Structural Models of the Dollar
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

This paper addresses several questions about the time series processes followed by dollar exchange rates. The stochastic process for exchange rates implied by structural models and the conditions under which they would be described by random walks are examined. Tests on the univariate time series for dollar exchange rates are undertaken to determine if there is evidence for departures from a random walk. Multivariate tests examine whether longer-run movements in the dollar are linked to those in other economic variables, and whether deviations from these long-run relationships contain information for predicting exchange rate movements.

Abstract

This paper addresses several questions about the time series processes followed by dollar exchange rates. The stochastic process for exchange rates implied by structural models and the conditions under which they would be described by random walks are examined. Tests on the univariate time series for dollar exchange rates are undertaken to determine if there is evidence for departures from a random walk. Multivariate tests examine whether longer-run movements in the dollar are linked to those in other economic variables, and whether deviations from these long-run relationships contain information for predicting exchange rate movements.

I. Introduction

The substantial nominal and real appreciation of the dollar during the first half of the 1980s and its subsequent depreciation after 1985 has been difficult to explain. 1/ According to one interpretation, the dollar’s strength in the early 1980s reflected a substantial shift in the U.S. policy mix toward relatively tight monetary policy and large budget deficits that pushed up nominal and real interest rates causing large inflows of capital. Subsequently, as the monetary stance was loosened and the fiscal deficit narrowed, these pressures reversed themselves leading to declines in the dollar. According to another interpretation, changes in the incentives to invest in U.S. real assets in the early 1980s together with political uncertainties and financial deregulation abroad led to a massive but temporary increase in the demand for the dollar during the first half of the 1980s. With the unwinding of these factors in the latter part of the decade, the dollar then reversed its previous gains.

Notwithstanding a large amount of empirical work it has been very difficult to distinguish between these alternative interpretations of the dollar’s movements during the 1980s. Most structural exchange rate models have explained only a small proportion of the dollar’s movements, and have typically exhibited periods of instability. 2/ Moreover, the building blocks for many of these models including interest rate parity and purchasing power parity, have been shown to have been violated in the 1970s and 1980s.

The empirical regularity that has emerged is that the nominal and real exchange rates of the dollar in terms of other major currencies are closely approximated by random walks, and that forecasts from a random walk outperform those from most structural exchange rate models. 3/ Some investigators see the random walk property of exchange rates as a natural outcome of efficient financial markets in which prices fully reflect all available information. 4/ According to others, the random walk carries strong implications for identifying the kinds of shocks that have driven exchange rates, with the presumption that most of these shocks have been expected to be permanent, and the models that are appropriate for analyzing their behavior. 5/

This paper addresses a number of questions about the time series processes followed by exchange rates with a view to shedding light on the factors behind the permanent or longer-run movements in the dollar. Rather than estimate a particular structural model, the paper focuses on the stochastic processes for exchange rates implied by wide classes of structural models, and whether the data are consistent with these processes. In addition, using cointegration techniques, the paper examines whether the long-run variance of exchange rates can be accounted for by the economic variables identified in structural exchange rate models.

The paper is organized as follows. Section II discusses the stochastic processes for exchange rates implied by structural exchange rate models and identifies two sources of systematic movement in exchange rates and hence departures from a random walk—those related to predictable movements in forcing variables (extrinsic dynamics) and those related to the internal dynamics of models (intrinsic dynamics). Section III examines the actual time series processes followed by nominal and real exchange rates of the dollar to see if there is evidence of mean reversion and departures from a random walk. Section IV identifies the time series properties of the forcing variables appearing in structural exchange rate models and tests for long-run cointegrating relationships between these variables and dollar exchange rates. Section V provides some concluding remarks.

II. Structural Exchange Rate Models

This section reviews several classes of exchange rate models with a view to identifying the time series processes they predict for exchange rates. The section begins by considering the solutions for exchange rates implied by flexible and fixed price monetary models, and then discusses the dynamic process contained in portfolio models. 6/ Two sources of systematic exchange rate movements are identified: those caused by predictability in the forcing variables included in exchange rate models (extrinsic dynamics) and those due to dynamic adjustment to shocks (intrinsic dynamics).

Monetary models. The monetary models are based on parity conditions linking interest rates on assets denominated in different currencies (equations (1) and (2)); a parity condition linking domestic and foreign prices, with the long-run real exchange rate constant under purchasing power parity (equation (3)); and equilibrium in financial markets, as represented by equality between the stock demands and supplies of domestic and foreign money (equations (4) and (5)). 7/ The monetary models assume residents of each country hold only their own money.

