Exchange Rate Dynamics and Multiple Structural Breaks: a Preliminary Investigation

Cem Ertur, LATEC UMR 5601, Université de Bourgogne1 Zaka Ratsimalahelo, CRESE, Université de Franche-Comté2

First draft: March 1999

Abstract: This paper considers issues related to multiple structural changes in the context of exchange rate dynamics. The most commonly used formal test for long-run Purchasing Power Parity (PPP) consists of testing whether the real exchange rate has a unit root. If the unit root hypothesis can be rejected, there is evidence of mean reversion of the real exchange rate or, equivalently, long-run PPP. But the empirical evidence for the post 1973 period has been mixed. Failure to reject the unit root hypothesis may be due to the presence of structural breaks in the real exchange rate series. It is indeed well known that standard unit root tests are biased in favor of nonstationarity when a structural change is present (Perron 1989, 1990). Moreover all unit root tests with endogenous determination of date of the break on the mean or trend (Perron and Vogelsang 1992 and Banerjee, Lumsdaine and Stock, 1992) suppose the presence of a single break. But it may be the case that more than one break is present in the series, so the results of these latter tests might be biased too. We therefore propose to test the number of structural breaks and determine endogenously their dates using the methodology developed by Bai and Perron (1998a,b) prior to unit root tests. The empirical application is based on five nominal and real exchange rates relative to the US Dollar (Deutsche Mark, French Franc, Japanese Yen, British Pound and Swiss Franc) in monthly and quarterly data over the period 1971-1998 and we find strong evidence in favor of multiple structural breaks.

1

LATEC, Pôle d’Economie et de Gestion, 2 bd Gabriel, 21066 Dijon Cedex – France. e-mail : [email protected]. 2 CRESE, U.F.R. Sciences Economiques, Université de Franche-Comté, 25030 Besançon. e-mail : [email protected].

Key words : exchange rates, purchasing power parity, multiple structural changes, break point determination

2

1. Introduction An important issue for exchange rate modeling is the relative importance of real versus nominal shocks in accounting for movements in real and nominal exchange rates. Disequilibrium models of exchange rate determination (e.g. Dornbusch, 1976) assume that prices in goods markets adjust sluggishly to shocks and that variation in real and nominal exchange rates is due primarily to nominal shocks (e.g. shocks to the money supply). Equilibrium models (e.g. Stockman, 1987) assume instantaneous price adjustment and rely on permanent real shocks (e.g. shocks to technology, preferences) to explain movements in real and nominal exchange rates. Isolating these sources of variation can assist policymakers in determining the extent of excess variability in exchange rates. The period of flexible exchange rates since 1973 has been characterized not only by high volatility of real exchange rates but also by apparent absence of mean-reversion. The most commonly used formal test for long-run purchasing power parity (PPP) consists of testing whether the real exchange rate has a unit root. If the unit root hypothesis can be rejected, there is evidence of mean reversion of the real exchange rate or, equivalently, long-run PPP. Alternatively, since cointegration between the exchange rate, domestic price level and foreign price level is a necessary condition for PPP, evidence of PPP can be found by testing the null hypothesis of no cointegration among these variables. The empirical evidence for the post 1973 period has been mixed. While several authors report evidence in favor of mean reversion, evidence against the unit root hypothesis is still elusive (e.g. Corbae and Ouliaris, 1988, 1991; Mark and Choi, 1997). Whether one uses conventional Augmented-Dickey-Fuller tests or a variety of more powerful alternatives, the unit root null is typically rejected only occasionally. The problem, of course, is that it is impossible to determine whether the failure to reject the unit root hypothesis is caused by the low power of the relevant statistical tests on relatively short spans of data, or whether the unit root null cannot be rejected because it is the correct hypothesis. The failure to reject the unit root hypothesis may also be caused by the presence of structural breaks in the foreign exchange markets. Indeed, events such that institutional changes, development of commodity and financial markets, fiscal and monetary regime changes, oil shocks etc. may be the source of structural breaks in the series. Perron (1989,1990) notes that unit root behavior may be mimicked by a series that contains a structural break. This is a potentially significant criticism for the more recent studies of exchange rates. Results from studies that find that the real exchange contains a unit root may thus be spurious if the series includes one or more structural breaks for which account is not taken. Cointegration tests, which examine whether the residual from a regression of the nominal exchange rate on the foreign-domestic price level differential contains a unit root but neglect to test for a structural break, may also be misleading. Dropsy (1996) find for five exchange rates that structural changes can be responsible for most exchange rate deviations from their equilibrium defined by purchasing power parity. He uses different tests such that CUSUM test by Ploberger and Kramer (1992) and Wrigth (1993), the sequential additive and innovative outlier tests by Perron and Vogelsang (1992) and the sequential trend-shift tests by Banerjee, Lumsdaine and Stock (1992). While these tests treat the date of break as unknown a priori, all consider only a single structural change. In this paper, we consider the problem of the multiple structural changes which is more attractive on

