On Interpreting the Random Walk and Unit Root in Nominal and Real Exchange Rates
Author:
Mr. Charles Adams https://isni.org/isni/0000000404811396 International Monetary Fund

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Mr. Bankim Chadha
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Publication Date:
01 Jan 1991
eISBN:
9781451930801
Language:
English
Keywords:
SP; exchange rate; price level; random walk property; driftless random walk; a number of time-series test; random walk finding; Real exchange rates; Exchange rates; Monetary base; Forward exchange rates; Spot exchange rates
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The random walk property of exchange rates is regarded as carrying implications for the kinds of shocks that have driven exchange rates and the models appropriate for analyzing their behavior. This paper describes the results of stochastic simulations of Dornbusch's (1976) sticky-price monetary model, calibrated for representative parameter values for the United States, The paper shows that when all shocks are nominal, the model generates time series for real and nominal exchange rates that are statistically indistinguishable from random walks and that cointegration tests can provide misleading information about long-run relationships between the variables in the model.

Abstract

The random walk property of exchange rates is regarded as carrying implications for the kinds of shocks that have driven exchange rates and the models appropriate for analyzing their behavior. This paper describes the results of stochastic simulations of Dornbusch's (1976) sticky-price monetary model, calibrated for representative parameter values for the United States, The paper shows that when all shocks are nominal, the model generates time series for real and nominal exchange rates that are statistically indistinguishable from random walks and that cointegration tests can provide misleading information about long-run relationships between the variables in the model.

On Interpreting the Random Walk and Unit Root in Nominal and Real Exchange Rates

THE representation of many economic time series, including nominal and real exchange rates, by unit-root processes has become increasingly common in the literature. (See, for example, Meese and Singleton (1982), Huizinga (1987), Kaminsky (1987), Meese and Rogoff (1985), and Adams and Chadha (1991).) An important empirical regularity that has emerged over the recent floating rate period is that both nominal and real exchange rates are closely approximated by random walks, which are a special case of unit-root processes.1 (See, for example, Mussa (1984), Levich (1979), and Adams and Chadha (1991).)

In the view of several investigators, the random walk carries strong implications for identifying the kinds of shocks that have driven exchange rates. (See, for example, Campbell and Clarida (1987) and Kaminsky (1987).) If exchange rates are viewed as being in equilibrium, the random walk property implies that all shocks driving them must be permanent. Further, the characterization of real exchange rates by random walks has been viewed as consistent with a primary role for real, rather than nominal, shocks, since the latter would be expected on the basis of most models to have only transitory effects on real variables. (See Stockman (1980, 1983), Huizinga (1987), and Campbell and Clarida (1987).) Moreover, the random walk finding has been used to argue 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 and would be expected to induce systematic or predictable movements in exchange rates.

Using simulation techniques, this paper shows that a stochastic version of Dornbusch's (1976) overshooting model, configured for representative parameter values for the United States and in which all shocks are nominal, is capable of generating time series for both real and nominal exchange rates that are statistically indistinguishable from random walks. This is the case even though the model generates a “true” process for the nominal exchange rate that has a unit root with systematic components (and, hence, deviates from a random walk), and the time series for the real exchange rate generated by the model follows a stationary stochastic process. The simulations serve, therefore, as a cautionary note against drawing strong inferences about the kinds of shocks affecting exchange rates on the basis of statistical tests that cannot reject a unit root or random walk in exchange rates. In short, relevant alternative hypotheses about exchange rate behavior—as typified in this paper by the Dornbusch model—may imply time-series processes for nominal and real exchange rates that are too close to a random walk to allow one to place much confidence in an inability to reject the null hypothesis of a random walk. In addition, the simulations raise questions as to the small-sample properties of some of the cointegration tests that have been applied to exchange rates.

I. Stochastic Simulations

The simulations were carried out using a stochastic version of Dorn-busch's original overshooting model, configured with representative parameter values for the United States.2 Under the assumption that the money supply follows a driftless random walk and given the assumed parameter values, the solutions to the model have the following key features.3 (1) All of the nominal variables in the model, with the exception of the nominal interest rate, are described by unit-root processes, but deviate from random walks as a result of the intrinsic dynamics associated with sticky-price adjustment.4 (2) The real exchange rate (and all other real variables in the model) are stationary and integrated of order zero— the real exchange rate follows an AR(1) process with a coefficient equal to 0.9375.5 (3) While the nominal variables contain both transitory (stationary) and permanent (random walk) components and may drift apart from the money supply in the short run, they are cointegrated with the money supply in the long run.6 (4) In response to innovations in the money supply, both the nominal and real exchange rate overshoot their long-run equilibrium values.7