Rt=Rt*+[ftt+1st](1)
Rt=Rt*+[Etst+1st](2)
qt=st+pt*pt(3)
mt=pt+βytαRt(4)
mt*=pt*+β*yt*α*Rt*(5)

Monetary models have been applied in different ways over the recent floating rate period. 8/ In ‘flexible-price’ applications, the models determine nominal exchange rates on the basis of given values for real variables such as relative outputs and the real exchange rate. 9/ In “sticky-price” applications, equations (1)-(5) describe the long-run behavior of nominal exchange rates, with departures from these relationships in the short run. 10/

Under the assumption of flexible prices, the solution to equations (1)-(5) has a familiar form in which the nominal exchange rate depends on the current and expected future values of its determinants: here, relative money supplies, relative outputs, and the real exchange rate 11/

st=1(1+α)Σi=0Et[α(1+α)]i[(mt+imt+i*)β(yt+iyt+i*)+qt+i].(6)

The significance of events in the future in equation (6) is determined by the size of the discount factor α/(1 + α), which depends on the (semi-interest) elasticity of money demand. If the elasticity of money demand is very high, events expected far into the future have almost the same weight as those in the present.

The forward-looking nature of equation (6) does not mean that exchange rates should be entirely unpredictable and follow driftless random walks (or martingales), conditions that are sometimes regarded as an implication of market efficiency. 12/ The only efficiency conditions in the monetary model relate to the forward exchange market and imply that forward exchange rates should be unbiased predictors of expected future spot exchange rates. Given the assumption that residents in each country only hold their own money, there is no efficiency condition for the spot exchange market and exchange rates can be characterized by systematic movements in the monetary model.

The amount of systematic exchange rate movement in the flexible-price model is determined by the size of the (semi) interest elasticity of money demand and the systematic movement in the forcing variables. Table 1 shows the solutions for the nominal exchange in the flexible-price model under alternative assumptions about the stochastic processes for relative money supplies, including those with inter-temporally correlated components. 13/ Interpretation of the solutions is facilitated by decomposing the change in the nominal exchange rate between any two periods into a component that is unpredictable as of period t-1 (unsystematic component) and a component that is predictable (systematic component) 14/

Table 1.

Flexible Price Monetary Model

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Note:

For each money supply process the systematic (or predictable) change in the exchange rate over the next period is given by

(Etst+1 - st)

stst1=(stEt1st)+(Et1stst1).(7)

Given that there are no intrinsic dynamics in the flexible-price monetary models, systematic exchange rate movements only occur when there are predictable elements in the forcing variables. The most obvious source of systematic movements arises from differences in monetary growth rates across countries: countries with faster monetary growth rates will tend to have continuously depreciating currencies (Table 1). 15/ But systematic movement can also be the result of temporary or inter-temporally correlated shocks to money supplies. 16/ For example, a temporary increase in the domestic money supply will lead to a temporary depreciation of the domestic currency, followed by expectations of appreciation. The only case in which the nominal exchange rate follows a driftless random walk in the flexible-price model is when the forcing variables are described by driftless random walks. 17/

The solutions in Table 1 also imply that any nonstationarity in nominal exchange rates derives from nonstationarity in the forcing variables. In each solution, the order of integration of the exchange rate matches that of the money supply process. 18/ If, for example, relative money supplies are integrated of order two, exchange rates have the same order of integration. The real exchange rate is exogenous in this version of the flexible-price monetary model. If purchasing power parity holds in the long run, the real exchange rate will be a stationary variable. However, if there are permanent real shocks, such as changes in tastes or productivity, it will be nonstationary.

An important extension of the monetary model allows for short-run price rigidities. In an early paper Dornbusch(1976) modified the flexible-price monetary model to allow for price stickiness and attempted to rationalize the high degree of volatility in nominal exchange rates over the floating rate period. The major features of Dornbusch’s sticky-price model are illustrated in Table 2 using a simple stochastic model that includes a Phillip’s curve and an aggregate demand equation. 19/ Solutions to the model are shown for the case in which the domestic money supply follows a first-order auto-regressive process and all other forcing variables are constant. 20/ In contrast to the flexible-price model, the Dornbusch model includes intrinsic dynamics associated with slowly adjusting commodity prices.

Table 2.

Sticky-Price Monetary Model

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The solutions to the sticky-price model presented in Table 2 have two important implications for short-run exchange rate behavior. The first is that nominal and real exchange rates may overshoot in response to innovations in the money supply. A sufficient condition for nominal exchange rate overshooting in the model presented in Table 2 is that the money supply follows a random walk so that all changes in the money supply are expected to be permanent. 21/ The second implication is that there can be predictable nominal and real exchange rate movements associated with the slow adjustment of commodity prices. These can arise with the money supply itself following a random walk (with ρ = 1 in Table 2), implying that deviations of the exchange rate from a random walk occur even if forcing variables are entirely unpredictable, and reflect the intrinsic dynamics of the model.

If purchasing power parity is assumed to hold in the long run, the real exchange rate is a stationary variable in the sticky-price model. It will, however, be closely correlated with the nominal exchange rate in the short run and display predictable movements. If there are permanent real shocks, changing for example the level of potential output, the real exchange rate will be nonstationary in this model.

Given the potential importance of real shocks for exchange rates, the flexible-price monetary model has been adapted to allow a more explicit role for real factors, giving rise to inter-temporal equilibrium models in which nominal and real exchange rates are simultaneously determined. 22/ These applications continue to assume price flexibility but allow for shocks such as productivity improvements or increases in real government spending to explicitly influence nominal and real exchange rates. In addition, the models include intrinsic dynamics associated with factors such as slow adjustment of investment spending and inter-temporal labor substitution.