3

longer spans of data. Note that, unlike some previous papers, our approach is directly oriented at the issue of testing for multiple structural changes in the mean of the exchange rate series per se and not of testing for a unit root in the presence of a structural change. Once the different regimes are identified it may be possible to test for unit root inside each of them without bias problems. While this approach cannot be considered as a test of PPP, it can still be considered as a first step before implementation of unit root tests. The problem of multiple structural changes has indeed received more attention recently. Andrews, Lee and Ploberger (1996) consider optimal tests in the linear model with known variance. Garcia and Perron (1996) study the sup Wald test for two changes in a dynamic time series. Liu, Wu and Zidek (1997) considered multiple structural changes in a linear model estimated by least squares and proposed an information criterion for the selection of the number of changes. Independently, Bai and Perron (1998a, b) consider a similar problem in a more general framework: a supWald type tests for the null hypothesis of no change versus an alternative hypothesis containing an arbitrary number of changes, a test for the null hypothesis of l changes versus the alternative hypothesis of l + 1 changes. Then the appropriate number of changes can be determined with a specific to general modeling strategy with unknown dates for the structural breaks. Their procedure allows also for general forms of serial correlation and heteroskedasticity in the errors, lagged dependent variables, and trending regressors. In this paper, we use Bai and Perron’s (1998a, b) methodology to test the multiple structural change hypothesis in exchange rates. The paper is organized as follows: in section 2 we consider the general model and underlying assumptions. Consistency, rate of convergence, asymptotic distribution and the method to compute global minimizers are briefly presented. The details can be found in Bai and Perron (1998a, b). In section 3 we present the different tests: a test of no break versus a fixed number of breaks, Double Maximum tests (UDmax and WDmax), a test of l versus l + 1 breaks and the problem of estimating the number of breaks. Section 4 presents and interprets the empirical results. Section 5 contains conclusions. 2. Econometric Methodology 2.1. The Model and Assumptions 2.1.1. The Model We consider the following multiple linear regression with m breaks (m+1 regimes): yt = x’tβ + z’tδ1 + εt yt = x’tβ + z’tδ2 + εt

t = 1,...,T1 t = T1+1,...,T2

.

(1)

.

. yt = x’tβ + z’tδm+1 + εt

t = Tm+1,...,T.

where yt is an observation on the dependent variable at time t; xt(px1) and zt(qx1) are vectors of regressors and εt is an unobservable disturbance. The vectors β and δj (j = 1,...,m+1) are unknown parameters. The indices (T1,...,Tm), or the break points, are explicitly treated as unknown. Our purpose is to estimate the unknown regression coefficients together with the break points when T observations on (yt, xt, zt) are available. Note that this is a partial structural change model in the sense that the parameter vector β is not subject to shifts and is

4

effectively estimated using the entire sample. When p = 0, we obtain a pure structural change model where all the coefficients are subject to change. To proceed, it is convenient to introduce some terminology. First, we call an m-partition of the integers (1,...,T), an m-uple vector of integers (T1,...,Tm) such that 1 0.That is, there exist nonnegative constants {ci: i ≥ 1} and {ψm: m ≥ 0} such that ψm ↓ 0 as m → ∞ and for all i ≥ 0 and m ≥ 0, we have (a) EE(εiℑi-m)r ≤ cirψmr. (b) Eεi - E(εiℑi+j) ≤ cirψri+j (c) maxi ci ≤ k < ∞ (d) ∑∞j= 0 j1+kψj < ∞ for some k > 0. (e) The disturbances εt are independent of the regressors ws for all t and s. ii) Let ℑt* = σ-field {...,wt-1,wt,..., εt-1, εt-2...) .We assume (a) that (εt) is a martingale difference sequence relative to ℑt* and suptEεt4+δ < ∞ (b) We have T-1 ∑[t Tv ] ztzt’ →p Q(v) uniformly in v ∈ [0, 1], where Q(v) is positive definite for v > 0 and strictly increasing in v. c) If the disturbances εt are not independent of the regressors {zt} for all t and s, the minimization problem defined by (3) is taken over all possible partitions such that Ti − Ti −1 > αT (i = 1,...,m+1) for some α > 0. (note that this not required under part (i). A5. Ti0 = [Tλi0], where 0 < λ10 < ... 1 as am = c(q,α,1)/c(q,α,m). This version is denoted: WD maxF*T(M,q) = max1≤m≤M am sup F*T((λ1,...,λm;q). Again, we use the asymptotically equivalent version: ) ) WDmaxFT(M,q) = max1≤m≤M am FT(( λ 1, ..., λ k;q) Note that, unlike the UDmaxFt(M,q) test, the value of the WDmaxFT(M,q) depends on the significance level chosen since the weights themselves depend on α. Asymptotic critical values are provided in Bai and Perron (1998a, table I, p. 58 and 1998b tables 1a, 1b and 1c) for M = 5 and ε = 0.05, 0.10, 0.15, 0.20 and 0.25. 3.3. A test of l versus l + 1 breaks Bai and Perron (1998a, p.59) propose a test of the null hypothesis of l breaks against the alternative that an additional break exists. This test is labeled supFT( l + 1 / l ). The test is ) ) ) applied to each segment containing the observations Ti -1 to Ti (i = 1,.. l + 1 ), the estimates Ti need not be the global minimizers of the sum of squared residuals, all that is required is that the ) ) break fractions λ i = Ti /T converge to their true value at rate T. They conclude for a rejection in favor of a model with l + 1 breaks if overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of squared residuals from the l breaks model. The break date thus selected is the one associated with this overall minimum. Asymptotic critcal values are tabulated in Bai and Perron (1998a table II, p. 61 and 1998b, tables 2a, 2b, 2c and 2d) for ε = 0.05, 0.10, 0.15, 0.20 and 0.25, values of l from 1 to 9 and values of q from 1 to 10. 3.4. Estimating the number of breaks A common procedure to select the dimension of a model is to consider an information criterion. Yao (1988) suggests the use of the Bayesian Information Criterion (BIC) defined as ) BIC (m) = ln σ 2 (m) + p*ln(T)/T, ) ) ) where p* = (m+1)q + m + p and σ 2 = T-1ST( T1 ,..., Tm ). The number of breaks can be consistently estimated at least for normal sequence of random variables with shifts in mean. An alternative proposed by Liu, Wu and Zidek (1997) (LWZ) is a modified Schwarz’criterion that takes the form: ) ) LWZ(m) = ln[ST( T1 ,..., Tm )/(T - p*)] + (p*/T)c0(ln(T))2+δ They suggest using δ = 0.1 and c0 = 0.299. However simulation experiments presented in Bai and Perron (1998b) shows that using BIC can imply surestimation of the number of breaks particularly when serial correlation is present and LWZ can imply underestimation. The method proposed is then based on the sequential application of the supFT( l + 1 / l ) tests. Start by estimating a model with no break. Then perform parameter-constancy tests for each subsamples (those obtained by cutting off at the estimated breaks), adding a break to a subsample associated with a rejection with the test supFT( l + 1 / l ). This process is repeated increasing l sequentially until the test supFT( l + 1 / l ) fails to reject the null hypothesis of no

