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Nominal exchange rates and unit roots: a reconsideration
Journal of International Money and Finance (1992), 11, 539-551
Nominal exchange rates and unit roots: a reconsideration JOSEPH
Federal
A. WHITT, JR.*
Reserve Bank of Atlanta, Atlanta, GA 30303, USA and Federal Reserve Board, Washington, DC 20551, USA
This paper provides empirical evidence that nominal exchange rates are stationary, and are not random walks. Other studies using classical tests suggest that exchange rates contain unit roots, thereby supporting the random walk hypothesis. The apparently permanent nature of shocks to exchange rates has been interpreted to mean that disequilibrium models such as the Dornbusch overshooting model are not useful in explaining exchange rate behavior. This paper applies a new statistical test for unit roots proposed by Sims and based on Bayesian posterior odds ratios; the results favor exchange rate stationarity for several countries. (JEL F31)
This paper presents the results of a new test for the presence of unit roots as applied to nominal exchange rates. The presence of a unit root in a variable means that it is subject to permanent stochastic shocks, not merely temporary shocks around a deterministic level or trend. As discussed in Meese and Singleton (1982), this question affects the validity of tests for efficiency in the foreign exchange market. In addition, Stockman (1987) argues that the evidence that nominal exchange rate movements are largely permanent implies that theoretical models that emphasize monetary disturbances as sources of transitory movements, such as the popular Dornbusch (1976) overshooting model, are inadequate. Empirical studies have generally supported the presence of a unit root in nominal exchange rate data (see Meese and Singleton, 1982; or Baillie and Bollerslev, 1987, 1989). Moreover, Meese and Rogoff (1983) show that the basic random walk model, which is a special case of a unit root, does as well or better than a variety of structural and more complex time series models in forecasting exchange rates. In a more general context, Sims (1988) argues that classical statistical tests for the presence of unit roots, such as the Dickey-Fuller (1979) tests used in Meese and Singleton (1982), are fundamentally flawed. As an alternative, he proposes * Federal Reserve Bank of Atlanta and Visiting Economist, Federal Reserve Board. I wish to thank Will Roberds for many helpful discussions. Sean Becketti, Curt Hunter, James Lothian, Leigh Riddick, Charles Whiteman, and two anonymous referees also made useful suggestions. The views expressed are those of the author and do not necessarily reflect those of the Federal Reserve Bank of Atlanta, or the Federal Reserve System. 0261P5606/92/06/0539-13
11; 1992 Butterworth-Heinemann
Ltd
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and unit roots
a test based on Bayesian posterior odds ratios that is designed to discriminate between a unit root and a large but stationary autocorrelation coefficient. This paper applies the Sims test to monthly nominal exchange rate data from the current period of flexible exchange rates. The results are mixed, but for several countries they favor the hypothesis that exchange rates are stationary and have substantial autocorrelation, but not a unit root. In Section I, Sims test and its differences with classical tests are discussed. Section II contains empirical results from applying various tests for the presence of unit roots to dollar exchange rates with five other major currencies. It also provides Monte Carlo results to compare the performance of classical tests with the Sims test. Section 111 presents the conclusions.
I. Sims test for the presence of a unit root Consider the following autoregressive (1)
model of the nominal exchange rate:
(6 - P) = P(%,
- P) + 5,
where e, is the exchange rate at time t; p is a constant term; and E, - N(0, 02) is the error term and is independent of past values of e,. In this model, the long-run behavior of the exchange rate is critically dependent on the value of the autoregressive coefficient p. If 0 < p < 1, the exchange rate is a stationary variable; it fluctuates in the vicinity of a constant unconditional mean, /L. In the absence of future shocks, the deviation (e, - cl) would shrink in subsequent periods when 0 < p < 1. In this case, the expected value of the exchange rate in the distant future is its unconditional mean, p. By contrast, if there is a unit root (p = I), the behavior of the exchange rate is quite different. In this case, the exchange rate is nonstationary; it has no fixed unconditional mean, and there is no tendency for (e, - p) to shrink. Instead, the exchange rate is a random walk; the expected value of the exchange rate in the distant future is always equal to its current value, e,. Accordingly, the problem for empirical work is to make statistical inferences about the value of p. Using a classical approach, Dickey and Fuller (1979) provide tests of the null hypothesis that p = 1, using statistics generated by an OLS regression of e, onto its own lagged value. The standard t-test is not appropriate, because under this null hypothesis the variance of the exchange rate is infinite. Sims (1988) and Sims and Uhlig (1988) argue that classical tests such as the ones proposed by Dickey and Fuller provide a misleading picture of the plausibility of unit roots. Using Bayesian methods, they show that the prior implicit in the classical tests not only gives excessive (in their view) weight to the unit root null, but also gives substantial and disproportionate weight to values of p above 1.’ As an alternative to the classical analysis, Sims (1988) proposes a Bayesian test based on a suggestion in Learner (1978, pp. 100-108). Learner argues that when testing a point-null hypothesis against a composite alternative, using the Bayesian posterior odds ratio is a more sensible approach than using a likelihood ratio test or the sampling theory approach. The posterior odds ratio can be interpreted as a weighted average of the likelihood function over all points
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consistent with the null hypothesis, divided by a similar weighted average of the likelihood function over all points in the alternative. The weights are derived from the prior distribution of the parameters. By contrast, the likelihood ratio test uses only the maximum values of the likelihood function for each alternative. To apply Learner’s suggestion to equation ( 1 ), Sims proposes a prior distribution for p which spreads probability a, 0 < c1< 1, uniformly on the interval (0, l), and gives the unit root (p = 1) probability (1 - a). This specification gives a clear but limited advantage to the unit root hypothesis, because any individual point between zero and one has essentially zero prior probability, while the point where p = 1 has probability (1 - CC). Suppose fsP = Jm z = (1 - fi)/ up is the conventional t-statistic for testing p = 1, (D(x) is the cumulative distribution function for the standard normal distribution evaluated at x, and 4(x) is its probability density function. Then Sims shows that in large samples the posterior odds ratio favors the null hypothesis (p = 1) if’
This criterion is different from classical hypothesis tests, not only because of the role of the prior distribution, but also because of the presence of rsp in the denominator. If the null hypothesis (p = 1) is true, oP should shrink much faster as sample size rises than when the alternative is true. This criterion tends to favor the null hypothesis when cp is small (for given values of a and 7). By contrast, likelihood ratio tests fail to use all the information in Do. In actual applications, Sims suggests that for annual economic data the alternative hypothesis can reasonably be limited to values of p between l/2 and 1. For more frequent data, the interval associated with this alternative hypothesis has a lower bound closer to 1. In the case of quarterly data, the interval is approximately (0.84, l), because 0.84 to the fourth power is equal to the lower bound for annual data; the interval is (0.94, 1) for monthly data. Therefore, Sims proposes the following revised criterion: the null hypothesis (p = 1) is favored if y > 0,
(3) where y=2log
1-a __ i CI
1
- log[a,2] + 2log[l
- 2-“s]
- 2log[Q(r)]
- log[2711 - r2,
and s is the number of periods per year (e.g., 12 for monthly data).3 In typical examples, cP < 1, implying that -log(o,2) is positive. Smaller values of (TVinduce larger values of - log( c,‘), thereby favoring the unit root hypothesis. However, larger values of T = (1 - ;)/a,, favor the alternative hypothesis. In empirical work, it is informative to present the Sims criterion in a different way, by calculating the marginal prior probability on the unit-root null, (1 - CI”), that on an ex post basis would be necessary to force the Sims criterion to favor the null hypothesis. The formula for (1 - CC*)can be derived by setting y equal to zero and solving for the value of (1 - cr*), taking z and cp as given by the
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sample information. (4)
Doing so produces the following:
(1 - a*) = 1 - l/{ 1 + exp[log(o,) + log(Q(r))
- log(1 - 2-i”)
+ (1/2)log(2n)
+w)~21},
where exp is the exponential function. The value of (1 - x*) depends only on the sample information. A large value of ( 1 - a*), i.e., a value near 1, implies that the sample information tends to reject the unit-root hypothesis; hence a large prior weight on the unit-root null is necessary to force the Sims criterion to favor a unit root. Alternatively, a small value of (1 - a*), i.e., a value near zero, implies that the sample information tends to favor the unit-root hypothesis. In an actual testing situation, if the value of (1 - u*) is larger than the reader’s own prior on the unit root, then the reader’s posterior odds ratio favors stationarity; if it is smaller, the test result favors the unit-root hypothesis. Another issue in applied work is whether to include a constant or trend term in the regression used to estimate 6. In the case of classical DickeyyFuller tests, critical values are affected by the presence or absence of a constant or trend.4 Moreover, the critical values provided by Fuller (1976) for the test that includes a constant are correct only if the true model is a random walk with zero drift (see Schmidt, 1988). The Bayesian test statistics presented in equations ( 3 ) and (4) can be derived from models containing a constant or trend if the prior on the constant or trend is flat.5 However, the performance of the test is affected by the treatment of the constant and trend terms; therefore, to help in evaluating empirical results using this test, Monte Carlo evidence is presented below.