Using the random number generator on PC-TSP, simulations were carried out for 100 runs of the model at three different sample sizes:8 a “small” sample size of 75 observations (approximately the number of quarterly observations since the beginning of the recent floating rate period); a “medium” sample size of 150 observations (almost 40 years); and a “large” sample size of 400 observations (100 years).9 Given the simulation results, a number of time-series tests were then applied to determine whether they were able to detect the small amount of systematic movement in the model's nominal variables and the stationarity of the real exchange rate. In addition, a number of cointegration tests were undertaken to examine whether they could identify the long-run relationships in the model.

The first set of tests that were applied are for unit roots in the stochastic processes for the variables in the model. To reduce the reliance on any one single test, three different unit-root tests were employed: Sargan-Bhargava, Dickey-Fuller, and augmented Dickey-Fuller tests.10 As discussed in the literature, the power of these tests depends on the form of the stochastic process describing the variables, with the possibility that in the case of variables with large moving-average errors, the tests may tend incorrectly to accept a unit root, even when a series is stationary (see Schwert (1988)). While this possibility may cause difficulties in the case of the nominal exchange rate, the forward rate, and the price level whose changes follow moving-average processes, there is no simple solution available, so we chose, given their widespread usage, to apply the standard tests.11 In any event, the real exchange rate follows a pure AR(1) process in this model, and the Dickey-Fuller and Sargan-Bhargava tests were set up for this case.

The results from the unit root-tests are summarized in Tables 1A, 1B, and 1C.12 The Dickey-Fuller and augmented Dickey-Fuller tests do not reject the null hypothesis of a unit root in the nominal variables in the model in a large number of cases at sample sizes of 75 or 150 observations (although the Sargan-Bhargava test shows a tendency to overreject a unit root with 75 observations.) The tests, however, tend to overreject the null hypothesis of a unit root in many of the nominal variables with 400 observations. Most worrisome is that with 75 and 150 observations, all of the test statistics are consistently unable to reject a unit root in the real exchange rate, even though its true process is stationary.13 The failure to reject a unit root in the real exchange rate reflects the fact that the deviations from a unit root implied by the model are “small.” As the number of observations is increased to 400, the Dickey-Fuller and augmented Dickey-Fuller correctly reject the null hypothesis of a unit root in the real exchange rate about 90 percent of the time. This result is tempered, however, by the fact that with 400 observations these tests also incorrectly reject the null of a unit root in the spot and forward exchange rate about 20 and 35 percent of the time. Given the results of these tests and with a small number of observations, one would correctly conclude that the model's nominal variables have unit roots. Most of the time, however, one would incorrectly conclude that the real exchange rate is a nonstationary unit-root process.14 Since one is unlikely to expect that nominal shocks could permanently influence the real exchange rate, one would turn naturally to real factors to explain their apparent nonstationarity.

The next set of tests we conducted are for the null hypothesis that each variable is described by a random walk under the maintained (and incorrect in the case of the real exchange rate) assumption that each series has a unit root. (For a recent application of these and other tests to exchange rates, see Adams and Chadha (1991).) The tests can be regarded as determining whether there are significant transitory components in the time series for each variable. If there are no transitory components, the series are judged to follow random walks. As noted earlier, the only series that is described by a random walk is the money supply. Exchange rates—spot and forward—contain systematic components as a result of the intrinsic dynamics of the model, but these may not, of course be large enough to be detected by the statistical tests. The price level contains large systematic and predictable components, given the assumption of price stickiness. The true process for the real exchange rate is characterized by systematic movements around a fixed mean, so the random walk should be rejected for this series.

Table 1A.

Tests for Null Hypothesis of a Unit Root: Sargan-Bhargava Statistic

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Note: See Note to Table 1C.
Table 1B.

Tests for Null Hypothesis of a Unit Root: Dickey-Fuller Statistic

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Note: See Note to Table 1C.
Table 1C.

