A new empirically weighted monetary aggregate for the United States.

Drake, Leigh ; Mills, Terence C.

I. INTRODUCTION

This article uses an approach to long-run econometric modeling proposed by Pesaran et al. (2001; hereafter PSS) to develop an empirically weighted broad monetary aggregate for the United States and to demonstrate the advantages of this type of aggregate from a monetary policy perspective. In particular, we examine the ability of this type of approach to deal with periods of significant financial innovation and money demand instability.

A key requirement for monetary aggregates to provide a useful role in guiding monetary policy is that they should be stably related to the objectives of policy, such as inflation or nominal income growth. In this context, the United States has exhibited periodic evidence of significant money demand instability. Most notably, these have been Goldfeld's (1976) case of the "missing M1" in 1973/74 and the "missing M2" episode of the early 1990s (Feldstein and Stock, 1996). In both cases, the aggregate in question experienced a significant velocity increase and, as a consequence, previously established and apparently stable money demand relationships began to seriously overpredict the growth in these aggregates.

The missing M1 episode in 1973/74, together with subsequent evidence that M2 was more predictable than M1, led many economists to use M2 as an indicator of nominal activity. Hence, during the 1980s, M2 became the primary intermediate target of monetary policy. Furthermore, the primacy of M2 appeared to be well supported by the available empirical evidence. Feldstein and Stock (1994), for example, established that the rate of change of M2 was a statistically significant predictor of the rate of change of nominal gross domestic product (GDP) over the period 1959-92. Furthermore, M2 remained statistically significant when short-term interest rates were added to the relationship. Subsequent research, such as Miyao (1996) and Estrella and Mishkin (1997), however, cast doubt on the robustness of this result. Carlson et al. (2000, p. 34), for example, summarize the current situation by arguing that "the promising empirical conclusions of Feldstein and Stock (1994) that established predictive content for M2 in a vector error correction setting do not seem to find support in data that extend through the mid-1990s."

The key factor behind the breakdown in the M2 relation appears to have been the substitution away from time deposits and into mutual funds, particularly stock and bond mutual funds, in the low interest rate environment of the early 1990s. Although the definition of M2 has been expanded by the Federal Reserve in the past to include money market mutual funds (MMMFs) and money market deposit accounts (MMDAs), for example, a number of analysts (Duca, 1995; Darin and Hetzel, 1994; Orphanides et al. 1994) have advocated that M2 should be expanded further to include these stock and bond mutual funds (M2+). Significantly, these studies typically use the simple sum aggregation approach, in which all component assets are given equal and constant weights over time. This approach does not seem consistent, however, with the accumulating evidence of a significant shift in wealth-holders preferences sometime in the early 1990s. Carlson et al. (2000), for example, add to the empirical evidence that had accumulated during the 1990s by suggesting that the instability was associated with a permanent upward shift in M2 velocity between 1990 and 1994. Furthermore, they argue that "our results support the hypothesis that households permanently reallocated a portion of their wealth from time deposits to mutual funds" (p. 381). Clearly, simple sum aggregation is not able to take account of these changes in preferences. More significantly, however, simple sum aggregation cannot take account of any changes in the relationship between component assets and nominal income over time.

Although Carlson et al. (2000) do manage to reestablish a stable money demand relationship for the MZM (money at zero maturity) and M2M (M2 minus small time deposits) aggregates through the 1990s, this is only possible by specifically accounting for the financial innovation that occurred in the early 1990s. Specifically, a linear shift variable is incorporated for the period 1990-94. Though this is an interesting result, it is of limited use to policy makers in the sense that this type of evidence is only available ex post, often with a considerable time lag. In other words, it is only with the benefit of hindsight that a particular period of instability can be rationalized in terms of financial innovation, permanent velocity shifts, and so on. In contrast, policy making is an ongoing process conducted in real time, and policy makers need to be assured that movements in a variable (such as M2) contains reliable information content in respect of policy objectives. For example, in response to the evident problems with M2 in the early 1990s, the Federal Open Market Committee (FOMC) downgraded its role, and no single variable has subsequently taken its place.

II. INCORPORATING FINANCIAL INNOVATIONS AND CHANGING PREFERENCES

What is required is a monetary aggregate that can endogenously respond to changes in wealth holder preferences, possibly caused by financial innovations, which impinge on the information content of monetary aggregates or their sub components. A possible theoretical solution to this problem is to employ the Divisia aggregation procedure, advocated by Barnett (1980, 1982) and adopted by many central banks around the world. This type of weighted monetary aggregate allows the composition of the aggregate to respond to financial innovations that impact on relative rates of return. Specifically, the component asset weights are derived as monetary expenditure shares that in turn are influenced by relative interest rates captured by the user costs or rental prices of the assets. Although a number of studies have produced evidence of stable broad money demand relationships using Divisia aggregation (Belongia and Chalfont 1989, for the United States; Belongia and Chrystal 1992, and Drake and Chrystal 1994, 1997, for the United Kingdom), the fact that the aggregate can only accommodate financial innovations that impact on relative interest rates, and thereby also rental prices, implies that Divisia aggregates may not fully reflect any changes in the relationship between "money" and target variables, such as nominal income or inflation.

Furthermore, although the Divisia approach is eminently appropriate for conventional monetary asset components such as M2 assets, the application of the Divisia index number methodology to the missing M2 episode of the early 1990s is nevertheless potentially problematic. Although an established methodology does exist (Barnett et al. 1997) for incorporating risky assets, such as stock and bond mutual funds, into Divisia monetary aggregates, the risk adjustments implied by the consumption capital asset pricing model (CCAPM) methodology employed tend to be relatively small. As is well established in the "equity premium puzzle" literature (see Mehra and Prescott 1985, and Drake et al. 1999), the large risk adjustments implied by the typical equity premiums cannot be produced in the absence of unreasonably high coefficients of relative risk aversion.

An alternative approach proposed by Feldstein and Stock (1996) is to prod uce empirically weighted monetary aggregates. Feldstein and Stock (1996) argue that "our objective is to develop a procedure that automatically adjusts the composition of the monetary aggregate in a way that makes the resulting measure of the money stock a stable leading indicator of nominal GDP and potentially a useful control instrument for altering nominal GDP" (p. 5). Feldstein and Stock (1996) employ two alternative methodologies to produce the empirically weighted monetary aggregates. The first is a switching regression methodology that attaches weights of either one or zero to monetary aggregate subcomponents and in which the switch dates are established on the basis of the ability of the aggregate to forecast GDP growth. The second is a time-varying parameter model in which the component weights evolve over time so as to produce an aggregate with a stable predictive relationship to nominal GDP. The switching regression approach of Feldstein and Stock (1996) has also been applied to Canadian monetary aggregates by Siklos and Barton (2001).

In contrast, we construct empirically weighted monetary aggregates based on a new approach to testing for the existence of a linear long-run relationship when the orders of integration in or the form of cointegration between the underlying regressors are not known with certainty. Hence, in contrast to Feldstein and Stock (1996), the component weights derived at any point in time are drawn from the cointegrating relationship between the component assets and nominal GDP. Unlike Divisia aggregation, this approach does not require any data on relative interest rates and can therefore incorporate stock and bond mutual funds without the need for any risk adjustment of returns, thereby avoiding the problem of the "equity premium puzzle." Furthermore, by using this approach in a recursive fashion we are able to analyze how the empirical weights evolve over time. This is particularly useful with respect to informing the debate over particular episodes of money demand instability.