II. Empirical
results with exchange
rate data
It has been known for some years that since the end of the Bretton Woods system
and the move to a flexible exchange rate system, nominal exchange rates between major currencies have approximately followed random walks (see, for example, Mussa, 1979). The volatility and unpredictability of exchange rate movements have often been attributed to the impact of ‘news’ in an institutional setting in which exchange rates are asset prices determined in highly efficient markets which are not dominated by government regulation or intervention (Mussa, 1976, 1979; Frenkel, 1981). In such a setting, new information is immediately embodied in the current spot exchange rate as well as expected future spot rates. Observed movements of the spot rate are for the most part unanticipated because they mainly reflect the impact of new information; if there were a large anticipated component of exchange rate changes, then the market would not be efficient because large profits could be gained at little risk. Even so, there are reasons to think that exchange rates are not true random walks. Mussa ( 1979, p. 11) observes that the anticipated rate of change of exchange rates may well be serially correlated but not constant, giving rise to a limited amount of serial dependence in actual exchange rate changes. In particular, the popular Dornbusch (1976) model of exchange rate behavior implies that monetary shocks can cause the exchange rate to overshoot its new long-run equilibrium in the short run, before gradually adjusting toward long-run equilibrium. During
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the period that the exchange rate is moving gradually toward long-run equilibrium, there would be serial dependence of exchange rate changes, at least in the absence of additional real or monetary shocks. Hakkio (1986, p. 223) argues that the exchange rate should follow a random walk only if the market fundamentals that determine the exchange rate also follow random walks. Theories of exchange rate behavior suggest that the market fundamentals include domestic and foreign money supplies, real incomes, interest rates, and current accounts; in his view, these variables are unlikely to be random walks. Empirical tests using exchange rate data from the current period of flexible exchange rates have strongly favored the random walk hypothesis6 In testing for market efficiency using autocorrelations, Cornell (1977) finds that a random walk model of the spot exchange rate fits the data very well. Meese and Singleton (1982) perform Dickey-Fuller tests using weekly data; their results favor the presence of a single unit root, which is consistent with the random walk hypothesis.’ Using a test proposed by Phillips and Perron (1988) that is designed to correct the Dickey-Fuller approach for general forms of serial correlation or heteroskedasticity, Baillie and Bollerslev (1987, 1989) also report evidence in favor of the presence of a single unit root in exchange rate data. As discussed in Meese and Singleton (1982), the finding of unit roots in exchange rate data implies that it is inappropriate to estimate models explaining the level ofexchange rates, as in Bilson (1978) or Frankel(1979). In addition, tests involving the relationship between the forward rate and the expected future spot rate should not be done in levels, as in Frenkel (1981). Unit root tests also have implications for theoretical modeling of exchange rates. As discussed in Nelson and Plosser (1982) and Schwert (1987), the presence of a unit root in a variable means that it is subject to permanent shocks that generate a stochastic long-run growth path, rather than merely temporary shocks around a deterministic level or trend. A random walk is a special case of a unit root, in which all innovations are permanent. Nelson and Plosser (1982) report that many macroeconomic series, including several that presumably are fundamental determinants of exchange rates, contain unit roots even after adjustment for time trends. Similar findings are reported by Harvey (1985) and Schwert (1987), among others.’ Nelson and Plosser (1982) argue that their finding of unit roots in a variety of macroeconomic time series (not including exchange rates, which they do not examine) implies that theoretical models that emphasize monetary disturbances as sources of transitory fluctuations are inadequate.9 Instead, they recommend theoretical models that emphasize real factors as sources of permanent shocks. With regard to exchange rates, Stockman (1987) argues in a similar vein that the evidence that nominal exchange rate movements are largely permanent implies that the Dornbusch (1976) overshooting mechanism cannot be an important source of observed exchange rate movements. In related work, Meese and Rogoff (1983) show that a simple random walk model does as well or better than a variety of structural and more complex time series models in forecasting the exchange rate during the period since the end of Bretton Woods. The structural models that they consider include a version of the Dornbusch ( 1976) model, and they allow for a number of possible determinants of exchange rates, including the money supply, real income, the short-run interest differential, the long-run expected inflation differential, and trade balances. The
Nominal exchange rates and unit roots
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inability of traditional structural models to explain the exchange rate movements of recent years has stimulated work on alternative hypotheses, such as the speculative bubbles examined in Meese (1986) or equilibrium exchange rate models that emphasize the importance of real shocks to technology or tastes, as recommended in Stockman (1987). How solid is the conclusion that exchange rates contain a unit root? As discussed earlier, the test proposed by Sims (1988) provides an alternative to conventional tests of the Dickey-Fuller variety. Using the United States as the base country, the test was applied to monthly data on the nominal exchange rate for five countries: the United Kingdom, France, Germany, Switzerland, and Japan.” The sample period began in June 1973, several months after the final breakdown of the Bretton Woods system, and ended in May 1988, thereby providing 180 observations. The tests were performed in two ways. For Table 1, fi was estimated by regressing the log of each exchange rate onto its own lagged value, plus a constant term. For Table 2, a linear time trend was added as an additional regressor.’ 1 To check the specification, residuals were examined for evidence of
TABLE
1. Tests for a unit root in the nominal using constant only (no time trend).
exchange
DickeyyFuller Country United Kingdom France Germany Switzerland Japan Nore:
The sample
period
P
r,l
0.9793 0.9895 0.9823 0.9840 1.0082
-1.76 - 1.09 ~ 1.13 - 1.24 0.74
Sims (I -x*) 0.7044 0.402 1 0.5356 0.5262 0.1297
is June 1973 to May 1988
TABLE 2. Tests for a unit root in the nominal with time trend included.
exchange
DickeyyFuller Country United Kingdom France Germany Switzerland Japan Nole:
The sample
period
rate
rate
?
rr
Sims (1 -X*)
0.983 1 0.989 1 0.9807 0.9752 0.9838
- 1.00 -0.78 - 1.21 - 1.51 -0.96
0.5122 0.3986 0.5674 0.6824 0.4974
is June 1973 to May 1988.
545
JOSEPHA. WHITT,JR.