11

additional break. The final number of breaks is thus equal to the number of rejections obtained with the parameter constancy tests (Bai and Perron, 1998b, p.26). The limiting distribution of the test, in this sequential setup, is the same as before because it still implies break fractions that converge at rate T (Bai, 1997). 4. Empirical Results In this section, we apply the testing procedures proposed by Bai and Perron (1998a,b) to five nominal and real exchange rates in logarithms relative to the US Dollar (namely Deutsche Mark DEM/USD, French Franc FRF/USD, Japanese Yen JPY/USD, British Pound GBP/USD and Swiss Franc CHF/USD) for the 1971-1998 period. We consider monthly nominal data from Pacific Exchange Rate Service (by W. Antweiler, University of British Columbia, Vancouver, Canada) for the period 1971:1-1998:12 (336 observations). The monthly real exchange rates are computed using Consumer Price Indexes from the UN Monthly Bulletin of Statistics for the period 1971:1-1998:8 (332 observations). The quarterly nominal data are computed as averages over monthly data for the period 1971:1-1998:4 (112 observations) and the real data cover the period 1971:1-1998:2 (110 observations). Figures 1 to 10 illustrate the fluctuations of these exchange rates over each period. We are interested in testing the presence of structural change in the mean of these series, more precisely the number of breaks (with a max of 5 breaks) and the dates of these breaks. It must be stressed that this cannot be interpreted as a test of the Purchasing Power Parity hypothesis when applied to logarithms of real exchange rates since we do not test for unit root, but simply as a test of constancy of PPP equilibrium. It can be considered like a first stage analysis since the presence of one or more structural changes in the series can cause serious bias in unit root testing procedures as indicated by Perron (1989, 1990). To that effect we apply the procedure proposed by Bai and Perron (1998a,b) to the simplest case of pure structural change with only a constant as regressor : z t = {1} and take into account the possible structure of serial correlation and heteroskedasticity using the procedure of Andrews (1991)3. We used a trimming of ε = 0.10 which corresponds to each segment having at least 33 observations for monthly data and 16 observations for quarterly data4. The results are presented in table 1 for monthly nominal and real exchange rates (sample size 336 and 332 observations respectively) and in table 2 for quarterly nominal and real exchange rates (sample size of 112 and 110 observations respectively). The first issue is the determination of the number of breaks. We note that in Table 1 the sup FT (k ) tests are all significant for k = 1,...,5 except for the FRF/USD real exchange rate where it is not the case for k = 1 even at the 10% significance level, but the tests are actually significant at the 5% level for k = 2,...,5 . So at least one break is present. The double maximum tests UDmax and WDmax which allow us to test the null hypothesis of no break versus an unknown number of breaks given the upper bound of 5 breaks are all significant at the 5% level indicating the presence of at least one break for all the series. Next we turn to the SupF (l + 1 / l ) and we note that the SupF (5 / 4) tests are all significant at the 5% level.

3

We wish to thank Pierre Perron for the Gauss program we used to implement all the estimation and testing procedures. 4

We also used a trimming of ε = 0.05 , but the results were quite similar and aren’t presented here.

