IIIIIIIIIIIIIIIIII II IIIIII IIIIII ”HHS? 3 1293 00620 7322 I Libfi.’-.-.. I Pafichigan State University This is to certify that the dissertation entitled ESSAYS IN EMPIRICAL INTERNATIONAL FINANCE presented by KABSOO HONG has been accepted towards fulfillment of the requirements for PH - D - degree in ECONOMICS OW Major professor Date (1/ E/(S/ci MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or baton ode duo. DATE DUE DATE DUE DATE DUE WI In s M: FEB 1 92007 o I T MSU Is An Affirmative ActioNEqual Opportunity Institution ____.._.———- ESSAYS IN EMPIRICAL INTERNATIONAL FINANCE BY Kabsoo Hang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1989 1 i in. J ,. 9M " I90 ABSTRACT ESSAYS IN EMPIRICAL INTERNATIONAL FINANCE BY Kabsoo Hong This dissertation consists of three essays: 1. "Impact of EMS Membership on its Nominal Exchange Rate Volatility" compares the exchange rate volatilities before and.after the advent of the EMS (European Monetary System), comparing members' currency volatilities with the non-EMS currency volatilities. Multivariate as well as univariate GARCH models indicate that the existence of the EMS has coincided with a marked reduction in the volatilities of intra-EMS exchange rates. However, in the EMS versus non-EMS cases, or between-non- EMS currency cases, some countries show at least constant volatilities. Hence, we cannot say that this stability results from the system itself. Furthermore, member countries' exchange rate volatilities against the US dollar show different patterns under their exchange-rate mechanisms. 2. "Multivariate Cointegration Tests and Long-Run Purchasing Power Parity Theory" reexamines the relationship between prices and exchange rates by multivariate cointegration tests developed by Johansen (1988). This method uses vector autoregressive processes. Cointegrating vectors between prices and exchange rates are determined simultaneously by maximum likelihood estimation. This study also reexamines the stationarity of the levels of price series and relative price indexes and find that some are 1(2). After analyzing purchasing power parity (PPP) doctrines and finding evidence that PPP holds even after 1973, results are compared with the conventional unit root tests for PPP which showed unfavorable, but low power, results. Short-run dynamics are analyzed in the last section. 3. "Multivariate Cointegration Tests for a Set of Foreign Exchange Rates and a Comparative Study of the Forecasting Accuracy of the Random- Walk and the Error-Correction.Models' begins with the work of Baillie and Bollerslev (1989), which showed the existence of one long-run equilibrium relation between a set of seven.daily exchange rate series by multivariate tests for unit roots. After eliminating some currencies redundant to this relationship, this study finds that the EMS currencies, with the German mark as a driving currency, contribute to such a long-run relationship. Although each exchange rate series has a univariate representation as a random walk, it appears that the random-walk model does not outperform our error-correction model in out-of—sample forecasting accuracy. ACKNOWLEDGEMENTS I wish to thank my dissertation committee chairman, Professor Robert T. Baillie, for his comments, advice, and guidance over the period of this projectu The other committee members, Professor Robert.H. Rasche, Yoonjae Choe and Owen F. Irvine Jr. also provided valuable comments that both improved the project and helped to complete it. I have benefitted from many others as well. VI wish especially to thank Dr. Timothy Lane of the International Monetary Fund for general encouragement and advice onI my topics. Professor Sam ‘Yoo in. the Pennsylvania State University gave me valuable comments in the field of cointegrations. Professor Peter J. Schmidt has given me advice on theoretical questions. ii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . iv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter I. INTRODUCTION . . . . .1 II. IMPACT OF EMS MEMBERSHIP ON ITS NOMINAL EXCHANGE RATE VOLATILITY: AN APPROACH WITH UNIVARIATE AND MULTIVARIATE GARCH MODELS. . . .6 1. INTRODUCTION . . . . 7 2. TESTS FOR A UNIT ROOT IN WEEKLY EXCHANGE RATE SERIES . . 9 3 MODELS WITH TIME DEPENDENT CONDITIONAL HETEROSKEDASTICITY: GARCH (1,1) . . . . . . . . . . . . . 12 3.1 Implication of the GARCH Model . . . . . . . . . . 12 3.2 Tests for the EMS Currency Volatility . . . . . . . 