It is important to recognize, however, that the empirically weighted aggregate proposed here will only be optimal, in the context of capturing the evolving relationship between the monetary components and nominal GDP, if the estimated model is appropriate. In particular, the use of a linear model, when the data generating function is nonlinear, could produce a suboptimal aggregate with an unstable relationship to the target variable. Indeed, many monetary asset demand studies explicitly recognize this issue by estimating nonlinear systems, such as the Fourier and asymptotically ideal model (AIM) systems (Drake et al. 1999, 2003, Fisher and Fleissig 1994, Fleissig and Swofford 1997). Hence, although this is clearly an important issue, our objective is to use the notion of a longrun cointegrating relationship between the monetary components and nominal income using the PSS approach. The trade-off, however, is that this implies that we must follow Feldstein and Stock (1996) and adopt a linear model specification as the literature on nonlinear cointegration is not yet sufficiently developed. In recognition of this potential trade-off, therefore, it is important that the empirical performance of the new weighted aggregate is assessed, both with respect to simple sum aggregates and an appropriate Divisia aggregate, as the latter also responds endogenously to aspects of financial innovation.

We focus initially on a sample period running from 1960:2 to 1977:4 to reexamine the missing M1 period of the early/mid-1970s. Subsequently, we use the full sample period and focus in particular on the missing M2 period of the early 1990s. Because our aim is to produce an empirically weighted monetary aggregate that is useful to policy makers making decisions in real time, we focus particularly on the out-of-sample properties of the new monetary aggregates.

The remainder of this paper is structured as follows. Section III outlines the modelling technique and the derivation of the empirically weighted monetary aggregate. Section IV then discusses the empirical results. Section V contrasts the out-of-sample nominal income and inflation forecasting performance of the new aggregate with the simple sum aggregates M2 and M2+ and with Divisia M2. Section VI concludes.

III. CONSTRUCTING A WEIGHTED MONETARY AGGREGATE FOR THE UNITED STATES

The data set used to construct an empirically determined weighted monetary aggregate contains quarterly observations from 1960:2 to 2001:2 on the logarithms of nominal GDP at factor cost, denoted [y.sub.t], and five monetary components:

[x.sub.1t]: M1 (currency, demand deposits, other checkable deposits)

[x.sub.2t]: Savings and money market deposit accounts plus small-denomination time deposits (approximately M2 [excluding retail MMMFs]-M1)

[x.sub.3t]: Retail MMMFs

[x.sub.4t]: Large-denomination time deposits, repurchase agreements, eurodollar deposits, and institutional MMMFs (approx M3 - [[x.sub.1] to [x.sub.3]])

[x.sub.5t]: Stock and bond mutual funds

The levels of the five aggregates are shown in Figure 1. Readily apparent is the enormous growth in the [x.sub.5] component (stock and bond mutual funds) from the early 1990s. This is typically argued to be responsible for the missing M2 puzzle of the time, and we can see the corresponding decline in the broad money components of M2 from the decline in [x.sub.2] between 1991 and 1995. A similar decline is also evident in [x.sub.4], which corresponds to the broad M3 money components. In contrast to the growth in stock and bond mutual funds, MMMFs have exhibited more modest growth dating from the early 1980s, as evidenced by the profile of [x.sub.3].

The approach taken to construct a weighted monetary aggregate is that proposed by PSS and has been used successfully in respect of U.K. monetary aggregates by Drake and Mills (2001). We thus begin by considering the following vector autoregressive model of order p(VAR[p]) in the vector of variables [z.sub.t] = ([y.sub.t], [x'.sub.t])', where [x.sub.t] = ([x.sub.1t], ..., [x.sub.kt])' is the vector of monetary components: (1)

(1) [z.sub.t] b + [c.sub.t] + [p.summation over i=1][[PHI].sub.i][Z.sub.t-1] + [[epsilon].sub.t],

where b and c are vectors of intercepts and trend coefficients and [[PHI].sub.i], i = 1, 2, ..., p, are matrices of coefficients. We assume that the roots of

[absolute value of [I.sub.5] - [p.summation over i=1] [[PHI].sub.i][z.sup.i]] = 0

are outside the unit circle [absolute value of z] = 1 or satisfy z = 1, so that the elements of [z.sub.t] are permitted to be either I(0), I(1), or cointegrated. The unrestricted vector error correction form of (1) is given by

(2) [DELTA][z.sub.t] = b + [c.sub.t] + [PI] [z.sub.t-1] + [p-1.summation over i=1] [[TAU].sub.i] [DELTA][z.sub.t-i] + [[epsilon'.sub.t],

where

[PI] = - [I.sub.5] - [p.summation over i=1] [[PHI].sub.i])

and

[[GAMMA].sub.i] = - [p.summation over j=i+1] [[PHI].sub.j], i = 1, ..., p -1

are matrices containing the long-run multipliers and the short-run dynamic coefficients, respectively.

Given the partition [z.sub.t] = [([y.sub.t], [x'.sub.t])' we define the conformable partitions [[epsilon].sub.t] = ([[epsilon].sub.1t], [[epsilon'.sub.2t]' and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

and make the standard assumption that [[epsilon].sub.t] = ([[epsilon].sub.1t], [[epsilon].sub.2t])' follows a multivariate i.i.d, process having mean zero, nonsingular covariance matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and finite fourth moments. We also assume that [[pi].sub.21] = 0, which ensures that there exists at most one (nondegenerate) long-run relationship between [y.sub.t] and [x.sub.t], irrespective of the level of integration of the [x.sub.t], process.

With this assumption and the partitioning given, (2) can be written in terms of the dependent variable [y.sub.t] and the forcing

variables [x.sub.t] as

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The contemporaneous correlation between el, and [[epsilon].sub.2t], can be characterized by the regression

(5) [[epsilon].sub.1t] = [omega]'[[epsilon].sub.2t] + [[xi].sub.t],

where [omega] = [[summation of].sup.-1.sub.22] [[sigma].sub.21], {[xi.sub.t]} is an i.i.d.(0, [[sigma].sup.2.sub.[xi]]) process with [[sigma].sup.2.sub.[xi]] = [[sigma].sub.11] - [[sigma].sub.12] [[summation of].sup.-1.sub.22][[sigma].sub.21], and the {[[xi].sub.t]} and {[[epsilon].sub.2t]} processes are uncorrelated by construction. Substituting (4) and (5) into (3) yields

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[a.sub.0] [equivalent to] [b.sub.1] - [omega]' [b.sub.2], [a.sub.1] [equivalent to] [c.sub.1] - [omega]'][c.sub.2], [theta] [equivalent to] [[pi].sub.11,] [delta]' = [[pi]'.sub.12] - [[PI]'.sub.22][omega], [[psi].sub.i] [equivalent to] [[gamma].sub.11,i] - [omega]'[[gamma].sub.21,i], [[phi].sub.0] [equivalent to] [omega]', and [[phi].sub.i] [equivalent to] [[gamma].sub.12,i] - [omega]'[[GAMMA].sub.22,i].