autocorrelation, using the Durbin (1970) h-statistic and the Box-Pierce Q-statistic (see Granger and Newbold, 1977); no evidence of autocorrelation was found.i2 As expected on the basis of previous empirical work on this issue, the regressions produced estimates of the autoregressive coefficient i, rather close to one; the first column of each table provides the estimates. In addition, statistics for testing the null hypothesis (p = 1) are presented in each table. For comparison purposes, the second column in each table presents the Dickey-Fuller test statistic, r E ((fi - 1)/o,). To reject the null hypothesis at the 90 per cent significance level would require that r be less than -2.57 in Table 1, and less than - 3.13 in Table 2.13 Clearly, none of the countries in this sample comes close to rejecting the unit root on the basis of these classical tests.14 These results using the DickeyyFuller test corroborate the results in Meese and Singleton (1982). What about the Sims test? Recall that the Sims test favors the null hypothesis (p = 1) if the test statistic (1 - x*) defined in equation (4) is less than the prior weight on the unit-root null. Values of (1 - a*) for each exchange rate are given in the third column of each table. If one’s prior gives even odds (1 - x = 0.5) then the results are consistent in both tables, favoring stationarity for the British pound, German mark, and Swiss franc, while favoring a unit root for the French franc and Japanese yen. Alternatively, if one gives more weight to stationarity, as favored by Sims (1988, p. 471), and chooses the prior weight on the unit root to be 0.2, then the test results would favor stationarity for one additional currency, the French franc, as well as for the Japanese yen in the results including time trend.’ 5 As mentioned above, the Sims test was derived on the assumption that the sample size is large, and like the Dickey-Fuller test it is affected by the treatment of possible constant and trend terms.16 To shed light on the properties of these tests, a Monte Carlo study was done using artificially-generated time series of the same length as the exchange rate data, 180 months. Four data-generating processes were used: a simple random walk, a random walk with drift, a stationary model with constant mean, and a stationary model with trend. All of the models can be represented as variants of the following equation: (5)
Yf - (Yo + Bt) = PC_&1 - (Yo + P(t -
I))1 + P + Et,
where y, is the initial value of y and is non-stochastic; and e, is a normally-distributed random error term. If p is set equal to 1, and the coefficient on the time trend (p) and the drift parameter (PC)are both equal to zero, equation (5) reduces to a simple random walk. Allowing p to be non-zero generates a random walk with drift. Alternatively, if p is positive but less than 1, y, is a stationary variable. With B and p both equal to zero, y, is stationary with a fixed unconditional mean, y,. If /I and p are equal to one another but non-zero, y, is stationary around a linear trend. In the Monte Carlo simulations, artificial time series of y, were generated using equation (5) with assumed values of p, p, p, and y,, plus values of e, created by the GAUSS 2.0 random number generator. For each repetition, 190 monthly observations were created. The DickeyyFuller and Sims test statistics were then calculated using only the last 180 observations to reduce any effects of the choice of starting value y,. To obtain rejection frequencies, this process was repeated
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TABLE3. Monte
Carlo results (rejection
A. B.
C.
D. E. F.
frequencies).
Dickey-Fuller with constant
Model p=l p=p=o p=l /f = -0.05 fl=o p=l p = -0.01 /I=0 p = 0.85 p=fl=o p = 0.85 p = /I = -0.00058 p = 0.85 p = p = 0.00358
10.40%
rates and unit roots
Dickey-Fuller with trend 10.08%
Sims with constant
Sims with trend
70.00%
93.76%
0.96
10.08
1.20
93.76
2.12
10.08
10.40
93.76
14.20
11.96
85.72
97.04
13.04
11.96
83.88
97.04
5.44
11.96
49.08
97.04
2500 times for each model. In all cases 10 per cent critical values of the Dickey-Fuller tests from Fuller (1976) were used, while in the Sims tests, the prior probability of the unit-root null was set at 50 per cent (even odds).” Table 3 presents rejection frequencies for the unit-root null hypothesis for six different data-generating models. The results in the first row of the table, labelled A, are for data generated by a simple random walk, with no drift. This is the model that represents the null hypothesis in classical Dickey-Fuller tests. As the table indicates, for this model the Monte Carlo results indicate that both Dickey-Fuller tests mistakenly reject the unit-root null about 10 per cent of the time, as they should given the choice of test size. For this model and sample size the Sims tests do much worse. The Sims test with a constant alone mistakenly rejects the unit-root null about 70 per cent of the time, while the Sims test with trend rejects over 93 per cent of the time. The next two rows give results when the data are generated by a random walk with drift. Two drift terms were used; they were chosen to approximate in size the drift implied by regressions of exchange rates on a constant and their own lagged values. Row B has a large (in absolute value) drift, as suggested by results using the Japanese yen; row C has a smaller drift, as suggested by results using the British pound. Adding drift causes the Dickey-Fuller test with constant alone to reject the unit-root null much less frequently; for example, for model B the rejection frequency is only 0.96 per cent, even though the critical value was chosen for a test of size 10 per cent.” The performance of the Sims test with constant alone improves substantially when drift is introduced to the random walk model. When the drift term is large (row B), the Sims test with constant mistakenly rejects the unit-root null only 1.2 per cent of the time; when the drift is smaller (row C), the rejection rate rises to 10.4 per cent, still far below the 70 per cent rate that occurred when there was
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no drift at all (row A). As for the tests with trend included, both the Dickey-Fuller and the Sims test generate the same rejection frequencies as they did when no drift was present. The last three rows of the table give rejection frequencies when the data are actually stationary. In these rows, the value of the autoregressive coefficient p was set at 0.85 in terms of annual data; this value is well inside the stationary region, but still implies substantial positive autocorrelation. In terms of monthly data, this value translates into a p of 0.9865, which is in the range of estimated values reported in Tables 1 and 2. In row D, there is no trend; the data are stationary around a constant unconditional mean. For this sample size and value of p, the Dickey-Fuller test with constant alone correctly rejects the unit-root null only 14.2 per cent of the time; the Dickey-Fuller test with trend rejects even less frequently, 11.96 per cent of the time. By contrast, the Sims test with constant alone rejects 85.72 per cent of the time, while the Sims test with trend rejects 97.04 per cent of the time. Rows E and F use stationary models with linear trends. The sizes of the trend parameters were chosen to be roughly consistent with actual exchange rate data.lg Adding a trend to the stationary model causes the Sims test with constant alone to reject the unit root less frequently; for the smaller trend (row E) its rejection frequency is 83.88 per cent, and for the larger trend (row F) it is down to 49.08 per cent. Even so, it correctly rejects far more often than the Dickey-Fuller test with constant alone, which is also affected by the presence of trend. By contrast, both the Dickey-Fuller and Sims tests that include a trend are not affected by the size of the trend in the artificial data. To summarize the Monte Carlo results, none of the tests is clearly superior to the others in all situations. Both of the classical Dickey-Fuller tests rarely reject the unit-root null when it is true, but they have little power against stationary alternatives.20 By contrast, for this sample size the Sims test with trend rarely accepts the unit-root null, even when it is true. The test that shows the greatest ability to discriminate between a unit-root model and stationarity is the Sims test with constant alone. It rarely rejects the unit-root null when the data are generated by a random walk with drift, as in rows B and C, and it does reject the unit-root null fairly frequently when the data are generated by a stationary model with p equal to 0.85 (in terms of annual data) and either a small or zero trend element, as in rows D, E, and F. However, the Sims test with constant can often lead to mistaken conclusions when the data are generated either by a random walk with zero or near-zero drift, as in row A, or by a stationary model with a large trend element, as suggested by a comparison of rows E and F.21 In light of the Monte Carlo evidence, what conclusions can be drawn about the presence of unit roots in exchange rate data? It is clear that in a classical testing framework, we cannot reject the unit-root null at standard significance levels for any of these currencies; however, this does not imply that a unit root is very likely, because the power of the classical tests is rather low. Using reasonable priors, a Bayesian approach indicates that the weight of the evidence favors stationarity for three currencies (the British pound, German mark, and Swiss franc) and possibly a fourth (the French franc), depending on the prior. The Japanese yen shows the strongest evidence in favor of a unit root, but even it might be stationary if a strong trend were allowed for.
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Conclusions
This study indicates that, by a preponderance of the evidence, though not beyond a reasonable doubt, nominal exchange rates do not contain unit roots. This conclusion is based on results using a Bayesian statistical test that is designed to discriminate between a unit root and a large but stationary autocorrelation coefficient. The test was applied to monthly exchange rate data for five industrial countries vis-ci-vis the United States. While classical tests are consistently unable to reject the unit root hypothesis, the new test does reject it consistently for three of the countries. For the other two countries, results were mixed, depending on the exact prior used and on whether a time trend was included. It should be emphasized that the Bayesian test proposed by Sims that is used in this paper is quite different from the familiar DickeyyFuller test, as shown by the Monte Carlo simulations reported in Table 3. In the language of classical testing, an ideal test would combine high power and small size. The Monte Carlo results, which used artificial data that were constructed to be similar to the available sample of exchange rate data from the current period of flexible exchange rates, show that the classical DickeyyFuller test with a nominal size of 10 per cent has rather low power, less than 15 per cent, even when the true autoregressive parameter is well below one. The Sims test has much greater power, but its size is problematic. When a time trend is included, the size of the Sims test is disconcertingly large. When only a constant is included, the size of the Sims test depends on the true parameters generating the data; the size is small when the artificial data contain the amount of drift indicated by regressions using actual exchange rate data, but it is disturbingly large if the data are generated by a random walk with little or no drift. Using what I believe are reasonable priors that have been suggested in the literature, I find with the Bayesian approach that, in most cases, the evidence favors the hypothesis that nominal exchange rates do not contain unit roots. The results in this paper suggest that exchange rates are close to, but not quite, random walks; shocks to the level of exchange rates are not all permanent. Therefore, it may be premature to dismiss standard models of exchange rate behavior because they allow for temporary exchange rate movements. An alternative interpretation is that for the countries in the sample, which have large, well-diversified economies and reasonably stable monetary policies, nominal exchange rates tend to move within some sort of relatively narrow bounds, because the movements of real and monetary factors are limited. After all, during the sample period none of these countries experienced hyperinflation or a drastic change in real income, either of which would cause a major permanent shift in at least the level of the exchange rate, according to standard exchange rate theory.
