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(.|I_t) denotes the mathematical conditional expectation operator, conditional on the information set available to agents at time t, I_t. 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.

Using equations (1) and (3) in (2), we obtain a familiar reduced form for the spot exchange rate:

where x_t = [m′ – γy′]_t. Solving (4) forward, we obtain

where the transversality condition, $\underset{i\to \infty }{\lim }{[\lambda /(1+\lambda )]}^{i}E\left(s_{t+i}\right|I_t)=0$, has been imposed. Equation (5) is the basic equation of the forward-looking monetary approach to the exchange rate (FMAER). Equation (5) makes clear that the monetary model with rational expectations involves solving for the entire expected future path of the forcing variables (that is, money supplies and income levels). An interesting implication of the present value model of the exchange rate, which has been exploited for stock prices and interest rates by Campbell and Shiller (1987), but not in the exchange rate literature, is that the exchange rate should be cointegrated with the forcing variables contained in x_t.² To see this, subtract x_t from both sides of equation (5) and rearrange

Now, if m_t, $m_t^{*}$, y_t, and $y_t^{*}$ are first-difference stationary I(1) variables, then the right-hand side of equation (6) must be I(0), subject only to the caveat that forecasting errors are stationary, which is clearly the case for rational expectations (see Taylor (1991)).³ Thus, the right-hand side of equation (6) must also be stationary, and so, if s_t is also an I(1) series, the exchange rate must be cointegrated with m_t, $m_t^{*}$, y_t, and $y_t^{*}$, with certain restrictions on the cointegrating parameters:

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

which follows from equations (1) and (2). This is because from (3)—the UIP condition—the interest rate differential, $i_t^{\prime }$, must be I(0) for s_t ~ I(1).

If cointegration in equation (7) is found, then a more stringent test may be carried out by testing the forward restrictions. Following Campbell and Shiller (1987), this may be done as follows.

If L_t, as defined in (7), is in fact I(0), we can obtain an estimate of it by cointegration methods. If x_t is I(1), then both Δx_t, and L_t, 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 z_t= [Δx_t, …, Δx_t–p+1, L_t, …, L_t–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

and

The standard multiperiod forecasting formula may be used to forecast z in any future period; that is

where H_t is a restricted information set consisting of current and lagged values of L_t and Δx_t.

Projecting both sides of equation (6) onto H_t, applying the law of iterated mathematical expectations,⁴ and using equations (11)–(13) yields

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 (L_t, Δx_t)′

Now, define the “theoretical spread” as the right-hand side of equation (14)

Testing the restrictions expressed in equation (16) is tantamount to testing $H_0:L_t=L_t^{*}$, for all t. However, Campbell and Shiller (1987) point out that (16) may be rejected because of economically unimportant deviations from the null hypothesis, such as data imperfections, which are nevertheless statistically significant. Thus, a less formal check on the validity of the restrictions is simply to compare the time series of the actual and theoretical spreads, L_t, and $L_t^{*}$; manifest differences in their behavior would be indicative of economically important deviations from the null hypothesis.

Now, consider the implications of the present analysis for the detection of foreign exchange market bubbles. Add a bubble term, b_t, to the right-hand side of equation (5)

where b_t satisfies

Subtracting x_t from both sides of equation (18) yields

Since b_t is explosive by construction, equation (20) implies that s_t and x_t cannot be cointegrated for x_t ~ I(1). Thus, testing for stationarity of the spread, L_t = s_t – x_t, 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 x_t) 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 X_t 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, for the purposes of exposition, we have excluded a constant.⁸ The long-run static equilibrium corresponding to equation (21) is⁹

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 ${\beta }_i^{\prime }$ is the ith row of β′

Johansen demonstrates that the likelihood function for this problem is proportional to

where ${\hat{\lambda }}_1,\mathrm{\dots },{\hat{\lambda }}_N$ denote the N squared canonical correlations between the X_t-k and ΔX_t, series (arranged in descending order, so that λ_i > λ_j for i < j), corrected for the effect of the lagged differences of the X process (for details of how to extract the ${\hat{\lambda }}_i$’s, see Johansen (1988, 1989), and Cuthbertson, Hall, and Taylor (1992)). Further, the number of distinct cointegrating vectors is shown to be equal to the number of nonzero λ_i’s. Thus, the likelihood ratio statistic for the null hypothesis of, at most, r cointegrating vectors—the TRACE statistic—is seen to be

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 $(s_t,m_t^{*},m_t,y_t^{*},y_t,i_t^{*},i_t)\prime$. This was done in order to obtain a preliminary estimate of λ. In the second exercise, we tested for cointegration among $(s_t,m_t^{*},m_t,y_t^{*},y_t)\prime$.

