In this paper we re-examine the monetary model of the exchange rate from a number of complementary perspectives. We begin by examining the validity of the model in its forward-looking, rational expectations formulation, since although standard reduced-form equations have received little empirical support for the recent floating experience (see MacDonald and Taylor (1992a)), the forward-looking monetary model has enjoyed relatively more success (see, among others, Hoffman and Schlagenhauf (1983), Woo (1985), and Finn (1986)).1 In this paper, however, we argue that previous tests of the forward model may have been incorrectly implemented. Our method of obtaining a forward-looking solution relies on the exploitation of the recently developed multivariate cointegration methodology (Engle and Granger (1987) and Johansen (1988)) and its application to present-value models (Campbell and Shiller (1987), MacDonald and Speight (1990), and Taylor (1991, 1992)).
Using data for the deutsche mark-U.S. dollar exchange rate over the period 1976–90, we demonstrate, first, that the static monetary approach to the exchange rate has some validity when considered as a long-run equilibrium condition; second, that when the exchange rate fundamentals suggested by the monetary model are assumed, the speculative bubbles hypothesis is rejected; and third, that the full set of rational expectations restrictions imposed by the forward-looking monetary model are rejected. Finally, however, we demonstrate that the monetary model can be used to generate a dynamic error-correction exchange rate equation that has robust in-sample and out-of-sample properties—including beating a random walk (the usual benchmark for exchange rate forecasting) in postsample forecasting.
I. The Monetary Model: Forward-Looking Restrictions, Bubbles, and Cointegration
The flexible price monetary model is now well known and requires only the briefest of descriptions here (see MacDonald and Taylor (1992a) for a comprehensive discussion). The model relies on a relative money market equilibrium condition, as in equation (1); an expression that links the exchange rate to the home and foreign price levels (purchasing power parity (PPP), as in equation (2); and uncovered interest rate parity (UIP), which links home and foreign interest rates and the expected rate of exchange rate change, as in equation (3):
where m denotes the logarithm of the domestic money supply; p, the logarithm of the domestic price level; y, the logarithm of domestic income; i, the domestic interest rate; s, the logarithm of the exchange rate (domestic price of foreign currency), and where, if an asterisk denotes a foreign variable, then a prime denotes a domestic minus a foreign variable; for example, m′ = m – m*; Δ is the first-difference operator; and E(.|It) denotes the mathematical conditional expectation operator, conditional on the information set available to agents at time t, It. As is standard in the literature on the monetary approach to the exchange rate, we assume common money demand parameters in the home and foreign country.
where xt = [m′ – γy′]t. Solving (4) forward, we obtain
where the transversality condition,
Now, if mt,
Previously, when researchers (see, for example, Hoffman and Schlagenhauf (1983), Meese (1986), and Finn (1986)) implemented the present-value exchange rate model, they used a first-difference transformation of all the variables. However, as Engle and Granger (1987) have demonstrated, if a vector of variables is cointegrated, an empirical model formulated in pure first differences omits the error-correction term and thus misspecifies the dynamics. Thus, a preliminary step in testing the FMAER is to test for a cointegration relation of the kind embodied in equation (7). Note that cointegration in (7) is not inconsistent with the existence of a cointegration relation corresponding to the basic flex-price monetary approach equation
If Lt, as defined in (7), is in fact I(0), we can obtain an estimate of it by cointegration methods. If xt is I(1), then both Δxt, and Lt, are each stationary, I(0), series. Thus, by the multivariate form of Wold’s decomposition (Hannan (1970)), there exists a Wold representation that may be approximated by a vector autoregression (VAR) of lag depth p, say. Let zt= [Δxt, …, Δxt–p+1, Lt, …, Lt–p+1]′. This VAR may be represented in companion form as
or, in a more compact notation
Further, define g′ and h′ as (1 × 2p) selection vectors with unity in the (p + 1)th and first elements respectively, so that
The standard multiperiod forecasting formula may be used to forecast z in any future period; that is
where Ht is a restricted information set consisting of current and lagged values of Lt and Δxt.