Notes 1. 2.
3.
Sims and Uhlig (1988) p. 8. In a full Bayesian analysis, the constant on the right-hand side of expression (2) might be a number other than one, depending on the loss function applying to the two hypotheses. However, this extension is not pursued here. Sims suggests ignoring the term ‘- 2log[@(r)]’ because it is likely to be small when ,6 < 1 and is asymptotically negligible; he also leaves out the term ‘-log(2n)’
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See Fuller (1976), p. 373. See Sims (1988), p. 467. If non-flat priors about the constant or trend terms are used, DeJong and Whiteman (1989) and Schotman and van Dijk (1990) have shown that inferences about the autoregressive parameter can be affected. 6. More precisely, previous empirical work has shown that classical tests fail to reject the null hypothesis of a unit root. Nevertheless, these results have been commonly interpreted as affirmative evidence in favor of a unit root, as in Baillie and Bollerslev (1989) or Lastrapes (1989). Meese and Singleton also estimated autoregressions of the exchange rate of order one through eight; the fact that the coefficients beyond the first lag were small was also favorable for the random walk hypothesis. Hakkio (1986) casts doubt on these results by demonstrating that standard tests such as those used by Cornell (1977) or Meese and Singleton (1982) have very low power against alternatives that are ‘nearly’ random walks. However, Hakkio’s results do not bear directly on the possibility that exchange rates do not contain a unit root, because the alternatives he considers are all ARIMA processes that contain a unit root. The conclusion that many macroeconomic time series contain unit roots and are not trend-stationary has recently been challenged by DeJong and Whiteman (1989) and DeJong et al. (1988). With regard to real GNP, West (1988) and Cochrane (1988) counter that finding a unit root in this series does not necessarily imply that monetary shocks are unimportant, because there are plausible theoretical models with both a unit root in real GNP and an important role for monetary shocks. Also see Christian0 and Eichenbaum (1989). 10. The use of monthly data is advantageous because it greatly reduces the problems of heteroskedasticity and departures from normality. As discussed in Baillie and Bollerslev (1987), conditional heteroskedasticity and departures from normality are pronounced in daily and weekly exchange rate data, but they virtually disappear when monthly data are used. In this paper, end-of-month data from International Financial Statistics were used. 11. Meese and Singleton (1982) include a time trend in all their equations. 12. In calculating the Box-Pierce statistic, 12 lags were used to coincide with monthly data; in addition, 20 lags were used as recommended in Granger and Newbold (1977, p. 93). For further discussion of this issue, see Hakkio (1986, footnote 9). 13. The critical values were obtained from Fuller (1976), p. 373. 14. As a check on the results presented here, the classical Phillips-Perron (1988) test that is designed to correct the Dickey-Fuller approach for general forms of serial correlation or heteroskedasticity was performed as well. The results coincided closely with the Dickey-Fuller results, indicating no rejections of the unit-root hypothesis. 15. The prior favored by Sims still gives some advantage to the unit-root hypothesis, because in terms of annual data the point null hypothesis has the same prior probability as the infinite number of points in various intervals that are consistent with the alternative hypothesis, for example (0.875 < p < 1). 16. As suggested by an anonymous referee, one reason for concern about the small-sample properties of the Sims test is the well-known downward bias in finite samples of the least-squares estimate of the coefficient on a lagged dependent variable, such as 3. 17. The same random numbers were used for each choice of parameter values, as in Hakkio (1986). In addition, the standard error of the random error terms was set at 0.035 for all models; this is similar to the standard errors obtained using actual exchange rate data. 18. The sensitivity of the Dickey-Fuller test to the presence of drift is discussed in Schmidt (1988). 19. The results presented here use the largest and smallest (in absolute value) trend terms found for the five currencies used in this study; they were obtained by regressing the log of each exchange rate onto a constant and a linear trend. 20. Hakkio (1986), DeJong and Whiteman (1989), and DeJong et al. (1988) have all discussed the low power of the Dickey-Fuller and similar classical tests of the unit-root hypothesis when faced with relevant alternatives. 21. Additional Monte Carlo results, not reported here, verify that for the stationary model, increasing the size of the trend coefficient beyond the value used in row F will, ceteris parihus, further shrink the rejection frequency of the Sims test with constant alone.
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References BAILLIE,RICHARD,ANDTIM BOLLERSLEV, ‘The Message in Daily Exchange Rates: A Conditional Variance Tale,’ mimeo, Michigan State University, 1987. BAILLIE,RICHARD,AND TIM BOLLERSLEV, ‘Common Stochastic Trends in a System of Exchange Rates,’ Journal of Finance, March 1989, 44: 1677181. BILSON, JOHN F.O., ‘Rational Expectations and the Exchange Rate,’ in Jacob A. Frenkel and Harry G. Johnson, eds, The Economics of Exchange Rates, Reading, Massachusetts: Addison-Wesley Publishing Co., 1978. CHRISTIANO,LAWRENCEJ., AND MARTIN EICHENBAUM, ‘Unit Roots in Real GNP: Do We Know, and Do We Care?,’ mimeo, Federal Reserve Bank of Minneapolis, 1989. COCHRANE,JOHN H., ‘How Big Is the Random Walk in GNP?,’ Journal of Political Economy, October 1988, 96: 893-920. CORNELL,BRADFORD,‘Spot Rates, Forward Rates and the Exchange Market,’ Journal of Financial Economics,
1977, 5: 55-65.
DEJONC;, DAVID N., JOHN C. NANKERVIS,N.E. SAVIN, AND CHARLESH. WHITEMAN,‘Integration Versus Trend-Stationarity in Macroeconomic Time Series,’ Department of Economics Working Paper, University of Iowa, 1988. DEJONG, DAVIDN., ANDCHARLESH. WHITEMAN,‘Trends and Random Walks in Macroeconomic Time Series: A Reconsideration Based on the Likelihood Principle,’ Department of Economics Working Paper, University of Iowa, 1989. DICKEY, DAVID A., AND WAYNE A. FULLER, ‘Distribution of the Estimators for Autoregressive Time Series With a Unit Root,’ Journal of the Americun Statistical Associution, 1979, 74: 427-431. DORNBUSCH,RUDIGER, ‘Expectations and Exchange Rate Dynamics,’ Journal of Political Economy, December 1976, 84: 1161~ 1176. DURBIN, J., ‘Testing for Serial Correlation in Least-Squares Regression when Some of the Regressors are Lagged Dependent Variables,’ Econometrica, 1970, 38: 410-421. FRANKEL,JEYFREYA., ‘On the Mark: A Theory of Floating Exchange Rates Based on Real Interest Differentials,’ American Economic Reciew, September 1979, 69: 610-622. FRENKEL,JACOBA., ‘Flexible Exchange Rates, Prices and the Role of ‘News’: Lessons from the 197Os,’ Journal of Politica/ Economy, August 1981, 89: 665-705. FULLER,WAY~F, Introduction to Statistical 7ime Series, New York: John Wiley & Sons, 1976. GRAK~ER, CLIVE W.J., ANU PA~JL NF.WBOL.D,Forecasting Economic Time Series, New York: Academic Press, 1977. HAKKIO, CRAIG S., ‘Does the Exchange Rate Follow a Random Walk? A Monte Carlo Study of Four Tests for a Random Walk,’ Journal qflnternutional Money and Finance, 1986, 5: 221-229.