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).

Table 1.

Unit-Root Tests

Note: See text for data definitions. An asterisk denotes a U.S. variable; those without asterisks are German variables; τ_μ and τ_τ are standard augmented Dickey-Fuller test statistics with allowance for a constant mean and for a trend in mean, respectively (Fuller (1976)); Z(τ_μ) and Z(τ_τ) are the Phillips-Perron transforms of these statistics (Phillips (1987)). The null hypothesis is that the variable in question is first-difference stationary, I(1); approximate 5 percent critical value for τ_μ and Z(τ_μ) is –2.89, with rejection region { ϕ| ϕ < –2.89}; the 5 percent rejection region for τ_τ and Z(τ_τ) is {ϕ| ϕ < –3.43} (Fuller (1976)).

Table 1.

Unit-Root Tests

Variable	τ μ	τ τ	Z(τμ)	Z(ττ)
S t	2.86	0.05	–1.96	–1.83
ΔSt	–9.37	–15.68	–10.47	–10.40
m t	–1.28	–1.48	1.28	–1.85
Δmt	–14.66	–14.64	–14.75	–14.71
y t	–1.90	–1.81	–1.83	–1.67
Δyt	–9.29	–9.29	–9.79	–9.26
i t	–0.11	–1.12	–0.52	–2.02
Δit	–4.19	–4.20	–5.13	–5.08
$m_t^{*}$	–0.56	–2.62	–0.70	–1.75
$\mathrm{\Delta }{m}_t^{*}$	–9.34	–8.32	–11.53	–11.51
$y_t^{*}$	–0.77	–3.03	–0.66	–2.57
$\mathrm{\Delta }{y}_t^{*}$	–2.71	–8.74	–8.77	–8.74
$i_t^{*}$	0.69	–2.36	–2.12	–2.21
$\mathrm{\Delta }{i}_t^{*}$	–3.45	–3.85	–10.93	–10.89

Note: See text for data definitions. An asterisk denotes a U.S. variable; those without asterisks are German variables; τ_μ and τ_τ are standard augmented Dickey-Fuller test statistics with allowance for a constant mean and for a trend in mean, respectively (Fuller (1976)); Z(τ_μ) and Z(τ_τ) are the Phillips-Perron transforms of these statistics (Phillips (1987)). The null hypothesis is that the variable in question is first-difference stationary, I(1); approximate 5 percent critical value for τ_μ and Z(τ_μ) is –2.89, with rejection region { ϕ| ϕ < –2.89}; the 5 percent rejection region for τ_τ and Z(τ_τ) is {ϕ| ϕ < –3.43} (Fuller (1976)).

View Table

Table 1.

Unit-Root Tests

Variable	τ μ	τ τ	Z(τμ)	Z(ττ)
S t	2.86	0.05	–1.96	–1.83
ΔSt	–9.37	–15.68	–10.47	–10.40
m t	–1.28	–1.48	1.28	–1.85
Δmt	–14.66	–14.64	–14.75	–14.71
y t	–1.90	–1.81	–1.83	–1.67
Δyt	–9.29	–9.29	–9.79	–9.26
i t	–0.11	–1.12	–0.52	–2.02
Δit	–4.19	–4.20	–5.13	–5.08
$m_t^{*}$	–0.56	–2.62	–0.70	–1.75
$\mathrm{\Delta }{m}_t^{*}$	–9.34	–8.32	–11.53	–11.51
$y_t^{*}$	–0.77	–3.03	–0.66	–2.57
$\mathrm{\Delta }{y}_t^{*}$	–2.71	–8.74	–8.77	–8.74
$i_t^{*}$	0.69	–2.36	–2.12	–2.21
$\mathrm{\Delta }{i}_t^{*}$	–3.45	–3.85	–10.93	–10.89