Portfolio models, The key assumptions of the portfolio models are that (interest-earning) assets denominated in different currencies are imperfect substitutes and residents of each country have a preference at the margin for assets denominated in their own currencies. 23/ An important source of systematic movement in nominal and real exchange rates in these models derives from intrinsic dynamics associated with current account imbalances. When residents of a country have a preference at the margin for their own commodities and assets, countries with current account surpluses may have continually appreciating real and nominal exchange rates due to the need to offset incipient excess demands for domestic assets and goods. 24/ More generally, these models suggest that nominal exchange rates will depend on the accumulation of outside domestic assets (money and bonds) and (net) foreign bonds. Given that outside domestic assets are created through the fiscal balance while (net) foreign assets are acquired by running current account surpluses, the models imply that the evolution of the exchange rate may be associated with fiscal and current account balances. The stochastic processes nominal and real exchange rates follow then depend on the stochastic processes describing asset supplies—money, public interest-bearing debt, and foreign assets, as well as the intrinsic dynamics of the models. 25/

Neither the monetary nor portfolio models reviewed in this section implies that nominal or real exchange rates should be characterized by random walks. Predictable movements in exchange rates can arise either from extrinsic dynamics in the forcing variables or from intrinsic dynamics that spread the adjustment to shocks over time. If purchasing power parity is assumed to hold in the long run, real exchange rates will display predictable fluctuations around a fixed mean. But if allowance is made for permanent real shocks, such as changes in tastes or technology, real exchange rates will be nonstationary. The stochastic process for nominal exchange rates also displays predictable movements in these models which can either be around a fixed or a varying mean, depending on the processes followed by forcing variables. Differences in monetary policy across countries are one reason why nominal exchange rates might be nonstationary. Divergences in fiscal policy as well as current account imbalances are another potential source of nonstationarity. Both nominal and real exchange rates are expected therefore to deviate from random walks on the basis of standard exchange rate models. 26/

The only models that predict a (driftless) random walk for nominal or real exchange rates, regardless of the stochastic process describing forcing variables, are the currency substitution and financial arbitrage models. Neither of these models, however, appears to have found its way into the mainstream of structural exchange rate models. In currency substitution models, the requirement that the expected pecuniary returns from holding domestic and foreign monies be equal when monies are perfect substitutes implies that the change in the nominal exchange rate should be unpredictable. 27/ In financial arbitrage models, the equalization of expected real interest rates on domestic and foreign bonds is interpreted as requiring that the expected change in the real exchange rate be unpredictable. 28/ It is unclear, however, why real interest rates measured in terms of different consumption baskets should be equalized, suggesting that the financial arbitrage models may not provide a firm basis for expecting real exchange rate changes to be entirely unpredictable.

III. Univariate Time Series Tests

This section presents evidence on the time series processes followed by nominal and real exchange rates of the dollar in terms of other major currencies. 29/ After briefly examining measures of the variability of dollar exchange rates over the recent floating rate period, the section presents standard tests for the stationarity of exchange rates and departures from a random walk. This is then followed by the application of new and more powerful tests intended to detect evidence of mean reversion in exchange rates over long periods. 30/

An examination of the time series processes followed by exchange rates is useful for a number of reasons. First, in the absence of speculative bubbles, any nonstationarity in exchange rates must ultimately be determined by nonstationarity in the forcing variables appearing in structural exchange rate models. Explaining the long-run behavior of exchange rates then requires identifying other economic variables with similar orders of nonstationarity. Second, particular processes such as the random walk are frequently regarded as carrying strong implications for the shocks that have driven exchange rates, and the economic models that are appropriate for analyzing their behavior. If one is to view the exchange rate as an equilibrium exchange rate, then the random walk property implies that all shocks driving them must be permanent. 31/ Further, for the real exchange rate such a finding argues for the prevalence of real rather than nominal shocks, since the latter would be expected to have only transitory effects. As noted earlier, moreover, the random walk finding argues against the applicability of models—such as the sticky-price monetary model of Dornbusch(1976)—that ascribe a major role to short-run disequilibrium dynamics, since such dynamics induce systematic movements in exchange rates. Third, specific hypotheses such as purchasing power parity can be tested by looking for evidence of mean reversion in real exchange rates. 32/

The data series for the study comprise the four bilateral spot exchange rates between the U.S. dollar and the deutsche mark, Japanese yen, pound sterling, and Canadian dollar. In order to ensure comparability with other studies, real exchange rates are measured using consumer prices. The data are sampled quarterly over the period 1974:1 through 1989:4 and are taken from International Financial Statistics.

Before applying the statistical tests, it is useful to consider some summary statistics for the data series. Confirming the view that exchange rates behave like asset prices, Tables 3a and 3b show that the variance of quarterly changes in nominal exchange rates has been considerably larger than that in relative consumer prices. 33/ For some bilateral rates, it is almost 75 times the variance of changes in commodity prices. As a result, the variance of changes in the real exchange rate is dominated by that of changes in the nominal exchange rate. In fact, the variance of changes in real and nominal exchange rates are essentially equal, and there is almost perfect contemporaneous correlation between them. 34/ The table also shows that while there is some evidence of drift in exchange rates, in no case is it statistically significant.

