1. Background

The theoretical framework underlying velocity models is well known and is exhaustively surveyed elsewhere (e.g. Laidler (1985)). Accordingly, only the main features will be highlighted. Specifically, velocity models can be interpreted as being derived from a stable money-demand function, where the demand for money is written as:

where M refers to the monetary aggregate under consideration, Y is a nominal scale variable, and $\overline{\mathrm{X}}$ is a vector referring to all other factors, such as interest rates, which might influence money demand.

The choices of which specific variables to use to represent these theoretical quantities are interdependent. For example, if the demand for M1, a relatively narrow definition of money, is under consideration, then since an important rationale for holding M1 lies in its ability to satisfy transactions needs, GNP might be the appropriate scale variable. ^2/ Alternatively, if a broader definition of money is the focus of analysis and if, as a result, portfolio allocation decisions across differing forms of financial wealth are considered to be important, then wealth might be a more appropriate scale variable. Since it has received the most attention in the empirical literature, and given its traditionally important role as an intermediate target variable for nominal GNP and inflation, the focus of this appendix will initially be on M1. The demand for other monetary aggregates will be analyzed as appropriate.

Given a number of technical conditions, equation (1) can be rewritten as:

whence:

where V is velocity. The importance of this equation lies in the fact that if V is stable over time then money will track changes in the scale variable. ^3/ In particular, M1 would then track nominal GNP, making M1 a useful intermediate-target variable.

2. Recent developments

Chart 1 characterizes the behavior of M1 velocity, measured as the ratio of nominal GNP to M1. For many years, even though it was not constant, velocity tended to move in a relatively predictable fashion—for the period 1946 through 1981, velocity grew at an annual average rate of about 3.6 percent per annum. ^4/ However, around 1981, a break occurred and since that time M1 velocity has tended to decline and has been unstable.

United States: Money Velocity

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

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United States: Money Velocity

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

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United States: Money Velocity

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

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A large and growing body of literature has been devoted to explaining this shift in the behavior of velocity. Loosely, the explanations that have been advanced can be classified by whether they assume the shift is due to a misspecification of the velocity model, to underlying structural changes, and/or to cyclical or volatility factors. To elaborate on the distinction between structural and cyclical/volatility factors, refer back to equation (2). In essence, structural changes would imply that the functional form g(·) has altered while cyclical/volatility changes imply unusual variability in the arguments of that function. To the extent that either of these two factors prove to be important, what is needed is a more comprehensive approach to estimating money demand.

An important question that arises in examining the possibility of model misspecification is that of the appropriate choice of monetary aggregate in terms of which the velocity model should be defined. At issue is whether M1 velocity models alone broke down in the early 1980s.

In this connection, the Federal Open Market Committee (FOMC) has argued that M1 is more affected than are the broader monetary aggregates by the continuing process of deregulation since M1 contains a disproportionately large volume of interest-earning accounts that serve not only as transactions balances but also as savings vehicles and yet is not sufficiently comprehensive to internalize shifts of funds between liquid assets motivated by rate of return considerations. ^5/ The relevant interest-bearing accounts in M1 are the NOW and Super NOW (SNOW) accounts which were introduced nationwide in the 1980s. The FOMC pointed out that since interest rates on these accounts have tended to be relatively sticky, large flows have occurred between these and other savings vehicles in response to small changes in market interest rates. As a result, the behavior of M1 velocity became volatile.

This might suggest that a broader monetary aggregate be substituted for M1 in the velocity model. However, the broader the monetary aggregate under consideration the less will be the control of the Federal Reserve over that aggregate. Alternatively, on the grounds that NOW and SNOW accounts may be held for savings purposes, thereby weakening the transactions based link between M1 and nominal output and prices, use of a definition of money which excludes these accounts might be indicated. ^6/

In his time-series analysis of velocity behavior, Rasche considers the empirical implications of using monetary measures which exclude NOW-type accounts (Rasche (1987)). He finds that velocity shifts similar to those in M1 occur in currency velocity measures, adjusted monetary base velocity measures, and to a lesser extent in M1A velocity measures. ^7/ Stone and Thornton arrive at a similar conclusion concerning the role of M1A. Against this, Mascaro, in his comment on Rasche’s work, argues that, in terms of Rasche’s own results, M1A appears clearly superior to M1 yielding a more stable measure of velocity (Mascaro (1987)).

