Broad Money Growth and Inflation in the United States

A U.S. inflation-forecasting model recently developed by the Federal Reserve—the so-called P* relationship—is analyzed. An innovation in that model is the significance of M2 velocity in predicting changes in inflation. However, this paper's empirical analysis indicates that an inflation equation in levels, rather than first differences, is more appropriate and reveals that the significance of Ml velocity is not robust to this alternative specification. Although there is a long-run relationship between M2 and the price level, the output gap in a Phillips curve model captures much of the short-run deviations from this relationship.

Abstract

A U.S. inflation-forecasting model recently developed by the Federal Reserve—the so-called P* relationship—is analyzed. An innovation in that model is the significance of M2 velocity in predicting changes in inflation. However, this paper's empirical analysis indicates that an inflation equation in levels, rather than first differences, is more appropriate and reveals that the significance of Ml velocity is not robust to this alternative specification. Although there is a long-run relationship between M2 and the price level, the output gap in a Phillips curve model captures much of the short-run deviations from this relationship.

Broad Money Growth and Inflation in the United States

IN recent years, the U.S. Federal Reserve System has relied less on monetary aggregates and more on a broad range of economic and financial variables to guide the implementation of monetary policy.1 This trend culminated in 1987 when the Federal Reserve decided to cease specifying an annual growth range for Ml,2 the variable that had been viewed as the most reliable monetary aggregate in indicating changes in nominal income growth, because of instability in empirical money demand equations for this narrow aggregate. More recently, however, the apparent long-run relationship between M2 and the price level, the so-called P* relationship proposed by Hallman, Porter, and Small (1989), has raised interest within the Federal Reserve in this broader aggregate as a guide to monetary policy.

The P* relationship combines two long-standing propositions in economics—the quantity theory of money and the lagged adjustment of prices.3 Specifically, P* is the long-run equilibrium price level defined as Pt*M2tV2*/Qt*, where V2* is the average value of M2 velocity for the period 1955 to 1988. and Qt* is the Federal Reserve's measure of potential output. By definition, the actual price level is given by PtM2t V2t/Qt, where V2t is actual M2 velocity, and Qt is real gross national product (GNP), Price disequilibrium in the system is measured by ptpt*, or equivalently, (v2tv2*) + (q*tqt), where lowercase variables represent the natural logarithms of their uppercase counterparts.4 The empirical analysis of Hallman, Porter, and Small found that deviations of pt from p*t are significant in predicting changes in inflation over a one- to three-year horizon and that (v2tv2*) and (q*tqt) are individually significant with approximately equal coefficients. Because the hypothesis of equal coefficients on the velocity and output gaps could not be rejected, they concluded that in the long run P* ties together the level of broad money and prices.

The P* relationship relies extensively on the statistical property that v2t is mean reverting or, more generally, stationary.5 This paper considers whether the specification of the Hallman-Porter-Small inflation equation, which is in first differences rather than in levels of the inflation rate, is consistent with the stationarity of v2t, and concludes that an inflation equation in levels is the more appropriate specification. Moreover, in the levels equation the addition of the velocity gap to a traditional Phillips curve model based on the output gap does not significantly improve that model's inflation forecasts.

The remainder of the paper is organized as follows: Section 1 provides a characterization of the time-series properties of the variables under consideration. Section II considers the stability of the demand function for M2 over the period 1959 to 1989.6 Section III examines alternative inflation equations and the robustness of Hallman, Porter, and Small's results to these specifications. Section IV presents some conclusions.

I. Time-Series Properties of the Variables

An important preliminary to the specification of the empirical equations for the demand for M2 and inflation is to examine the order of integration of each of the variables to be included in the regressions, and, in particular, to test for the presence of a unit root.7 Previous studies, such as Nelson and Plosser (1982), Schwert (1987), and Perron (1988), generally concluded that most macroeconomic time series are I(l), although some tests found price indices to be stationary only in second differences. These results are confirmed for the data set used in this paper.