Table 3a.

Summary Statistics for the Log of U.S. Dollar Exchange Rates and Relative Price Levels Quarterly, 1974:1 to 1989:4

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Table 3b.

Correlation of Changes in U.S. Dollar Real and Nominal Exchange Rates

Quarterly, 1974:1 to 1989:4

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The reported correlations are between contemporaneous values of the first difference of real exchange rate and 0 to 8 lags of the first difference of nominal exchange rates.

The first set of tests we apply are for unit-root nonstationarity in the time series processes for nominal and real exchange rates and relative prices (Table 4). 35/ As was noted earlier, none of the structural exchange rate models carries strong predictions as to whether we would expect to find stationarity in nominal exchange rates, with the issue an empirical one. Differences in monetary growth rates across countries as well as current account imbalances, however, suggest that it is quite likely nominal exchange rates will be nonstationary. Whether real exchange rates would be expected to be stationary is also an empirical question. Only in the case of models built on purchasing power parity is there a presumption that real exchange rates will be stationary.

Table 4.

Test Statistics for the Null Hypothesis of a Unit Root 1/

Quarterly, 1974:1 to 1989:4

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All regressions include a constant and three seasonals.

Regression includes two lags.

For residuals from the augmented Dickey-Fuller regression.

Based on the test statistics recorded in Table 4, the null hypothesis of a unit root cannot be rejected for any nominal or real exchange rate. Given that movements of the real exchange rate are dominated by those of the nominal exchange rate, we also examined whether the nonstationarity in real exchange rates reflected a single source of nonstationarity in nominal exchange rates. Accordingly, we performed unit root tests on the relative price series separately. These tests were unable to reject the null hypothesis of nonstationarity in relative prices with the exception of those for the United Kingdom. Overall, the results are consistent with non-stationarity in the bilateral nominal and real exchange rates of the dollar with the yen, deutsche mark, and Canadian dollar. Only in the case of the real exchange rate with the United Kingdom is there some ambiguity, with tests on the price component of the real exchange rate suggesting that nonstationarity in the real rate derives only from nonstationarity in the nominal exchange rate.

An inability to reject unit root nonstationarity in real and nominal exchange rates does not imply that exchange rates have no predictable components. Even when these series have unit roots, the changes in these series could be serially correlated implying that the history of each series would contain information for predicting its future. 36/ In addition, the series could be described by drift. The next set of tests are for the null hypothesis that exchange rates follow random walks. 37/ As indicated earlier, most exchange rate models do not predict that exchange rates will follow random walks except under very special conditions. Essentially all the forcing variables must follow random walks and there must be no intrinsic dynamics associated with factors such as sticky prices and asset accumulation.

The tests presented in Table 5 are for the null hypothesis that each series (with the exception of relative prices with the United Kingdom 38/) is a random walk, under the maintained hypothesis that each series has a unit root. 39/ They are based on the autocorrelations of the first difference of each series and testing whether these sample autocorrelations are statistically different from zero. 40/ When these autocorrelations are significantly different from zero, changes in the variable are judged to have systematic components, and the null hypothesis of a random walk is rejected. 41/ According to these tests, the null hypothesis of a random walk cannot be rejected for any bilateral exchange rate, with the exception of that with the pound sterling. In all reported cases, however, there is strong and statistically significant evidence against a random walk in relative price levels. Although relative prices are nonstationary, they appear to have strong systematic components that are easily detected by the statistical tests. 42/

Table 5.

Q - Statistic for Sample Autocorrelations in First Differences 1/2/

Quarterly, 1974:1 to 1989:4

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The Ljung-Box Q-statistic is constructed as

Q(N)=T(T+2)Σj=1N1Njρ^j2

where T is the number of observations, N the number of lags employed and ρ^j is the estimated autocorrelation coefficient at lag j. Q(N) is distributed approximately as a chi-square with N degrees of freedom.

All first differences were first regressed on a constant and three seasonals, and the Q-statistic was then computed for the residuals.

The results from these tests suggest that in all cases other than the bilateral rates with the United Kingdom, nominal and real exchange rates are indistinguishable from a random walk. This is the case even though there is evidence suggesting systematic movements in relative prices. The failure to reject the hypothesis that the real exchange rate follows a random walk reflects the fact that the time series properties of relative prices are swamped by those of nominal exchange rates.

Given the potential importance of a random walk in nominal and real exchange rates, the next set of tests attempt to determine whether there is any evidence of departures from a random walk in the long run. The central advantage of the tests is that—by focusing on the longer-run components of the data—they should be able to detect small and slowly evolving deviations from a random walk. The tests thus address the problem that while exchange rates may not be exactly described by random walks they may be sufficiently close to be indistinguishable in the short run. The use of long-run data has to be weighed, of course, against the small number of observations at long lags. Nevertheless, the tests have been applied with some success in measuring the size of the unit root component in real GNP and in stock prices, and some investigators have applied the tests to measuring the permanent component of real exchange rates. 43/

The tests are based on estimating the ratio of 2π times the spectral density of the first differences of a series at frequency zero, to the variance of its first differences. This ratio or ‘normalized’ density can be written in terms of the autocorrelations of the series as

f(0)=1+2Σi=1ρi,(8)

where ρi is the autocorrelation at lag i of the first difference of the series. The spectral density cannot become negative by construction, and can assume any positive value or zero. When the spectral density exceeds one in value there is a preponderance of positive correlations, and conversely when it is below one negative correlations dominate.