16 3.2.1 Test Method . . . . . . . . . . . . . . . 16 3. 2.2 Test Results . . . . . . . . . . l7 4. THE MULTIVARIATE GENERALIZED ARCH APPROACH . . . . . . . 20 4.1 Estimation of Multivariate GARCH Models . . . . . . 21 4.2 Model Estimates . . . . . . . . . . . . . . . . . . 23 5. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . .27 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . 41 III. MULTIVARIATE COINTEGRAITON TESTS AND LONG- RUN PURCHASING POWER PARITY THEORY. . . . . . . . . . . . . . . . . . . . . . . . . .44 1. INTRODUCTION . . . . . . 45 2. UNIVARIATE TESTS FOR UNIT ROOTS IN THE MONTHLY EXCHANGE RATES AND CONSUMER PRICE INDEXES . . . . . . 47 3. HYPOTHESIS TESTING OF COINTEGRATION VECTORS BETWEEN EXCHANGE RATES AND RELATIVE PRICES AND LONG RUN PPP THEORY . . . . S3 4. SHORT-RUN DYNAMICS . . . . . . . . . . . . . . . . . . . 60 5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 65 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . 84 IV. MULTIVARIATE COINTEGRATION TESTS FOR A SET OF FOREIGN EXCHANGE RATES AND A COMPARATIVE STUDY OF THE FORECASTING ACCURACY OF THE RANDOM- WALK AND THE ERROR- CORRECTION MODELS. . . . . . . . . . . . . . 87 1. INTRODUCTION . . . . . . 88 2. MULTIVARIATE TESTS FOR UNIT ROOTS IN A SET OF EXCHANGE RATES . . . . . . . . . . . 90 3. ARIMA MODELS OF EXCHANGE RATE SERIES . . . . . . 96 4. A COMPARATIVE STUDY OF THE FORECASTING ACCURACY OF THE ERROR- CORRECTION AND THE RANDOM- WALK MODELS . . . . . . . . . . . 98 5. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . .l00 REFERENCES . . . . . . . . . . . . . . . . . . . . . .'. . . 110 iii CHAPTER II. 2-1. 2-2. CHAPTER III. LIST OF TABLES Phillips-Perron Unit Root Tests on Exchange Rate Series . . 28 Estimation of GARCH models with D-mark and US dollar as Base Currency . . . . . . . . . . . . . . . . . . . . . . . . . 29 Estimation of GARCH models with EMS Currencies against the US dollar . . . . . . . . . . . . . . . . . . . . . . . . . 30 Summary Statistics with the Implications of GARCH (1,1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . 31 Summary of t-Statistics for Shift in Volatility after March, 1979 . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Estimation of GARCH models with the Italian lira and the Pound sterling as Base Currency . . . . . . . . . . . . . . . . . 33 Estimation of GARCH models with the Japanese yen and the Canadian dollar as Base Currency . . . . . . . . . . . . . 34 Logged Monthly Spot Exchange Rates . . . . . . . . . . . . 67 Unit Root Tests on Logged Monthly Exchange Rate Series . . 68 Logged Consumer Price Indexes A. Sample Autocorrelations for Level . . . . . . . . . . 69 B. Sample Partial Autocorrelations . . . . . . . . . . . 69 A. Sample Autocorrelations for First Differenced Logged Consumer Price Indexes . . . . . . 70 B. Sample Autocorrelations for Second Differenced CPI . 70 A. Phillips- -Perron Unit Root Tests on Logged Consumer Price Indexes of West Germany . . 71 3. Phillips- -Perron Unit Root Tests on Logged Consumer Price Indexes of Switzerland . . . . 72 C. Phillips- -Perron Unit Root Tests on Logged Consumer Price Inexes of the United States . . . . 73 D. Phillips- -Perron Unit Root Tests on Logged Consumer Price Indexes of Italy . . . . . . . . . . . . . . . . . . 74 iv 10. 11. 12. 13. 14. CHAPTER IV. Phillips-Perron Unit Root Tests on Logged Consumer Price Indexes . . . . . . . . . . . . . . . . . . . . . . . . . 75 Second Unit Root Tests on Logged Consumer Price Indexes . 76 Phillips-Patron Tests for One Unit Root on Logged Relative Pricgfiatio........................77 Second Unit Root Tests on Logged Relative Price Ratios . 78 Vector Autoregressive Estimates for the U.S.A.-Germany . 79 Test Statistics for the Hypothesis for various Values of Cointegration Vectors between Log of Exchange Rates and Log of Relative Prices . . . . . . . . . . . . . . . . . . . 80 The Eigenvalues A and Eigenvectors V and Estimated Alphas 81 Tests of PPP in ast+¢(P-P')c-¢¢, where ‘t. is stationary . . 82 Univariate Unit Root Tests to Deviations from PPP . . . . 83 Multivariate Tests for Cointegration vectors in the Logarithms of Daily Spot Exchange Rates . . . . . . . . . . . . . 103 Eigenvalues, Eigenvectors and Estimated Alphas with seven Daily Exchange Rates. . . . . . . . . . . . . . . . . . 