The tests used to detect the presence of unit roots in autoregressions of the individual time series are the augmented Dickey-Fuller (ADF) test and three developed by Phillips and Perron: Z(α); Z(tα); and Z3).8 The testing strategy followed is essentially that proposed by Perron (1988), in which a modified F-test, Z2), is used to assess the significance of the drift terms in the initial autoregressions that allow for a time trend. If the null hypothesis of a unit root for a series cannot be rejected by the initial tests, and if the modified F-test indicates the absence of a significant drift term, a second and more powerful set of tests for the presence of a unit root is used. These tests, which are based on autoregressions that include only a constant, are not invariant with respect to a significant drift parameter.

The results based on the first test for the sample period 1959:1 to 1989:4 indicate that m2t and qt are I(1), in common with much of the empirical literature. The results for the remaining variables are reported in Table 1. For inflation, πt, v2t, and the opportunity cost of holding M2 balances, denoted by (rtb3trm2t) (the logarithm of the difference between the after-tax rate of return on three-month Treasury bills and the weighted-average rate of return on M2-only assets after taxes), the results are borderline, because the ADF and Phillips-Perron tests lead to conflicting inferences.9 With respect to v2t, the modified F-test Z2) indicates the absence of a significant drift term. However, a significant drift term is indicated for πt, and (rtb3trm2t). The second set of tests is undertaken for v2t, in which the alternative hypothesis is the mean reversion of this series. The tests again do not provide an unambiguous indication of the presence of a unit root in this series; the null hypothesis of a unit root can be rejected on the basis of the ADF test, but not on the basis of the Z(α) and Z(tα) tests.

Table 1.

Tests for Unit Roots

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Note: For the model with a time trend, the critical value at the 5 percent confidence level for the augmented Dickey-Fuller (ADF) and Z(t) statistics is -3.41; that for Z(α) is -21.8; that for Z3) is 6.25; and that for Z2) is 4.68. For the model with no time trend, the critical value at the 5 percent confidence level for the ADF and Z(tα) statistics is 2.86, and that for the Z(α) statistic is -14.1. See Perron (1988).

Lag length refers to the order of the autoregression used to calculate the ADF statistic. With respect to the Phillips-Perron tests, lag length refers to the length of the Bartlett window used to estimate the variance of the partial sum of squared residuals from the first-order autoregression upon which these tests are based. The estimator is from Newey and West (1987).

To summarize, the hypothesis that m2t and qt, are integrated of order one cannot be rejected. With respect to πt, v2t, and (rtb3trm2t), the ADF and Phillips-Perron tests lead to conflicting inferences, with the series being either I(0) or I(1). Thus, while the ambiguity about the stationarity of v2t is reflected in the time-series properties of the opportunity cost of holding M2 balances, as would be predicted by money demand theory, no firm conclusions about the long-run relationship among m2t, pt, and qt can be drawn from these univariate tests.10

II. Estimation of the Demand for M2

Since the univariate tests do not provide a clear indication about the stationarity of v2t, a dynamic model of the demand for M2 is estimated with an imposed long-run relationship among m2t, pt, and qt to identify the money market's long-run equilibrium. The imposed relationship is the assumed mean reversion of M2 velocity, (v2t − 1v2*), which forms the error-correction term in a short-run equation for Δ(m2tpt).11 The level of output lagged one period is included to test the restriction that the income elasticity of the demand for money is unity. In addition, the short-run model includes the level and changes in the opportunity cost of holding M2 balances. Finally, a dummy variable for the first quarter of 1983 allows for the introduction of money market deposit accounts at the beginning of 1983.

The final estimation results are as follows:12

Δ(m2tpt)=1.781(0.495)+0.546Δ(0.051)(m2t1pt1)+0.127Δqt(0.050)+0.770πt(0.085)
0.601(0.082)πt10.337(0.093)(rtb3trm2t)0.555Δ(0.112)(rtb3t1rm2t1)+0.052(0.016)(v2t1v2*)+0.035d(0.001)
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The heteroscedasticity-consistent standard errors are shown in parentheses under the estimated coefficients. Directly under the regression are the sample period, multiple correlation coefficient, standard error of the regression, and a set of misspecification tests (for autocorrelation, non-normality, and heteroscedasticity of the residuals, and for parameter instability).13