12

Given the simulation results presented by Bai and Perron (1998b), we will favor the sequential procedure which seems to perform better than procedures based on information criteria like BIC and the modified Schwarz criterion of Liu, Wu and Zidek (1997) (LWZ). The sequential procedure using a 5% significance level generally selects 5 breaks except for FRF/USD real exchange rate and GBP/USD nominal and real exchange rates. For the FRF/USD real exchange rate, it selects 0 break. We present in this case the results for procedures using information criteria, i.e. BIC and LWZ, which select both 5 breaks. The sequential procedure selects 3 breaks for GBP/USD nominal exchange rate and 1 break for GBP/USD real exchange rate. We note that in all the models, the estimated coefficients are significantly different from zero. The first break date generally selected corresponds to the switch to flexible exchange rate regime on March 1973 and the first oil shock. The dates vary from 73:9 to 78:6 for France, except for GBP/USD real exchange rate for which the unique break detected is on 87:9. The date for the second break corresponds to the revision of IMF’s Charter on January 1976, the dates vary from 77:9 to 78:12. The date for the third break corresponds to the second oil shock of 1981: the dates vary from 81:4 to 82:5. The date for the fourth break may be linked to the Plaza Accord on September 1985, the dates vary from 86:2 to 86:12. The date for the fifth break for DEM/USD and FRF/USD 89:12 may be linked to the German reunification, economic reconversion of Eastern Europe and European currency capital market deregulation. The dates vary from 89:12 to 95:11 for Japan. We note that 95% confidence intervals are most of the time relatively small indicating that breaks are estimated rather precisely. In Table 2 we implement the same procedures to quarterly nominal and real exchange rate series. Concerning the determination of the number of breaks, the picture is similar is some aspects but we note however some important differences. The results seem sensitive to sampling interval. The sup FT (k ) tests are all significant for k = 1,...,5 except for the DEM/USD and FRF/USD real exchange rate where it is not the case for k = 1 even at the 10% significance level, but the tests are actually significant at the 5% level for k = 2,...,5 . So at least one break is present. The double maximum tests UDmax and WDmax are all significant at the 5% level indicating the presence of at least one break for all the series. The results of the SupF (l + 1 / l ) are different from the previous ones: the SupF (5 / 4) tests are not significant even at the 10% level (except for CHF/USD nominal exchange rate) and can’t be computed for GBP/USD and JPY/USD given the location of the breaks from the global optimization, there was no more place to insert an additional break that satisfy the minimal length requirement. But the SupF (4 / 3) tests are significant at the 5% level except for FRF/USD nominal and real and GBP/USD real exchange rates. In these latter cases we note that the SupF (3 / 2) tests are significant at the 5% level. The sequential procedure using a 5% significance level selects 4 breaks except for FRF/USD and GBP/USD nominal and real exchange rates. It selects 3 breaks for the FRF/USD nominal exchange rate and 0 break for the FRF/USD real exchange rate. We present in this case the results for procedures using information criteria, i.e. BIC and LWZ, which select both 3 breaks. The sequential procedure selects 3 breaks for GBP/USD nominal exchange rate and 1 break for GBP/USD real exchange rate. We note again that in all the models, the estimated coefficients are significantly different from zero.

13

The first break date generally selected corresponds again to the switch to flexible exchange rate regime on March 1973 and the first oil shock. The dates vary from 74:4 to 77:3, except for GBP/USD real exchange rate for which the unique break detected is on 87:3. The date for the second break corresponds to the second oil shock of 1981, the dates vary from 81:1 to 82:1. The date for the third break may be linked to the Plaza Accord on September 1985: the dates vary from 85:4 to 86:2. The date for the fourth break may be linked to the German reunification and economic reconversion of Eastern Europe. These dates vary from 89:4 to 94:2. We note again that 95% confidence intervals are most of the time relatively small indicating that breaks are estimated rather precisely.

5. Conclusion We have applied a new methodology elaborated by Bai and Perron (1998a, b) to test for multiple structural change in five nominal and real exchange rates relative to the US Dollar (Deutsche Mark, French Franc, Japanese Yen, British Pound and Swiss Franc). This issue is of major interest in the context of the PPP hypothesis. Unit root tests are widely used to test this hypothesis and it is well known (Perron 1989, 1990) that standard tests are biased in favor of unit root when a structural change is present. All unit root tests with endogenous determination of date of the break on the mean or trend (Perron and Vogelsang 1992 and Banerjee, Lumsdaine and Stock, 1992) suppose the presence of a single break, except those of Lusdaine and Pappell (1997) which allow for two breaks. But it may be the case that more than one break is present in the series. So the results of these tests might be biased too. We therefore proposed to test the number of structural breaks and determine endogenously their dates using the methodology developed by Bai and Perron (1998a,b). We have noted that it is possible to detect most of the time 5 breaks in monthly series and 4 breaks in quarterly series for nominal and real exchange rates. The date of these breaks corresponds to major events in the International Monetary System, to the two oil shocks of 1973 and 1981, and economic problems encountered by Eastern Europe and European currency capital market deregulation. Yet much of the work remains to be done, further research should attempt to investigate more deeply the theoretical and fnancial aspects of thsi issue and precise the exact nature of these shocks. Unit root tests for each of the regimes identified could also be performed. Finally, multiple structural breaks in more general exchange rates models could be investigated using the same methodology.

References Andrews, D.W.K. (1991), “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation”, Econometrica, Vol 59, pp. 817-858. Andrews, D.W.K. (1993), “Tests of Parameter Instability and Structural Change with Unknown Change Point” Econometrica, Vol 61, pp. 821-856. Andrews, D.W.K., and W. Ploberger (1994), “Optimal Tests when a Nuisance Parameter is Present only under the Alternative”, Econometrica, Vol 62, pp. 1386-1414. Andrews, D.W.K., I., Lee and W. Ploberger (1996), “Optimal Change Point Tests for Normal Linear Regression”, Journal of Econometrics, Vol 70, pp. 9-38. 14