Mascaro’s strong support of the M1A approach is based in part on results reported in earlier research work. In particular, Darby, Mascaro, and Marlow (1987) find that M1A has been superior to M1 and M2 as an indicator of the prospective pattern of real growth since 1982. Further, in their analysis of the behavior of quarterly inflation, they argue that M1A remains a superior indicator while M1 is no longer useful. Finally, contrary to Rasche, they conclude that the changed behavior of M1 velocity is traceable to the other checkable deposit (OCD) component of M1 demand.

Given the attention it has received, a few brief observations on some aspects of their work are in order. First, in their analysis of the indicator properties of the various monetary aggregates, it is not entirely clear why any monetary aggregate in and of itself would be a particularly good indicator of the course of real GNP. It is more conventional to find monetary aggregates being targeted on nominal GNP. As noted above, the importance of the velocity equation ^8/ derives from the theoretical observation that a monetary aggregate would track nominal GNP if its velocity is stable over time. Second, when looking at their regression equations “explaining” real GNP growth, it is noteworthy that, with few exceptions, the coefficients of the monetary aggregates tend to be insignificant at the 5 percent level. In particular, the coefficients on M1A are insignificant. It would be interesting if the out-of-sample forecasts had included results based on an equation estimated without any monetary regressor.

Third, the fact that the M1 equations in both inflation and real growth exhibit evidence of structural shifts does not necessarily invalidate the use of M1 as an intermediate target variable. In fact Rasche argues that it is still possible to find a parsimonious parameterization of the change in money demand which produces a function that is stable in all other respects and which validates velocity and aggregate money demand functions as useful macroeconomic concepts. ^9/

Fourth, the supposed superior performance of M1A over such a short period as the 1980s could be amenable to alternative explanations. ^10/ For example, it might be due to two counteracting effects. ^11/ At any rate, the shifts in velocity, while they can be parameterized by the judicious use of dummy variables, may still be fundamentally unexplained. Therefore, one cannot rule out the possibility that the superior performance of M1A is statistically spurious. In this connection, it is worth noting that many of the noninterest bearing deposits in M1A are held by businesses under compensating balance arrangements. It is not clear that this important component of M1A can be rationalized in terms of a transactions demand motive. ^12/

Fifth, and of relevance for the empirical work later, the regressions in the Darby et al. paper were run in levels rather and first differences. ^13/

Trends in the velocity of M1A and M2, respectively, are presented in Chart 1. The chart reinforces some of the observations that have already been made concerning alternative velocity measures. While it exhibits no obvious trend, M2 velocity is clearly volatile—though this volatility could be greatly reduced in a fully articulated demand for money equation. Concerning M1A velocity, as argued by Darby et al., unlike M1 velocity, it did not shift in the early 1980s. However, when one considers its most recent behavior, it seems to be exhibiting a pattern not unlike that of M1 velocity. Further, its most recent trend is not unlike the pre-1980 trend in M1 velocity. This possibility was foreseen by Darby et al ^14/ and is consistent with their view that M1A has inherited the transactions demand role that M1 had prior to the 1980s. ^15/

Granger-Causality tests provide some further information on the potential targeting properties of the alternative monetary aggregates within the context of simple velocity models. ^16/ The results for whether any monetary aggregate Granger-causes nominal GNP or inflation (measured by the GNP deflator) are presented in Tables 1 and 2, respectively. The results are based on quarterly data and are presented for two periods, specifically 1960 to 1980 and 1960 to 1987, so as to help evaluate the impact of the 1980s on the underlying relations. ^17/ Finally, the tests are conducted in both level and first difference terms. The latter tests are most likely of greater relevance, because, for the reported F-tests to be meaningful, it is necessary that autocorrelation be eliminated from the regressions. ^18/

Table 1.