where ψ = (λ/1+λ). If equation (14) is to hold nontrivially, the following 2p parameter restrictions are imposed on the VAR:
Postmultiplying equation (15) by (I – ψA), we may then obtain a set of 2p linear restrictions, which the FMAER imposes on the VAR for (Lt, Δxt)′
Now, define the “theoretical spread” as the right-hand side of equation (14)
Testing the restrictions expressed in equation (16) is tantamount to testing
Now, consider the implications of the present analysis for the detection of foreign exchange market bubbles. Add a bubble term, bt, to the right-hand side of equation (5)
where bt satisfies
Subtracting xt from both sides of equation (18) yields
Since bt is explosive by construction, equation (20) implies that st and xt cannot be cointegrated for xt ~ I(1). Thus, testing for stationarity of the spread, Lt = st – xt, is equivalent to testing for the presence of bubbles.5 In contrast to Meese’s (1986) test for foreign exchange market bubbles, however, we are able to distinguish between testing for bubbles (equivalent to testing nonstationarity of the spread) and testing the restrictions implied by the FMAER (that is, equation (16)).
Note, however, that the procedure just outlined for testing the FMAER is nonoperational as it stands, since both the income elasticity γ (needed to form xt) and the interest rate semi-elasticity λ (needed to form ψ) are unknown. With respect to γ, an estimate can be obtained from the cointegration estimate of equation (7) and, because of the super-consistency of cointegration parameter estimates (Stock (1987)), can be treated as known in testing the set of equations (16). With respect to λ, we pursued two options. First, an estimate of λ can be obtained from the cointegration parameter in the basic flex-price equation (equation (8)). Again, by appeal to the super-consistency result, this can then be treated as known in constructing tests of (16). The second option was to use extraneous estimates of λ from the literature. Bilson (1978), in his Bayesian estimation of the monetary model, used a prior for λ of 0.015 with a 95 percent confidence interval ranging from zero to 0.03. Accordingly, we tried three values of λ within this interval: 0.015,0.001, and 0.03.
II. Econometric Methods
As MacDonald and Taylor (1991) have indicated, previous tests of the long-run relationship between the exchange rate and the monetary variables, which rely on the Engle-Granger (1987) two-step methodology, suffer from a number of deficiencies. In order to test for cointegration, we use the multivariate cointegration technique proposed by Johansen (1988, 1989) and Johansen and Juselius (1990). This technique is superior to the simpler regression-based technique because it fully captures the underlying time-series properties of the data, provides estimates of all of the cointegrating vectors that exist within a vector of variables (that is, using ordinary least squares to estimate a cointegration relationship for an N-dimensioned vector does not clarify whether one is dealing with a unique cointegrating vector or simply a linear combination of the potential N – 1 distinct cointegrating vectors that may exist within the system), and offers a test statistic for the number of cointegrating vectors (again, this contrasts with the regression-based methodology). This test may therefore be viewed as more discerning in its ability to reject a false null hypothesis. We now present a brief discussion of the Johansen technique.6
Let Xt be an N × 1 vector of I(1) variables, and assume that this vector has a kth order VAR representation with Gaussian errors ∈t:7
where the long-run coefficient matrix, Π, is defined
Π is an N × N matrix whose rank determines the number of distinct cointegrating vectors that exist between the variables in X. Define two N × r matrices, α and β, such that
The rows of β′ form the r distinct cointegrating vectors, such that, if
Johansen demonstrates that the likelihood function for this problem is proportional to
Additionally, the likelihood ratio statistic for testing, at most, r cointegrating vectors against the alternative of r + 1 cointegrating vectors—the maximum eigenvalue statistic—is given by
TRACE and λMAX will have nonstandard distributions under the null hypothesis, although approximate critical values have been generated by Monte Carlo methods and tabulated by Johansen (1988), Johansen and Juselius (1990), and Osterwald-Lenum (1990).10
An additional advantage of using the Johansen methodology is that it allows direct hypothesis tests on the coefficients entering the cointegrating vectors. We may therefore test the hypothesized values of the coefficients noted above and, additionally, the implicit restrictions that the coefficients on the money and income terms are equal and opposite. Such tests have, in fact, been conducted previously in the exchange rate literature (see, among others, Haynes and Stone (1981) and Rasulo and Wilford (1980)). The novel feature of the present tests is that they are robust to the nonstationarity of the data; previous tests, which use the levels of the variables and standard t-tests or F-ratios, are not (see MacDonald and Taylor (1991) for a further discussion).