Bai, J. (1997), “Estimation of Change Point in Multiple Regression Models”, The Review of Economics and Statistics, Vol 79, pp. 551-563. Bai, J.and P. Perron (1998a), “Estimating and Testing Linear Models with Multiple Structural Changes”, Econometrica, Vol 66, pp. 47-78. Bai, J.and P. Perron (1998b), “Computation and Analysis of Multiple Structural Change Models” Manuscript, Boston University. Banerjee, A., Lumsdaine, R. and J. Stock (1192), “Recursive and Sequential Tests of the Unit Root and Trend-Break Hypotheses: Theory and International Evidence”, Journal of Business and Economic Statistics, Vol 10, pp. 271-288. Corbae, P.D. and Ouliaris, S. (1988), “A Test of Long-Run Purchasing Power Parity, Allowing for Strucutral Breaks”, The Economic Record, Vol 70, pp. 508-511. Corbae, P.D. and Ouliaris, S. (1990), “Cointegration and Tests of Purchasing Power Parity”, Review of Economics and statistics, March 1990, pp. 26-33. Dornbusch, R. (1976), “Expectations and Exchange Rate Dynamics” Journal of Political Economy 84, 1161-74. Dropsy, V. (1996), “Real exchange rates and Structural breaks”, Applied Economics,Vol 29, pp. 209-219. Garcia, R. and P. Perron (1996), “An Analysis of the Real Interest Rate under Regime Shifts”, The Review of Economic and Statistics, Vol 78, pp. 111-125. Liu, J., S. Wu, and J.V. Zidek (1997), “On Segmented Multivariate Regressions”, Statistica Sinica, Vol 7, pp. 497-525. Lumsdaine, R.L. and D.H. Papell (1997), “Multiple Trend Breaks and the Unit Root Hypothesis”, Review of Economics and Statistics, 1996, vol. 54, p.212-218. Mark, N.C. and D. Choi (1997), “Real Exchange Rate Prediction over Long Horizons”, Journal of International Economics, Vol 43, pp. 29-60. Perron, P. (1989), “The Crash, the Oil Shock and the Unit Root Hypothesis”, Econometrica, Vol 57 pp. 1361-1401. Perron, P. (1990), “Testing for a Unit Root in a Time Series with a Changing Mean”, Journal of Business and Economic Statistics, Vol 8, pp.. 153-162. Perron., P. and T.J. Vogelsang (1992), “Nonstationarity and Level Shifts with an Application to Purchasing Power Parity”, Journal of Business and Economic Statistics, Vol 10, pp.321-335. Ploberger , W. and W. Kramer (1992), “The CUSUM test with OLS residuals”, Econometrica, Vol 60, pp. 271-385. Stockman, A.C. (1987), “The Equilibrium Approach to Exchange Rates”, Economic Review, Federal Reserve Bank of Richmond, March/April, pp..12-30. Wright, J.H. (1993), “The CUSUM Test based on Least Squares Residuals in Regressions with Integrated Variables”, Economic Letters, Vol 41, pp. 353-358. Yao, Y. (1988), “Estimating the Number of Change-Points via Schwarz’Criterion”, Statistical Probability Letters, Vol 6, pp. 181-189.

15

Terminal date

1 2 3 a a a a a a

4 a a a a

5 x a a a a

6 x x a a a a

7 x x x a a a a

8 x x x x a a a a

9 10 11 12 13 14 15 16 17 18 19 20 x x x x x x x b b b b b x x x x x x x b b b b b x x x x x x x b b b b b x x x x x x x b b b b b x x x x x x x b b b b b a x x x x x x x x x x x a a x x x x x x x x x x a a a x x x x x x x x x a a a a x x x x x x x x a a a a x x x x x x x a a a a x x x x x x a a a a x x x x x a a a a x x x x a a a a x x x a a a a x x a a a a x a a a a a a a a a

Figure A: Example of the triangular matrix of sums of squared residuals with T = 20, h = 5 and m = 2 Source: Bai and Perron (1998b)

Notes: The column number indicates the initial date of a segment while the horizontal number indicates the terminal date for example, the entry (4, 10) indicate a segment that starts at date 4 and ends at date 10, hence having 7 observations. a: indicates a segment not considered since it must be at least of length 5, b: indicates a segment not considered since otherwise there would be no place for 3 segments of length 5. x: indicates an admissible segment

16

DEM/USD-LN

Breaks selected

z t = {1}

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

50.09 *

232.92 *

90.40 *

BIC

5

LWZ

δˆ1

δˆ2

δˆ3

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

214.25 *

284.27 *

284.27 *

481.73 *

176.68 *

36.83 *

139.89 *

42.36 *

5

Sequential

5

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

Estimates seq.

1.15

0.91

0.70

0.97

0.60

0.48

73:9

77:11

81:4

86:6

89:12

Standard errors

(0.015)

(0.014)

(0.012)

(0.001)

(0.013)

(0.014)

(73:6–73:11)

(77:8–78:2)

(80:12–81:5)

(86:3–86:8)

(88:8–90:8)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 332 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

10.09 *

49.83 *

82.67 *

212.58 *

4198.33 *

4198.33 *

7114.49 *

11.26 *

149.20 *

48.68 *

187.64 *

BIC

5

LWZ

5

Sequential

5

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

0.86

0.68

0.58

1.01

0.76

0.68

73:9

77:10

81:4

86:6

89:12

(0.008)

(0.008)

(0.011)

(0.027)

(0.016)

(0.011)

(73:7-73:10)

(77:2-78:2)

(80:12-81:5)

(86:3-87:2)

(87:12-91:5)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

20.67 *

208.89 *

224.37 *

583.88 *

1280.61 *

1280.61 *

2170.12 *

37.666 *

422.87 *

145.05 *

20.36 *

BIC

5

LWZ

5

Sequential

5

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

DEM/USD-LR

Breaks selected

Estimates seq. Standard errors FRF/USD-LN

Breaks selected

z t = {1}

z t = {1}

Estimates seq.