HARVEY,ANDREWC., ‘Trends and Cycles in Macroeconomic Time Series,’ Journal qf Business and Economic Statistics, July 1985, 3: 216-227. LASTRAPES,WILLIAM D., ‘Exchange Rate Volatility and U.S. Monetary Policy: An ARCH Application,’ Journal of Money, Credit and Banking, February 1989, 21: 66-77. LEAMER.EDWARDE., Specification Searches, Ad Hoc Ir$erence Gth Nonexperimental Data, New York: John Wiley & Sons, 1978. MFESE, RICHARE)A., ‘Testing for Bubbles in Exchange Markets: A Case of Sparkling Rates?,’ Journal qf Political Economy, April 1986, 94: 345-373. MEESF, RICH~KUA., ANDKEN~IETHRoc,ot.t, ‘Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?,‘Journa/ oflnternational Economics, February 1983,14: 3-24. MEFSE, RICHARDA., ANDKFNNETH J. SINGLETON,‘On Unit Roots and the Empirical Modeling of Exchange Rates,’ Journal of Finance, September 1982, 37: 1029- 1035. Muss4, MICHAEL, ‘The Exchange Rate, the Balance of Payments and Monetary and Fiscal Policy Under a Regime of Controlled Floating,’ Scandinacian Journa/ of Economics, 1976, 78: 229-248. MUSSA, MICHAEL,‘Empirical Regularities in the Behavior of Exchange Rates and Theories of the Foreign Exchange Market,’ Carnegie-Rochester Conference Series on Public Policy, Autumn 1979, 11: 9-57. NELSON, CHARLES R., !\NU CHARLESI. PLOSSER,‘Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications,’ Journal qf Monetary Economics, September 1982. 10: 139-162.
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PHILLIPS,PETERC.B., AND PIERRE PERRON,‘Testing for a Unit Root in Time Series Regression,’ Biometrika, 1988, 75: 335-346. SCHMIDT,PETER, ‘Dickey-Fuller Tests with Drift,’ Working Paper, Michigan State University, 1988. SCHOTMAN,PETER, AND HERMANK. VAN DIJK, ‘A Bayesian Analysis of the Unit Root in Real Exchange Rates,’ Working Paper, Erasmus University, 1990. SCHWERT,G. WILLIAM,‘Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data,’ Journal of Monetary Economics, 1987, 20: 73-103. SIMS, CHRISTOPHER A., ‘Bayesian Skepticism on Unit Root Econometrics,’ Journal qf Economic Dynamics and Control, 1988, 12: 463-474. SIMS, CHRISTOPHERA., ANDHAROLDUHLIG, ‘Understanding Unit Rooters: A Helicopter Tour,’ Discussion Paper 4, institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis and University of Minnesota, 1988. STOCKMAN,ALAN C., ‘The Equilibrium Approach to Exchange Rates,’ Federal Reserve Bank of Richmond Economic Review, March/April 1987, 73: 12-30. WEST, KENNETHD., ‘On the Interpretation of Near Random-Walk Behavior in GNP,’ American Economic Review, March 1988, 78: 202-209.
Nominal exchange rates and unit roots: a reconsideration JOSEPH
Federal
A. WHITT, JR.*
Reserve Bank of Atlanta, Atlanta, GA 30303, USA and Federal Reserve Board, Washington, DC 20551, USA
This paper provides empirical evidence that nominal exchange rates are stationary, and are not random walks. Other studies using classical tests suggest that exchange rates contain unit roots, thereby supporting the random walk hypothesis. The apparently permanent nature of shocks to exchange rates has been interpreted to mean that disequilibrium models such as the Dornbusch overshooting model are not useful in explaining exchange rate behavior. This paper applies a new statistical test for unit roots proposed by Sims and based on Bayesian posterior odds ratios; the results favor exchange rate stationarity for several countries. (JEL F31)
This paper presents the results of a new test for the presence of unit roots as applied to nominal exchange rates. The presence of a unit root in a variable means that it is subject to permanent stochastic shocks, not merely temporary shocks around a deterministic level or trend. As discussed in Meese and Singleton (1982), this question affects the validity of tests for efficiency in the foreign exchange market. In addition, Stockman (1987) argues that the evidence that nominal exchange rate movements are largely permanent implies that theoretical models that emphasize monetary disturbances as sources of transitory movements, such as the popular Dornbusch (1976) overshooting model, are inadequate. Empirical studies have generally supported the presence of a unit root in nominal exchange rate data (see Meese and Singleton, 1982; or Baillie and Bollerslev, 1987, 1989). Moreover, Meese and Rogoff (1983) show that the basic random walk model, which is a special case of a unit root, does as well or better than a variety of structural and more complex time series models in forecasting exchange rates. In a more general context, Sims (1988) argues that classical statistical tests for the presence of unit roots, such as the Dickey-Fuller (1979) tests used in Meese and Singleton (1982), are fundamentally flawed. As an alternative, he proposes * Federal Reserve Bank of Atlanta and Visiting Economist, Federal Reserve Board. I wish to thank Will Roberds for many helpful discussions. Sean Becketti, Curt Hunter, James Lothian, Leigh Riddick, Charles Whiteman, and two anonymous referees also made useful suggestions. The views expressed are those of the author and do not necessarily reflect those of the Federal Reserve Bank of Atlanta, or the Federal Reserve System. 0261P5606/92/06/0539-13
11; 1992 Butterworth-Heinemann
Ltd
540
Nominal
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rates
and unit roots
a test based on Bayesian posterior odds ratios that is designed to discriminate between a unit root and a large but stationary autocorrelation coefficient. This paper applies the Sims test to monthly nominal exchange rate data from the current period of flexible exchange rates. The results are mixed, but for several countries they favor the hypothesis that exchange rates are stationary and have substantial autocorrelation, but not a unit root. In Section I, Sims test and its differences with classical tests are discussed. Section II contains empirical results from applying various tests for the presence of unit roots to dollar exchange rates with five other major currencies. It also provides Monte Carlo results to compare the performance of classical tests with the Sims test. Section 111 presents the conclusions.
I. Sims test for the presence of a unit root Consider the following autoregressive (1)
model of the nominal exchange rate:
(6 - P) = P(%,
- P) + 5,
where e, is the exchange rate at time t; p is a constant term; and E, - N(0, 02) is the error term and is independent of past values of e,. In this model, the long-run behavior of the exchange rate is critically dependent on the value of the autoregressive coefficient p. If 0 < p < 1, the exchange rate is a stationary variable; it fluctuates in the vicinity of a constant unconditional mean, /L. In the absence of future shocks, the deviation (e, - cl) would shrink in subsequent periods when 0 < p < 1. In this case, the expected value of the exchange rate in the distant future is its unconditional mean, p. By contrast, if there is a unit root (p = I), the behavior of the exchange rate is quite different. In this case, the exchange rate is nonstationary; it has no fixed unconditional mean, and there is no tendency for (e, - p) to shrink. Instead, the exchange rate is a random walk; the expected value of the exchange rate in the distant future is always equal to its current value, e,. Accordingly, the problem for empirical work is to make statistical inferences about the value of p. Using a classical approach, Dickey and Fuller (1979) provide tests of the null hypothesis that p = 1, using statistics generated by an OLS regression of e, onto its own lagged value. The standard t-test is not appropriate, because under this null hypothesis the variance of the exchange rate is infinite. Sims (1988) and Sims and Uhlig (1988) argue that classical tests such as the ones proposed by Dickey and Fuller provide a misleading picture of the plausibility of unit roots. Using Bayesian methods, they show that the prior implicit in the classical tests not only gives excessive (in their view) weight to the unit root null, but also gives substantial and disproportionate weight to values of p above 1.’ As an alternative to the classical analysis, Sims (1988) proposes a Bayesian test based on a suggestion in Learner (1978, pp. 100-108). Learner argues that when testing a point-null hypothesis against a composite alternative, using the Bayesian posterior odds ratio is a more sensible approach than using a likelihood ratio test or the sampling theory approach. The posterior odds ratio can be interpreted as a weighted average of the likelihood function over all points