There are two values of the spectral density function that are of particular interest. One arises when a series follows a random walk, and all the autocorrelations of the first differences are zero. In this case the spectral density will equal unity, suggesting that departures from a value of unity can be used to detect deviations from a random walk. 44/ The other arises when the level of a series follows any stationary stochastic process. The spectral density in this case will be zero, implying that the statistic can also be used to detect mean reversion and stationarity. 45/

Estimation of the spectral density requires estimates of the autocorrelation parameters at different numbers of lags. Ideally, the number of lags should be large enough to pick up mean reversion over very long periods, but this has to be weighed against the small number of observations at long lags. In what follows we work with a maximum lag length of 44 quarters, that is 11 years. Given a finite number of autocorrelations, the estimates are based on constructing the statistic

f(0)=1+2Σj=1nw(n,j)ρ^j,

where n denotes the number of autocorrelations and w(n,j) represent a set of weights given by

w(n,j) = (n+1-j)/(n+1).

Use of these weight has the effect of giving smaller weight to autocorrelations at long lags. For further details, see Huizinga (1987).

The estimates of the spectral density function are presented in Charts 1 and 2. There are a number of rather interesting features of the estimates. First, reflecting the high correlation between nominal and real exchange rates, the estimates of the spectral density functions are essentially the same for nominal and real exchange rates. Second, all exchange rates with the exception of those with the Canadian dollar show a distinct hump-shaped pattern, with the spectral density first rising above unity and then falling below it. This pattern implies a preponderance of positive correlations at short lags and negative correlations at long lags, and suggests that exchange rate changes in the short run tend to be reinforced, while in the long run they tend to be reversed.

Chart 1.
Chart 1.

Estimates of f(0) for bilateral dollar nominal exchange rates with the United Kingdom, Germany, Canada, and Japan.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

Chart 2.
Chart 2.

Estimates of f(0) for bilateral dollar real exchange rates with the United Kingdom, Germany, Canada, and Japan.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

The possibility of positive autocorrelation in short-run exchange rate changes was noted by Mussa (1979) in his characterization of empirical regularities under floating exchange rates. Mussa suggested that such correlations were consistent with bandwagon effects in exchange markets, whereby once a rate started moving in a certain direction speculators would move it further. More recently, it has been suggested that the correlations may reflect very short-term bubbles in exchange markets. 46/ The preponderance of negative correlations at long lags, on the other hand, suggests mean reversion in nominal and real exchange rates over very long periods. 47/ The exceptions are the Canadian dollar exchange rates where the finding of values for the spectral density above unity at short and long lags suggests a dominance of positive autocorrelations and no evidence of mean reversion.

Given the spectral density function declines below one, the results suggest that there is evidence for mean reversion over the long run for three of the four exchange rates examined. As such there may be grounds for believing that the apparent random walk in nominal and real exchange rates is masking a rather slow adjustment process, and that the use of observations at long lags allows the random walk to be rejected. Somewhat interestingly, the results suggest that there is mean reversion in both nominal and real rates. As far as we know, the possibility of nominal rates being stationary has not been noted in the literature. Taken literally, stationarity in nominal rates implies the absence of any permanent differences in nominal shocks across countries.

There is, however, a rather basic difficulty in establishing the statistical significance of the deviations of f(0) from unity as is evident from Tables 6-7 which present (two) standard errors for the estimates in charts 1-2. 48/ Given the small number of degrees of freedom at long lags and the large variance of the nominal exchange rate, the estimates have large standard errors and it is virtually impossible to reject the null hypothesis that the spectral density is equal to unity. This is the case even in those instances where the spectral density is close to 0.3 in the long run. The only cases for which the null hypothesis could be rejected are the nominal and real rates with the Canadian dollar. In these cases, however, the finding that the spectral density function is above one suggests positive rather than negative autocorrelations, and no evidence of mean reversion.

Table 6.

Estimates of f^N(0) and Their Standard Errors, σ^, for Bilateral Dollar Nominal Exchange Rates 1/

Quarterly, 1974:1 to 1989:4

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Standard errors are constructed under the null hypothesis of the nominal exchange rate being a random walk, so that there is no autocorrelation in the first differences.

Table 7.

Estimates of f^N(0) and Their Standard Errors, σ^, for Bilateral Dollar Real Exchange Rates 1/

Quarterly, 1974:1 to 1989:4

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Standard errors are constructed under the null hypothesis of the real exchange rate being a random walk, so that there is no autocorrelation in the first differences.