104 Eigenvalues, Eigenvectors and Estimated Alpha with three Daily Exchange Rates (D-mark, French franc and Italian lira) 105 Vector Autoregressive Estimates with lag l for three Daily Exchange Rates: D-mark, French franc and Italian lira 106 Autocorrelation Functions (ACF) and Partial Autocorrelation Functions (PACF) . . . . . , . . . . . . . . . . . . . 107 Percentage Forecasting Accuracy . . . . . . . . . . . . 108 CHAPTER II. 1B. 2A. 2B. 3A. 3B. AA. AB. SA. SB. 6A. 6B. CHAPTER IV. Charts: US dol A. B. LIST OF FIGURES Exchange Rate Volatility of Netherlands guilder against the Deutsche mark . . . . . . . . . . . . . . . . . 35 Exchange Rate Volatility of Belgian franc against the Deutsche mark . . . . . . . . . . . . 35 Exchange Rate Volatility of Netherlands guilder against the Italian lira . . . . . . . . . . . . . . . . . . 36 Exchange Rate Volatility of Belgian franc against the Italian lira . . . . . . . . . . . . . . . 36 Exchange Rate Volatility of Netherlands guilder against the US dollar . . . . . . . . . . . . . . . . . . . . . 37 Exchange Rate Volatility of Deutche mark against the US dollar . . . 37 Exchange Rate Volatility of Canadian dollar against the US dollar . . . . . . . . . . . . . . . . . . . . . . . . . 38 Exchange Rate Volatility of Swiss franc against the US dollar . . . . . . . . . 38 Exchange Rate Volatility of Netherlands guilder against the UK pound . . . . . . . . . . . . . . . . . . . . . . . . . 39 Exhcange Rate Volatility of Italian lira against the UK pound . . . . . . . . . . . . . . . . . . 39 Exchange Rate Volatility of Canadian dollar against the UK pound . . . . . . . . . . . . . . . . . . . . . . . . . 40 Exchange Rate Volatility of Swiss franc agaisnt the UK pound . . . . . . . 40 lar . Movements of the European Currency Unit (ECU) Against the . 109 US dollar per ECU, Monthly Average. . 109 ECU per US dollar, Quarterly Average. . 109 vi CHAPTER I INTRODUCTION In March 1973 the Bretton Woods system of fixed but adjustable exchange rates collapsed and the economies moved to a system of floating exchange rates. This change was a reflection of the failure of the Bretton Woods system to deal effectively with the fundamental current account imbalances. During the early 1970s, the prevailing academic view was that flexible exchange rates would solve the increasingly obvious problems of the Bretton Woods system and thereby create a far less difficult environment for the management of domestic monetary and fiscal policies. What, then, of the case made for flexible exchange rates by its proponents? There were essentially five claims made by' the early advocates of flexible exchange rates, that is, such writers as: Friedman (1968), Sohman (1969), Johnson (1970) and Machlup (1970). First, flexible exchange rates move to offset a country's relative price level; under flexible exchange rates, if we let the exchange rate depreciate, it compensates for the price increase to maintain the country's competitive position. This is known as purchasing power parity (PPP). Second, such a regime would be a relatively stable one, in contrast with the supposed inherent instability of the Bretton Woods system. Third, both Friedman (1968) and Sohmen (1969) argued that floating exchange rates would isolate a country from shocks emanating from the rest of the world. The fourth argument for floating rates is the independence they give a country in 2 pursuing monetary policy, and the final justification for the regime is that, inuprinciple, central banks need.not hold.foreign.exchange reserves, since official intervention will be zero. However, the recent history of currency gyrations under the prevailing floating exchange rate regime suggests that most of the propositions advanced in the articles by the early advocates are doubtful since fllexible exchange rates have not performed as expected. First, many authors have found little evidence of the empirical validity of PPP after 1973 (e.g., Frenkel, 1981). Furthermore, large and. frequent exchange rate changes have produced a range of unforseen and generally disruptive side effects throughout the economies of the industrialized countries. Two widely-cited papers by Meese and Rogoff (1983a, 1983b) were the first studies to provide extensive and fairly'convincing«evidence that existing models of systematic exchange rate behavior could not outperform a random-walk model, even when the forecasts of systematic behavior were based on the ex post realized value of the explanatory variables. Therefore, to reduce such an exchange rate volatility, even among member countries, the Exchange Rate Mechanism.