Note: See text for data definitions. An asterisk denotes a U.S. variable; those without asterisks are German variables; τ_μ and τ_τ are standard augmented Dickey-Fuller test statistics with allowance for a constant mean and for a trend in mean, respectively (Fuller (1976)); Z(τ_μ) and Z(τ_τ) are the Phillips-Perron transforms of these statistics (Phillips (1987)). The null hypothesis is that the variable in question is first-difference stationary, I(1); approximate 5 percent critical value for τ_μ and Z(τ_μ) is –2.89, with rejection region { ϕ| ϕ < –2.89}; the 5 percent rejection region for τ_τ and Z(τ_τ) is {ϕ| ϕ < –3.43} (Fuller (1976)).

Table 2a.

Cointegration Results

(System involving s, m, m^*, y, y^*, i, i^*)

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 18.77 ~ χ²(12); restricted cointegrating relationship (largest eigenvalue only): $s_t=(m_t-m_t^{*})-(y_t-y_t^{*})+0.049i_t-0.050i_t^{*}\mathrm{.}$ In Tables 2a and 2b, r denotes the number of distinct cointegrating vectors; critical values for the TRACE and λMAX statistics are taken from Osterwald-Lenum (1990).

Table 2a.

Cointegration Results

(System involving s, m, m^*, y, y^*, i, i^*)

Null Hypothesis	TRACE Statistic	5 Percent Critical Value	Null Hypothesis	λMAX Statistic	5 Percent Critical Value
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

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 18.77 ~ χ²(12); restricted cointegrating relationship (largest eigenvalue only): $s_t=(m_t-m_t^{*})-(y_t-y_t^{*})+0.049i_t-0.050i_t^{*}\mathrm{.}$ In Tables 2a and 2b, r denotes the number of distinct cointegrating vectors; critical values for the TRACE and λMAX statistics are taken from Osterwald-Lenum (1990).

View Table

Table 2a.

Cointegration Results

(System involving s, m, m^*, y, y^*, i, i^*)

Null Hypothesis	TRACE Statistic	5 Percent Critical Value	Null Hypothesis	λMAX Statistic	5 Percent Critical Value
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

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 18.77 ~ χ²(12); restricted cointegrating relationship (largest eigenvalue only): $s_t=(m_t-m_t^{*})-(y_t-y_t^{*})+0.049i_t-0.050i_t^{*}\mathrm{.}$ In Tables 2a and 2b, r denotes the number of distinct cointegrating vectors; critical values for the TRACE and λMAX statistics are taken from Osterwald-Lenum (1990).

Table 2b.

Cointegration Results

(System involving s, m, m^*, y, y^*)

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 6.13 ~ X²(4).

Table 2b.

Cointegration Results

(System involving s, m, m^*, y, y^*)

Null Hypothesis	TRACE Statistic	5 Percent Critical Value	Null Hypothesis	λMAX Statistic	5 Percent Critical Value
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

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 6.13 ~ X²(4).

View Table

Table 2b.

Cointegration Results

(System involving s, m, m^*, y, y^*)

Null Hypothesis	TRACE Statistic	5 Percent Critical Value	Null Hypothesis	λMAX Statistic	5 Percent Critical Value
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

Note: Imposing homogeneity on relative money and relative income yields a likelihood ratio statistic of 6.13 ~ X²(4).

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, L_t—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 (Δx_t, L_t)′, for $L_t=s_t-m_t^{\prime }+y_t^{\prime }$. For each of the values of λ discussed in the previous section, namely λ = 0.05, λ = 0.03, λ = 0.015, and λ = 0.001, we constructed heteroscedastic-robust linear Wald statistics of the null hypothesis (16). In each case, the restrictions are strongly rejected.

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 L_t and $L_t^{*}$ (for all of the chosen values of λ) also revealed important differences and evidence of “excess volatility” in L_t.¹⁴ Figure 1 graphs L_t and $L_t^{*}$ for the case λ = 0.05.

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Actual and Theoretical Spread

Citation: IMF Staff Papers 1993, 004; 10.5089/9781451956986.024.A004

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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 u_t denotes a disturbance term; z_t, denotes the cointegrating vector normalized on S_t; and ρ is expected to be negative. Thus, a positive value of z_t, implies that S_t 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: