] and [R.sub.t] into real total rates of return, 1 + [Mathematical Expression Omitted] and 1 + [Mathematical Expression Omitted] such that

(2) [Mathematical Expression Omitted]

and

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] defined in this way can be termed real rates of excess return, and [Mathematical Expression Omitted] is the consumer goods price index.

Using a quadratic utility function or assuming [Mathematical Expression Omitted] is a bivariate Gaussian process for each asset i, where [c.sub.t+1] is real aggregate consumption at time t+l, they derive the risk adjusted user cost for the ith risky asset at time t([[Pi].sub.it])

(3) [Mathematical Expression Omitted],

where

(4) [Mathematical Expression Omitted],

where E is the expectations operator, [Z.sub.t] is the Arrow-Pratt measure of relative risk aversion, and [c.sub.t+1]/[c.sub.t] is the consumption growth rate. Hence, the risk premia is the product of the coefficient of relative risk aversion and the covariance between the asset returns and real per capita consumption growth. The risk premia increases when either the relative risk aversion or covariance increases. In using the conventional CAPM assumption that the benchmark rate process is deterministic or risk adjusted, [Mathematical Expression Omitted] is a risk-free rate. This amounts to assuming that the risk premium has been extracted from the benchmark rate. In practice, however, the benchmark rate is automatically risk adjusted because it is the upper envelope of the risk adjusted component rates.

It is worth stressing at this point that the "equity premium puzzle" literature suggests that the covariance in equation (4) above will typically be close to zero and hence the size of the consumption risk adjustment in the CCAPM will be negligible.(5) Indeed, this has been found in some U.S. studies. Liu [1995], for example, found that the estimated covariances ranged from 1.45 x [10.sup.-5] for currency to 2.66 x [10.sup.-6] for time deposits and -3.96 x [10.sup.-7] for commercial paper. In turn, this implied that the risk premia for the assets studied was typically around 0.1% of the size of the asset returns for Z = 1. In contrast, the covariance terms for equities, government bonds, and unit trusts in this study are, respectively, 1.7 x [10.sup.-4]; 1.4 x [10.sup.-4], and 3.5 x [10.sup.-5]. This implies that, as a percentage of mean real excess returns, the risk premium ranges between 0.51% and 3.57% for equities (for Z = 1 and Z = 7), between 0.501% and 3.507% for government bonds, and between 0.12% and 0.84% for unit trusts. While these adjustments are still relatively small (and hence something of an equity premium puzzle remains), they are by no means insignificant. To put the figures into perspective, for example, management charges on U.K. unit trusts (which would be expected to impact on asset holders' decision making) typically range between 0.5% and 4% and hence are comparable to the levels of risk premia adjustment found in this study. Furthermore, it is clear from the subsequent results that the covariance terms are sufficiently large for the variation in Z between 1 and 7 to have a marked impact on the derived substitution elasticities.

Since the covariance between [Mathematical Expression Omitted] and [c.sub.t+1]/[c.sub.t] in (4) may also be nonzero for capital certain assets, the user costs for these assets should also, in principle, be risk adjusted before computing Divisia monetary aggregates.6 In practice, however, Barnett, Kirova, Liu, and Zhou [1994] and Barnett and Xu [1998] find negligible empirical differences between risk adjusted and nonrisk adjusted Divisia capital certain monetary aggregates. They state, however, that risk adjustment may be important if assets contribute substantially to household risk, such as common stock or bond funds. Hence, for the purposes of this paper, we assume that risk adjustment is required only in the case of noncapital certain assets. For capital certain assets we continue to use the standard rental price formula given in equation (1).

III. THE AIM SPECIFICATION

In the demand systems approach, the objective is to derive and estimate demand share equations and elasticities of substitution between assets from the indirect utility function. The AIM specification is particularly suitable for this purpose because it can globally approximate any continuous (utility) function and give arbitrarily accurate estimates of elasticities of substitution. The AIM specification of Barnett and Jonas [1983] approximates the indirect utility function using a Kth order Muntz-Statz series expansion:

(5) [Mathematical Expression Omitted],

where [a.sub.O], [a.sub.ik], and [a.sub.ijkm] ..., are the parameters to be estimated, [less than](k) = [2.sup.-k] is the exponent function used in Barnett and Yue [1988], and n is the number of assets. Here v = p/p[prime]x is a vector of expenditure normalized user costs and x and p are vectors of quantities and user costs respectively. In this case, the vector x consists of both capital certain and risky assets.