(ERM) of the European Monetary System (EMS) was established in.March 1979. The purpose of this dissertation is to review and evaluate empirically some parts of international finance which are related to the arguments mentioned above. It consists of three essays: the first essay examines the volatility of bilateral exchange rates of the EMS currencies to see whether they stabilized exchange rates among member countries compared with non-EMS currency volatilities. In the second essay, purchasing power parity after the establishment of the floating exchange rate system is tested to see 3 whether it exists after 1973. In the last essay, the random-walk model of exchange rate determination is compared with the error-correction model. In the first essay (Chapter II), the EMS currency volatility is tested for its stability. The EMS came into operation in March 1979 to create "a zone of monetary stability in Europe," comprising "greater stability at ‘home and abroad." It is a. system. of fixed, though adjustable, exchange rates. However, the dynamics of the EMS represent a considerable challenge to economists. Among many arguments against the EMS, this essay examines exchange rate volatilities before and after the advent of the EMS, comparing members' currency volatilities with the non- EMS currency volatilities. GARCH (Generalized.Autoregressive Conditional Heteroskedasticity) models are introduced to stress the importance of the stylized leptokurtic characteristic in the exchange rate series with the student t-distribution, and also the possibility of a time-dependent conditional heteroskedasticity. After estimating the univariate GARCH models, multivariate GARCH approaches are given, since nonzero covariance among exchange rate innovations requires a joint estimate of sets of regressions and.since exchange rates are bilateral rates, it should affect all rates if a new information comes to the foreign exchange market. In the second essay (Chapter III), the PPP theory after the 19703 is examined“ The general view is that a currency's equilibrium level is best associated with its international purchasing power parity. Such a relationship between prices and exchange rates is generally rejected after the 19703 [ e.g., Frenkel, 1981; Dornbusch, 1980]. However, the conventional tests disregard the fact that levels of price indexes (and 4 also some first°differenced CPIs) and exchange rates are nonstationary. This essay carefully tests for unit roots in the price indexes and relative price indexes to determine whether they have two unit roots by Dickey and Pantula (1987) tests. Then PPP hypothesis is tested in the multivariate context developed by Johansen (1988) . This method gives more efficient estimates, since it not only allows for general dynamic properties of the structure of the underlying process, but it also gives maximum likelihood estimates. The third essay (Chapter IV) starts with the work of Baillie and Bollerslev (1989), which showed, by means of multivariate tests for unit roots, the existence of one long-run equilibrium relation between a set of seven daily exchange rate series. This result indicates a perceptible deviation from weak-form efficiency for each of the exchange rates, because, in the first-order error-correction model, if two or more exchange rates are cointegrated, part of the changes will usually be predictable. First, after eliminating some currencies redundant to this relation, we find a driving currency for such a long-run relations, and then we analyze this long-run equilibrium with the remaining currencies. Second, as we confirm in our study, each exchange rate series has a univariate representation as a random walk, but since a vector of the first differenced exchange rates should have a lagged error-correction term applied to it, we compare the forecasting accuracy of an error- correction model with that of a random-walk model. References Friedman, M. , "The Case for Flexible Exchange Rtaes," in Richard Caves 5 and Harry Johnson. eds.. MW Internagign§1_figggimig§, Homewood, Ill., Irwin, 1968, Pp. 413-40. Johnson, Harry G., "The Case for Flexible Exchange Rates - 1969", in Bergsten, Halm, Machlup, and Roosa, eds., Apn;ggghgs_52_§xga§g1_ MW, Princeton, N.J. , Princeton University Press, 1970. Machlup, F., "The Case for Floating Exchange Rates", in Bergsten, Helm, Machlup. and Reese. eds.. WWW £3;h§3g§_g§£g§, Priceton, N.J., Princeton