It follows from (6) that, if [phi] [not equal to] 0 and [delta] [not equal to] 0, there exists a long-run relationship between the levels of [y.sub.t] and [x.sub.t], given by

(7) [y.sub.t] = [[theta].sub.0] + [[theta].sub.1] t + [theta]'[x.sub.t] + [[upsilon].sub.t]

where [[theta].sub.0] [equivalent to] - [a.sub.0]/[phi], [[theta].sub.1] [equivalent to] - [a.sub.1]/[phi], [theta] [equivalent to] -[delta]/[phi] is the vector of long-run response parameters and {[[upsilon].sub.t]} is a mean zero stationary process. If [phi] 0 then this long-run relationship is stable and (6) can be written in the error correction model (ECM) form

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

If [phi] = 0 in (8) then no long-run relationship exists between [y.sub.t] and [x.sub.t]. However, a test for [phi] = 0 runs into the difficulty that the long-run parameter vector [theta] is no longer identified under this null, being present only under the alternative hypothesis. Consequently, PSS test for the absence of a long-run relationship and avoid the lack of identifiability of [theta] by examining the joint null hypothesis [phi] = 0 and [delta] = 0 in the unrestricted ECM (6). Note that it is then possible for the long-run relationship to be degenerate, in that [phi] [not equal to] 0 but [delta] = 0, in which case the long-run relationship involves only [y.sub.t], and possibly a linear trend.

PSS consider the conventional Wald statistic of the null [phi]= 0, [sigma] = 0 and show that its asymptotic distribution involves the nonstandard unit root distribution and depends on both the dimension and cointegration rank (0 [less than or equal to] r [less than or equal to] k) of the forcing variables [x.sub.t]. This cointegration rank is the rank of the matrix [[pi].sub.22] appearing in (4). PSS obtain this asymptotic distribution in two polar cases: (1) when 1123 is of full rank, in which case [x.sub.t] is an I(0) vector process, and (ii) when the [x.sub.t] process is not mutually cointegrated (r-0 and [[PI].sub.22] = 0) and hence is an I(1) process. They point out that the critical values obtained from stochastically simulating these two distributions must provide lower and upper critical value bounds for all possible classifications of the forcing variables into I(0), I(1), and cointegrated processes. A bounds procedure to test for the existence of a long-run relationship within the unrestricted ECM (6) is thus as follows. If the Wald (or related F-) statistic falls below the lower critical value bound, then the null [phi]= 0, [sigma] = 0 is not rejected, irrespective of the order of integration or cointegration rank of the variables. Similarly, if the statistics are greater than their upper critical value bounds, the null is rejected and we conclude that there is a long-run relationship between [y.sub.t] and [x.sub.t]. If the statistics fall within the bounds, inference is inconclusive, and detailed information about the integration--cointegration properties of the variables is then necessary to proceed further. It is the fact that we may be able to make firm inferences without this information, and thus avoid the severe pretesting problems usually involved in this type of analysis, that makes this procedure attractive in applied situations. PSS provide critical values for alternative values of k under various situations. The two that are relevant here are Case 1: [a.sub.0] [not equal to] 0, [a.sub.l] = 0 (with an intercept but no trend in [6]), and Case 2: [a.sub.0] [not equal to] 0, [a.sub.l] [not equal to] 0 (with both an intercept and a trend in [6]).

PSS show that this testing procedure is consistent and that the approach is applicable in quite general situations. For example, equation (6) can be regarded as an autoregressive distributed lag model in [y.sub.t] and [x.sub.t], having all lag orders equal to p. Differential lag lengths can be used without affecting the asymptotic distribution of the test statistic.

IV. EMPIRICAL RESULTS

The Missing MI Episode

Our first exercise is to consider the period up to the end of 1977. During this period, the aggregate underlying [x.sub.3] was either zero, or almost so, and hence is excluded from the analysis. Thus we focus on using [x.sub.1t] , [x.sub.2t], and [x.sub.4t] so that k= 3 and attention is initially concentrated on the period up to the end of 1972, that is, before the episode of the missing M1.

In implementing the PSS approach, our first task is to check that the assumptions required for attention to focus solely on equation (6) are satisfied. One underlying assumption, implicit in the discussion, is that the maximal order of integration of the {[z.sub.t]} process is unity. Unit root tests of the individual series making up {[DELTA][z.sub.t]} show that a unit root is rejected at the 5% level in each case. A second assumption, explicitly discussed above, is that [[pi].sub.21] =0 in (the partitioned form of) the unrestricted vector error correction (2).

Estimation of this equation with p set equal to 5 produced t-statistics on the coefficients of [y.sub.t-1] in the equations for [DELTA][x.sub.it] i= 1, 2, 4, of 0.27, -0.22, and 0.46, thus producing no evidence against the null hypothesis [[pi].sub.21] = 0. A setting of p = 5 was thought to be an appropriate trade-off between the need to account for any seasonal dynamics and the degrees of freedom available given the length and dimension of [z.subt],. Having thus ascertained that the conditions required for (6) to be considered in isolation are satisfied, the following parsimonious specification of this equation was eventually arrived at:

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The standard diagnostic checks (probability values are shown in brackets) indicate no evidence of misspecification. The Wald statistic for testing whether there exists a long-run relationship between [y.sub.t], and [x.sub.t] produces an F-statistic of 20.32. This is well beyond the 1% significance level upper bound in both Cases 1 and 2: with three regressors these upper bounds are 5.61 and 5.23, respectively (note that the trend was found to be insignificant and hence has been omitted from the chosen specification). We must therefore conclude that such a long-run relationship certainly exists and, given our estimates, the long run relationship (7) is

[y.sub.t] = 0.09 + [??][x.sub.t] = 0.09 + 1.24(0.76[x.sub.l.t] + 0.20[x.sub.2,t] + 0.04 [x.sub.4,t]).

Having thus demonstrated that a long-run relationship exists between nominal income and these monetary components up to 1972, we now use this model to investigate M1 instability around the missing money period. The stability of the component weights over time is investigated by estimating the model recursively up to the end of 1977: that is, the model is recursively fitted through the estimation sample period up to end-1972, and then recursive estimation is continued for the next five years.

It is clear from Figure 2 that there is a relative constancy in the recursive weights prior to 1970. More significantly, the "optimal" monetary aggregate (based on the relationship with nominal income) would have a weight on [x.sub.1] (M1 assets) of around 0.85 to 0.9, and relatively low weights on both [x.sub.2] and [x.sub.4]. Hence, it is perhaps not surprising that the standard Goldfeld money demand specification, which focused on M1, performed well prior to the early 1970s. It is clear, however, that from 1970 the implied optimal weight on [x.sub.1] begins to decline while that on [x.sub.2] begins to increase. This trend continues in the out-of-sample period and begins to accelerate from late 1973, so that by 1977 the implied optimal weights are around 0.5 on both [x.sub.1] and [x.sub.2]. This is highly suggestive because this is precisely the time when the standard Goldfeld model began to seriously overpredict M1 growth.