The AIM specification is a semi-nonparametric function that is dense in a Sobolev norm and can give an arbitrarily accurate asymptotic approximation to the levels and derivatives of any continuous function.? This requires iteratively increasing the degree of expansion K until the function converges asymptotically to the elasticity estimates. Consequently, asymptotic statistical tests will have the correct level of significance. In contrast, locally flexible forms like the Taylor-series Translog and AIDS models do not possess these properties and can only approximate the data at a single point and may fail to converge to the true utility function, as demonstrated by Barnett and Yue [1988]. Seminonparametric functions such as the Fourier flexible form and AIM are frequently used in money/asset demand studies. The Fourier has been used by Ewis and Fisher [1985], Barnett, Fisher, and Serletis [1992], Fisher [1992], and Fisher and Fleissig [1994; 1997]. The AIM specification has been used by Barnett and Yue [1988], Fleissig, and Swofford [1996; 1997] and Drake, Fleissig, and Swofford [1997].

The AIM demand share equations, Sit are derived by applying Roy's identity to the indirect utility function (5) and can be expressed as

(6) [S.sub.it] = h([v.sub.it], [Alpha])

and examples can be found in Barnett and Yue [1988]. The parameter estimates from the AIM share equations are used to calculate substitution elasticities between capital certain and capital uncertain assets.

IV. ELASTICITIES OF SUBSTITUTION: THEORETICAL CONSIDERATIONS

In theoretical models with only two variables (inputs or goods), the elasticity of substitution between these variables is unambiguous; they must be substitutes. When there are more than two variables, however, measuring substitution between variables [x.sub.i] and [x.sub.j] becomes more complex as the curvature of a function in higher dimensions can be measured in infinitely many directions. Following Mundlak [1968] and Blackorby and Russell [1975; 1981; 1989], we consider three wellknown measures of substitution, the Allen-Uzawa elasticity, the Morishima elasticity, and the McFadden shadow elasticity. The McFadden elasticity is also known as the Hicks elasticity of substitution.

The Allen-Uzawa elasticity is defined as:

(7) [A.sub.ij] (y, P) = C(y, P), [C.sub.ij] (y, P)

/ [C.sub.i] (y, P) [C.sub.j] (y, P),

where C([center dot]) is a twice differentiable continuous cost function, [C.sub.i] and [C.sub.ij] are the first and second partial derivatives of the cost function with respect to [p.sub.i], and y is output. The Allen-Uzawa elasticity is symmetric by construction and, as Chambers [1988] and Blackorby and Russell [1975; 1989] argue, is of limited use, because it fails to give information about how relative demands [x.sub.o]/[x.sub.j] change in response to a price variable [p.sub.i].

A measure that takes into account the possibility of nonsymmetry is the Morishima elasticity of substitution:

(8) [M.sub.ij] (y, P) = [P.sub.i] [C.sub.ij] (y, P) / [C.sub.j] (y, P)

- [P.sub.i] [C.sub.ii] (y, P) / [C.sub.i] (y, P)

In our multi-asset framework, substitution between assets [x.sub.i] and [x.sub.j] depends on which user cost changes, [p.sub.i] or [p.sub.j]. Hence, the Morishima elasticity can capture the lack of symmetry that seems possible in such calculations. Since [C.sub.j] is the jth share, we can write [C.sub.j] = [p.sub.j][x.sub.j]/expenditure and by substituting [C.sub.j] into equation (8), the Morishima elasticity tends to negative or positive infinity when [p.sub.j], [x.sub.j] or [x.sub.i] tend to zero. The user cost [p.sub.j] tends to zero when the own rate tends to the benchmark rate and agents treat asset j more like a "free" good and therefore the utility surface and demand for asset j become difficult to approximate.(8) Thus, in cases of very small user costs, we may expect to obtain imprecisely estimated, and possibly very large elasticities.