1.57

1.47

1.91

2.10

1.81

1.70

78:6

81:4

84:1

86:12

89:12

Standard errors

(0.022)

(0.005)

(0.011)

(0.01)

(0.014)

(0.012)

(78:5-78:7)

(81:2-81:5)

(83:11-84:3)

(86:9-87:1)

(88:8-90:6)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 332 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

6.88

107.09 *

130.41 *

826.69 *

6786.82 *

6786.82 *

9769.38 *

40.76 *

186.30 *

89.74 *

303.19 *

BIC

5

LWZ

5

Sequential

0

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

1.76

1.60

1.46

1.79

1.95

1.60

73:9

78:3

81:1

83:10

86:7

(0.001)

(0.008)

(0.01)

(0.033)

(0.007)

(0.015)

(X)

(77:12-78:5)

(80:8-81:2)

(83:9-84:2)

(86:5-86:8)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

71.36 *

114.69 *

392.17 *

397.46 *

389.87 *

397.46 *

660.68 *

78.25 *

47.47 *

16.42 *

16.42 *

FRF/USD-LR

Breaks selected

Estimates BIC-

z t = {1}

LWZ Standard errors JPY/USD-LN

z t = {1}

17

Breaks selected

Estimates seq. Standard errors JPY/USD-LR

Breaks selected

Estimates seq. Standard errors GBP/USD-LN

Breaks selected

Estimates seq. Standard errors GBP/USD-LR

Breaks selected

Estimates seq. Standard errors CHF/USD-LN

Breaks selected

BIC

5

LWZ

5

Sequential

5

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

5.73

5.70

5.39

5.47

4.94

4.71

73:9

77:9

82:1

86:2

92:8

(0.013)

(0.01)

(0.007)

(0.011)

(0.023)

(0.026)

(72:7-75:3)

(77:7-77:10)

(80:12-82:6)

(85:11-86:3)

(91:7-93:5)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 332 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

50.54 *

929.12 *

253.61 *

530.71 *

585.78 *

929.12 *

1067.55 *

184.40 *

15.62 *

670.96 *

19.60 *

BIC

5

LWZ

5

Sequential

5

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

5.80

5.45

5.58

5.16

4.96

5.15

73:9

82:1

86:2

92:8

95:11

(0.006)

(0.035)

(0.001)

(0.015)

(0.031)

(0.007)

(X)

(81:12-82:2)

(X)

(91:8-93:1)

(95:9-96:3)

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

77.09 *

135.56 *

412.90 *

386.21 *

6366.30 *

6366.30 *

10788.33 *

70.44 *

57.31 *

56.88 *

56.88 *

BIC

5

LWZ

5

Sequential

3

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

-0.88

-0.62

-0.76

-0.46

-

-

75:6

78:12

82:1

-

-

(0.011)

(0.013)

(0.005)

(0.025)

()

()

(75:3-75:7)

(78:10-79:5)

(X)

()

()

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 332 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

27.81 *

89.58 *

237.40 *

177.46 *

196.18 *

237.40 *

332.40 *

6.59

245.74 *

19.46 *

67.58 *

BIC

5

LWZ

5

Sequential

1

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

-0.52

-0.67

-

-

-

-

87:9

-

-

-

-

(0.021)

(0.019)

()

()

()

()

(85:11-91:8)

()

()

()

()

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

43.20 *

366.92 *

188.49 *

183.02 *

341.87 *

366.91 *

579.34 *

102.50 *

36.33 *

60.78 *

33.08 *

BIC

5

LWZ

5

Sequential

5

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

z t = {1}

z t = {1}

z t = {1}

z t = {1}

Estimates seq.

1.29

0.95

0.58

0.79

0.43

0.31

74:3

77:10

82:5

86:7

90:4

Standard errors

(0.024)

(0.022)

(0.026)

(0.0001)

(0.01)

(0.016)

(73:11-74:6)

(77:5-77:12)

(82:4-82:6)

(86:6-86:8)

(88:3-90:7)

18

CHF/USD-LR

Breaks selected

z t = {1}

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

19.50 *

270.34 *

53.00 *

BIC

5

LWZ

δˆ1

δˆ2

δˆ3

M =5 SupFT (5)

T = 336 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

398.67 *

275.80 *

398.67 *

601.64 *

65.86 *

22.72 *

466.05 *

26.80 *

5

Sequential

5

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

Estimates seq.

1.00

0.68

0.53

0.84

0.54

0.44

73:9

77:10

82:5

86:7

90:4

Standard errors

(0.007)

(0.017)

(0.024)

(0.001)

(0.015)

(0.01)

(73:6-73:10)

(76:4-78:5)

(82:4-82:6)

(X)

(89:2-91:5)

Table 1: Empirical results: logarithms of monthly nominal and real exchange rates relative to the US Dollar for five countries: Germany, France, Japan, UK and Switzerland. LN means logarithm of nominal exchange rate, LR means logarithm of real exchange rate. The SupFT (k ) tests and reported standard errors and confidence intervals allow for the possibility of serial correlation in the disturbances. The heteroskedasticity and autocorrelation consistent covariance matrix is constructed following Andrews (1991) using a quadratic kernel with automatic bandwith selection based on an AR(1) approximation. We use a 5% size for the sequential test SupF ( k + 1 / k ) . In parentheses are the standard errors (robust to serial correlation) for δˆi ( i = 1,...,6 ) and the 95% confidence intervals for Tˆk ( k = 1,..., 5 ). A * indicates significance at the 5% level. Asymptotic critical values can be found in Bai and Perron (1998b) for a trimming of ε = 0.10 : for the SupFT (k ) tests, the critical values at the 5% level for k = 1,..., 5 are respectively 9.63; 8.78; 7.85; 7.21; 6.69. For the UDmax and WDmax tests, the critical values at 5% level are respectively 10.17 and 10.91. For the SupF ( k + 1 / k ) tests the critical values at the 5% level for k = 1,..., 5 are respectively 9.63; 11.14; 12.16; 12.83; 13.45. (X) indicates that confidence intervals could not have been computed.