The estimation results indicate that much of the variation in Δ(m2tpt) is explained by its lagged value, the inflation rate, and movements in the opportunity cost of holding M2 balances. Though significant, the error-correction term, (v2t − 1v2*), has relatively little explanatory power, possibly because the transactions services yielded by M2 balances are small relative to their pecuniary yield.14 The mean of the difference between the after-tax rate of return on three-month Treasury bills and the weighted-average rate of return on M2-only assets after taxes for the full sample period is only 40 basis points. Finally, the velocity gap term was entered in first differences, but was not significant in this form; moreover, the lagged level of output was insignificant and thus was omitted from the final estimation.

The overall explanatory power of the equation is similar to the results obtained by Ebrill (1988); Moore, Porter, and Small (1990); Boughton (1991); and Boughton and Tavlas (1991). Moreover, the equation passes a range of misspecification tests. With respect to the residuals, there is no significant evidence of serial correlation, nonnormality, or heteroscedasticity. A Chow test for parameter stability for the subperiods 1959–77 and 1978–89 reveals no indication of a structural break in the equation.

III. Estimation of Inflation Equations

Given the empirical equation for the demand for M2, we turn to the estimation of inflation equations similar to those of Hallman, Porter, and Small (1989). These equations relate inflation to its lagged values, to deviations of output from its potential level, and to deviations of velocity from its mean value. Essentially, the approach is to augment a Phillips curve model of price determination with a measure of departures from long-run money market equilibrium. Thus, this model does not require the equality of coefficients on the output and velocity gap terms. In addition, the change in the relative price of gasoline and oil, Δrpgast, is included in the model to allow for supply shocks.15 This relative price is measured as the difference between the log of the implicit deflator for the consumption of gasoline and oil and the log of the implicit GNP deflator.

The inflation equations are estimated in both first differences and in the levels of the inflation rate to determine the more appropriate specification. One reason to specify the inflation equation in first differences is to ensure that the dependent variable is stationary, given the time-series properties of the inflation rate. However, if m2t, πt, and qt are I(1), then v2t cannot be stationary, since no linear combination of m2t, pt, and qt would exist that is cointegrated in the sense of Engle and Granger (1987), which requires that the individual series or linear combinations thereof have the same order of integration.16 In general, a linear combination of two nonstationary time series with different orders of integration has the same order of integration as the component with the higher order.17 Thus, if the inflation equation is specified in first differences to ensure stationarity of the dependent variable, the velocity gap, (v2tv2*), would also require differencing, and failure to do so could lead to incorrect statistical inferences.18

An alternative reason to specify the inflation equation in first differences is that the inflation rate is stationary but the velocity and output gaps are functions of the inflation rate. For example, the past rate of inflation may persist unless the stance of monetary policy is changed, while monetary policy may aim to offset changes in the inflation rate. If the relationship between broad money growth and inflation is completely summarized by the velocity and output gaps, the restriction of unity on the coefficient of the lagged inflation rate would be warranted. However, if the transmission mechanism of monetary growth is not fully reflected in this specification, and if the Federal Reserve acts to stabilize the inflation rate, the levels equation would be the appropriate specification.

The final estimation results are as follows:

πt=0.001(0.001)+0.348πt1(0..082)+0.554(0.099)[(πt2+πt3+πt4+πi5)/4]+0.37(0.013)(0.67Δrpgast2+0.33Δrpgast4)0.030(0.016)(v2t1v2*)0.038(0.012)(qt1*qt1)
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and

Δπt=0.002(0.004)0.631(0..079)Δπt10.447(0.094)πt20.303(0.132)Δπt30.139(0.102)Δπt4+0.036(0.014)(0.67Δrpgast2+0.33Δrpgast4)0.042(0.012)(v2t1v2*)0.036(0.014)(qt1*qt1)
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With respect to the first-differences equation, the results are similar to those obtained by Hallman, Porter, and Small. Changes in inflation exhibit considerable persistence and appear to be well represented by a fourth-order autoregressive process, although the inflation rate according to this specification exhibits considerable overshooting. The coefficients on the (q* t − 1qt − 1) and (v2t − 1v2*) are significant and approximately equal in magnitude. The velocity gap term was entered in first differences rather than in levels, but was not significant in this form. Finally, the coefficient on Δrpgast is approximately equal to the share of oil consumption in GNP.