A third measure of substitution is the symmetric Hicks, or McFadden, shadow elasticity:

(9) [H.sub.ij] (y, P) = ([S.sub.i]/([S.sub.i] + [S.sub.j])) [M.sub.ij]

+ [S.sub.j] / ([S.sub.i] + [S.sub.j])) [M.sub.jt],

which is a share weighted average of the Morishima elasticity showing the percentage adjustment in demand ratios associated with changes in a price ratio. Because of the problems associated with the Allen-Uzawa elasticity, the subsequent analysis of substitution across assets is conducted using both the Morishima elasticities and the McFadden shadow elasticities.(9)

V. DATA

All estimation is performed using quarterly monetary and financial data from 1979 q1 to 1994 q2. Series for expected returns for risky assets, used in equation (3), are also calculated. Monetary services are assumed to be proportional to the real per household stock of monetary asset holdings. In the case of capital certain assets, data on the personal sector nominal asset holdings, together with the corresponding interest rates (yields) were provided by the Bank of England. These assets are NC (notes and coins); NIBS (noninterest-bearing sight deposits), IBS (interest-bearing sight deposits), TD (bank time deposits), BSD (building society deposits), and NS (national savings). The own rates of return on NC and NIBS are taken to be zero, although clearly the opportunity cost and hence the rental prices of these assets is not zero. Fisher, Hudson, and Pradhan [1993] provide details on the own rates of return for the interest-bearing assets IBS, TD, and BSD. In the case of NS, the representative own rate is taken to be the national savings investment account rate.

For noncapital certain assets, data for the personal sector holdings are provided for three categories by Datastream: EQ (equities), GB (government bonds), and UT (unit trusts). The data are in nominal terms and are converted into real per household holdings. Quarterly total returns on U.K. unit trusts were provided by Micropal. For equities, the total return is made up of the quarterly dividend yield together with the quarterly percentage capital gain or loss. These are calculated on the basis of the Financial Times all share index. Similarly, for government bonds the total return is measured by the quarterly yield on Long Term Government bonds together with the quarterly percentage capital gain or loss as measured by the Financial Times actuaries long-term government bond price index.

The risk-adjusted user cost formula (3) requires an estimate of expected total returns on risky assets ([Mathematical Expression Omitted]) for our period of estimation, 1979 q1 to 1994 q2. Single equation VAR models, estimated over the sample 1971 q1 to 1978 q4, are used to give one-period-ahead estimates of ([Mathematical Expression Omitted]) for 1979 q1. The VAR equation sample periods are then sequentially updated by one period to give a series for ([Mathematical Expression Omitted]) as a sequence of one-period-ahead forecasts.

Empirical estimates for the coefficient of relative risk aversion ([Z.sub.t]) in equation (3) are generally close to unity but may be larger. Instead of selecting an arbitrary value for [Z.sub.t], a goal of this paper is to determine the effect on substitution elasticities between financial assets as the degree of asset holders' relative risk aversion increases ([Z.sub.t] = 1, 3, 5, 7).

In order to get precise parameter estimates, it is necessary to reduce the dimension of the study by grouping assets with similar characteristics. This can by done using formal separability tests, which have some limitations. Parametric tests of separability are sensitive to the choice of the utility function. Hence, if the utility function is misspecified, then so are the separability tests. An alternative is to use nonparametric tests that are independent of the utility function. However, the popular nonparametric test of Varian [1982; 1983] has been found to have low power as shown in Barnett and Choi [1989].(10) In the absence of formal separability testing, the most natural grouping of assets for this study of substitution between risky and capital certain assets appears to be A1: cash assets (NC, NIBS); A2: interest-bearing deposit assets (IBS, TD, BDS, NS); and A3: risky assets (EQ, GB, UT).

Group A1 consists of assets whose yield is zero. The remaining capital certain assets all have nonzero yields and we label group A2 as interest-bearing deposit assets. Group A3 consists of noncapital certain assets. A chained Divisia index of the assets in each group is used to form the three aggregates. Since the rental prices on NC and NIBS are identical, these assets are perfect substitutes for each other and aggregating with the Divisia index is equivalent to using a simple sum index.