[FIGURE 2 OMITTED]

To illustrate this missing M1 episode, we estimate a standard Goldfeld partial adjustment model for M1 up to end-1972. (Goldfeld-type models using both M1 and composite M3 are reported in Appendix A). Figure 3 clearly shows that the model fits the data very well prior to 1972:4, whereas the out-of-sample evidence shows that the model progressively overpredicted M1 growth after 1973. Subsequent studies have attempted to explain the missing M1 puzzle in terms of problems with the partial adjustment model (inadequate dynamic specification) and the impact of inflation and financial innovation. Our results suggest that the instability can be explained very simply by a failure to adequately account for the switch out of M1 assets and into M2 assets, which may have been prompted by the impact of high inflation and high nominal interest rates on the zero yielding M1 assets. This conjecture is confirmed by fitting a Goldfeld demand function to [??][x.sub.t], up to end-1972 and forecasting out-of-sample until end-1977. Figure 4 illustrates quite clearly that the optimally weighted aggregate based on [x.sub.1], [x.sub.2], and [x.sub.4] (approximately M3) produces a good fit based on the standard Goldfeld model, both within and out of sample. It should be noted, however, that unlike the M1 specification, the significant coefficient on prices in the context of a real money demand equation for weighted M3 suggests, not surprisingly, that there is some misspecification inherent in the simple Goldfeld-type partial adjustment model.

[FIGURE 3 & 4 OMITTED]

It is interesting to note from Figure 2 that [x.sub.4] has a weight that is relatively low and stable for most of the sample period. Hence, the optimal weighted monetary aggregate would be composed mainly of M2 assets. Furthermore, as the weights on [x.sub.1] and [x.sub.2] converge toward 0.5 after 1973, the weight on [x.sub.4] trends toward zero. Figure 5 illustrates that this is also the case when stock and bond mutual funds ([x.sub.5]) are included in the model (the estimated model is shown in Appendix B). Although the decline in the [x.sub.1] weight and the increase in the [x.sub.2] weight occurs slightly later than in Figure 2, both [x.sub.4] and [x.sub.5] trend toward a weight of zero after 1974.

[FIGURE 5 OMITTED]

Again, this is a significant result from a policy perspective because it explains the favorable empirical results obtained for simple sum M2 and the evolution of M2 as the primary intermediate target variable by the 1980s. More specifically, the use of a simple sum monetary aggregate that accorded equal weights (of unity) to both M1 and (M2 - M1) assets but a weight of zero to any broader monetary assets accords reasonably well with the optimal weighting scheme implied by our analysis. Hence, because this optimal weighting scheme is derived from the implied nominal income relationship, it would be expected that the M2 aggregate would perform well both empirically and in a policy context. In essence, given the almost zero weight on [x.sub.4], this is illustrated very clearly by the out-of-sample performance of the weighted aggregate (effectively M2) shown in Figure 4.

Clearly, however, the continued success of simple sum M2 as a key monetary policy variable through the 1980s and 1990s would depend crucially on two factors. First, the stability and equality of the weights on [x.sub.1] and [x.sub.2] are important. If the optimal weights were to deviate significantly from the weights of 0.5 evident in the late 1970s, then the simple sum M2 aggregate (which imposes equal weights of unity on all components) would be expected to increasingly diverge from the "optimal" aggregate. It should also be noted that if either [x.sub.1] or [x.sub.2] were to exhibit periods of very rapid growth, this would produce excessive growth in simple sum M2 (relative to the optimally weighted aggregate) by virtue of the weights of unity on both these component assets. A case in point is the very rapid growth in M2 as a result of the significant increase in the [x.sub.2] component after 1983. This is discussed further subsequently.

Second, if the implied optimal weights on assets such as [x.sub.4] and [x.sub.5] were to increase over time, then the M2 aggregate would again be expected to diverge increasingly from the optimal aggregate over time. As was stressed previously, a particularly serious episode of M2 instability occurred during the early 1990s. Hence, in the next section we use the full sample period of 1960.2 to 2001.2 and focus in particular on the out-of-sample properties of the model post 1990.

Full Sample Results

Due to the fact that MMMF data is only available from 1973 and that MMMF holdings do not begin to exhibit rapid growth until the 1980s, we combine MMMFs with the [x.sub.2] assets to form the aggregate [x.sub.23] = [x.sub.2] + [x.sub.3] (in levels). As emphasized previously, we are particularly interested in examining the period of M2 instability in the early 1990s, known as the period of missing M2. This has largely been attributed to the substitution away from M2 assets and into mutual funds, particularly stock and bond mutual funds, in the low-interest-rate environment of the early 1990s. Hence, we first examine the recursively estimated weights from a model estimated up to the end of 1989 so that the estimated weights during the 1990s are genuine out-of-sample recursive weights.

The long-run equation was developed using the techniques outlined earlier. After the underlying assumptions required for the approach to be used were found to be satisfied, the Wald statistic for testing whether there exists a longrun relationship between [y.sub.t] and [x.sub.t] produced an F-statistic of 7.00. This is again well beyond the 1% significance level upper bound in both Cases 1 and 2, with k = 4 these upper bounds are 5.06 and 4.92, respectively (note that the trend was found to be insignificant and is excluded from the specification shown below)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Three dummy variables are included to deal with outlying residuals in 1978:2, 1980:4, and 1982:1. After their inclusion, the standard diagnostic checks indicate no evidence of misspecification. We therefore conclude that a long-run relationship certainly exists and, given our estimates, the long-run relationship (7) is

[y.sub.t] = 0.12 + [??][x.sub.t] = 0.12 = 1.05 + (0.44[x.sub.l.t] + 0.58[x.sub.2,3,t] + 0.03[x.sub.4,t] - 0.05 [x.sub.5,t]).

As before, this specification was estimated recursively, and the calculated weights are shown in Figure 6 (note that the presence of the dummies only allows the recursions to be estimated after 1982). The in-sample recursive weights confirm the trend decrease in the [x.sub.1] weight and the trend increase in the [x.sub.2] weight ([x.sub.23] weight in this case) that was apparent from the early 1970s in Figures 2 and 5. From 1987 until 1992, however, these weights are relatively stable at around 0.4 and 0.6, respectively. With respect to [x.sub.4] and [x.sub.5], the implied weights are both close to zero, with the weight on [x.sub.4] being slightly positive and that on [x.sub.5] slightly negative. From the perspective of the late 1980s/early 1990s, therefore, there was no strong rationale for the inclusion of either broad M3 monetary assets ([x.sub.4]) or stock and bond mutual funds ([x.sub.5]) in official monetary aggregates. Similarly, these results confirm the evidence presented earlier in the sense that M2 would be expected to outperform MI given the increased weight on [x.sub.23] and the decreased weight on [x.sub.1] implied by the long-run nominal income relationship.

Turning now to the "missing money" period after 1991, however, it is clear from Figure 6 that considerable variation is apparent in the optimal monetary aggregate component weights after 1992. In particular, the implied weight on [x.sub.1] increases substantially to around 0.7 between 1992 and 1996. This may well reflect the gradual impact of the low-inflation/low-interest-rate environment of the early 1990s on the willingness of wealth holders to hold M1 balances and use these for transactions. Indeed, in Figure 1 we can see the sharp increase in [x.sub.1] balances between 1991 and 1995. Hence, this period may represent the opposite scenario to that of the Goldfeld missing money episode in which wealth holders had switched away from M1 due to the impact of high inflation and high interest rates in the mid-1970s.