The significance of (v2t − 1v2*)and the overshooting of the inflation rate appear sensitive to the specification of the inflation equation, however. In the levels equation, the velocity gap is only marginally significant, while the lagged coefficients of inflation sum to 0.90. The coefficient on Δrpgast is again approximately equal to the share of oil consumption in GNP.

The explanatory power of the two equations is similar, as measured by the standard error of the regression. However, the levels equation has a somewhat better fit. With respect to the first-differences equation, Hallman, Porter, and Small (1989), Cristiano (1989), Pecchenino and Rasche (1990), and Kuttner (1990) obtained comparable standard errors. Both the levels and first differences equations pass the set of misspecification tests. With respect to the residuals, there is no significant indication of serial correlation, nonnormality, or heteroscedasticity. A Chow test for parameter stability for the subperiods 1959–73 and 1974–89 reveals no indication of a structural break in either equation. However, in the first differences equation there is evidence of second-order and fourth-order serial correlation at the 10 percent level.

A possible explanation for the discrepancy between the two equations regarding the significance of (v2t − 1v2*) is an overdifferencing of the inflation rate in the first-differences equation. As Plosser and Schwert (1978) demonstrated, the differencing of the inflation series if it is stationary introduces a moving-average process in the residuals of the regression.19 In which case, the residuals of the equation would be serially correlated and the standard errors of the estimated coefficients could be biased. The direction of the bias, however, is difficult to predict a priori in a multivariate regression.

The J-test and the JA-test from Davidson and MacKinnon (1981) and Fisher and McAleer (1981), respectively, permit formal comparisons of the alternative specifications.20 These artificial nesting models essentially involve including predicted values (^πt − πt-1) from a levels equation as an additional regressor in the first-differences equation and predicted values (Δ^πT + πt − 1) from a first-differences equation in the levels equation. If in these augmented regressions the predicted values of one model are significant but not those of the competing model, the competing model would be rejected. The J-statistic of the predicted values from the levels equation included in the first-differences equation is 1.342, while the JA-statistic is 10.100.21 The J-statistic and JA-statistic of the predicted values of the first-differences equation included in the levels equation are 0.188 and 0.011, respectively. Thus, the JA-test leads to a rejection of the first-differences specification; however, the results of the J-test are inconclusive.22

Although neither specification of the inflation equation is clearly better, several considerations tend to favor the levels equation. First, the standard error of the levels equation is lower than that of the first-differences equation, which is consistent with the weak explanatory power of (^πT − πT-1) in the augmented first-differences equation. Second, there is evidence of second-order and fourth-order serial correlation in the residuals of the first-differences equation at the 10 percent level, indicating that it may be overdifferenced. Third, the velocity gap is significant only in levels in the first-differences equation, while it would be expected to enter in differences if πt were I(1). One interpretation of this result is that the inflation rate is stationary, but the velocity and output gaps do not completely summarize the relationship between broad money growth and inflation.

The above result, pointing to a relatively weak explanatory power of (v2t − 1v2*) with respect to the inflation rate, is consistent with the empirical equation for the demand for M2. However, this result does not refute the primary conclusion of Hallman, Porter, and Small that M2 and the price level are related in the long run. The coefficients on the velocity and output gaps differ by less than one standard deviation, which suggests that imposing the restriction of equality on the coefficients would not significantly worsen the fit of the model.23 Thus, while we find some support for the P* relationship, our main conclusion is that the output gap summarizes much of the relationship between broad money growth and inflation, such as in the Phillips curve models of Braun (1984), Gordon (1985), and Adams and Coe (1990), and the velocity gap contains relatively little additional information.24

To confirm the point, we consider whether the forecast errors of a Phillips curve model can be explained by the forecasts of either the unconstrained version of the levels equation or its constrained version, in which the velocity and output gaps are replaced by (pt − 1pt − 1*). Specifically, the forecast-encompassing test proposed by Chong and Hendry (1986) is applied to the one-step-ahead forecasts of these three equations for the period 1971:1 to 1989:4. Table 2 reports the constrained-levels inflation equation and the Phillips curve equation.