VI. ESTIMATION AND RESULTS

The AIM specification is estimated over the sample period 1979 q1 - 1994 q2 using the maximum likelihood routine of TSP International version 4.2. As the AIM parameters are homogenous of degree zero, the normalization a1 + a2 + a3 = 1 was imposed. Convergence was set at 1 x [10.sup.-5]. The AIM specification is based on an infinite order Muntz-Statz expansion, and Eastwood [1991] shows that the order of a series expansion is best determined using an upward F-test truncation rule. Therefore, we first test to see whether the second-order expansion (K = 2), that is, AIM(2), is statistically preferred to AIM(l). The AIM(l) specification was statistically rejected at the 1% level, using a Wald test. The residuals from the AIM(2) specification, however, were found to be serially correlated. Serial correlation is a common problem in estimating a system of static demand equations and is possibly due to misspecified dynamics. Using a Monte Carlo experiment, however, Fleissig and Swofford [1997] find that a static AIM(2) with an AR(1) correction generally gives an excellent approximation to data generated by a dynamic process. Since it is unlikely that agents will completely adjust portfolio holdings within one period, the AR(1) correction is probably approximating partial [TABULAR DATA FOR TABLE I OMITTED] adjustment of portfolio holdings. In money demand systems, Barr and Cuthbertson [1991a], Fisher and Fleissig [1994; 1997], Fleissig and Swofford [1996;1997], and others also use an AR(1) correction. Details of the AIM(2) estimation results with an AR1 correction, and where the relative risk aversion parameter (Zt) is set equal to unity, are provided in Appendix Table A1. For reasons of brevity, the remaining AIM estimation results are not detailed but are available from the authors on request.

The economic content of the AIM parameter estimates is evaluated using the Morishima and McFadden-Hicks elasticities of substitution. We first analyze substitution between capital certain assets, which were labeled "cash assets" (A1) and "interest-bearing deposit assets" (A2). Subsequently, the effect of risky assets on substitution elasticities is analyzed. Finally, we examine how the degree of risk aversion alters the substitution relationship between assets. The terminology ME12 is used to indicate the Morishima elasticity between assets A1 and A2 when the rental price of A1 changes and ME31 indicates the Morishima elasticity between assets A3 and A 1 when the rental price of A3 changes. Similarly, H12 indicates the McFadden-Hicks elasticity of substitution between assets A1 and A2 when the rental price ratio of these assets changes and is symmetric by construction. While traditional log-linear models assume a constant elasticity of substitution, the more flexible nonlinear AIM specification allows the slope of the utility surface and hence substitution elasticities to change over time. Thus, in a nonlinear environment, stable asset demands can be associated with variable substitution elasticities.

Substitution between Capital Certain Assets (A1 and A2)

Table I shows that many of the elasticities of substitution between cash and deposit assets are not significantly different from zero. In the case of Z = 1, for example, the Morishima elasticity is significant at 25% of the data points for ME12 (16 of 62) and at 16% for ME21 (10 of 62). For the McFadden-Hicks measure (H12), 17% (11 of 62) of the elasticities are significant. These results are difficult to compare to previous studies, which use the Allen-Uzawa measure and often fail to report standard error estimates or produce estimates (and standard errors) only at a given point in the sample (the sample mean). Results from Fleissig and Swofford [1996] and Fisher and Fleissig [1997], which are the first papers to estimate substitution between monetary assets using the McFadden and/or Morishima elasticities, generally find that capital certain monetary assets are substitutes for each other having (absolute) elasticities less than unity. Since noncapital certain assets are excluded from their studies, substitution between financial assets may well change when risk is introduced. Clearly, our results suggest that substitution between capital certain assets is often not significantly different from zero. We now examine the magnitude of the elasticities of substitution which are significantly different from zero.