Conversely, the implied optimal weight on [x.sub.23] declines from 0.6 to around 0.2 by 1998, and this decline is mirrored by an increase in the weight on [x.sub.4] from just over zero in 1992 to around 0.3 by the end of the sample, broadly equivalent to the weight on [x.sub.23]. Perhaps more significantly, however, the implied optimal weight on [x.sub.5] remains remarkably stable and slightly negative throughout the so-called missing M2 period. Furthermore, although the weight increases somewhat after 1996, the implied optimal weight is only just positive by the end of the sample.

These results have important policy implications. First, they confirm that simple sum M2 would be expected to perform poorly during and after the missing money episode in the context of money demand and velocity instability. As mentioned, the increase in the optimal weight on [x.sub.1] balances probably reflects the increased willingness of wealth holders to hold M1 balances and use them for transactions/nominal spending. This would tend to be naturally reflected in a decline in the optimal weight on [x.sub.23] given the substitution between [x.sub.1] and [x.sub.23] assets with respect to transactions balances, and the fact that the recursively estimated optimal weights are derived from a long-run nominal income relationship. This substitution would not be adequately reflected by simple sum M2, however, given the equal weights of unity applied to both [x.sub.1] and [x.sub.23]. At the same time, however, the low-interest-rate environment of the early 1990s encouraged individuals holding broad M2 monetary components ([x.sub.23]) on the basis of an "asset motive" to substitute them for higher-yielding assets, such as stock and bond mutual funds. This asset motive hypothesis is supported by the fact that the recursively estimated weights on [x.sub.5] do not increase significantly either during the missing money episode or subsequently. Hence, the substantial substitution out of M2 balances and into stock and bond mutual funds was not significant from the perspective of transactions balances and subsequent nominal spending.

In summary, therefore, the problems of M2 instability and unreliability (in a policy context) during the missing M2 episode cannot simply be attributed to the rapid shift into stock and bond mutual funds ([x.sub.5] assets) from the early 1990s. This will undoubtedly have created problems for policy makers in the context of M2 providing misleading signals regarding future nominal income growth and inflation. In the terminology of Estrella and Mishkin (1997), official simple sum M2 at this time will have been characterized by a low signal to noise ratio. Nevertheless, our results make it quite clear that the appropriate policy response was not to redefine M2 to include the [x.sub.5] assets. Although the shift from M2 to [x.sub.5] assets was substantial our results suggest that this had little significance in the context of future spending (nominal income) and inflation. Hence, had U.S. policy makers at the time shifted their attention away from M2 toward a so-called M2+ aggregate (including stock and bond mutual funds), the growth in this aggregate would have overstated potential future inflationary pressures because most of the growth appears to be associated with an asset motive rather than a transactions motive. This is evidenced by the stable and low (slightly negative) weight on [x.sub.5] assets throughout the period of rapid growth from the early 1990s.

An important point to reiterate from a monetary policy perspective, therefore, is that the results reported in Figure 6 are genuinely forward-looking. The form of the PSS model was specified using data up to end-1989, and the recursively estimated weights are postsample weights thereafter. Hence, in the context of real-time policy making, this type of technique could provide valuable ongoing information regarding the information content of monetary aggregates and components during periods of financial innovation and turbulence. The results would have confirmed, quarter by quarter, that the rapid shift into stock and mutual funds was not in itself a cause for concern, but that the weights accorded to [x.sub.1] and [x.sub.23] should have been increased and decreased respectively. The results would also have suggested that consideration be given to monitoring M3 (including [x.sub.4] assets) as well as the optimally weighted M2. As emphasized previously, Carlson et al. (2000) do manage to reestablish a stable money demand relationship for MZM and M2M through the 1990s by specifically accounting ex post for the financial innovation that occurred in the early 1990s. In a policy context, however, these financial innovations persuaded the FOMC to downgrade the role of M2 in 1993.

[FIGURE 6 OMITTED]

Finally, having established the "optimal" monetary weights in an out-of-sample context, it is interesting to analyze the implied recursive weights when the PSS model is specified over the full sample period, 1960:2 to 2001:2 (this model is reported in Appendix C). It is clear from Figure 7 that the trends in the optimal weights are broadly consistent with those observed in Figure 6, although the variability in the weights is somewhat less pronounced. Nevertheless, we again see the period of relative stability in the late 1980s/early 1990s, followed by the subsequent increase in the implied weight on [x.sub.1] and the decrease in the weight on [x.sub.23] Interestingly, Figure 7 confirms that the most substantial variations in the optimal weights did not take place until after 1994. This, combined with the very small and slightly negative weights on [x.sub.4] and [x.sub.5], suggests that M2 would have provided a reasonable leading indicator of nominal spending/inflation for much of the missing money period of the early 1990s. This is confirmed in Figure 8, where the growth in M2 and the optimally weighted monetary aggregate are broadly equivalent in the early 1990s, but diverge significantly after the mid-1990s, with M2 exhibiting much faster growth than the optimally weighted aggregate. It is also interesting to note that the growth rates of M2 and the optimally weighted aggregates diverge considerably over the period 1983-1987, with the former exhibiting much higher annualized growth rates.

[FIGURES 7-8 OMITTED]

The evidence provided in Figure 7 confirms that there is no strong rationale for expanding M2 to include [x.sub.5] assets, as in the so-called M2+ aggregate, given the consistently low and slightly negative optimal weights on [x.sub.5] in the recursively estimated long-run income relationship. Furthermore, Figures 9 and 10 indicate that had policy makers focused on an aggregate such as M2+ during the missing M2 episode, this aggregate would have provided highly misleading signals in respect of subsequent inflationary pressures in the U.S. economy. To take some account of the lags inherent in the monetary transmission mechanism, we plot the growth rates of these monetary aggregates lagged eight quarters against current inflation. Figure 9 illustrates that the correspondence between U.S. inflation and prior weighted money growth is generally very good, particularly during the 1970s and 1990s. In contrast, Figure 10 indicates very clearly that M2+ significantly overestimated U.S. inflationary pressure during the missing M2 period of the early 1990s. As we have seen, M2 would have proved to be a more reliable monetary indicator at that time. Furthermore, M2+ continued to overestimate future nominal spending and inflationary pressure in the United States throughout the 1990s.

[FIGURES 9-10 OMITTED]

It is also interesting to note that prior M2+ growth would have provided extremely misleading monetary policy signals over the period 1984-1989. This is also a feature we noted in respect of M2 growth over the corresponding period in Figure 8 (1982-1987). Although optimally weighted money also seems to overstate the inflationary pressures in the mid-1980s somewhat, the leading indicator properties appear to be good in terms of signaling a shift from declining to rising inflation. Furthermore, as alluded to previously, the annualized growth rates of weighted money over the period 1982-1987 were much lower than either M2 or M2+ and much lower than the peaks associated with the 1970s.