From a statistical point of view, the unavoidable conclusion is that the null hypothesis of a random walk cannot be rejected at standard significance levels. Based on the estimates of the spectral density function, however, it could be conjectured that there may be mean reversion in the data but the tests applied simply do not have the power to reject the random walk. Nevertheless, we proceed under the assumption that exchange rates are statistically indistinguishable from a random walk and are nonstationary, while noting that it could be misleading to place too much weight on the random walk in trying to identify the kinds of shocks that have moved exchange rates, or the models that are appropriate for analyzing them. The tests that are available possibly do not have the power to discriminate between a random walk and interesting alternative hypotheses that might imply ‘small’ deviations from a random walk.

IV. Multivariate Time Series Tests

A univariate random walk in exchange rates only implies that exchange rates cannot be predicted from their own history. This section takes an alternative approach by investigating whether movements in the dollar against other major currencies can be accounted for by the forcing variables included in exchange rate models. The approach thus seeks to determine whether the longer-run movements in exchange rates are linked to long-run movements in these variables, and whether deviations of exchange rates from any long-run cointegrating relationships contain information for predicting future exchange rate changes.

The approach is based on the methodology of cointegration and error-correction as developed by Engle and Granger (1987). Cointegration can be defined as follows. The components of the vector Zt are said to be cointegrated of order d,b (denoted Zt ~ CI(d,b)) if all components of Zt are I(d) and there exists a vector α ≠ 0 such that Xt = α’Zt - I(d-b), b>0. The vector α is then called the cointegrating vector. 49/ Several observations can be made about the concept of cointegration. Cointegration applies to the long-run relationships between variables, and describes a situation in which variables that may drift apart in the short run have a tendency to move together in the long run. As such it is well suited for determining the long-run link between the forcing variables in structural exchange rate models and exchange rates. Second, a necessary condition for exchange rates to be cointegrated with a set of variables is that all these variables are integrated of order one given that the exchange rate is an I(1) variable. It is possible, however, to have variables that are integrated of a higher order, provided that there are cointegrating relationships among these variables that reduce their collective order of integration to that of the exchange rate. 50/ Finally, if cointegrating relationships can be found it is possible to specify an error-correction mechanism in which either the exchange rate or the forcing variables can be used to predict the rate at which the economy will return to long-run equilibrium. 51/

The methodology of cointegration is applied in this section in three steps. In the first, the forcing variables identified in structural exchange rate models are individually tested to determine their order of integration. In the second, tests are undertaken to determine whether exchange rates are cointegrated with these variables, yielding a number of cointegrating relationships. Finally, at the last stage, we test to see whether deviations of exchange rates from long-run values implied by the cointegrating vectors contain information for predicting future movements in exchange rates.

Based on the review of structural exchange rate models, four sets of forcing variables are tested: money supplies and interest rate differentials; fiscal variables, including government spending, fiscal deficits, and stocks of outstanding public sector debt; current account balances, in level terms and cumulated to give net foreign asset positions; and measures of economic activity or productivity, including real GNP/GDP, industrial production, labor productivity, and—as a proxy for capital productivity—real share prices. Given the large number of possible combinations of these variables, some simplification is obviously necessary. To make the number of tests manageable, all domestic variables are measured relative to their foreign counterparts. In the case of money supplies or interest rates this is a reasonable approximation: the (logarithmic) difference between domestic and foreign money supplies appears in many structural models as does the differential between domestic and foreign interest rates. It is less apparent, however, how variables such as government expenditures, fiscal deficits, and public debt stocks should be treated. The approach adopted was to use the difference between domestic and foreign government expenditures as shares of GNP, and differences in the ratio to GNP of fiscal deficits and public debt across countries. 52/ Details on how other variables are measured are provided in the accompanying tables.

The results for the order of integration of forcing variables are presented in Table 8. For each variable, Dickey-Fuller and augmented Dickey-Fuller tests were used to test the null hypothesis that a given series was either integrated of order one or two. The results can be summarized as follows. Among the monetary variables, relative money supplies in all cases are integrated of order one, implying that they can potentially account for the nonstationarity in exchange rates. Interest differentials, both nominal and real, on the other hand, appear to be stationary and cannot therefore be cointegrated with exchange rates. The finding that real interest rate differentials are stationary is consistent with the findings of Meese and Rogoff (1985). 53/ They interpreted this result to imply that the variance of the real exchange rate reflects changes in the (expected) long-run real exchange rate. 54/

Table 8.

Estimated Order of Integration for Forcing Variables from Dickey-Fuller and Augmented Dickey-Fuller tests 1/

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Note:

1. All data are from International Financial Statistics.

2. All variables are measured as the the value for the United States minus the value for the foreign country.

3. Variable Notation: G denotes nominal government consumption; Y denotes Gross National Product at market prices; D denotes the central government balance; ∫ is used to denote cumulative value of; Y’ denotes Industrial Production; RV denotes real Gross National Product; RV denotes real industrial share prices; (X/L) denotes output per man hour in manufaturing; ULC denotes unit labor costs in manufacturing; CAB denotes the domestic currency value of the multilateral current account balance.

4. Table entries. “I(n)” denotes variable is integrated of order n, i.e., needs to be differenced n times to obtain a stationary variable; “n.a.” denotes data was unavailable.