Nominal GNP and Money. Granger–Causality Tests.

Notes:

^*designates significance at the 1 percent level.

^**designates significance at the 5 percent level.

Table 1.

Nominal GNP and Money. Granger–Causality Tests.

		Number of Lags in Distributed Lag Part	1960-1980	1960-1987
MIA
	Levels	4	F(4,75) = 3.52	F(4,103) = 4.88
		8	F(8,67) = 4.51**	F(8,95) = 2.93
First differences		4	F(4,74) = 5.00	F(4,102) = 4.66
		8	F(8.66) = 4.46**	F(8,94) = 2.78
Ml
	Levels	4	F(4,75) = 5.17	F(4,103) = 1.92
		8	F(8,67) = 4.13**	F(8,95) = 1.37
First differences		4	F(4,74) = 5.92**	F(4,102) = 4.62
		8	F(8,66) = 4.26**	F(8,94) = 2.67
M2
	Levels	4	F(4,75) = 7.62**	F(4,103) = 2.89
		8	F(8,67) = 5.88**	F(8,95) = 2.19
First differences		4	F(4,74) = 7.67**	F(4,102) = 7.23**
		8	F(8,66) = 8.89**	F(8,94) = 3.97**
M3
	Levels	4	F(4,75) = 7.83**	F(4,103) = 0.96
		8	F(8,67) = 6.20*	F(8,95) = 1.29
First differences		4	F(4,74) = 9.53**	F(4,102) = 5.80**
		8	F(8,66) = 8.06*	F(8,94) = 3.10**

Notes:

^*designates significance at the 1 percent level.

^**designates significance at the 5 percent level.

View Table

Table 1.

Nominal GNP and Money. Granger–Causality Tests.

		Number of Lags in Distributed Lag Part	1960-1980	1960-1987
MIA
	Levels	4	F(4,75) = 3.52	F(4,103) = 4.88
		8	F(8,67) = 4.51**	F(8,95) = 2.93
First differences		4	F(4,74) = 5.00	F(4,102) = 4.66
		8	F(8.66) = 4.46**	F(8,94) = 2.78
Ml
	Levels	4	F(4,75) = 5.17	F(4,103) = 1.92
		8	F(8,67) = 4.13**	F(8,95) = 1.37
First differences		4	F(4,74) = 5.92**	F(4,102) = 4.62
		8	F(8,66) = 4.26**	F(8,94) = 2.67
M2
	Levels	4	F(4,75) = 7.62**	F(4,103) = 2.89
		8	F(8,67) = 5.88**	F(8,95) = 2.19
First differences		4	F(4,74) = 7.67**	F(4,102) = 7.23**
		8	F(8,66) = 8.89**	F(8,94) = 3.97**
M3
	Levels	4	F(4,75) = 7.83**	F(4,103) = 0.96
		8	F(8,67) = 6.20*	F(8,95) = 1.29
First differences		4	F(4,74) = 9.53**	F(4,102) = 5.80**
		8	F(8,66) = 8.06*	F(8,94) = 3.10**

Notes:

^*designates significance at the 1 percent level.

^**designates significance at the 5 percent level.

Table 2.

Inflation and Money. Granger-Causality Tests.

Notes:

Inflation is measured by changes in the GNP deflator.

^**designates significance at the 5 percent level.

Table 2.

Inflation and Money. Granger-Causality Tests.

		Number of Lags in Distributed Lag Part	1960-1980	1960-1987
MIA
	Levels	4	F(4,75) = 1.45	F(4,103) = 6.47**
		8	F(8,67) = 1.15	F(8,95) = 3.77**
First differences		4	F(4,74) = 1.81	F(4,102) = 4.22
		8	F(8,66) = 1.57	(F(8.94) = 3.03**
M1
	Levels	4	F(4,75) = 2.76	F(4,103) = 1.31
		8	F(8,67) = 1.93	F(8,95) = 1.30
First differences		4	F(4,74) = 3.03	F(4,102) = 1.33
		8	F(8,66) = 2.78	F(8,94) = 1.60
M2
	Levels	4	F(4,75) = 3.62	F(4,103) = 1.06
		8	F(8,67) = 2.37	F(8,95) = 1.29
First differences		4	F(4,74) = 0.99	F(4,102) = 0.16
		8	F(8,66) = 2.44	F(8,94) = 1.73
M3
	Levels	4	F(4,75) = 3.50	F(4,103) = 0.63
		8	F(8,67) = 2.45	F(8,95) = 1.72
First differences		4	F(4,74) = 2.70	F(4,102) = 0.04
		8	F(8,66) = 3.49**	F(8,94) = 2.96