Table II provides summary statistics for the statistically significant elasticities only. It is clear that when the elasticities are statistically significant they show unambiguously that cash and deposit assets are strong substitutes in demand. ME12, for example, has a mean of 2.05, with values ranging from 1.54 to 2.57, while ME21 exhibits a mean value of 2.84 and ranges from 1.57 to 5.03. These results are echoed by the McFadden-Hicks elasticity (H12), which has a mean of 2.04 and ranges from 1.21 to 2.74. A further interpretation of the results depends on whether the statistically significant elasticities appear randomly over time and for which intervals the standard errors of the substitution elasticities increase. TABLE II Summary Statistics Mean Std Min Max Morishima Elasticities z = 1: ME12 2.049 0.297 1.540 2.573 ME21 2.843 0.962 1.573 5.026 ME13 2.458 0.579 1.671 4.157 ME31 1.806 0.308 1.154 2.738 ME23 2.736 0.638 1.672 4.030 ME32 2.448 0.450 1.806 3.080 z = 3: ME12 2.145 0.894 1.538 3.737 ME21 2.316 0.567 1.640 2.990 ME13 1.894 0.368 1.416 3.337 ME31 1.533 0.291 1.179 2.706 ME23 2.333 0.481 1.488 2.902 ME32 1.989 0.286 1.735 2.442 z = 5: ME12 1.592 0.095 1.511 1.697 ME21 1.860 0.125 1.637 2.006 ME13 1.896 0.246 1.386 2.510 ME31 1.498 0.172 1.226 2.041 ME23 1.796 0.110 1.635 1.871 ME32 1.681 0.000 1.681 1.681 z = 7: ME12 1.758 0.378 1.349 2.161 ME21 1.437 0.136 1.341 1.533 ME13 1.996 0.227 1.397 2.424 ME31 1.533 0.168 1.305 1.970 ME23 1.532 0.000 1.532 1.532 ME32 0.000 0.000 0.000 0.000 McFadden Elasticities z = 1: H12 2.035 0.497 1.207 2.735 H13 1.724 0.314 1.170 2.610 H23 2.413 0.444 1.831 3.019 z = 3: H12 2.068 0.681 1.400 3.684 H13 1.524 0.351 1.098 2.662 H23 1.965 0.291 1.664 2.418 z = 5: H12 1.697 0.179 1.508 1.931 H13 1.483 0.210 1.188 1.868 H23 1.688 0.000 1.688 1.688 2 = 7: H12 1.865 0.462 1.332 2.478 H13 1.555 0.282 1.153 1.867 H23 0.000 0.000 0.000 0.000

The elasticities of substitution between capital certain assets together with their standard errors are now examined. Any insignificant elasticity over 6.0 is omitted from the graphical analysis to prevent the scale from being unnecessarily affected by insignificant values. For reasons of brevity we elect to show the full range of elasticities only for Z = 1 (the full set of elasticities and standard errors are available from the authors on request). It is clear from Figure 1 that assets A1 and A2 are generally substitutes in demand although the positive elasticities are only occasionally significant as was indicated in Table I. Interestingly, the associated standard errors increase substantially following the rapid growth of interest-bearing sight deposits in 1984 q4 and remain relatively high until after sterling's exit from the ERM in September 1992. Thereafter, the Morishima elasticity remains statistically significant at just over 2. Thus, it appears that the precision of the elasticity estimates depends on the period analyzed.

In summary, the results from Tables I and II show that 16%-25% of the elasticities of substitution between capital certain assets exceeds 2.0 on average, when the elasticities are statistically significantly different from zero. This is in contrast to previous studies, which have tended to find evidence of low substitutability or even complementarity, when the Allen-Uzawa elasticity of substitution is used. These results clearly violate the implicit assumption behind simple sum monetary aggregation that the component assets are all perfect substitutes for each other, which would theoretically imply infinite elasticities of substitution.

Substitution between Risky Assets (A3) and Capital Certain Assets (A1, A2)

Results from Table I show that many of the estimated substitution elasticities between deposit assets (A2) and risky assets (A3) are not statistically significantly different from zero. Hence, most of the elasticities relating to interest-bearing deposit assets (A2) are not significantly different from zero. In contrast, most of the substitution elasticities between cash assets (A1) and risky assets (A3) are significantly different from zero. This is most evident with respect to the Morishima elasticities where at worst 41/62 (66%) of the elasticities are significant and at best 54/62 (87%) are significant, depending on the value of the coefficient of relative risk aversion ([Z.sub.t]) specified.