University Press, 1970. Sohman, E., £1231b1g_§35h§ngg_335g§,2xu1ed., Chicago, University of Chicago Press, 1969. I. IHPACT OF EMS MEMBERSHIP OR ITS NOMINAL HCHANGE RATE VOLATILITY: AN APPROACH WITH UNIVARIATE AND HULTIVARIATE GARCH MODELS l . - Introduction A number of studies considered the evidence that the EMS has reduced exchange rate volatility. Ungerer, M (1983) noted that 'the exchange rate variability of EMS currencies has diminished since the introduction of the system----'1 and updated the conclusion with a later paper (1986). The European Commission (1982), Ungerer (1983), Dennis and Nellis (1984), Bank of England (1984) , and Rogoff (1985) also studied the variability of EMS currencies. In the notable study by Ungerer, 9L3], (1983, 1986), variable approaches to this question were used with various choices of exchange rates (bilateral, effective, nominal, and real), data frequency (daily, weekly, and monthly), and the level and change in exchange rates. However, all of these studies which have tested for a downward shift in exchange-rate volatility for members of the EMS post-March, 1979 have generally relied on the unconditional distribution, independently and identically drawn from a normal distribution. It is by now an accepted fact that exchange-rate distributions tend to be leptokurtic (fat-tailed, highly-peaked) and that the variance shifts through time with new information available at time t-l, as noted by Taylor and Artis (1988). They applied non-parametric tests for volatility shifts which do not require actual estimation of the distribution parameters as well as tests for a shift in the conditional variance with a random walk with Autoregressive Conditional Heteroskedasticity (ARCH) disturbances. They found a significant reduction in the conditional variance of exchange rate 1 Ungerer, g§_§l, (1983), pp 3-9. 8 for the EMS currencies against the D-mark and signs of a significant rise in the conditional variance against the US dollar (see Taylor and Artis, 1988 p. 12) However, they didn't demonstrate how to derive the likelihood ratio test which played.a key role in their tests for shift in volatility. To derive a likelihood ratio test is not easy, given different observations and/or different distributions in each period ifigi, Pre-EMS and Post—EMS. Also, after discussing the leptokurtic distribution in the exchange-rate change in one section, they ignored this distribution in their ARCH model and estimated. the parameters under the normality assumption. This paper will stress the importance of this stylized leptokurtic characteristic with the student t-distribution and also the possibility of a time-dependent conditional heteroskedasticity with multivariate GARCH (generalized ARCH) models as well as univariate models. I will test intra-EMS volatility against the Italian lira instead of the Deutsche mark (D-mark) to eliminate any possible impact of the role of the D-mark as a reserve currency or leading currency in the EMS. The US dollar will be used as a base currency to test the volatility change for non-EMS currencies, and the Pound Sterling will also be used to see whether there will be any difference in measuring volatility with a choice of a base currency. In Section 1 I use unit root tests to check the stationarity in the weekly exchange-rate series. In Section 2 the univariate GARCH models are used to explain how the time-dependent conditional heteroscedasticity is built after diagnostic tests with LJung-Box Q (k) and Qz(k) statistics to check serial correlations. In Section 3 the test results of the EMS 9 currency volatility after March 13, 1979 are analyzed, and in Section a the multivariate GARCH models are estimated. Some conclusions are given in Section 5. 2. Tests for a Unit Root in Weekly Exchange Rate Series Autoregressive time series with a unit root have been the subject of much recent attention in the econometrics literature. In part, this is because the unit root hypothesis is of considerable interest, not only with data from financial and commodity markets where it has a long history, but also with macroeconomic time series. Initially, the research was confined to cases where the sequence of innovations driving the model is independent with common variance. Frequently, it was assumed that the innovations were iid(0,az) or, further, that they were iid N(0,az). However, independence and homoskedasticity are rather strong assumptions to make about the errors in.most empirical econometric workm There is now a substantial body of research that exchange-rate series exhibit time- dependent heteroskedasticity (see Baillie and Bollerslev (1989), Bollerslev (1987), Milhoj (1987), McCurdy and Morgan (1983) and the references therein.). I have used the unit root test methods of Phillips (1987) and Phillips and Perron (1986, 1988) which are robust to a wide variety of serial correlation and time-dependent heteroskedasticity. These tests involve computing the OLS regressions: s, - r. + “bu-172) + 'as,-, + a, (2-1) s, - ,.