Tests for Null Hypothesis of a Unit Root: Augmented Dickey-Fuller Statistic

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Note: The test statistics were computed for each of 100 runs of the model with the indicated number of observations. The mean, median, standard deviation, and extreme values describe the values of the respective statistics obtained over the 100 runs. The Sargan-Bhargava statistic (SB) was computed by regressing the variable on a constant and computing the Durbin-Watson for the residual. The Dickey-Fuller statistic (DF) was computed as the t-ratio of the coefficient on the lagged level, in a regression of the first difference of the variable on the lagged level. The augmented Dickey-Fuller statistic (ADF) was computed as the i-ratio of the coefficient on the lagged level, in a regression of the first difference of the variable on the lagged level and four lags of the first difference of the variable. All three test statistics have nonstandard distributions, and in practice various critical values have been emploved. We impose a uniform significance level of 5 percent, and as critical values: 0.334 for SB from Hall and Henrv (1988); -2.89 for both the DF and the ADF from Schwert (1988).

The results for the random walk tests are summarized in Tables 2A and 2B. The tests are based on the autocorrelations of the first difference of each series at successive lags and testing whether these sample autocorrelations are significantly different from zero, using the Box-Ljung Q-statistic. (See Granger and Newbold (1977).) The Q-statistic is computed at a lag length of 20 quarters and a longer lag length of 40 quarters to allow for longer-run systematic movements in the series. When the autocorrelations are significantly different from zero at these lag lengths, a series is judged to have systematic components, and the null hypothesis of a random walk is rejected. Because the tests are based on the first differences of the series (rather than their levels), standard distributional assumptions apply to the tests.

The results in Tables 2A and 2B suggest that under the maintained assumption that each series has a unit root, the null hypothesis of a random walk cannot from 90 to 95 percent of the time be rejected for either the nominal or real exchange rate. Furthermore, there is no tendency for the null hypothesis to be rejected more often when a larger number of observations is used. It is only in the case of nominal prices (which exhibit considerable inertia) that the tests consistently (and correctly) reject the random walk. The null hypothesis that the money supply follows a random walk is rejected about 5 percent of the time. Based on the Box-Ljung tests, one would conclude that both nominal and real exchange rates are subject only to permanent shocks. As Stockman (1980) has argued, such findings are consistent with a dominant role for permanent shocks that affect nominal and real exchange rates in the same way. One would be correct in assuming that the data are consistent with a primary role for permanent shocks (the money supply follows a random walk), but would of course be wrong in identifying the shocks as real.

Table 2A.

Tests for Null Hypothesis of No Autocorrelation: in First Differences: Box-Ljung Q-Statistic at 20 Lags

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Note: See Note to Table 2B.
Table 2B.

Tesis for Null Hypothesis of No Autocorrelation: in First Differences: Box-Ljung Q-Statistic at 40 Lags

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Note: The Box-Ljung Q-statistics were computed for each of 100 runs of the model with the indicated number of observations. The mean, median, standard deviation, and extreme values describe the values of the Q-statistics obtained over the 100 runs. The Box-Ljung Q-statistic is distributed approximately as a chi-squared variable, with the degrees of freedom equal to the number of lags employed in its construction. We impose a significance level of 5 percent, and the critical values are: at 20 lags 31.41; at 40 lags, 55.75.

The last set of tests are for cointegration between the nominal exchange rate and the money supply, and between the nominal exchange rate and the (one-period) forward exchange rate. As noted above, the unit-root nonstationarity in the true processes for spot and forward rates derives from a single source in the model (the unit root in the money supply). It is interesting to determine whether standard tests for cointegration are able to detect this common unit root. Following Engle and Granger (1987), we test for cointegration by estimating a cointegrating regression of the form yt = Axt + vt and testing to see whether the residual, vt, contains a unit root. Under the null hypothesis that yt, and xt are not cointegrated, vt will have a unit root. The Sargan-Bhargava, Dickey-Fuller, and augmented Dickey-Fuller tests are then used to test for a unit root in the residuals. If the null hypothesis of a unit root can be rejected, the variables are said to be cointegrated.15

The reintegrating regressions also provide estimates of the reintegrating parameter A in the equation yt = Axt + vt. Given the use of logarithms, this parameter represents the long-run elasticity of yt with respect to xt. As argued by Stock (1987), when series are cointegrated, estimates of the cointegrating parameter should be highly efficient and converge rapidly to their true values (super consistency). The cointegrating regression between the exchange rate and the money supply should therefore deliver a super-consistent estimate of the long-run elasticity of the exchange rate with respect to the money supply of unity. The cointegrating regression between the spot and forward exchange rates should also converge rapidly to its true long-run value of unity. Unfortunately, because the test statistics for the cointegrating parameters have nonstandard distributions, standard /-tests cannot be applied to test whether coefficients are significantly different from their “long-run” values (see Stock and Watson (1988)).