JOSEPH
A. WHIR, JR.
541
consistent with the null hypothesis, divided by a similar weighted average of the likelihood function over all points in the alternative. The weights are derived from the prior distribution of the parameters. By contrast, the likelihood ratio test uses only the maximum values of the likelihood function for each alternative. To apply Learner’s suggestion to equation ( 1 ), Sims proposes a prior distribution for p which spreads probability a, 0 < c1< 1, uniformly on the interval (0, l), and gives the unit root (p = 1) probability (1 - a). This specification gives a clear but limited advantage to the unit root hypothesis, because any individual point between zero and one has essentially zero prior probability, while the point where p = 1 has probability (1 - CC). Suppose fsP = Jm z = (1 - fi)/ up is the conventional t-statistic for testing p = 1, (D(x) is the cumulative distribution function for the standard normal distribution evaluated at x, and 4(x) is its probability density function. Then Sims shows that in large samples the posterior odds ratio favors the null hypothesis (p = 1) if’
This criterion is different from classical hypothesis tests, not only because of the role of the prior distribution, but also because of the presence of rsp in the denominator. If the null hypothesis (p = 1) is true, oP should shrink much faster as sample size rises than when the alternative is true. This criterion tends to favor the null hypothesis when cp is small (for given values of a and 7). By contrast, likelihood ratio tests fail to use all the information in Do. In actual applications, Sims suggests that for annual economic data the alternative hypothesis can reasonably be limited to values of p between l/2 and 1. For more frequent data, the interval associated with this alternative hypothesis has a lower bound closer to 1. In the case of quarterly data, the interval is approximately (0.84, l), because 0.84 to the fourth power is equal to the lower bound for annual data; the interval is (0.94, 1) for monthly data. Therefore, Sims proposes the following revised criterion: the null hypothesis (p = 1) is favored if y > 0,
(3) where y=2log
1-a __ i CI
1
- log[a,2] + 2log[l
- 2-“s]
- 2log[Q(r)]
- log[2711 - r2,
and s is the number of periods per year (e.g., 12 for monthly data).3 In typical examples, cP < 1, implying that -log(o,2) is positive. Smaller values of (TVinduce larger values of - log( c,‘), thereby favoring the unit root hypothesis. However, larger values of T = (1 - ;)/a,, favor the alternative hypothesis. In empirical work, it is informative to present the Sims criterion in a different way, by calculating the marginal prior probability on the unit-root null, (1 - CI”), that on an ex post basis would be necessary to force the Sims criterion to favor the null hypothesis. The formula for (1 - CC*)can be derived by setting y equal to zero and solving for the value of (1 - cr*), taking z and cp as given by the
542
Nominal exchange rates and unit roots
sample information. (4)
Doing so produces the following:
(1 - a*) = 1 - l/{ 1 + exp[log(o,) + log(Q(r))
- log(1 - 2-i”)
+ (1/2)log(2n)
+w)~21},
where exp is the exponential function. The value of (1 - x*) depends only on the sample information. A large value of ( 1 - a*), i.e., a value near 1, implies that the sample information tends to reject the unit-root hypothesis; hence a large prior weight on the unit-root null is necessary to force the Sims criterion to favor a unit root. Alternatively, a small value of (1 - a*), i.e., a value near zero, implies that the sample information tends to favor the unit-root hypothesis. In an actual testing situation, if the value of (1 - u*) is larger than the reader’s own prior on the unit root, then the reader’s posterior odds ratio favors stationarity; if it is smaller, the test result favors the unit-root hypothesis. Another issue in applied work is whether to include a constant or trend term in the regression used to estimate 6. In the case of classical DickeyyFuller tests, critical values are affected by the presence or absence of a constant or trend.4 Moreover, the critical values provided by Fuller (1976) for the test that includes a constant are correct only if the true model is a random walk with zero drift (see Schmidt, 1988). The Bayesian test statistics presented in equations ( 3 ) and (4) can be derived from models containing a constant or trend if the prior on the constant or trend is flat.5 However, the performance of the test is affected by the treatment of the constant and trend terms; therefore, to help in evaluating empirical results using this test, Monte Carlo evidence is presented below.
II. Empirical
results with exchange
rate data
It has been known for some years that since the end of the Bretton Woods system
and the move to a flexible exchange rate system, nominal exchange rates between major currencies have approximately followed random walks (see, for example, Mussa, 1979). The volatility and unpredictability of exchange rate movements have often been attributed to the impact of ‘news’ in an institutional setting in which exchange rates are asset prices determined in highly efficient markets which are not dominated by government regulation or intervention (Mussa, 1976, 1979; Frenkel, 1981). In such a setting, new information is immediately embodied in the current spot exchange rate as well as expected future spot rates. Observed movements of the spot rate are for the most part unanticipated because they mainly reflect the impact of new information; if there were a large anticipated component of exchange rate changes, then the market would not be efficient because large profits could be gained at little risk. Even so, there are reasons to think that exchange rates are not true random walks. Mussa ( 1979, p. 11) observes that the anticipated rate of change of exchange rates may well be serially correlated but not constant, giving rise to a limited amount of serial dependence in actual exchange rate changes. In particular, the popular Dornbusch (1976) model of exchange rate behavior implies that monetary shocks can cause the exchange rate to overshoot its new long-run equilibrium in the short run, before gradually adjusting toward long-run equilibrium. During
JOSEPH A. WHITT, JR.
543
the period that the exchange rate is moving gradually toward long-run equilibrium, there would be serial dependence of exchange rate changes, at least in the absence of additional real or monetary shocks. Hakkio (1986, p. 223) argues that the exchange rate should follow a random walk only if the market fundamentals that determine the exchange rate also follow random walks. Theories of exchange rate behavior suggest that the market fundamentals include domestic and foreign money supplies, real incomes, interest rates, and current accounts; in his view, these variables are unlikely to be random walks. Empirical tests using exchange rate data from the current period of flexible exchange rates have strongly favored the random walk hypothesis6 In testing for market efficiency using autocorrelations, Cornell (1977) finds that a random walk model of the spot exchange rate fits the data very well. Meese and Singleton (1982) perform Dickey-Fuller tests using weekly data; their results favor the presence of a single unit root, which is consistent with the random walk hypothesis.’ Using a test proposed by Phillips and Perron (1988) that is designed to correct the Dickey-Fuller approach for general forms of serial correlation or heteroskedasticity, Baillie and Bollerslev (1987, 1989) also report evidence in favor of the presence of a single unit root in exchange rate data. As discussed in Meese and Singleton (1982), the finding of unit roots in exchange rate data implies that it is inappropriate to estimate models explaining the level ofexchange rates, as in Bilson (1978) or Frankel(1979). In addition, tests involving the relationship between the forward rate and the expected future spot rate should not be done in levels, as in Frenkel (1981). Unit root tests also have implications for theoretical modeling of exchange rates. As discussed in Nelson and Plosser (1982) and Schwert (1987), the presence of a unit root in a variable means that it is subject to permanent shocks that generate a stochastic long-run growth path, rather than merely temporary shocks around a deterministic level or trend. A random walk is a special case of a unit root, in which all innovations are permanent. Nelson and Plosser (1982) report that many macroeconomic series, including several that presumably are fundamental determinants of exchange rates, contain unit roots even after adjustment for time trends. Similar findings are reported by Harvey (1985) and Schwert (1987), among others.’ Nelson and Plosser (1982) argue that their finding of unit roots in a variety of macroeconomic time series (not including exchange rates, which they do not examine) implies that theoretical models that emphasize monetary disturbances as sources of transitory fluctuations are inadequate.9 Instead, they recommend theoretical models that emphasize real factors as sources of permanent shocks. With regard to exchange rates, Stockman (1987) argues in a similar vein that the evidence that nominal exchange rate movements are largely permanent implies that the Dornbusch (1976) overshooting mechanism cannot be an important source of observed exchange rate movements. In related work, Meese and Rogoff (1983) show that a simple random walk model does as well or better than a variety of structural and more complex time series models in forecasting the exchange rate during the period since the end of Bretton Woods. The structural models that they consider include a version of the Dornbusch ( 1976) model, and they allow for a number of possible determinants of exchange rates, including the money supply, real income, the short-run interest differential, the long-run expected inflation differential, and trade balances. The
Nominal exchange rates and unit roots
544
inability of traditional structural models to explain the exchange rate movements of recent years has stimulated work on alternative hypotheses, such as the speculative bubbles examined in Meese (1986) or equilibrium exchange rate models that emphasize the importance of real shocks to technology or tastes, as recommended in Stockman (1987). How solid is the conclusion that exchange rates contain a unit root? As discussed earlier, the test proposed by Sims (1988) provides an alternative to conventional tests of the Dickey-Fuller variety. Using the United States as the base country, the test was applied to monthly data on the nominal exchange rate for five countries: the United Kingdom, France, Germany, Switzerland, and Japan.” The sample period began in June 1973, several months after the final breakdown of the Bretton Woods system, and ended in May 1988, thereby providing 180 observations. The tests were performed in two ways. For Table 1, fi was estimated by regressing the log of each exchange rate onto its own lagged value, plus a constant term. For Table 2, a linear time trend was added as an additional regressor.’ 1 To check the specification, residuals were examined for evidence of