III. Data and Results
The data for this study relate to the deutsche mark-U.S. dollar exchange rate; they are taken from the International Monetary Fund’s International Financial Statistics database, and run from January 1976 through December 1990. In particular, the exchange rate used (s) is line ag (expressed as home currency per unit of foreign currency); the monetary aggregates (m) are M1, line 34; the income measure (y) is industrial production, line 66c; and the short-term interest rate (r) is line 60c. The money supply and income measures are seasonally adjusted. In the remainder of the paper, an asterisk denotes a series corresponding to the United States; those without asterisks correspond to Germany. All series except interest rates were put into logarithmic form.
We conducted two cointegration exercises. The first involved testing for cointegration within the vector
In order to implement the Johansen procedure, a lag length must be chosen for the VAR, and the orders of integration of the series entering the VAR must be determined. Our procedure for choosing the optimal lag length was to test down from a general 13-lag system until reducing the order of the VAR by 1 lag could be rejected using a likelihood ratio statistic. The residuals from the chosen VAR were then checked for whiteness. If the residuals in any equation proved to be nonwhite, we sequentially chose a higher lag structure until they were whitened. For the system involving interest rates, we found that an eighth-order lag satisfied these criteria. For the system excluding interest rates, a twelfth-order system was necessary. The orders of integration of the series were determined using the standard Dickey-Fuller and Phillips-Perron statistics.11
In Table 1 our tests for a unit root indicate that all series are I(1) processes.12 In Tables 2a and 2b we report the trace and maximum eigenvalue statistics derived from the full system, including interest rates. These indicate the presence of three statistically significant cointegrating vectors among the series. Imposing homogeneity on both relative money and income produces, moreover, an insignificant test statistic. With these restrictions imposed, the cointegration coefficients on German and U.S. interest rates (taking the eigenvector corresponding to the largest eigenvalue of the system) are opposite in sign and almost equal in absolute magnitude: 0.049 and 0.05, respectively. However, imposing the additional restriction of equal and opposite interest rate coefficients on all three significant cointegrating vectors (as is necessary with the Johansen procedure—Johansen (1988, p. 236)) led to a rejection of the null hypothesis at the 5 percent level (test statistic not reported). Since, however, this restriction is close to being satisfied on the most significant cointegrating vector (as reported in Table 2b), we inferred a value of λ of 0.05 as the cointegration estimate of this parameter. Note that these results provide evidence supportive of the flex-price monetary model, interpreted as a long-run equilibrium model.13 This is in contrast to much previous work on the monetary model, which has utilized the inferior two-step cointegration methodology (see, among others, Boothe and Glassman (1987), Meese (1986) and McNown and Wallace (1989)), but is consistent with the results of MacDonald and Taylor (1991).
(System involving s, m, m*, y, y*, i, i*)
|r ≤ 6||1.43||9.09||r = 6 | r = 1||1.43||9.09|
|r ≤ 5||6.33||19.96||r = 5 | r = 6||4.90||15.67|
|r ≤ 4||18.17||34.91||r = 4 | r = 5||11.84||22.00|
|r ≤ 3||39.19||53.12||r = 3 | r = 4||21.02||28.14|
|r ≤ 2||77.10||76.07||r = 2 | r = 3||35.81||34.40|
|r ≤ 1||115.87||102.14||r = 1 | r = 2||40.87||40.30|
|r = 0||159.87||131.70||r = 0 | r = 1||43.99||46.46|
(System involving s, m, m*, y, y*)
|r ≤ 4||0.14||9.09||r = 4 | r = 5||0.14||9.09|
|r ≤ 3||5.58||19.96||r = 3 | r = 4||5.44||15.67|
|r ≤ 2||19.95||34.91||r = 2 | r = 3||14.38||22.00|
|r ≤ 1||44.40||53.12||r = 1 | r = 2||24.45||28.14|
|r = 0||79.08||76.07||r = 0 | r = 1||34.67||34.40|
Next, we tested for cointegration among the vector of variables excluding interest rates, as suggested by our analysis in Section I. The results are given in Table 2b. These demonstrate the existence of a unique cointegrating vector among the series, in which the restrictions of homogeneity of relative money and of relative incomes cannot be rejected at the 5 percent level. The finding of cointegration of these series—and hence, the stationarity of the spread, Lt—is thus tantamount to a rejection of the speculative bubbles hypothesis, independently of whether or not the full set of FMAER restrictions can be imposed.