Given the inevitable volatility in the returns on risky assets (even after risk adjustment) and the occasional presence of very small rental prices (associated with the envelope approach used to construct the benchmark rate), it is to be expected that some of the elasticities would be poorly determined. Nevertheless, there is a clear disparity between the general significance of the elasticities of substitution between cash (A1) and risky assets (A3) and those involving deposit assets (A2). Furthermore, it is clear from Table II that, when considering only the statistically significant elasticities, there is considerable substitutability between these assets. Specifically, the elasticities of substitution ME31, ME13 and H13 are generally above unity and often greater than 2.0. The statistically significant elasticities relating to deposit assets (A2) are clearly of a similar magnitude and indicate that all the assets in question are unambiguously substitutes.

Turning now to the evidence provided by the full time series of elasticities, Figure 2 illustrates that cash assets (A1) and risky assets (A3) are substitutes in demand with the Morishima elasticity generally significantly different from zero. This is particularly evident in the early to mid-1980s, and following sterling's departure from the ERM. In both these cases, the elasticities range between 2.0 and 4.0 and were extremely well determined in the context of relatively narrow standard error bands. Again, during periods of significant and rapid changes in relative user costs the Morishima elasticities are somewhat less well determined. Obvious examples of this are the period 1986 to 1988 (which encompassed both the rapid increase in equity prices and the stock market crash of October 1987) and the periods of entry into and exit from the ERM in October 1990 and September 1992, respectively. With respect to the latter, this probably reflects the impact of the changes in the monetary regime on expectations and the yield curve, whose shape, and thus the relative own rates and yields, altered dramatically as a result of the abrupt changes in monetary policy. It is quite clear, for example, that there is a general widening of the standard error bands around both these dates.

Figure 3 shows that the Morishima elasticity between deposit assets and risky assets (ME23) exhibits either modest substitutability or modest complementarity. While it was evident in Table II that most of the significant elasticities were in fact positive (indicating substitutability), ME23 tends to be better determined in the post ERM period. Furthermore, although the elasticities over this period are only marginally significant at the 10% level, there appears to be stronger evidence in favor of complementarity than of substitutability over this period.

For completeness, Figures 4, 5, and 6 show the McFadden-Hicks elasticities together with their standard error bands. These broadly confirm the evidence provided by the Morishima elasticities in the sense that the elasticities H 12

and H23 are often insignificant while the elasticity H13 is generally significant aside from the period 1986 to 1988 and the time around ERM entry. As with ME13, H13 is precisely estimated in the post-ERM period with a mean value of around 1.8. In contrast, the elasticities H12 and H23 are not significant during this period, although the latter does indicate possible complementarity, which is consistent with the evidence provided by ME23.

Some Hypotheses about Substitution and Relative Risk Aversion

Transactions costs (brokerage costs, shoe leather costs, and so forth) may in practice inhibit substitution between assets. That being the case, it may take quite substantial changes in relative user costs to induce consumers/households to switch between substitutable assets. Clearly, the fact that cash assets (A1) have a zero return while risky assets (A3) tend to offer relatively high returns (even after risk adjustment), suggests that substitutions at the margin are most likely to occur between assets A1 and A3. This argument is supported by the results that the mean values of the significant elasticities between assets A1 and A3 tends to fall appreciably as the coefficient of risk aversion increases (see Table II). This suggests that, as the perceived riskiness of capital uncertain assets increases, and consequently the risk adjusted return falls relative to other nonrisky assets, households become less willing to overcome any transactions costs/inertia and the like and substitute between assets.

Results from Table II show that the mean values of all the significant elasticities for ME12, ME23, and ME13 tend to fall as the degree of risk aversion increases. For ME12, this cannot be directly attributable to risk, since neither of the relevant assets is risky. This result is consistent with the presence of adjustment costs, however, since a higher coefficient of relative risk aversion will tend to imply a lower risk adjusted benchmark rate of return. Consequently, the incentive to overcome costs of adjustment in order to take advantage of relative user cost changes will be reduced.

Finally, the evidence presented in Tables I and II is potentially consistent with some well-established theories of money demand. The (micro) buffer stock models of the precautionary demand-type of Miller and Orr [1966] and the transactions-type of Akerlof and Milbourne [1980], for example, rely on the notion of narrow money being used as a relatively low cost buffer asset. Hence, the presence of any costs of adjustment (such as brokerage costs, inertia, shoe leather costs, and so on) can help to explain why transactions balances will not always be optimally adjusted relative to interest bearing assets as is implicitly assumed in the Baumol/Tobin inventory theoretic models.

The presence of an opportunity cost of holding "money" in these buffer stock models (transactions balances pay a lower return than do alternative assets such as bonds) explains why transactions/precautionary balances will be permitted only to build up to some threshold before it will pay an individual to transfer surplus accumulated balances to interest bearing assets and reduce transactions balances down to some return point (the anticipated transactions/precautionary balances required in the next period). In this type of model, however, the particular type of alternative interest bearing asset is often left unspecified. It is not unreasonable to assume that since any accumulated surplus balances are not expected to be required as future transactions or precautionary balances, they will be transferred to those assets yielding the highest risk-adjusted return. According to the results presented in this paper, this asset would generally be in the A3 category. Furthermore, for any given costs of adjustment, an increase in the user cost of transactions balances relative to the risk-adjusted returns on other assets would be expected to induce individuals to lower their target threshold level, which would imply, on average, a shift out of transactions balances and into the highest risk-adjusted interest-bearing asset. In the context of the present paper, this would be picked up as a generally high elasticity of substitution between cash assets and risky assets and a relatively low elasticity between cash assets and deposit assets.

Implications for Monetary Aggregation

If people regard risky assets as substitutes for money, as is generally assumed to be the case for deposit assets, then there may be an argument for including them as part of the official monetary aggregates. This approach has been applied to U.S. data by Duca [1995] and Orphanides et al. [1994] who add bond funds and bond and equity funds respectively to M2 using simple sum aggregation and without any risk adjustments. The results in this paper show that while U.K. risky assets (A3) are closer substitutes for capital certain assets (A1) than are deposit assets (A2), they are not perfect substitutes.(11) This is evidence against forming monetary aggregates using simple summation that requires that the assets be perfect substitutes for each other. In addition, adjusting for risk is important because the degree of risk changes the magnitudes of the elasticity estimates. Note that to form an admissible monetary aggregate that includes capital uncertain assets requires testing if these risky assets, together with other "monetary assets," form a weakly separable subutility function. This is beyond the scope of this current paper but remains an important issue for future research.

Finally, given that risky assets are consistently found to be substitutes for cash assets, and if they are indeed regarded as money, then their exclusion from broad money aggregates carries with it the risk that the latter may be periodically destabilized by substitutions between assets within the official (narrow or broad) aggregate and risky assets outside it. As Barnett [1980] has pointed out, these destabilizing substitutions would be completely internalized by specifying an appropriately wide aggregate and aggregating across the assets using a suitable aggregation procedure, such as the Divisia index number approach. The traditional simple sum aggregates that assume perfect substitution between assets are clearly inappropriate, given the evidence of variable degrees of substitution between assets over time.

VII. CONCLUSIONS

This paper uses the risk adjusted index number procedure of Barnett and Liu [1995] and Barnett, Jensen, and Liu [1997] to analyze substitution between U.K. capital certain and noncapital certain 'risky' assets. Clearly, central banks regard near money (interest-bearing) assets as substitutes for cash assets and include them in broad monetary aggregates. Thus, if recent financial innovation significantly increases monetary service flows from risky assets, then these risky assets may also become close substitutes for cash assets. Whether agents regard risky assets as substitutes for cash assets is an empirical issue that we evaluate using the seminonparametric AIM specification.