TABLE
1. Tests for a unit root in the nominal using constant only (no time trend).
exchange
DickeyyFuller Country United Kingdom France Germany Switzerland Japan Nore:
The sample
period
P
r,l
0.9793 0.9895 0.9823 0.9840 1.0082
-1.76 - 1.09 ~ 1.13 - 1.24 0.74
Sims (I -x*) 0.7044 0.402 1 0.5356 0.5262 0.1297
is June 1973 to May 1988
TABLE 2. Tests for a unit root in the nominal with time trend included.
exchange
DickeyyFuller Country United Kingdom France Germany Switzerland Japan Nole:
The sample
period
rate
rate
?
rr
Sims (1 -X*)
0.983 1 0.989 1 0.9807 0.9752 0.9838
- 1.00 -0.78 - 1.21 - 1.51 -0.96
0.5122 0.3986 0.5674 0.6824 0.4974
is June 1973 to May 1988.
545
JOSEPHA. WHITT,JR.
autocorrelation, using the Durbin (1970) h-statistic and the Box-Pierce Q-statistic (see Granger and Newbold, 1977); no evidence of autocorrelation was found.i2 As expected on the basis of previous empirical work on this issue, the regressions produced estimates of the autoregressive coefficient i, rather close to one; the first column of each table provides the estimates. In addition, statistics for testing the null hypothesis (p = 1) are presented in each table. For comparison purposes, the second column in each table presents the Dickey-Fuller test statistic, r E ((fi - 1)/o,). To reject the null hypothesis at the 90 per cent significance level would require that r be less than -2.57 in Table 1, and less than - 3.13 in Table 2.13 Clearly, none of the countries in this sample comes close to rejecting the unit root on the basis of these classical tests.14 These results using the DickeyyFuller test corroborate the results in Meese and Singleton (1982). What about the Sims test? Recall that the Sims test favors the null hypothesis (p = 1) if the test statistic (1 - x*) defined in equation (4) is less than the prior weight on the unit-root null. Values of (1 - a*) for each exchange rate are given in the third column of each table. If one’s prior gives even odds (1 - x = 0.5) then the results are consistent in both tables, favoring stationarity for the British pound, German mark, and Swiss franc, while favoring a unit root for the French franc and Japanese yen. Alternatively, if one gives more weight to stationarity, as favored by Sims (1988, p. 471), and chooses the prior weight on the unit root to be 0.2, then the test results would favor stationarity for one additional currency, the French franc, as well as for the Japanese yen in the results including time trend.’ 5 As mentioned above, the Sims test was derived on the assumption that the sample size is large, and like the Dickey-Fuller test it is affected by the treatment of possible constant and trend terms.16 To shed light on the properties of these tests, a Monte Carlo study was done using artificially-generated time series of the same length as the exchange rate data, 180 months. Four data-generating processes were used: a simple random walk, a random walk with drift, a stationary model with constant mean, and a stationary model with trend. All of the models can be represented as variants of the following equation: (5)
Yf - (Yo + Bt) = PC_&1 - (Yo + P(t -
I))1 + P + Et,
where y, is the initial value of y and is non-stochastic; and e, is a normally-distributed random error term. If p is set equal to 1, and the coefficient on the time trend (p) and the drift parameter (PC)are both equal to zero, equation (5) reduces to a simple random walk. Allowing p to be non-zero generates a random walk with drift. Alternatively, if p is positive but less than 1, y, is a stationary variable. With B and p both equal to zero, y, is stationary with a fixed unconditional mean, y,. If /I and p are equal to one another but non-zero, y, is stationary around a linear trend. In the Monte Carlo simulations, artificial time series of y, were generated using equation (5) with assumed values of p, p, p, and y,, plus values of e, created by the GAUSS 2.0 random number generator. For each repetition, 190 monthly observations were created. The DickeyyFuller and Sims test statistics were then calculated using only the last 180 observations to reduce any effects of the choice of starting value y,. To obtain rejection frequencies, this process was repeated
546
Nominal exchange
TABLE3. Monte
Carlo results (rejection
A. B.
C.
D. E. F.
frequencies).
Dickey-Fuller with constant
Model p=l p=p=o p=l /f = -0.05 fl=o p=l p = -0.01 /I=0 p = 0.85 p=fl=o p = 0.85 p = /I = -0.00058 p = 0.85 p = p = 0.00358
10.40%
rates and unit roots
Dickey-Fuller with trend 10.08%
Sims with constant
Sims with trend
70.00%
93.76%
0.96
10.08
1.20
93.76
2.12
10.08
10.40
93.76
14.20
11.96
85.72
97.04
13.04
11.96
83.88
97.04
5.44
11.96
49.08
97.04
2500 times for each model. In all cases 10 per cent critical values of the Dickey-Fuller tests from Fuller (1976) were used, while in the Sims tests, the prior probability of the unit-root null was set at 50 per cent (even odds).” Table 3 presents rejection frequencies for the unit-root null hypothesis for six different data-generating models. The results in the first row of the table, labelled A, are for data generated by a simple random walk, with no drift. This is the model that represents the null hypothesis in classical Dickey-Fuller tests. As the table indicates, for this model the Monte Carlo results indicate that both Dickey-Fuller tests mistakenly reject the unit-root null about 10 per cent of the time, as they should given the choice of test size. For this model and sample size the Sims tests do much worse. The Sims test with a constant alone mistakenly rejects the unit-root null about 70 per cent of the time, while the Sims test with trend rejects over 93 per cent of the time. The next two rows give results when the data are generated by a random walk with drift. Two drift terms were used; they were chosen to approximate in size the drift implied by regressions of exchange rates on a constant and their own lagged values. Row B has a large (in absolute value) drift, as suggested by results using the Japanese yen; row C has a smaller drift, as suggested by results using the British pound. Adding drift causes the Dickey-Fuller test with constant alone to reject the unit-root null much less frequently; for example, for model B the rejection frequency is only 0.96 per cent, even though the critical value was chosen for a test of size 10 per cent.” The performance of the Sims test with constant alone improves substantially when drift is introduced to the random walk model. When the drift term is large (row B), the Sims test with constant mistakenly rejects the unit-root null only 1.2 per cent of the time; when the drift is smaller (row C), the rejection rate rises to 10.4 per cent, still far below the 70 per cent rate that occurred when there was
JOSEPH
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no drift at all (row A). As for the tests with trend included, both the Dickey-Fuller and the Sims test generate the same rejection frequencies as they did when no drift was present. The last three rows of the table give rejection frequencies when the data are actually stationary. In these rows, the value of the autoregressive coefficient p was set at 0.85 in terms of annual data; this value is well inside the stationary region, but still implies substantial positive autocorrelation. In terms of monthly data, this value translates into a p of 0.9865, which is in the range of estimated values reported in Tables 1 and 2. In row D, there is no trend; the data are stationary around a constant unconditional mean. For this sample size and value of p, the Dickey-Fuller test with constant alone correctly rejects the unit-root null only 14.2 per cent of the time; the Dickey-Fuller test with trend rejects even less frequently, 11.96 per cent of the time. By contrast, the Sims test with constant alone rejects 85.72 per cent of the time, while the Sims test with trend rejects 97.04 per cent of the time. Rows E and F use stationary models with linear trends. The sizes of the trend parameters were chosen to be roughly consistent with actual exchange rate data.lg Adding a trend to the stationary model causes the Sims test with constant alone to reject the unit root less frequently; for the smaller trend (row E) its rejection frequency is 83.88 per cent, and for the larger trend (row F) it is down to 49.08 per cent. Even so, it correctly rejects far more often than the Dickey-Fuller test with constant alone, which is also affected by the presence of trend. By contrast, both the Dickey-Fuller and Sims tests that include a trend are not affected by the size of the trend in the artificial data. To summarize the Monte Carlo results, none of the tests is clearly superior to the others in all situations. Both of the classical Dickey-Fuller tests rarely reject the unit-root null when it is true, but they have little power against stationary alternatives.20 By contrast, for this sample size the Sims test with trend rarely accepts the unit-root null, even when it is true. The test that shows the greatest ability to discriminate between a unit-root model and stationarity is the Sims test with constant alone. It rarely rejects the unit-root null when the data are generated by a random walk with drift, as in rows B and C, and it does reject the unit-root null fairly frequently when the data are generated by a stationary model with p equal to 0.85 (in terms of annual data) and either a small or zero trend element, as in rows D, E, and F. However, the Sims test with constant can often lead to mistaken conclusions when the data are generated either by a random walk with zero or near-zero drift, as in row A, or by a stationary model with a large trend element, as suggested by a comparison of rows E and F.21 In light of the Monte Carlo evidence, what conclusions can be drawn about the presence of unit roots in exchange rate data? It is clear that in a classical testing framework, we cannot reject the unit-root null at standard significance levels for any of these currencies; however, this does not imply that a unit root is very likely, because the power of the classical tests is rather low. Using reasonable priors, a Bayesian approach indicates that the weight of the evidence favors stationarity for three currencies (the British pound, German mark, and Swiss franc) and possibly a fourth (the French franc), depending on the prior. The Japanese yen shows the strongest evidence in favor of a unit root, but even it might be stationary if a strong trend were allowed for.