DEM/USD-LN

Breaks selected

Estimates seq. Standard errors DEM/USD-LR

Breaks selected

z t = {1}

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 112 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

41.23 *

72.84 *

121.60 *

116.99 *

120.90 *

121.60 *

265.31 *

26.70 *

49.91 *

26.70 *

0.47

BIC

4

LWZ

3

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

1.08

0.77

0.97

0.58

0.47

-

74:4

82:2

86:3

90:3

-

(0.043)

(0.021)

(0.0004)

(0.012)

(0.017)

-

(74:1-75:4)

(82:1-82:3)

(X)

(68-81)

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 110 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

8.21

47.10 *

39.13 *

34.07 *

31.90 *

47.09 *

70.00 *

9.55 *

18.81 *

18.81 *

3.09

BIC

3

LWZ

3

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

z t = {1}

Estimates seq.

0.80

0.62

1.02

0.78

0.67

-

74:4

81:1

85:4

89:4

-

Standard errors

(0.05)

(0.014)

(0.046)

(0.025)

(0.013)

-

(74:2-78:2)

(80:1-81:2)

(85:1-88:1)

(88:1-92:3)

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 112 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

17.84 *

145.03 *

95.87 *

258.28 *

220.81 *

258.28 *

484.54 *

26.88 *

30.21 *

3.85

0.48

FRF/USD-LN

z t = {1}

19

Breaks selected

Estim BIC-LWZ Standard errors FRF/USD-LR

Breaks selected

Estim BIC-LWZ Standard errors JPY/USD-LN

Breaks selected

BIC

3

LWZ

2

Sequential

3

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

1.54

2.00

1.78

1.70

-

-

81:1

86:4

90:4

-

-

(0.027)

(0.045)

(0.007)

(0.016)

-

-

(80:1-81:3)

(X)

(86:3-91:1)

-

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 110 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

6.01

253.59 *

52.78 *

43.88 *

38.72 *

253.59 *

301.35 *

45.47 *

21.38 *

6.67

1.84

BIC

3

LWZ

3

Sequential

0

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

5.66

4.74

6.91

4.98

-

-

74:4

82:1

86:1

-

-

(0.19)

(0.012)

(0.19)

(0.087)

-

-

(74:3-75:1)

(X)

(85:2-86:3)

-

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 112 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

62.19 *

110.88 *

282.11 *

292.90 *

1056.44 *

1056.44 *

2318.23 *

64.42 *

31.71 *

57.68 *

-

BIC

3

LWZ

3

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

z t = {1}

z t = {1}

Estimates

5.69

5.39

5.47

4.93

4.71

-

77:3

81:4

86:1

92:3

-

Standard errors

(0.023)

(0.009)

(0.002)

(0.021)

(0.035)

-

(77:1-78:1)

(81:3-82:1)

(X)

(90:3-93:2)

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 110 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

41.22 *

79.93 *

102.63 *

575.19 *

235.55 *

575.19 *

989.00 *

50.43 *

50.43 *

50.43 *

-

BIC

4

LWZ

3

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

5.74

5.43

5.58

5.16

5.04

-

74:4

81:4

86:1

92:4

-

(0.041)

(0.029)

(0.009)

(0.015)

(0.007)

-

(73:4-75:4)

(81:2-83:1)

(85:3-86:2)

(92:2-94:2)

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 112 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

49.00 *

92.04 *

154.36 *

159.91 *

264.74 *

264.74*

580.94 *

37.06 *

14.25 *

18.71 *

-

BIC

4

LWZ

4

Sequential

2

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

JPY/USD-LR

Breaks selected

Estimates seq. Standard errors GBP/USD-LN

Breaks selected

z t = {1}

z t = {1}

Estimates seq.

-0.88

-0.69

-0.46

-

-

-

75:2

81:4

-

-

-

Standard errors

(0.014)

(0.029)

(0.028)

-

-

-

(73:2-75:3)

(78:2-83:1)

-

-

-

20

GBP/USD-LR

Breaks selected

z t = {1}

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

17.14 *

55.50 *

94.27 *

BIC

3

LWZ

δˆ1

δˆ2

M =5 SupFT (5)

T = 110 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

71.49 *

72.65 *

94.27 *

159.41 *

6.19

173.51 *

3.36

-

3

Sequential

1

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

Estimates

-0.52

-0.67

-

-

-

-

87:3

-

-

-

-

Standard errors

(0.029)

(0.02)

-

-

-

-

(85:3-95:2)

-

-

-

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 112 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

35.09 *

250.38 *

171.20 *

147.19 *

271.04 *

271.04 *

594.76 *

34.86 *

26.46 *

13.15 *

12.33 *

BIC

4

LWZ

2

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

1.12

0.58

0.79

0.39

0.28

-

77:3

82:1

86:2

94:2

-

(0.058)

(0.031)

(0.011)

(0.017)

(0.024)

-

(77:1-78:3)

(81:3-83:2)