5. The Dickey-Fuller and Augmented-Dickey-Fuller Regressions include a constant and three seasonals.

6. Real interest rates are computed using an AR(2) process for predicting inflation.

All the fiscal variables, with the exception of the (relative) U.K. fiscal deficit to GNP ratio, are integrated of order one and thus potential candidates for explaining the long-run variance of exchange rates. We interpret this to mean that differences in fiscal policy across countries hold the potential for explaining the long-run variance of exchange rates, independently of any short-run impact on interest rate differentials. The current account and cumulative net foreign asset positions of major industrial countries are also integrated of order one. All the measures of economic activity and productivity with the exception of relative unit labor costs for the United Kingdom are also characterized by the same order of integration as exchange rates. Explanations for the dollar’s movements based on real factors such as productivity developments may therefore be able to explain long-run exchange rate movements.

The tests for cointegration are based on the maximum likelihood procedures developed by Johansen (1988), and Johansen and Juselius (1989). Unlike the Granger-Engle procedure, which presumes the (potential) presence of exactly one cointegrating vector among a set of variables, estimated by regressing one of the variables on the contemporaneous levels of the other variables, the Johansen procedure allows for as many cointegrating vectors as the number of variables.

Before considering the relationships between exchange rates and other economic variables, it is interesting to determine whether the nonstationarity in dollar exchange rates derives from a single source such as U.S. economic policies, i.e., there is a “dollar phenomenon”, or reflects a more complex interaction with economic developments in particular countries. Accordingly, we tested whether the bilateral exchange rates of the dollar were themselves cointegrated and reflected a common source of nonstationarity. The results from these tests suggested that dollar exchange rates are not cointegrated, which means that long-run movements in the dollar are not dominated by developments in the United States and hence that if there are explanations of long-run movements in dollar exchange rates, they must be of a (relative) country specific nature. 55/

The results from individual bivariate cointegration tests of each exchange rate and other economic variables are summarized in Table 9. In order to highlight the central results, the table does not show the values of individual test statistics and only indicates whether cointegrating relationships were found for particular variables. 56/ The results suggest that little of the nonstationarity in exchange rates mirrors that of other economic variables in the long run. Nevertheless a number of cointegrating vectors are identified which carry implications for understanding exchange rate behavior.

Table 9.

Results of Bivariate Cointegration Tests for Exchange Rates and Other Economic Variables

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Notes:

1. All variables are measured as the the value for the U.S. minus the value for the foreign country.

2. Variable Notation: G denotes nominal government consumption; Y denotes Gross National Product at market prices; D denotes the central government balance; ∫ is used to denote cumulative value of; Y’ denotes Industrial Production; RY denotes real Gross National Product; RV denotes real industrial share prices; (X/L) denotes output per man hour in manufaturing; ULC denotes unit labor costs in manufacturung; CAB denotes the domestic currency value of the multilateral current account balance.

3. Table entries:

  • “—”: denotes no cointegrating vector found;

  • “Yes”: denotes at least one cointegrating vector found at the 5 percent significance level,

  • “n.a.”: denotes either data unavailable or that the variable was found to be stationary (see Table 8).

4. A prior constant and three seasonals were first removed from the data.

Confirming results found with structural exchange rate models, monetary variables do not account for the long-run variance of nominal and real exchange rates. 57/ The lack of cointegration between real exchange rates and monetary variables is to be expected on the basis of most structural models. The inability to find cointegration between nominal exchange rates and relative money supplies suggests that differences in monetary policy across major industrial countries have not been a major factor behind trends in nominal exchange rates over the recent floating rate period. This finding is also consistent with the observation in Section III that nominal exchange rates show some tendency for mean reversion, which implies an absence of permanent differences in nominal shocks across countries.

Fiscal variables explain some of the long-run variance in real and nominal exchange rates in all cases other than the U.S. dollar-yen exchange rate. In cases where cointegrating relationships are found, they either reflect the ratio of government spending to GNP or fiscal deficit positions. No cointegrating relationships were found between public sector debt stocks and nominal or real exchange rates. Current account variables account for some of the long-run variance in exchange rates, particularly in the case of the exchange rate with the Canadian dollar. There is also some evidence of cointegrating relationships for the deutsche mark and Japanese yen exchange rates but, contrary to the predictions of portfolio models, current account variables do not systematically account for the longer-run variance of most nominal and real exchange rates. Among the variables used to measure economic activity and productivity, there is a small number of cointegrating vectors. The real exchange rates with Canada and Japan appear to be cointegrated with labor productivity; those with Germany and the United Kingdom are cointegrated with the proxy for capital productivity. These relationships are open to a number of interpretations, but one possibility is that they reflect the impact of productivity shocks on the real exchange rate. 58/

In order to allow for the possibility that exchange rates could be cointegrated with a combination of the other economic variables, we also estimated a number of multivariate cointegrating vectors in which the nominal exchange rate was related to a complete set of monetary, fiscal, current account, and real productivity measures. For each bilateral nominal exchange rate at least two cointegrating vectors were found that were significant at the one percent level. One of these was picked and employed to measure the long-run level of the exchange rate as accounted for by the forcing variables. 59/ Deviations of the actual exchange rate from the long-run cointegrating vector can then be interpreted as measuring the equilibrium error in the system. Charts 3-6 plot the actual exchange rate and their long-run values as implied by the cointegrating vectors. As indicated in the charts, there are at times substantial and long-lived deviations from the long-run values, which tend at times to display greater variability than exchange rates.