The most significant result in this study is that risky assets are often substitutes for the highly liquid monetary (cash) assets. Furthermore, risky assets during the post ERM period were generally found to be stronger substitutes for cash assets than were interest bearing deposits. Also, accounting for risk aversion is important and the results show that on average, substitution tends to fall as the degree of relative risk aversion increases. As risky assets become closer substitutes for monetary assets, this suggests that agents may regard these noncapital certain assets, like deposit assets, as money. Future research should perform weak separability tests, which are beyond the scope of this paper, to establish whether risky assets should be included as part of the official monetary aggregates. The results also suggest that if central banks do include noncapital certain assets, then the simple-sum procedure of forming monetary aggregates should be abandoned in favor of an alternative aggregation technique that allows for less than perfect substitution, such as the Divisia index. APPENDIX TABLE A1 Parameter Estimates for the Asymptotically Ideal Model(*) AIM(2) with AR1 Correction and Relative Risk Aversion Z(t) = 1 Sample 1979q1-1994q2 SSE MSE Adjusted R-square Equation 1 0.014515 0.000250 0.98883 Equation 2 0.008971 0.000155 0.99542 Parameter Estimate Standard Error t-statistic B1 -0.115044 0.207777 -0.55369 B2 0.638442 0.161968 3.94177 B6 -0.174173 0.030198 -5.76762 B11 -1.97913 1.83861 -1.07643 B12 2.12274 1.35293 1.569 B13 -0.061932 0.472246 -0.131144 B14 -0.451536 0.469276 -0.962197 B15 8.38493 2.97701 2.81656 B16 -3.64013 1.11472 -3.26552 B17 -2.85572 0.829922 -3.44095 B18 1.08941 0.319203 3.41292 B19 -2.71756 22.0273 -0.123372 B20 -15.9786 10.8578 -1.47163 B21 16.026 8.68517 1.84521 B22 4.97774 10.2166 0.487223 B23 -11.5792 7.24991 -1.59715 B24 8.12263 4.67305 1.73819 B25 9.11E-02 1.44E-03 0.630805 B26 0.864753 2.38115 0.363166 B5 -0.212072 0.053086 -3.99486 B7 5.5262 1.64715 3.35501 B8 -1.06439 0.996667 -1.06795 B9 -3.86132 0.686614 -5.62371 B10 1.0775 0.266925 4.0367 B4 -7.86E-03 0.076598 -0.102555 * The notation for the parameter estimates is from Barnett and Yue [1992].

1. Mutual funds are called unit trusts or investment trusts in the United Kingdom.

2. Liquidity given uncertainty over expenditure flows and income receipts.

3. There is a large literature on demand systems for monetary assets. Much of the earlier work was based on the highly restrictive Translog model, as in Ewis and Fisher [1984], Serletis [1988; 1991a; 1991b], and Serletis and Robb [1986]. More recent publications use the flexible Fourier form, as in Ewis and Fisher [1985], Fisher and Fleissig [1994].

4. On the AIDS model, see Deaton and Muellbauer [1980]; on the Translog model, see Christensen, Jorgensen, and Lau [1973].

5. See Mehra and Prescott [1985] for a discussion of the equity premium puzzle.

6. This is because contemporaneous interest rates are not known at the start of the current period, as they are not paid in advance. Hence, at the start of the period they are random variables rather than deterministic and this can produce a nonzero covariance with real consumption growth.

7. El Badawi, Gallant, and Souza [1983] and Gallant [1981] define a seminonparametric function as the use of a truncated series expansion that is dense in a Sobolev norm.

8. We set [p.sub.j] = epsilon = 1 x [10.sup.-5] when the own rate of return equals the benchmark rate.

9. The traditionally used Allen Uzawa elasticities of substitution are symmetric and are incorrect when there are more than two variables; see Drake, Fleissig, and Swofford [1997].

10. The applications of these tests to monetary assets are discussed in detail in Swofford and Whitney [1987;1988;1994] and Swofford [1995].

11. In the post ERM period (93 q1 to 94 q2), when both ME 12 and ME13 are well determined and statistically significant, the mean value of ME12 is 2.2 while that of ME13 is 2.5.

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Drake: Professor of Monetary Economics, Loughborough University, United Kingdom

Phone 44-1509-222707, Fax 44-1509-223910

E-mail l.m.drake@lboro.ac.uk

Fleissig: Professor, University of California at Fullerton

Mullineux: Professor of Money and Banking, University of Birmingham, United Kingdom

Phone 44-121-414-6742, Fax 44-121-414-7377

E-mail mullinaw@css.bham.ac.uk