[FIGURE 8 OMITTED]

A possible explanation for the apparent overstatement of future inflationary pressures by optimally weighted money in the early to mid-1980s is that the recovery from the very severe recession of the early 1980s produced a period of above-trend growth. Hence, given that the monetary component weights are derived from a nominal income relationship, the relatively strong growth in weighted money would be manifested in a relatively strong subsequent growth in the real income component of nominal income and rather less in the growth of prices than would be the case when the economy was exhibiting trend growth. From a monetary policy perspective, however, the leading indicator properties of the optimally weighted aggregate, combined with continually updated forecasts for the real economy, should have provided a reasonable indicator of future inflationary pressures at the time.

Finally, it is interesting to note from Figure 7 that the full-sample model produces a much more moderate increase in the implied weight on [x.sub.4] to that suggested in Figure 6. It must be recognized, however, that the latter represents out-of-sample weights derived over more than a decade. In this context, therefore, the comparability of the general trends (if not the magnitudes) of the weights is remarkable. Furthermore, if this technique were used in a genuine policy-making context, the long-run model would be continually updated to provide sequentially updated recursive weights, rather than using the same long-run model to provide recursive weights up to 11 years ahead. Hence, in reality we would expect there to be less discrepancy through time between these two sets of recursive weights than is evident in the contrast between the weights in Figures 6 and 7.

V. FORECASTING TESTS

In this article we have intimated that by virtue of their construction, empirically weighted monetary aggregates should exhibit better leading indicator properties than simple sum aggregates such as M2, which have component weights fixed at unity through time. Furthermore, the previous analysis has clearly shown that the optimal weight on [x.sub.5] (stock and bond mutual funds) is consistently very low. This, combined with the very rapid growth in [x.sub.5], and consequently M2+, in certain time periods (as is evident from Figure 10), implies that M2+ may have a particularly poor forecasting record with respect to nominal income and inflation.

In contrast, however, the weights for Divisia monetary aggregates are not fixed and can change over time. In addition, it was emphasized previously that the Divisia monetary aggregate is a solution obtained from an optimization and can respond to some aspects of financial innovation. Hence, it is not clear a priori whether the empirical solution to the problem of financial innovation and changing wealth holder preferences proposed herein is superior to the theoretical solution proposed by the Divisia aggregation approach. This is an issue that can only be resolved empirically.

To assess the leading indicator properties of the new empirically weighted aggregate more formally, we now evaluate the relative performance of M2, M2+, Divisia M2, and the empirically weighted aggregate ([w.sub.t] = 0.44[x.sub.1,t] + 0.58[x.sub.23,t] + 0.03[x.sub.4,t] - 0.005[x.sub.5,t]) in the context of an out-of-sample forecasting analysis. We elect to use Divisia M2 due to the problems associated with incorporating risky assets (such as stock and bond mutual funds) in Divisia aggregations and due to the fact that the previous results indicated a very low weighting on the broad M3 assets. It is also the case that M2 was the preferred monetary aggregate for policy purposes during the 1980s and 1990s. Hence, our use of Divisia M2 is logical alongside the simple sum aggregates M2 and M2+.

An obvious variable to use, from a monetary policy perspective, in the context of the out-of-sample forecasting tests is nominal income. However, it could be argued that, as the component weights in the empirically weighted aggregate are derived from the long-run nominal income relationship, this would give the latter aggregate an intrinsic advantage over the other aggregates. Hence, in addition to the use of nominal income, we also conduct the out-of-sample forecasting tests with respect to inflation. This could be argued to offer a fairer and more discriminating test across the four alternative monetary aggregates and is consistent with the trend toward inflation targeting witnessed in many countries since the early 1990s.

With respect to both the nominal income and inflation forecasting tests, we use a modified version of the general approach adopted by Stock and Watson (1999) in their inflation forecasting tests. With respect to inflation, therefore, we use the forecasting model

[[pi].sup.k.sub.t+k] = a + [q.summation over i = 1] [b.sub.i][[pi].sup.k.sub.t-i] + [r.summation over i=1] [c.sub.i][x.sup.k.sub.t-i] + [e.sub.t+k]

where [[pi].sup.k.sub.t] (4/k)([p.sub.t] - [p.sub.t-k]) is k-period CPI inflation and [x.sup.k.sub.t] is a similarly defined growth rate of the indicator variable, which is either M2, M2 +, Divisia M2 or w. In the case of the nominal income forecasting test, the dependent variable is k-period nominal income growth. This modifies the Stock and Watson approach by using k-period growth rates as regressors rather than one-period rates. The lag lengths were set at q = r = 4, for k = 4, 6, 8, 12.

Although it could be argued that the recursive long-run weights may be more appropriate for w, rather than the fixed full-sample long-run weights, the initial volatility in the recursive weights combined with sample size constraints means that this was not feasible. It is important to note, however, that the use of fixed rather than variable weights may actually understate the true leading indicator properties of the empirically weighted aggregates, as the weights will not necessarily reflect the "optimal weights" (in the context of the nominal income or inflation relationship) at all points in time. In a real-time policy-making context, however, the long-run relationship and the implied optimal weights could be continually updated, prior to the growth rate of [w.sub.t] being used in a forward-looking inflation forecasting exercise.

In Table 1 we provide details of the out-of-sample forecasting tests for nominal income over the maximum possible sample period, 1991-2001. It is clear from this table that the forecasting accuracy of w is clearly superior to that of Divisia M2 and M2+ over all the forecast horizons. Somewhat surprisingly, the empirically weighted aggregate is outperformed by simple sum M2. This result is in line with our previous results, however, in the sense that despite the substitutions away from M2 assets and into stock and bond mutual funds during the 1990s, the weights on the former dominate the nominal income relationship. The M2 component assets would therefore be expected to exhibit a strong relationship with respect to nominal income, although the simple sum M2 weights would differ from the "optimal" weights implied by the empirically weighted aggregate. Hence, it is interesting to note that, with the exception of the 12-period horizon, the difference in the forecast accuracy of M2 and w is fairly marginal, and the superior performance of simple sum M2 may reflect at least in part the use of fixed rather than variable ("optimal") weights for the empirically weighted aggregate.

The fact that M2+ has by far the worst forecasting record of any of the alternative monetary aggregates tends to confirm the evidence provided previously that notwithstanding the dramatic substitution out of M2 assets and into stock and bond mutual funds during the 1990s, there was no rationale for redefining M2 to include these assets.

If we refer to Table 2, however, it is clear that the empirically weighted monetary aggregate w outperforms all of the other aggregates, in terms of out-of-sample inflation forecasts, at all forecast horizons. Furthermore, this superior performance is particularly evident at the longer forecasting horizons. Whereas the forecasting accuracy of the other aggregates tends to deteriorate at longer forecasting horizons, the accuracy of w actually improves between k = 8 and k = 12. Hence, at the 12-quarter forecasting horizon the RMSE (x [10.sup.6]) of the empirically weighted aggregate is only 18,090, in contrast to figures of 33,261, 28,458, and 33,384 for M2, M2+, and Divisia M2, respectively.