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Conclusions
This study indicates that, by a preponderance of the evidence, though not beyond a reasonable doubt, nominal exchange rates do not contain unit roots. This conclusion is based on results using a Bayesian statistical test that is designed to discriminate between a unit root and a large but stationary autocorrelation coefficient. The test was applied to monthly exchange rate data for five industrial countries vis-ci-vis the United States. While classical tests are consistently unable to reject the unit root hypothesis, the new test does reject it consistently for three of the countries. For the other two countries, results were mixed, depending on the exact prior used and on whether a time trend was included. It should be emphasized that the Bayesian test proposed by Sims that is used in this paper is quite different from the familiar DickeyyFuller test, as shown by the Monte Carlo simulations reported in Table 3. In the language of classical testing, an ideal test would combine high power and small size. The Monte Carlo results, which used artificial data that were constructed to be similar to the available sample of exchange rate data from the current period of flexible exchange rates, show that the classical DickeyyFuller test with a nominal size of 10 per cent has rather low power, less than 15 per cent, even when the true autoregressive parameter is well below one. The Sims test has much greater power, but its size is problematic. When a time trend is included, the size of the Sims test is disconcertingly large. When only a constant is included, the size of the Sims test depends on the true parameters generating the data; the size is small when the artificial data contain the amount of drift indicated by regressions using actual exchange rate data, but it is disturbingly large if the data are generated by a random walk with little or no drift. Using what I believe are reasonable priors that have been suggested in the literature, I find with the Bayesian approach that, in most cases, the evidence favors the hypothesis that nominal exchange rates do not contain unit roots. The results in this paper suggest that exchange rates are close to, but not quite, random walks; shocks to the level of exchange rates are not all permanent. Therefore, it may be premature to dismiss standard models of exchange rate behavior because they allow for temporary exchange rate movements. An alternative interpretation is that for the countries in the sample, which have large, well-diversified economies and reasonably stable monetary policies, nominal exchange rates tend to move within some sort of relatively narrow bounds, because the movements of real and monetary factors are limited. After all, during the sample period none of these countries experienced hyperinflation or a drastic change in real income, either of which would cause a major permanent shift in at least the level of the exchange rate, according to standard exchange rate theory.
Notes 1. 2.
3.
Sims and Uhlig (1988) p. 8. In a full Bayesian analysis, the constant on the right-hand side of expression (2) might be a number other than one, depending on the loss function applying to the two hypotheses. However, this extension is not pursued here. Sims suggests ignoring the term ‘- 2log[@(r)]’ because it is likely to be small when ,6 < 1 and is asymptotically negligible; he also leaves out the term ‘-log(2n)’
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See Fuller (1976), p. 373. See Sims (1988), p. 467. If non-flat priors about the constant or trend terms are used, DeJong and Whiteman (1989) and Schotman and van Dijk (1990) have shown that inferences about the autoregressive parameter can be affected. 6. More precisely, previous empirical work has shown that classical tests fail to reject the null hypothesis of a unit root. Nevertheless, these results have been commonly interpreted as affirmative evidence in favor of a unit root, as in Baillie and Bollerslev (1989) or Lastrapes (1989). Meese and Singleton also estimated autoregressions of the exchange rate of order one through eight; the fact that the coefficients beyond the first lag were small was also favorable for the random walk hypothesis. Hakkio (1986) casts doubt on these results by demonstrating that standard tests such as those used by Cornell (1977) or Meese and Singleton (1982) have very low power against alternatives that are ‘nearly’ random walks. However, Hakkio’s results do not bear directly on the possibility that exchange rates do not contain a unit root, because the alternatives he considers are all ARIMA processes that contain a unit root. The conclusion that many macroeconomic time series contain unit roots and are not trend-stationary has recently been challenged by DeJong and Whiteman (1989) and DeJong et al. (1988). With regard to real GNP, West (1988) and Cochrane (1988) counter that finding a unit root in this series does not necessarily imply that monetary shocks are unimportant, because there are plausible theoretical models with both a unit root in real GNP and an important role for monetary shocks. Also see Christian0 and Eichenbaum (1989). 10. The use of monthly data is advantageous because it greatly reduces the problems of heteroskedasticity and departures from normality. As discussed in Baillie and Bollerslev (1987), conditional heteroskedasticity and departures from normality are pronounced in daily and weekly exchange rate data, but they virtually disappear when monthly data are used. In this paper, end-of-month data from International Financial Statistics were used. 11. Meese and Singleton (1982) include a time trend in all their equations. 12. In calculating the Box-Pierce statistic, 12 lags were used to coincide with monthly data; in addition, 20 lags were used as recommended in Granger and Newbold (1977, p. 93). For further discussion of this issue, see Hakkio (1986, footnote 9). 13. The critical values were obtained from Fuller (1976), p. 373. 14. As a check on the results presented here, the classical Phillips-Perron (1988) test that is designed to correct the Dickey-Fuller approach for general forms of serial correlation or heteroskedasticity was performed as well. The results coincided closely with the Dickey-Fuller results, indicating no rejections of the unit-root hypothesis. 15. The prior favored by Sims still gives some advantage to the unit-root hypothesis, because in terms of annual data the point null hypothesis has the same prior probability as the infinite number of points in various intervals that are consistent with the alternative hypothesis, for example (0.875 < p < 1). 16. As suggested by an anonymous referee, one reason for concern about the small-sample properties of the Sims test is the well-known downward bias in finite samples of the least-squares estimate of the coefficient on a lagged dependent variable, such as 3. 17. The same random numbers were used for each choice of parameter values, as in Hakkio (1986). In addition, the standard error of the random error terms was set at 0.035 for all models; this is similar to the standard errors obtained using actual exchange rate data. 18. The sensitivity of the Dickey-Fuller test to the presence of drift is discussed in Schmidt (1988). 19. The results presented here use the largest and smallest (in absolute value) trend terms found for the five currencies used in this study; they were obtained by regressing the log of each exchange rate onto a constant and a linear trend. 20. Hakkio (1986), DeJong and Whiteman (1989), and DeJong et al. (1988) have all discussed the low power of the Dickey-Fuller and similar classical tests of the unit-root hypothesis when faced with relevant alternatives. 21. Additional Monte Carlo results, not reported here, verify that for the stationary model, increasing the size of the trend coefficient beyond the value used in row F will, ceteris parihus, further shrink the rejection frequency of the Sims test with constant alone.
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