Notes:

Inflation is measured by changes in the GNP deflator.

^**designates significance at the 5 percent level.

View Table

Table 2.

Inflation and Money. Granger-Causality Tests.

		Number of Lags in Distributed Lag Part	1960-1980	1960-1987
MIA
	Levels	4	F(4,75) = 1.45	F(4,103) = 6.47**
		8	F(8,67) = 1.15	F(8,95) = 3.77**
First differences		4	F(4,74) = 1.81	F(4,102) = 4.22
		8	F(8,66) = 1.57	(F(8.94) = 3.03**
M1
	Levels	4	F(4,75) = 2.76	F(4,103) = 1.31
		8	F(8,67) = 1.93	F(8,95) = 1.30
First differences		4	F(4,74) = 3.03	F(4,102) = 1.33
		8	F(8,66) = 2.78	F(8,94) = 1.60
M2
	Levels	4	F(4,75) = 3.62	F(4,103) = 1.06
		8	F(8,67) = 2.37	F(8,95) = 1.29
First differences		4	F(4,74) = 0.99	F(4,102) = 0.16
		8	F(8,66) = 2.44	F(8,94) = 1.73
M3
	Levels	4	F(4,75) = 3.50	F(4,103) = 0.63
		8	F(8,67) = 2.45	F(8,95) = 1.72
First differences		4	F(4,74) = 2.70	F(4,102) = 0.04
		8	F(8,66) = 3.49**	F(8,94) = 2.96

Notes:

Inflation is measured by changes in the GNP deflator.

^**designates significance at the 5 percent level.

A number of tentative conclusions emerge. First, confirming the occurrence of a shift in velocity in the early 1980s, money is more likely to Granger-cause nominal GNP in the sample period 1960 to 1980 than in the sample period 1960 to 1987, irrespective of the monetary aggregate under consideration. In particular, Granger-causality breaks down in the case of M1 in the full sample period.

Second, concentrating on the first-difference results for nominal GNP for the full sample period, both M2 and M3 Granger-cause nominal GNP, with the significance levels being somewhat higher in the case of M2. M1A does not Granger-cause nominal GNP at the chosen significance levels.

Third, concerning inflation, the results in Table 2, when compared with those in Table 1, suggest that monetary aggregates are less likely to Granger-cause inflation than nominal GNP. However, although it does not Granger-cause inflation in the subperiod 1960 to 1980, M1A does Granger-cause inflation when the full sample period is under consideration. This is somewhat puzzling suggesting that some factor unique to the early 1980s may be generating the result. ^19/

A final comment on the choice of an appropriate monetary aggregate—there is an alternative approach which relies on aggregation and index number theory to construct monetary aggregates. In this connection, much attention has focused on the Divisia approach which relies on the user costs of basic monetary assets to calculate the contributions of those basic assets to a constructed consistent monetary aggregate. The user costs are proportional to the difference between the yield on a “benchmark” asset, such as human capital, and the relevant component’s own yield. ^20/ On balance, this approach appears to make more of a difference the broader the monetary aggregate under consideration. A potential problem arises, however, when one considers how such a variable would be controlled by the Federal Reserve. The value of the Divisia approach may, therefore, lie more in indicating that an important source of instability in velocity equations derives from portfolio shifts within the monetary aggregates precipitated by shifts in relative rates of return than in providing an alternative intermediate-variable target.