The relatively poor performance of Divisia M2, in both the nominal income and inflation forecasting tests, suggests that the Divisia aggregate is not fully able to capture all financial innovations and changes in wealth holders' preferences. This may be because these changes are not fully reflected in relative interest rates and hence user costs (rental prices). It may also reflect a problem with the way Divisia aggregates, such as the Divisia M2 aggregate produced by the Federal Reserve Bank of St. Louis and used in this study, are typically constructed. Specifically, the benchmark rate of interest is generally taken to be a corporate bond yield or a long-term government bond yield. Arguably, given the extent of substitutions between M2 assets and stock and bond mutual funds during the 1990s, the return on the latter assets would be a more appropriate benchmark rate to be used in Divisia aggregation. Furthermore, when noncapital certain (risky) assets are used to generate benchmark rates of return, these should be risk-adjusted prior to the generation of the component asset rental prices (see Barnett et al. 1997). This is somewhat problematic, however, given the problem of the "equity premium puzzle" alluded to previously. Hence, such risk adjustment of benchmark returns is rarely conducted in practice, although Drake et al. (1998) demonstrate that the empirical performance of Divisia aggregates tends to improve if such risk adjustment is carried out.

Clearly, the issues discussed above do not provide a rationale for the fact that Divisia M2 is generally outperformed by simple sum M2 in both the out-of-sample nominal income and inflation forecasting tests. The answer to this puzzle might be somewhat more prosaic, however. If we examine the implied recursive "optimal" weights for the M2 component assets evident in Figure 6, for example, it is clear that for much of the 1990s the weight on the broader [x.sub.23] assets, although lower than that on the [x.sub.1] assets, is nevertheless relatively large. In the early 1990s, for example, the [x.sub.1] assets tended to have a weight slightly below 0.6, whereas the [x.sub.23] assets correspondingly exhibited a weight just above 0.4. In the context of Divisia aggregation, however, the [x.sub.1] assets are non-interest-bearing and would correspondingly be accorded a high rental price and hence a relatively large weight in the aggregate. Conversely, many of the [x.sub.23] assets will earn rates of return close to the benchmark return and will have very low rental prices. In turn, these assets will be accorded relatively little weight in the monetary aggregate.

A consequence of the Divisia methodology, therefore, is that the weights on the various component assets may deviate substantially from the "optimal" weights implied by the empirical nominal income relationship. In turn, it is also possible that in some periods, the fixed and equal weights of unity imposed under simple sum aggregation may actually be a closer approximation to these "optimal" weights, than the weights used in the Divisia M2 aggregation. This may provide a partial explanation for the mixed empirical evidence regarding the out of sample performance of Divisia vis-a-vis simple sum aggregates. Barnett et al. (1984), for example, contrast simple sum and Divisia aggregates in the context of out-of-sample forecasts of nominal GDP using U.S. data. They find that, although Divisia aggregates generally produced forecasts with lower RMSEs, the simple sum aggregates tended to exhibit lower average forecast errors. The empirical performance of Divisia aggregates has also been relatively poor in Canada (see Longworth and Atta-Mensah 1995, for example).

VI. SUMMARY AND CONCLUSIONS

This article uses an innovative approach to long-run modeling to develop new empirically weighted monetary aggregates for the United States. The empirical results shed important new light on two periods of severe monetary instability in the United States, the "missing money puzzles" of the early/mid-1970s and the early/mid-1990s. With respect to the former Goldfeld missing MI episode, the initial success of the Goldfeld partial adjustment type money demand function is easily explained by the dominant optimal weight associated with M1 balances and the relative stability of the optimal weights prior to 1973. Similarly, the subsequent period of money demand instability can be rationalized in terms of the significant decline in the implied empirical weight associated with M1 balances ([x.sub.1]) and the corresponding increase in the empirical weight accorded to M2 balances, as reflected in the implied recursive weights on [x.sub.2].

Turning now to the missing M2 episode of the early 1990s, this was a further period of money demand instability in respect of an aggregate that had previously appeared to be highly stable and valuable in a policy context. Again, the deterioration in the performance of the aggregate has been attributed to financial innovation and changes in wealth holder preferences, this time associated with the low-interest-rate environment of the early 1990s and the rapid growth of stock and bond mutual funds. Although there clearly was a considerable substitution away from broad M2 assets ([x.sub.23]) and into stock and bond mutual funds ([x.sub.5]) in the early to mid-1990s, our results clearly show that this shift had little implication with respect to future nominal spending (income), as evidenced by the low and relatively stable empirical weight accorded to [x.sub.5] in both Figures 6 and 7. Hence, there is little rationale for broadening the M2 aggregate to include stock and bond mutual funds, as in M2+. Rather, the recursively estimated weights evident in Figures 6 and 7 indicate that the fundamental problem (in respect of money demand instability) with the M2 aggregate after the early/mid-1990s relates to the constant and equal weights applied to the components of the simple sum M2 aggregate. Our results suggest that the optimal weight on [x.sub.1] was increasing in the early/mid-1990s while the optimal weight on [x.sub.23] was decreasing.

[FIGURES 6-7 OMITTED]

From a policy perspective, the difficulty with a period such as the missing M2 episode is that the official simple sum aggregate is likely to provide extremely noisy signals in respect of policy variables, such as nominal income and inflation. For example, the M2 aggregate itself would exhibit a sharp reduction in growth associated with the switch into stock and bond mutual funds, whereas the redefined M2+ type aggregate would exhibit much more rapid growth, because it includes stock and bond mutual funds. Hence, the fundamental problem at a time of significant financial innovation is that the relationship between key monetary aggregates (assets) and the economy is clearly changing, but it is not apparent at the time how the relationship is changing. This is graphically illustrated in Figure 10 by the misleading inflationary signals provided by M2+ throughout the 1990s. Furthermore, although the FOMC downgraded the status of M2 as a policy variable after 1993, our evidence suggests that M2 continued to perform relatively well (at least in a nominal income forecasting context) throughout the 1990s. This suggests that the downgrading was largely associated with the real-time uncertainty (low signal to noise ratio) associated with the monetary signals being provided to policy makers by the M2 aggregate.

[FIGURE 10 OMITTED]

The results presented here, however, suggest that the application of the PSS technique can produce an empirically weighted monetary aggregate that can help policy makers interpret the often confusing signals coming from the growth of monetary aggregates and their components. Furthermore, our results support the preeminence of the M2 component assets in the United States and suggest that there is currently no strong rationale for shifting to an aggregate, such as M3 or M2+. However, the fundamental message of this article is that financial innovations and changes in preferences do occur through time. Hence, it is imperative that monetary aggregates are constructed using the appropriate empiriccally determined monetary component weights and that these are monitored and updated on an ongoing basis to ensure that the aggregate continues to provide valuable forward-looking information with respect to the formulation of monetary policy.

The importance of using empirically determined monetary component weights is illustrated very clearly by the superior out-of-sample inflation forecasting performance of [w.sub.t] relative to M2, M2+, and Divisia M2. Furthermore, this superior forecasting performance is particularly evident at the longer forecasting horizons, which are arguably more important to central banks/policy makers in an inflationary targeting context.

Finally, an important potential issue for future research is to recognize that the underlying relationship between monetary components and policy variables may be nonlinear rather than linear, as assumed in this article. Hence, once the application of cointegration analysis to nonlinear relationships is adequately developed, this is a possibility that could be tested for, and if appropriate, the technique of empirically weighted monetary aggregates could be applied in an appropriately specified nonlinear model.