The forecast-encompassing test involves regressing the one-step-ahead forecast errors of equation i, ϵti, on the differences between the one-step-ahead forecasts of equations j and i, (y^tjy^ti). If this difference is significant in explaining ϵti, as measured by the t-statistic, t(i,j), of (y^tjy^ti), and conversely, if t(j,i) is not significantly different from zero, the forecasts of equation j are said to encompass those of equation i.

Table 3 shows the t(i,j) statistics for the three equations. To interpret the table, read down the columns to see if the forecasts of that model explain the forecast errors of any other model and across the rows to see if that model's forecast errors are explained by the forecasts of any other model. Only the unrestricted-levels inflation equation encompasses the forecasts of the Phillips curve model, although the t-statistic of 2.31 is not very large. Neither the forecasts of the constrained equation, which is based on the P* relationship, nor those of the Phillips curve equation explain the forecast errors of any other model.

Table 2.

Alternative πt Equations

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The heteroscedasticity-consistent standard errors are shown under the estimated coefficients.

The other misspecification tests do not differ significantly from those reported for the unconstrained πT equation.

IV. Conclusion

To analyze the P* relationship, this paper adopts a narrow interpretation of the quantity theory of money as a long-run money demand equation. From this perspective, (v2tv2*) measures the short-run departures from long-run money market equilibrium, while (qt‪ − qt*) represents such deviations in the goods market. This approach does not require the restriction of equality of the coefficients on the velocity and output gaps, although it does require the existence of a stable demand function for M2. Section II examines the stability of the empirical equation for the demand for M2.

Table 3.

Forecast Encompassing Statistics for Alternative πt Equations

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Note: The i, jth element of the matrix is the (t-statistic of the regression, ϵti=β(y^tjy^ti)+ηt, where y^ti and ϵti are the forecasts and forecast errors of model i. respectively, and y^tj are the forecasts of model j.

Given the M2 demand equation, the robustness of the Hallman-Porter-Small inflation equation, which is specified in first differences to an alternative specification in levels of the inflation rate is considered. The estimations indicate that the significance of the velocity gap in explaining variations in inflation is not robust to the equation's specification in levels. Moreover, the lack of significance of the velocity gap in the levels inflation equation is consistent with the weak explanatory power of the velocity gap term in the M2 demand equation.

While neither specification of the inflation equation is unambiguously preferable, several considerations tend to favor the levels equation. First, the standard error of the levels equation is lower than that of the first-differences equation. Second, there is weak evidence of second-order and fourth-order serial correlation in the first-differences equation, indicating that it may be overdifferenced. Third, the velocity gap is significant only in levels in the first-differences equation, while it would be expected to enter in differences if πt were I(1). One interpretation of this result is that the inflation rate is stationary, but the velocity and output gaps do not completely summarize the relationship between broad money growth and inflation.

The results of this paper, however, do not refute the existence of a long-run relationship between M2 and the price level, a primary conclusion of Hallman, Porter, and Small. Rather, it appears that the output gap summarizes much of the relationship between monetary growth and inflation, and the addition of velocity to a Phillips curve model adds little to its ability to forecast inflation accurately.

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*

Liam P. Ebrill, a Deputy Division Chief in the Western Hemisphere Department, received his Ph.D. from Harvard University.

Steven M. Fries is an Economist in the Research Department and holds a D.Phil. from Oxford University.

The authors thank Charles Adams, James Boughton, and Yusuke Horiguchi for useful comments.

1

Heller (1988) provides a recent description of the implementation of monetary policy in the United States.

2

The Federal Reserve publishes three official monetary aggregates: Ml consists of currency and checkable deposits; M2 includes in addition to M1 a variety of small-denomination savings-type deposits issued by depository institutions; and M3 consists of M2 plus certain large-denomination instruments such as certificates of deposit in excess of $ 100,000.