(85:4-86:3)

(91:3-96:3)

-

q =1

p=0

SupFT (1)

SupFT (2)

SupFT (3)

ε = 0.10 SupFT (4)

M =5 SupFT (5)

T = 110 UDmax

WD max

SupF (2 / 1)

SupFT (3 / 2)

SupFT (4 / 3)

SupFT (5 / 4)

15.68 *

63.11 *

54.64 *

84.29 *

94.19 *

94.19 *

206.68 *

24.61 *

82.12 *

24.61 *

8.53

BIC

3

LWZ

3

Sequential

4

δˆ1

δˆ2

δˆ3

δˆ4

δˆ5

δˆ6

Tˆ1

Tˆ2

Tˆ3

Tˆ4

Tˆ5

CHF/USD-LN

Breaks selected

Estimates seq. Standard errors CHF/USD-LR

Breaks selected

z t = {1}

z t = {1}

Estimates seq.

0.93

0.55

0.82

0.54

0.44

-

74:4

81:1

86:2

90:2

-

Standard errors

(0.03)

(0.025)

(0.026)

(0.016)

(0.011)

-

(74:1-75:2)

(80:1-81:4)

(85:4-87:1)

(88:4-91:3)

-

Table 2: Empirical results: logarithms of quarterly nominal and real exchange rates relative to the US Dollar for five countries: Germany, France, Japan, UK and Switzerland. LN means logarithm of nominal exchange rate, LR means logarithm of real exchange rate. The SupFT (k ) tests and reported standard errors and confidence intervals allow for the possibility of serial correlation in the disturbances. The heteroskedasticity and autocorrelation consistent covariance matrix is constructed following Andrews (1991) using a quadratic kernel with automatic bandwith selection based on an AR(1) approximation. We use a 5% size for the sequential test SupF ( k + 1 / k ) . In parentheses are the standard errors (robust to serial correlation) for δˆi ( i = 1,...,6 ) and the 95% confidence intervals for Tˆk ( k = 1,..., 5 ). A * indicates significance at the 5% level. Asymptotic critical values can be found in Bai and Perron (1998b) for a trimming of ε = 0.10 : for the SupFT (k ) tests, the critical values at the 5% level for k = 1,..., 5 are respectively 8.58; 7.22; 5.69; 4.99; 3.91. For the UDmax and WDmax tests, the critical values at 5% level are respectively 8.88 and 9.91. For the SupF ( k + 1 / k ) tests the critical values at the 5% level for k = 1,..., 5 are respectively 8.58; 10.13; 11.14; 11.83; 12.25. (X) indicates that confidence intervals could not have been computed.

21

1,4

7

1,2

6

1

5

0,8

4

0,6

3

0,4

2

0,2

1 ln(DEM/USD) ln(DEM/USD R)

ln(JPY/USD) ln(JPY/USD R)

0

0 0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

0

Figure 1: Logs of monthly nominal and real exchange rates: Deutsche Mark / US Dollar for the period 1971:1-1998:12

10

20 30 40

50 60 70 80

90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

Figure 3: Logs of monthly nominal and real exchange rates: Japanese Yen / US Dollar for the period 1971:1-1998:12 0

2,5

0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

-0,2 2

-0,4 1,5

-0,6

1 -0,8

0,5 -1 ln(FRF/USD) ln(GBP/USD)

ln(FRF/USD R)

ln(GBP/USD R)

0 0

-1,2

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

Figure 2: Logs of nominal monthly and real exchange rates: French Franc / US Dollar for the period 1971:1-1998:12

Figure 4: Logs of monthly nominal and real exchange rates: British Pound / US Dollar for the period 1971:1-1998:12 22

1,6

2,5 ln(CHF/USD) ln(CHF/USD R)

1,4 2 1,2

1

1,5

0,8

1 0,6

0,4 0,5 0,2 ln(FRF/USD) ln(FRF/USD R) 0

0 0

0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

Figure 5: Logs of monthly nominal and real exchange rates: Swiss Franc / US Dollar for the period 1971:1-1998:12

10

20

30

40

50

60

70

80

90

100

110

120

Figure 7: Logs of quarterly nominal and real exchange rates: French Franc / US Dollar for the period 1971:1-1998:2

1,4

7

1,2

6

1

5

0,8

4

0,6

3

0,4

2

0,2

1 ln(JPY/USD)

ln(DEM/USD)

ln(JPY/USD R)

ln(DEM/USD R) 0

0 0

10

20

30

40

50

60

70

80

90

100

110

120

0

Figure 6: Logs of quarterly nominal and real exchange rates: Deutsche Mark / US Dollar for the period 1971:1-1998:2

10

20

30

40

50

60

70

80

90

100

110

120

Figure 8: Logs of quarterly nominal and real exchange rates: Japanese Yen / US Dollar for the period 1971:1-1998:2 23

0 0

10

20

30

40

50

60

70

80

90

100

110

120 1,6 ln(CHF/USD) ln(CHF/USD R)

-0,2

1,4

1,2 -0,4

1 -0,6 0,8

0,6

-0,8

0,4 -1 0,2

ln(GBP/USD) ln(GBP/USD R) -1,2

0 0

Figure 9: Logs of quarterly nominal and real exchange rates: British Pound / US Dollar for the period 1971:1-1998:2

10

20

30

40

50

60

70

80

90

100

110

120

Figure 10: Logs of quarterly nominal and real exchange rates: Swiss Franc / US Dollar for the period 1971: 1-1998:2

24