Below, we report tests of the FMAER restrictions on the VAR for (Δxt, Lt)′, for
Tests of Forward-Looking Restrictions
a) Assuming λ= 0.05
Linear Wald statistic = 0.29E + 07 ~ χ2(12)
b) Assuming λ = 0.015
Linear Wald statistic = 0.33E + 08 ~ χ2(12)
c) Assuming λ = 0.001
Linear Wald statistic = 0.73E + 10 ~ χ2(12)
d) Assuming λ = 0.03
Linear Wald statistic = 0.81E + 07 ~ χ2(12)
Moreover, the deviations from the null hypothesis appear to be economically as well as statistically significant: the ratio of the variance of the actual spread to the theoretical spread (computed according to equation (17)) is in each case massively different from unity. Moreover, plots of the time-series behavior of Lt and
Figure 1.Actual and Theoretical Spread
IV. Forecasting with the Monetary Model
We demonstrated in the previous section that, for the deutsche mark-dollar exchange rate during the period of investigation, a cointegration relation exists that corresponds to the static monetary approach exchange rate equation. We suggested, therefore, that the monetary model can be interpreted as having at least long-run validity. According to the Granger representation theorem, however, if a cointegrating relationship exists among a set of I(1) series, then a dynamic error-correction representation of the data also exists. That is to say, there should exist a stable VAR in the first differences of the variables, augmented by one lag of the cointegrating vector itself—which represents the “equilibrium error.” This suggests that there should exist an exchange rate equation of the form
where ut denotes a disturbance term; zt, denotes the cointegrating vector normalized on St; and ρ is expected to be negative. Thus, a positive value of zt, implies that St is above its long-run equilibrium, and will tend to reduce the change in the exchange rate next period. In long-run equilibrium, when all of the first-differenced series in equation (29) have settled down to their steady-state values, the cointegrating relationship is recovered (subject only to a constant intercept) as the steady-state solution.
In practice, not all of the coefficients in equation (29) may be statistically significant, and greater efficiency may be gained by eliminating insignificant coefficients or imposing other statistically insignificant restrictions.
Using the estimated cointegrating vector for the dollar-mark exchange rate (for the system including interest rates), we estimated a dynamic error-correction form in this fashion. In carrying out this exercise, we reserved the last 24 data points—corresponding to the period 1989:1 through 1990:12—for postsample forecasting tests. Our final preferred parsimonious equation was as follows:
In this equation, R2 denotes the coefficient of determination; DW is the Durbin-Watson statistic; σ is the standard error of the regression; AR is a Lagrange multiplier autocorrelation test statistic for up to seventh-order autocorrelation; ARCH is a test statistic for up to seventh-order autoregressive conditional heteroscedasticity (Engle (1982)); HETX2 is a test statistic for heteroscedasticity based on the squares of the regressors; and RESET is a test for functional form misspecification. All of these diagnostics are distributed as central F under the null hypothesis with degrees of freedom as indicated in parentheses. Subscripted figures in parentheses are estimated (heteroscedastic-consistent) standard errors; those in square brackets are marginal significance levels. CHOW and PF denote, respectively, Chow’s (1960) and Hendry’s (1979) tests for postsample predictive failure, when the model is used to forecast over the period 1989:1–1990:12;15 the CHOW statistic is distributed as central F if the model is stable, while the PF statistic is χ2.
The model’s performance is impressive. Not only does it pass a wide range of in-sample diagnostic tests, it also appears to forecast well out of sample. Note, however, that the CHOW and PF statistics are based on static forecasts; that is, the forecasts of the exchange rate generated by the model are not fed back in, as in dynamic forecasting. Accordingly, we also carried out a dynamic forecasting exercise over the same remaining 24 data points, which followed a similar procedure to that of Meese and Rogoff (1983); that is, we sequentially re-estimated the model for every data point from 1989:1 onward, computing dynamic forecasts for a number of forecasting horizons. When this had been done for the whole sample, we then computed the root mean squared error (RMSE) of the forecast at each horizon. As a point of comparison, we also computed the RMSEs generated by a naive random walk model.
