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:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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.
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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.