APPENDIX A

Goldfeld Model for M1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Sample period 1959:2-1972:4

Goldfeld Model for Composite M2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Sample period: 1967:1-1972:4.

APPENDIX B

Model Containing [x.sub.5]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Sample period : 1960 : 2-1972:4,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

APPENDIX C

Model Estimated Over the Full Sample Period

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Sample period : 1960 : 2-2001 : 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ABBREVIATIONS

GDP: Gross Domestic Product

MMMF: Money Market Mutual Fund

ECM: Error Correction Model

TABLE 1
Nominal Income Growth Forecasts
RMSE (x [10.sup.6]): 1991-2001

K M2 M2+ w Divisia M2

 4 16,929 26,457 18,532 23,640
 6 15,388 21,983 16,981 19,836
 8 12,096 17,238 13,920 16,831
12 9,270 19,438 13,016 13,225

TABLE 2
Inflation Forecasts RMSE (x [10.sup.6]):
1991-2001

K M2 M2+ w Divisia M2

 4 17,545 19,147 14,677 23,565
 6 25,857 23,309 21,556 31,053
 8 33,019 27,603 24,336 36,668
12 33,261 28,458 18,090 33,384

(1.) In common with Feldstein and Stock (1996), we do not include interest rates in this vector. In the context of this type of leading-indicator approach, it could be argued that because there may well be a feedback rule from leading indicators of nominal income growth to the short-term interest rates set by central banks (especially in an inflation-targeting environment), the inclusion of interest rates may well bias the derived optimal weights. When the resultant empirically weighted aggregates are used in the context of a money demand analysis, however, it would be appropriate to include an interest rate as an explanatory variable.

REFERENCES

Barnett, W. A. "Economic Monetary Aggregates: An Application of Index Number Theory and Aggregation Theory." Journal of Econometrics, 14, 1980, 11-48.

--. "The Optimal Level of Monetary Aggregation." Journal of Money Credit and Banking, 14, 1982, 687-710.

Barnett, W. A., E. K. Offenbacher, and P. A. Spindt. "'The New Divisia Monetary Aggregates." Journal of Political Economy, 92, 1984, 1049-84.

Barnett, W. A., M. C. Jensen, and Y. Liu. "The CAPM Risk Adjustment for Exact Aggregation over Financial Assets." Macroeconomic Dynamics, 1, 1997. 485-512.

Belongia, M. T., and J. A. Chalfont. "The Changing Empirical Definition of Money: Some Estimates from a Model of the Demand for Money Substitutes." Journal of Political Economy, 97, 1989, 387-97.

Belongia, M. T., and K. A. Chrystal. "An Admissible Monetary Aggregate for the United Kingdom." Review of Economics and Statistics, 73, 1991, 497-503.

Carlson, J. B., D. L. Hoffman, B. D. Keen. and R. H. Rasche. "Results of a Study of Cointegrating Relations Comprised of Broad Monetary Aggregates." Journal of Monetary Economics. 46, 2000, 345-83.

Darin, R.. and R. L. Hetzel. "A Shift Adjusted M2 Indicator for Monetary Policy." Federal Reserve Bank of Richmond Economic Quarterly, 80, 1994, 25-47.

Drake, L., and K. A. Chrystal. "Company-Sector Money Demand: New Evidence on the Existence of a Stable Long-Run Relationship for the United Kingdom." Journal of Money, Credit and Banking, 26, 1994. 479-94.

--. "Personal Sector Money Demand in the UK." Oxford Economic Papers, 49, 1997, 188-206.

Drake, L. and T. C. Mills. "Empirically Weighted Monetary Aggregates for the UK." International Journal of Finance and Economics, 6, 2001, 217-34.

Drake, L., A. Mullineux. and J. Agung. "Incorporating Risky Assets in Divisia Monetary Aggregates." Loughborough University Economic Research Paper No. 15, 1998.

Drake, L., A. R. Fleissig, and A. Mullineux. "Are Risky Assets Substitutes for Monetary Assets." Economic Inquiry, 37, 1999, 510-40.

Drake, L., A. Fleissig, and J. Swofford. "A Semi-Nonparametric Approach to Neoclassical Consumer Theory and the Demand for UK Monetary Assets." Economica, 70, 2003, 99-120.

Duca, J. V. "Should Bond Funds Be Added to M2?" Journal of Banking and Finance. 19. 1995, 131-52.

Estrella. A.. and F. S. Mishkin. "Is There a Role for Monetary Aggregates in the Conduct of Monetary Policy?" Journal of Monetary Economics, 40, 1997. 279-304.

Feldstein, M., and J. H. Stock. "The Use of a Monetary Aggregate to Target Nominal GDP," in Monetary Policy, edited by N. G. Mankiw. Chicago: University of Chicago Press for the NBER, 1994. 7-62.

--. "Measuring Money Growth when Financial Markets are Changing." Journal of Monetary Economics. 37. 1996. 3-27.

Fisher, D., and A. R. Fleissig. "Money Demand in a Dynamic Fourier Expenditure System. Federal Reserve Bank of St. Louis Economic Review, 76, 1994, 117-28.

Fleissig, A. R., and J. Swofford. "Dynamic Asymptotically Ideal Models and Finite Approximation." Journal of Business Economic Statistics, 15, 1997, 482-92.

Goldfeld, S. M. "The Case of the Missing Money." Brookings Papers on Economic Activity. 3, 1976, 683-730.

Longworth, D., and J. Atta-Mensah. "The Canadian Experience with Weighted Monetary Aggregates." Working Paper 95-10, Bank of Canada. 1995.

Mehra, R., and E. Prescott. "The Equity Premium: A Puzzle." Journal of Monetary Economics, 15, 1985, 145-61.

Miyao, R. "Does a Cointegrating M2 Demand Relation Really Exist in the United States?" Journal of Money, Credit and Banking, 28, 1996, 365-80.

Orphanides, A., B. Reid, and D. H. Small. "The Empirical Properties of a Monetary Aggregate that Adds Bond and Stock Mutual Funds to M2." Federal Reserve Bank of St. Louis Review, 76, 1994, 31-52.

Pesaran, B., Y. Shill, and R. J. Smith. "Bounds Testing Approaches to the Analysis of Level Relationships." Journal of Applied Econometrics, 2001, 16, 289-326.

Siklos, P. L., and A. G. Barton. "Monetary Aggregates as Indicators of Economic Activity in Canada: Empirical Evidence." Canadian Journal of Economics, 34(1), 2001.

Stock, J. H., and M. W. Watson. "Forecasting Inflation." Journal of Monetary Economics, 44, 293-335.

LEIGH DRAKE AND TERENCE C. MILLS *

* Two anonymous referees made helpful suggestions on the revision of this article.

Drake." Professor, Nottingham University Business School, Jubilee Campus, Nottingham, NG8 1BB, UK. Phone 44 (0) 1159 846 7415, Fax 44 (0) 115 846 6667, E-mail leigh.drake@nottingham.ac.uk

Mills: Professor, Department of Economics, Loughborough University, Loughborough, LE11 3TU, UK.