Tables 3A, 3B, and 3C summarize the results from the cointegration tests. Two features of the results stand out. First, the null hypotheses of no cointegration between the nominal exchange rate and money supply, and between the nominal exchange rate and (one-period) forward rate, are only rejected in a small number of cases at sample sizes of 75 and 150 observations. In short, the tests do not find much evidence suggesting cointegration between these variables. It is only with 400 observations in the case of the cointegration tests between the nominal exchange rate and the money supply that the null hypothesis of no cointegration is rejected least 50 percent of the time. Second, only in the case of spot and forward exchange rates—for which the null hypothesis of no cointegration cannot be rejected—do the cointegrating regressions give a “good” estimate of the true cointegrating parameter of unity. The estimate of the cointegrating parameter for the nominal exchange rate and money supply is heavily influenced by overshooting effects; it deviates substantially from its true value of unity, particularly with sample sizes of 75 and 150 observations.16 As the number of observations increases, however, it becomes closer to its true value. An unwary investigator using a small sample set would incorrectly conclude that there is, on average, little evidence consistent with cointegration between exchange rates and money and that long-run homogeneity was not supported. This investigator would no doubt be puzzled by the lack of cointegration between the nominal exchange rate and forward rate, since both series appear to have large permanent components and they move closely together—even in the short run,17

Table 3A.

Engle-Granger Tests for Null Hypothesis of No Cointegration: 100 Runs with 75 Observations

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Note: See Note to Table 3C.
Table 3B.

Engle-Granger Tests for Null Hypothesis of No Cointegration: 100 Runs with 150 Observations

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Note: See Note to Table 3C.
Table 3C.

Engle-Granger Tests for Null Hypothesis of No Cointegration: 100 Runs with 400 Observations

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Note: The Engle-Granger tests for cointegration were carried out on the data from each of 100 runs of the model with the indicated number of observations. The mean, median, standard deviation, and extreme values describe the values of the respective statistics obtained over the 100 runs. The Sargan-Bhargava statistic (SB), the Dickey-Fuller statistic (DF), and the augmented Dickey-Fuller statistic (ADF) were computed as in Table 2 on the residuals of a regression of the nominal exchange rate on the money supply, and the nominal exchange rate on the forward exchange rate, respectively. Since this is a two-step procedure, the critical values used were different from those in Table 2; at the 5 percent significance level they were 0.334 for the SB from Hall and Henry (1988): -3.37 for the DF; and -3.17 for the ADF from Hall and Henry (1988).

II. Conclusions

The purpose of this paper has been to show that a model in which prices are sticky is capable of generating time series for nominal and real exchange rates that are difficult to distinguish statistically from random walks in small- and medium-sized samples. This is the case even though these series do not follow random walks, and, by construction, the real exchange rate is stationary. The failure to reject the random walk hypothesis reflects the fact that the deviations from a random walk implied by the model are small. Under these conditions, statistical tests have little ability to detect small systematic movements in nominal and real exchange rates. The tests are also unable—except in very large samples—to reject a unit root in the real exchange rate, even though this variable, by construction, is stationary. In addition, certain difficulties were noted in the application of cointegration tests and the estimation of long-run cointegrating parameters in small samples.

Most important, the results suggest that considerable care should be used in drawing inferences from the random walk behavior of exchange rates or the apparent unit-root nonstationarity in real exchange rates that has been found over the recent floating rate period. The simulations illustrate that a sticky-price model is capable of generating time-series processes for nominal and real exchange rates that are sufficiently close to random walks and unit-root processes as to be—for all practical purposes—indistinguishable.

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*

Charles Adams, Assistant to the Director, Western Hemisphere Department, obtained his doctorate from Monash University, Australia.

Bankim Chadha is an Economist in the External Adjustment Division of the Research Department. He holds a doctorate from Columbia University.

The authors would like to thank Tamim Bayoumi, Eduardo Borensztein, Michael Dooley, Guy Meredith, and Mark Stone for helpful comments and discussions.

1

Using the Beveridge and Nelson (1981) decomposition, any unit-root process can be decomposed into a random walk and a stationary component. The random walk component measures the permanent component of the series, and the stationary component measures the transitory component. A random walk is thus a special case of a unit-root process in which there is no stationary component.