* + a's,-, + .4 (2-2) 10 and 3t. " ass-1 4' at (2'3) where st is the log of spot exchange rates, T denotes the sample size, and the innovation sequences 1'5, u; and In, are allowed to follow a wide variety of stochastic behavior including conditional heteroskedasticity. The testing strategy recommended by Phillips and Perron is to start Eq. (2-1) and to test the null hypothesis H01: 75-0, 18-0, 'c'i-l and H02: 3-0, 5-1 by means of the statistics 2(02) and Z(Q3) respectively. If Ho1 and Ho2 can be rejected, then one should next test Ho3 :'5 - l by means of the Z(t;) statistic. If Ho} and Ho2 can not be rejected (i.e. they show both random walk and random walk with drift), then the strategy is to proceed to exclude the time trend and to test Ho‘ : u"- 0 and a':- 1 by the use of the Z(¢1) test statistic for testing a unit root without drift. Individual unit root tests of the null hypothesis on (2-2) and (2- 3) of the form H05: :3. - 1 and Ho“: 21-1 are tested by the statistics Z(t¢*) and 2(t5) ; see Phillips and Perron (1988) for the precise formula for each test statistic. In our analysis, I took weekly spot exchange rate data from the New York Foreign Exchange Market between January 3, 1973 and September 28, 1988. The series were constructed.by taking observations every Wednesday, and in the event of the market being closed, an observation on the next business day (i.e. Thursday; if the market was closed on that Thursday also, then Friday, and so on) was used. The data provided by the Federal Reserve System, are bid prices taken at noon, constituting a total sample ll of 827 observations for the EMS currencies and 8 major countries2 against the US dollar. Six different unit root test statistics were estimated for all currencies. Calculating the test statistics requires that consistent estimates of the variances of the sum of the disturbances fit, u; and 0‘ in (2-1) to (2-3) and a truncation lag, 2, corresponding to the maximum order of non-zero autocorrelation in the disturbances be chosen; see Phillips and Perron (1986) and Newey and West (1987) for details. Hence, the statistics were computed for 2 - 0,2,4,6 and 10, but were found to be remarkably similar for different values of'l. The results with lag 10 are reported in Table 1. Both simple unit root tests of the t-statistic type, Z(td*) and Z(t;), confirm the unit root with drift. At the same time, the 2(01) statistics accepts the random walk without drift, and the inclusion of a time trend and use of the 2015) statistics show the same results. However, the 2(t5) statistics reject the random walk without a drift at the usual 958 level for the Swiss franc. The overall indication is that there is strong evidence for the presence of unit root with a drift for all currency series, and.hence, all the series appear to be stationary in their first differences. zThe EMS currencies include West German D-mark, French franc, Italian lira, Belgian franc, Netherlands guilder, Irish pound, and Danish krone. The other major currencies include the US dollar with weighted value, Canadian dollar, Pound sterling, Austrian shilling, Swiss franc, Japanese yen, Swedish krona, and Norwegian krone. 12 3. Models with Time-Dependent Conditional Heteroskedasticity; GARCH (1,1) For time series analysis, the autoregressive heteroskedastic process (ARCH) type of model has proven useful in several different economic applications. Among many others, see Engle (1982), Engle and Kraft (1982), Coulson and Robins (1985), Engle, Lilien and Robins (1987), and Weiss (1984). However in this paper, the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) is considered for empirical study of the EMS currency volatility, since it allows for a much flexible lag structure (see Baillie and Bollerslev (1987) and Bollerslev (1986, 1987) for its applications with conditional t-distributed errors.). 