The results of this exercise are reported in Table 3, and are very interesting indeed: the dynamic error-correction model outperforms the random walk forecast at every forecast horizon. The results of this section thus suggest that, treated as a long-run equilibrium condition, the monetary model of the exchange rate may still be useful in forecasting the exchange rate.
|RMSE from Random|
In this paper we re-examined the monetary model of the exchange rate for the deutsche mark-U.S. dollar exchange rate over the period January 1976 to December 1990 and generated a number of new results.
Our results indicated rejection of the speculative bubbles hypothesis for the dollar-mark exchange rate over this period. However, tests of the full set of restrictions, which the forward-looking monetary model imposes on the relevant time-series processes, resulted in an overwhelming rejection of the model. The deviations of the data from the forward model were shown, moreover, to be economically as well as statistically important. This finding differs substantially from the results of other researchers who have empirically tested the forward-looking monetary model. We attribute the difference to the (more appropriate) way in which we have implemented the model.
However, some support for the flex-price monetary model—interpreted as a long-run model—could, nevertheless, be adduced from our results, since we found evidence of cointegration between the exchange rate, relative money supply, relative income, and relative interest rates. This result contrasts with many other cointegration tests of the monetary model, but confirms the findings of MacDonald and Taylor (1991).
Finally, and perhaps most significantly, we showed that imposing the monetary model as a long-run equilibrium condition on a dynamic, error-correction model led to dynamic exchange rate forecasts that were better than the random walk forecast at every horizon considered.
Given that the monetary model appears to hold as a long-run equilibrium condition, it may be that our rejection of the forward model is due to the presence of mean-reverting deviations from the fundamentals that are generated by the presence of speculators who do not conform to the rational expectations hypothesis—such as technical analysts or chartists (Frankel and Froot (1986), MacDonald and Young (1986), Allen and Taylor (1990), and Taylor and Allen (1992)). This would explain why allowing the data to determine the form of the short-run dynamics, while imposing theoretically consistent long-run constraints—that is, estimating an error-correction form—proved fruitful.
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The forward-looking monetary model was used, for example, as the maintained hypothesis in Meese’s (1986) study of foreign exchange market bubbles.
A time series is said to be integrated of order d, denoted I(d), if it must be differenced d times in order to achieve covariance stationarity.
That is, E(E(w | It,) | Ht) = E(w| Ht), for Ht ⊆ It.
Note that bubbles were implicitly ruled out in the earlier discussion of the present-value model by the imposition of the transversality condition,
For a more complete exposition, see Cuthbertson, Hall, and Taylor (1992).
Phillips (1987) suggests that the Johansen technique may also be applicable in the presence of heterogeneously distributed error processes. In our empirical analysis, an intercept term was included in the VARs, as in Johansen and Juselius (1990).
A constant term was included in our empirical work.
Dynamic steady-state equilibrium simply involves the addition of a term in the constant vector of steady-state growth rates to equation (7), which we omit here for expositional purposes; this does not affect the subsequent discussion.
The critical values recorded in Johansen’s 1988 paper are for a VAR without an intercept term. Johansen (1989) reports critical values for VAR systems with a constant for systems of up to 5 variables. These critical values have been extended by Osterwald-Lenum (1990) for systems of up to 11 variables. We use these latter critical values in the present study.
We report the former in addition to the latter, since as Schwert (1987) has noted, the latter statistics may reject the null of a unit root too often in the presence of a first-order moving average process.
For U.S. industrial production, there is some sign that the series may be stationary about a trend in mean. However, given the power of these tests and the values of the statistics obtained, we would argue that this evidence is slight, and we treat this series as a unit-root process. This view was confirmed by including a time trend in the VARs used to generate the Johansen results: the trend term was insignificant and the Johansen results were qualitatively unaffected.
This confirms the findings of MacDonald and Taylor (1991) for the deutsche mark-U.S. dollar exchange rate. The results reported in that study differ slightly from those reported here since long-term as opposed to short-term interest rates are used in the former.
Our results thus echo the excess volatility finding of Huang (1981), although in implementing the Campbell-Shiller technique and thereby allowing for nonstationarity and long-run constraints, our analysis is technically superior to Huang’s.
A guide to all of the diagnostics discussed in this section can be found in Cuthbertson, Hall, and Taylor (1992).