2

The parameters were selected to be consistent with data at a quarterly frequency. The key parameters in a simple version of the Dornbusch model are the semi-interest elasticity of money demand, the (reduced-form) elasticity of aggregate demand with respect to the real exchange rate, and the degree of price stickiness. (Obstfeld and Stockman (1985) discuss various extensions of the original model.) As discussed in a longer version of this paper, for the semi-interest elasticity of money demand, a value of minus 4 was used; at an average annual short-term interest rate of 8 percent, this implies an interest elasticity of minus 0.08, which is at the midpoint of Friedman's (1978) range for the elasticity of Ml demand with respect to short-term interest rates. There are few direct estimates of the elasticity of aggregate demand with respect to the real exchange rate, and a value of 0.25 was used. For the degree of price stickiness, a value of 0.05 was used, which is consistent with estimates by Taylor (1980) and Rotemberg (1982); it implies that around 20 percent of the gap between actual prices and their flexible equilibrium level is dissipated in a year (see Chadha (1989)).

3

References to all variables in the model, except nominal interest rates, are to their natural logarithms.

4

In fact, under the assumption that the money supply follows a driftless random walk, changes in the nominal exchange rate, the forward rate, and the price level are described by infinite-order, moving-average processes.

5

This coefficient corresponds to the stable eigenvalue of the model; the other eigenvalue equals 1.05.

6

Two unit-root processes, y, and xt, are cointegrated if there exists a nonzero constant, A, such that Zt = yt − Axt is stationary and integrated of order zero. In the Dornbusch model, the cointegrating parameters for the nominal variables (except the nominal interest rate) is unity, given that the money supply follows a random walk.

7

The assumed parameter values imply that a 1 percent increase in money supply leads on impact to 5 percent depreciation of the nominal and real exchange rates.

8

Innovations in the money supply were assumed to have a 1 percent standard deviation.

9

Full details on the simulation results and the variability of the simulated data are available from the authors on request.

10

The Sargan-Bhargava test examines whether the Durbin-Watson statistic for each series is significantly above zero, its value under the null of a unit root. The Dickey-Fuller test is based on regressing the first difference of each series on its one-period lagged level; under the null hypothesis of a unit root, the coefficient on the lagged level is zero. The augmented Dickey-Fuller test adds lagged changes in the series to the Dickey-Fuller regression to soak up any residual serial correlation. Under the null hypothesis of a unit root, all the test statistics have nonstandard distributions. (See Tables 1–3 for critical values employed.)

11

One possibility would have been to apply the tests proposed by Phillips and Perron (1988). Given that these tests have not been widely used, we conserve space by focusing on the Sargan-Bhargava, Dickey-Fuller, and augmented Dickey-Fuller tests.

12

Given that the changes in all variables have a zero mean by construction, constant terms are not included in the Dickey-Fuller or augmented Dickey-Fuller regressions.

13

Given that the AR(1) representation for real output contains the same root as that for the real exchange rate (0.9375), the results suggest that we would also be unable to reject a unit root in output. West (1988) has argued that under certain kinds of monetary policy and with inertia in wage-price adjustment, real gross national product in the United States may have a root sufficiently close to unity to make it indistinguishable from a unit-root process.

14

Krugman (1990) argues that based on an AR(1) representation of the (annual) real exchange rate with an AR parameter of 0.8, approximately 40 years of data would be required for the standard error on the parameter to decline sufficiently to reject a unit root at a given significance level. His argument relies on knowing the true value of the parameter. As our simulations have demonstrated, across finite samples, estimates of this parameter will vary, introducing an additional source of variation in the test statistic. Numerical simulation, based on representative parameter values, thus provides a way of characterizing the behavior of the test statistics at different sample sizes.

15

An alternative procedure would be to use the Johansen (1988) maximum-likelihood procedure for estimating the cointegrating vector as in Adams and Chadha (1991). The Engle-Granger procedure was adopted simply because it was computationally more feasible, given the large number of simulations carried out.

16

This finding confirms the results presented in Banerjee and Dolado (1986), who found large small sample biases in estimates of cointegrating parameters.

17

The difference between the spot rate and the one-period-ahead forward rate—that is, the forward premium—can be written as an AR(1) process with a coefficient of 0.9375—exactly the process for the real exchange rate. Given the results of the unit-root tests for the real exchange rate discussed above, it is not surprising that the spot and forward rate are not found to be cointegrated.

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Cover IMF Staff Papers
IMF Staff papers: Volume 38 No. 4
Author:
International Monetary Fund. Research Dept.
Volume/Issue:
Volume 1991: Issue 003
Publisher:
International Monetary Fund
ISBN:
9781451930801
ISSN:
1020-7635
Pages:
276
DOI:
https://doi.org/10.5089/9781451930801.024