Chart 3.
Chart 3.

(Log of) Actual and Long-run Value of Nominal Dollar-Japanese Yen Exchange Rate.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

Chart 4.
Chart 4.

(Log of) Actual and Long-run Value of Nominal Dollar-Deutsche Mark Exchange Rate.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

Chart 5.
Chart 5.

(Log of) Actual and Long-run Value of Nominal Dollar-Canadian Dollar Exchange Rate.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

Chart 6.
Chart 6.

(Log of) Actual and Long-run Value of Nominal Dollar-Pound Sterling Exchange Rate.

Citation: IMF Working Papers 1990, 102; 10.5089/9781451948349.001.A001

There are two possible interpretations of the equilibrium errors shown in Charts 3-6. One is that they reflect bubbles in exchange markets, suggesting that exchange rates have at times deviated substantially from long-run economic fundamentals. An alternative explanation is that they reflect the systematic short-run movements in exchange rates identified in exchange rate models. In order to test this latter hypothesis, error-correction equations for exchange rates were estimated in which the change in each bilateral nominal exchange rate was related to its own equilibrium error and the changes in the forcing variables. Consistent with the methodology of cointegration and error-correction, these equations also included nominal interest rate differentials as a potential variable explaining short-run exchange rate movements. 60/

Ds(t)=Σi=1kaiUti+Σi=0kbiRtiST+Σi=0kciRtiLT+Σi=1kdiDZti+εt(9)

As indicated in equation (9) each error-correction equation relates the change in the exchange rate to an error-correction term that measures the deviation of the exchange rate from the long-run value implied by the forcing variables, U, short and medium-run nominal interest differentials, RST and RLT, and lagged changes in the forcing variables, Z. The estimates of this equation for each bilateral nominal exchange rate are recorded in Table 10. Several features of the estimates are noteworthy. First, all coefficients on the equilibrium error are negative, so that the current change in the exchange rate is a negative function of the deviation of last period’s exchange rate from it’s long-run equilibrium value. The estimated coefficient is, however, insignificant for the bilateral dollar rate with the pound sterling; for the deutsche mark and the Canadian dollar, the coefficients are only one standard error away from zero implying a marginal significance level of 30 percent; for the yen the coefficient is significant at 1 percent. The coefficient of 0.2 for the dollar-yen rate implies that 20 percent of any deviation of the exchange rate from its long-run level can be expected to be dissipated in the next quarter. Second, the coefficients on short and medium-term interest differentials are insignificant for all rates with the exception of the yen, where both differentials are significant. Third, the fit of the equations is on the whole rather poor with between 29 percent—of the Canadian dollar’s—to 59 percent—of the yen’s—movements being explained by the error correction equations.

Table 10.

Estimates of Error Correction Equation for Nominal Dollar Exchange Rates 1/

Dst=a1ut1+Σi=01biRtiST+Σi=01ciRtiLT+Σi=13diDZti+εt

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t-ratios are in parentheses.

V. Conclusions

This paper has addressed a number of questions about the time series processes dollar exchange rates are likely to follow, and the conditions under which they would be expected to be described by random walks. Most structural exchange rate models were shown to generate departures from a random walk, even when all shocks were expected to be permanent. The random walk in exchange rates can therefore be viewed neither as an implication of market efficiency, nor as a guide to the kinds of shocks that have been important in moving exchange rates.

An examination of the univariate time-series properties for exchange rates showed that while there was some evidence for departures from a random walk and for the presence of mean reversion, there was a basic difficulty in establishing the statistical significance of the deviations. Under these conditions, we are forced to conclude that exchange rates are statistically indistinguishable from random walks, though we are reluctant to draw strong conclusions from this result.

Finally, we tested whether the forcing variables in structural exchange rate models were characterized by a similar order of nonstationarity as exchange rates. While many of these variables were found to be integrated of the same order as exchange rates, and hence could potentially account for their nonstationarity, only a small number of cointegrating relationships were found. Moreover, deviations from these cointegrating relationships were found to play only a limited role in predicting changes in dollar exchange rates.

Structural Models of the Dollar
Author: Mr. Charles Adams and Mr. Bankim Chadha
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    Estimates of f(0) for bilateral dollar nominal exchange rates with the United Kingdom, Germany, Canada, and Japan.

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    Estimates of f(0) for bilateral dollar real exchange rates with the United Kingdom, Germany, Canada, and Japan.

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    (Log of) Actual and Long-run Value of Nominal Dollar-Japanese Yen Exchange Rate.

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    (Log of) Actual and Long-run Value of Nominal Dollar-Deutsche Mark Exchange Rate.

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    (Log of) Actual and Long-run Value of Nominal Dollar-Canadian Dollar Exchange Rate.

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    (Log of) Actual and Long-run Value of Nominal Dollar-Pound Sterling Exchange Rate.