In the remainder of this section, some alternative explanations of the velocity shift are considered. Still under the heading of model misspecification is an important issue already alluded to, namely, that of the appropriate choice of a scale variable. The common use of nominal GNP as the variable of choice has been rationalized in terms of its being a useful proxy variable for the level of all transactions. With large inventory changes and with the growth in U.S. current account deficits, some have suggested using final sales to domestic purchasers instead. Alternatively, personal income might be preferable. Rasche (1987) rejected these solutions on the grounds that the shift in M1 velocity occurs irrespective of the scale variable selected. Likewise, Rasche finds that substituting wealth or net worth for GNP does not remove the break in the velocity trend.

A potential structural explanation of the shift in M1 velocity concerns the 1979 changes in the Federal Reserve’s operating procedures. Although money-stock targeting had been the strategic focus of monetary policy since 1975, the tactics or operating procedures whereby that policy was effected were changed in 1979 from a federal-funds-rate operating strategy to a reserves-oriented strategy. As Rasche points out, the difficulty with using this procedural shift to explain the velocity breakdown lies in the fact that, after accounting for the nationwide introduction of NOW accounts in January 1981, the shift in M1 velocity takes place in late 1981, later than the change in operating procedures.

Another structural explanation could be that an increase in interest rate volatility might have precipitated the shift in velocity. Again, Rasche argues out that the observed increase in interest rate velocity, since it occurred in late 1979 and early 1980, antedates the velocity shift. ^21/

As far as cyclical explanations are concerned, Stone and Thornton (1987) identify the possibility that exogenous changes in the supply of M1 could through lags induce cyclical swings in measured velocity. ^22/ However, as they themselves point out, such an explanation is unlikely to explain a sustained shift in velocity. At a more general level, however, this point is important since it cautions against automatically assuming that a demand rather than a supply function underlies the velocity relationship.

As an alternative cyclical explanation, Stone and Thornton discuss the role of changes in anticipated inflation in explaining shifts in velocity. By this view, inflation is a proxy for the opportunity cost of holding money. When inflation and presumably inflationary expectations are declining, the demand for money should rise with a concomitant decline being observed in velocity. Stone and Thornton are not persuaded that this variable has a significant role to play in explaining the shift in velocity. ^23/ However, others, notably Judd (1983) and Baba, Hendry, and Starr (1987) assign inflation an important role.

A range of explanations have, therefore, been advanced to explain the velocity shift of the early 1980s. While there is a continuing debate on precisely how to resolve the velocity puzzle, it is clear that velocity equations such as equation (2) are overly simplistic. It is therefore important to consider the underlying money demand function in which the velocity models are implicitly embedded.

a. Scale variable

While many variables have been used in the role of a scale variable, for the purposes of this paper the choice can be reduced to deciding between a wealth variable and an income variable and, among income variables, between using real GNP or some other variable such as real consumption. ^32/

On the matter of wealth versus income, the issue is whether the motivation for including the scale variable is based on the inventory-theoretic transactions demand approach to money demand, or on a Tobinesque portfolio balance approach. In this paper it is presumed that M1 remains sufficiently narrow to warrant the use of an income variable. ^33/ As for the choice of real GNP as opposed to real consumption, the real GNP variable performed significantly better in the regression equations for M1. The variable is designated as Y.

b. Opportunity cost variables

The choice of appropriate opportunity cost variables raises a number of distinct issues. First, there is the question of whether a short-term or a long-term interest rate should be included as a measure of the opportunity cost of holding money in terms of other financial assets. However, this choice appears not to be crucial to the empirical results, undoubtedly because of the tendency for interest rates on a wide variety of assets to move together. ^34/ In this paper, the three-month Treasury bill rate, designated as r, is included as the opportunity cost variable.

Hamberger (1966) included the dividend-price ratio as a proxy variable for the rate of return on equities and thus on physical capital. While the reliability of his empirical results has subsequently been questioned, nonetheless, his approach raises the possibility that the range of assets competing with M1 could be quite broad. ^35/ In this paper, the expected rate of inflation, designated ${\dot{\mathrm{p}}}^{\mathrm{e}}$, is introduced to perform this broader role. After trying various formulations of how this variable might be specified, a simple measure based on the concurrent difference of the logarithms of the index of the quarterly implicit GNP deflator was selected.

With the arrival of NOW and Super NOW accounts, the issue of the own-rate of return on M1 has received more attention. How to calculate this return accurately is, however, problematical, since returns can be paid in forms other than explicit interest rates. One approach has been to assume that the interest rate paid on demand deposits varies with the charges levied on checking accounts. ^36/ Another approach takes as its starting point the view that the banking system is competitive and that therefore the yield on noninterest bearing demand deposits will equal the return the bank gets when investing the deposits. ^37/ In the context of this paper, an attempt was made to accommodate the own-rate of return by adjusting the Treasury bill interest rate for the NOW account interest rate weighted by the share of NOW accounts in M1. The variable, however, did not perform well. Given the success others have experienced with more sophisticated forms of this variable, this is most likely a reflection of the relatively crude manner in which the variable was calculated. ^38/

c. Other variables

Two variables should be mentioned under this heading. First, an attempt is made to accommodate the potential impact the increased interest-rate volatility of the 1980s might have had on money demand. Note that it is difficult to rationalize the inclusion of a volatility variable into a model with complete capital markets—in such a world, investors can always hedge against risk. However, in a world with capital market imperfections, this type of variable may be theoretically justifiable. The expectation is that the coefficient on this variable would be positive—an increase in volatility would entice agents to remain more liquid than they would otherwise be. The actual variable has been constructed by taking the standard deviation of the two-period moving average of the ten-year government bond rate. ^39/ The variable is designated as VAR.

Finally, following Gordon (1984), whose work supports the conjecture that the credit controls sharply reduced the money supply in 1980:Q2 and contributed to a roughly equivalent rebound in money supply in 1980:Q3, a dummy variable, D, is included with the values -1, 1, in those two quarters, respectively, and zero elsewhere.

The following baseline “error-correction” model for M1 demand using quarterly data (1960:2 to 1988:1) was developed.

LM Test for Autocorrelation ^40/

η₁(1,100) = 0.80 η₂(1,100) = 1.57 η₃(4,97) = 0.83

LM Test for Autocorrelated Squared Residuals. ^41/

F(4,96) = 0.34 F(4,96) Critical Value = 2.47

Tests for Heteroskedasticity. ^42/

F(20,79) = 0.83 F(20,79) critical value = 1.70

The t-ratios are in parentheses. This equation satisfies a range of diagnostic tests. Particularly noteworthy is the absence of autocorrelation. To interpret this equation, consider the properties of the equilibrium solution. Setting the terms in differences equal to zero, the derived equilibrium solution is:

The income elasticity of M1 demand is 0.65. (As can be seen from equation (4), 0.65 is the value on Y/P in the error-correction term.

This value was not imposed but was suggested by the ratio of the estimated coefficients of a prior regression equation in which ln(M1/P)_-2 and ln(Y/P)_-2 were entered separately.) This is consistent with the view that there should be economies of scale in M1 holdings. The (competing yield) interest elasticity of money is:

E_M1/P,r = -0.038r

where this equals 0.38 at an interest rate of 10 percent per annum. Analogously, the elasticity of real M1 to changes in the expected rate of inflation is 0.37 at an expected annual rate of inflation of 10 percent. ^43/ Finally, at its mean value for the 1980s, the elasticity of real M1 to changes in the volatility variable is 0.128.

Given the concern with the capacity of monetary aggregates to track macroeconomic variables, the real issue is whether the equation resolves the instability issues discussed above. In Chart 2, the one-step residuals (${\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{X}}_{\mathrm{t}}^{\mathrm{\prime }}{\mathrm{\beta }}_{\mathrm{t}}\mathrm{=}{\overline{\mathrm{u}}}_{\mathrm{t}}$ where β_t is the estimated β using data up to and including t) are graphed together with their current standard errors (± 2σ_t). This graph, as might be expected, indicates that a period of instability in the real M1 equation occurred in the early 1980s. In other words, the representative regression equation presented here, though performing well by a number of criteria, does not fully “explain” the velocity decline. ^44/ (However, when Chart 2 is compared with analogous charts of residual behavior for alternative traditional real M1 demand equations, the evidence suggests that the results presented here represent a significant improvement.)

United States: Real M1 Equation: Residuals ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ One step residuals (${\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}}^{\mathrm{\prime }}{\hat{\mathrm{\beta }}}_{\mathrm{t}}\mathrm{=}{\overline{\mathrm{u}}}_{\mathrm{t}}$ where ${\hat{\mathrm{\beta }}}_t$ is the estimated $\hat{\mathrm{\beta }}$ using data up to and including t) bounded by the current standard errors ($\pm 2{\hat{\mathrm{o}}}_{\mathrm{t}}$)

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United States: Real M1 Equation: Residuals ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ One step residuals (${\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}}^{\mathrm{\prime }}{\hat{\mathrm{\beta }}}_{\mathrm{t}}\mathrm{=}{\overline{\mathrm{u}}}_{\mathrm{t}}$ where ${\hat{\mathrm{\beta }}}_t$ is the estimated $\hat{\mathrm{\beta }}$ using data up to and including t) bounded by the current standard errors ($\pm 2{\hat{\mathrm{o}}}_{\mathrm{t}}$)

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United States: Real M1 Equation: Residuals ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ One step residuals (${\mathrm{Y}}_{\mathrm{t}}\mathrm{-}{\mathrm{x}}_{\mathrm{t}}^{\mathrm{\prime }}{\hat{\mathrm{\beta }}}_{\mathrm{t}}\mathrm{=}{\overline{\mathrm{u}}}_{\mathrm{t}}$ where ${\hat{\mathrm{\beta }}}_t$ is the estimated $\hat{\mathrm{\beta }}$ using data up to and including t) bounded by the current standard errors ($\pm 2{\hat{\mathrm{o}}}_{\mathrm{t}}$)

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More interesting is the indication that the regression equation began again to exhibit some signs of instability in more recent years. This impression is confirmed by Chart 3, in which is presented an analysis of one-step ahead forecasts for the period 1986:1 to 1988:1. The bounds around the forecast paths are set for the 5 percent confidence level. In particular, the chart indicates that, in the third quarter of 1987, the equation was overpredicting real M1 demand. Further a χ² test comparing within and post-sample residual variances for parameter constancy rejects the hypothesis of parameter constancy at the 5 percent confidence level. This is a further indication that the ongoing process of financial reform and innovation may be continuing to have an unsettling impact on established empirical relationships.

United States: Real M1 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M1 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M1 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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To explore this issue of parameter constancy further, consider the same equation estimated over the truncated period 1960:2 to 1980:3.

η₁(1,70) = 0.02 η₂(1,70) = 0.15 η₃(4,68) = 0.49

ARCH: F(4,66) = 0.93 F(4,67) Critical Value = 2.51

Heteroskedasticity: F(20,50) = 1.32 F(20,50) Critical Value = 1.78

The obvious differences between this and the previous regression are first that, not surprisingly, the variability variable is less significant and, second, the competing rate of return variable performs poorly.

Some indication of why the competing rate of return variable (r) does not perform so well can be found in Chart 4. This chart presents the recursive least squares coefficient for r_-3 (intuitively, the chart tracks the estimated coefficient for that variable as more observations are added to the regression) which shows that until the late 1970s, the coefficient was not significantly different from zero, but became significantly negative thereafter. ^45/ It is evidence such as this that has led many to conclude that the interest elasticity of demand for real money balances has increased. That result is increasingly being cited as an additional reason for downplaying the potential targetting role of M1. Even if the M1 demand relationship were to be found to be stable, the increased interest elasticity of demand for money may be sufficient to negate the usefulness of M1 as an intermediate target variable. The target band established for M1 would have to be very broad to accommodate the range of shocks that might reasonably be expected to occur. Such a wide bandwould be of little use in the conduct of monetary policy or in communicating the stance of that policy to the public. ^46/

United States: Real M1 Equation: Bounded Recursive Least Squares Coefficient for r_-3^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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United States: Real M1 Equation: Bounded Recursive Least Squares Coefficient for r_-3^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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United States: Real M1 Equation: Bounded Recursive Least Squares Coefficient for r_-3^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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For comparison, equations for real M1A and real M2 were also estimated. Equation (7) presents an illustrative regression equation for M1A, where the modelling strategy pursued was analogous to that underlying equations (5) and (6). The sample period is 1960:3 to 1988:1.

η₁(1,102) = 0.29 η₂(1,102) = 0.01 η₃(4,99) = 0.37

ARCH: F(4,98) = 0.58 F(4,98) Critical Value = 2.46

Heteroskedasticity: F(14,88) = 18.89 F(14,88) Critical Value = 1.81

A technical point should be noted at the outset. Real consumption (C/P) is the regressor of choice for the scale variable after equations were fitted using both GNP and consumption. This may Lend support to the view that M1A is a better gauge of transactions demand than are other monetary aggregates. This point should not be accorded too much weight because the differences between the regressions in the alternative scale variables are not large and because, theoretically, it is difficult to see why demand deposits should be preferred to interest-bearing checking accounts as transactions balances.

As can be seen by the positive coefficient on real M1A lagged two periods, the equation is inconsistent with an “error-correction” model ^47/ Instead, the implied relationship is explosive and it is impossible to solve for a long-run solution. Some insight into the source of the problem can be found in Chart 5 in which the recursive least squares coefficient for the lagged value of real M1A is presented. Following a period in which there was a tendency for its value to be negative, the coefficient became significantly positive in the 1980s.

United States: Bounded Recursive Least Squares Coefficient for Ln(M1A/P)_-2^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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United States: Bounded Recursive Least Squares Coefficient for Ln(M1A/P)_-2^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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United States: Bounded Recursive Least Squares Coefficient for Ln(M1A/P)_-2^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The estimated value of the coefficient of the variable calculated by recursive least squares and bounded by plus or minus two standard errors of that estimated value.

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This outcome supports the point made in the context of the Granger-Causality tests—caution should be exercised in interpreting the superior tracking ability of M1A through 1985/86 since that superior performance could be spurious. This point is reinforced by a consideration of the equation’s forecasting characteristics which are displayed in Chart 6. In recent quarters, the equation has been showing signs of going off-track. ^48/

United States: Real M1A Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M1A Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M1A Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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A representative equation for the demand for real M2 balances is:

η₁(1,98) = 2.06 n₂(1,98) = 2.31 n₃(4,95) = 2.18

ARCH: F(4,94) = 0.96 F(4.94) Critical Value = 2.47

Heteroskedasticity: F(20,78) = 0.73 F(20,78) Critical Value = 1.71

A distinguishing feature of this equation is that the error-correction term suggested by the “testing-down” approach implies an income elasticity of unity. This may reflect the broader nature of M2, weakening the link between M2 and an underlying inventory-theoretic transactions based approach to money demand. In many other respects, this equation is similar to the estimated equation for real M1 demand. In fact, by the criterion of its forecasting ability, this equation outperforms the preceding equations. In particular, Chart 7 shows that the forecasts track actual developments closely through 1986:1 to 1988:1. Further, the equation satisfies tests for parameter constancy over that period. ^49/

United States: Real M2 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M2 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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United States: Real M2 Equation: Forecast Performance ^1/

Citation: IMF Working Papers 1988, 086; 10.5089/9781451953602.001.A001

^1/ The error bars show plus or minus two standard errors of the estimated value for the dependent variable, yielding an approximately 95 percent confidence interval for the one-step forecast.

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However, these results do not necessarily mean that M2 should or could replace M1 as an intermediate-variable target. M2 is a relatively broad aggregate and, as has already been argued, this raises questions concerning the Federal Reserve’s ability satisfactorily to control its path. ^50/ Further, as with other monetary aggregates, the continuing process of deregulation is altering the nature of M2 over time. For example, in 1980 the deposit rates on most instruments were regulated whereas this is no longer the case. ^51/