3

Humphrey (1989) surveys the antecedents to the P* model from the writings of David Hume to Milton Friedman.

4

This equivalence is obtained by taking the natural logarithms of the two equations that define P* and P and subtracting the second from the first. The P* specification of the equilibrium price level is essentially a reduced-form equation in which the inflationary effects of an increase in the aggregate demand for goods is exactly offset by the deflationary effects from the corresponding increase in the demand for money. See Kuttner (1990).

5

A series is stationary if it tends to revert over time either to its mean value or to a trend.

7

To say that a variable has a unit root is equivalent to stating that the variable is integrated of order one, I(l). A variable, xt, is integrated of order one if it is nonstationary and can be written as xt = α + xt-1 + ∈t, where ∈t has mean zero and a constant variance. Unit root refers to the coefficient on xt-1.

6

The basic data set begins in the first quarter of 1959, the initial period for which the official M2 series is available.

9

The logarithm of the opportunity cost cannot be used for all observations, since at times the spread is negative. Therefore, the logarithm of the opportunity cost is replaced by a second-order Taylor series expansion when the spread falls below 50 basis points. Money demand theory predicts that the opportunity cost of holding money and velocity should have the same time series properties. See. for example, McCallum and Goodfriend (1987).

10

One reason for the differing test results may be the presence of moving-average components in these series. See Schwert (1987, 1989) for a discussion of this issue.

11

Δ is a first-differences operator.

12

The empirical results in this and the subsequent section were obtained using PC-GIVE, version 6.1, by David Hendry and the Oxford Institute of Economics and Statistics.

13

See Hendry (1989) for a description of the misspecification tests.

14

This explanation suggests that a proxy for wealth may be a better scale variable. One alternative explanation proposed by Goodfriend (1985) is that the relevant variables are measured with error. A second alternative is the restrictive form of the error-correction term. See, for example, Boughton (1991).

15

See. for example, Adams and Coe (1990). The ADF and Phillips-Perron tests indicate that Δrpgast is I(1).

16

Two or more series are cointegrated if there exists a linear combination of the series that has a lower order of integration than that of the individual series.

17

Box and Jenkins (1970, pp. 121–22) show that the linear combination of nonstationary series with an order of integration d1 and a stationary series is nonstationary with an order of integration d1. This example readily generalizes the case where the stationary series is replaced by a nonstationary series with an order of integration d2 < d1. This linear combination is also nonstationary with an order of integration d1.

18

An example of the pitfalls that can arise in performing statistical inferences using nonstationary time series is the spurious regression problem. See, for example, Granger and Newbold (1974).

19

While Plosser and Schwert examine the case of a univariate regression, the results can be readily extended to the present case. Suppose that the regression yt = α + βxt + yzt + ϵt is the correct specification of the relationship. If the regression were estimated in first differences of xt and yt and the levels of zt, the residuals of this equation would contain the processes Δϵt and zt − 1.

20

Hallman, Porter, and Small (1989) test for the appropriate specification by nesting the first differences and levels equations, which in this case involves including the lagged level of the inflation rate in the first-differences equation. The conclusion from this test is that the null hypothesis of a first-differences specification cannot be rejected. However, they do not test whether the null hypothesis of a levels equation could be rejected. Cristiano (1989), Pecchenino and Rasche (1990), and Kuttner (1990) estimate inflation equations only in first differences.

21

The critical values of these statistics are the same as those of the t-statistic.

22

In small samples, the J-test tends to reject a true null hypothesis too frequently. For Monte Carlo evidence, see Godfrey and Pesaran (1983).

23

A Chow test of the restriction does not reject the hypothesis that the coefficients on the velocity and output gaps are equal.

24

Gordon (1985) reached a similar conclusion by decomposing “excess” nominal income growth, an analogue to the output gap, into the sum of excess money growth and changes in velocity. If money's influence on inflation arises solely through an output gap, the money growth and velocity terms, when entered in place of the output gap in an inflation equation, should have the same coefficients. This restriction is not rejected by the estimation results that he reported.