3.1 Implication of GARCH Model The first set of data consists of weekly exchange rates of EMS currencies against the US dollar and EMS currencies against the Mark from January 3, 1973 until September 28, 1988 for a total of 827 observations. The log of spot rates, 3,, are converted to continuously compounded rates of returna, y,’ - 1000 x (st - std). The dependent variable y,’ denotes the change in the logarithm of the exchange rates between time t and t-l and is shown to be stationary in its first difference from the results of Section 1. The full model is then: 3'; - b. + u. “t. I ot-l " D(0,ht) 2 h, ' ”o + a1 ut-l + 51hu-1 3For convenience of calculation, I multiplied 1000 by A 3,, which doesn't change the statistical results. 13 where “on is the set of all relevant and available information at time t-l, and where D(0,ht) represents some distribution with mean 0 and variance ht. The assumed process is a regression model with innovations that have either conditional normal or student t densities with time- dependent variance. The conditional-variance equation is assumed to follow a generalized ARCH (or GARCH) model. Before estimating the coefficients of GARCH models, the serial correlations are checked for implications of the GARCH model. First, most currencies were found to have moving average terms with significant levels. For example, the D-mark against the US dollar shows the value of Q(lO) - 22.5 in the Ljung-Box (1978) portmanteau test statistic} for up to tenth-order serial correlation in (yt - Ibo), which is very significant at any reasonable level in the corresponding asymptotic xfio distribution. After adding those moving average disturbance terms, the value of Q(lO) is reduced to 8.5, which is not significant at any reasonable level (see Table 3, Column. 1). This Q(lO) reduction is the same for other currencies, with some exceptions, for example, the D-mark against the ‘This is a test of the joint hypothesis that all autocorrelation coefficients are zero and as such as chi-square with M-p-q degrees of freedom. MA A 2 2 Q(r) - n(n+2) 3 ra(k) / (n-k) - x (M‘P'Q) k-l A n A A n A2 where r k) - 2 a a / 2 a (k-1,2,---) g t-k+l t t'k t-l t and (1-¢,L- -¢,LP)w, - ( 1-¢,L- -4.qu ) E.,, where (at) - iid N(0,az) with a discrete time series w1,.,.,w,,. In the case of Q(10), critical values of those yield 18.307 and 15.987 at 5% and 10% level, respectively. 14 Italian lira and the D-mark against the Netherlands guilder need no moving average disturbance terms at all. After considering these moving average disturbance terms, we have y, - b.3 + 0(L)e; (3-1) 9a)., - a, + 91%-, + 92%-, <3-2) e, | n,., ~ D(o,h,,) (3-3) E<¢2tIOt-1) " halt-1 ' 0% + 0162-1 + 511‘“ (3'4) On the other hand, (y, - In“)2 is clearly not uncorrelated over time to all currencies, as reflected by the significant Ljung-Box test statistic for absence of serial correlation in the square, 02(10), which is distributed asymptotically as a X30 distribution (see McLeod and Li, 1983). For example, when we don't use GARCH model, the D-mark against the US dollar shows Q2(10)-21.2, a very significant indication of the presence of serial correlation (see Table 3). The null hypothesis of no ARCH effects can be decisively tested with the Q2(k) statistic. Some series could have the squared residuals which appear to be autocorrelated even though the residuals do not (for our example, the Swiss franc against the US dollar; see Table 2-1). This absence of serial dependence in the conditional first moments, along with the dependence in the conditional second moments, is one of the implications of the ARCH or GARCH (p,q) model given by Eqs. (3-1) to (3-4) (see Bollerslev, 1987). With the GARCH model we estimated the parameters by the Berndt, Hall, Hall and Hausman (1974) algorithm. The maximum likelihood estimates of the parameters are presented in Tables 2-1 and 2-2 with asymptotic standard errors in parentheses. The summary of the relevant test statistics are shown in Table 3; for example, the Ljung-Box test statistic 15 for the standardized residuals, 6,11? and the standardized squared A residuals, “$113, from the estimated GARCH (1,1) model takes the values Q(10)-6.l3 and Q2(10)-3.48, respectively, for the D-mark against the US dollar, which doesn't indicate any further serial dependence. 0n the other hand, the hypothesis of the constant conditional variance fails with LR,” test statistics (see Table 4), which is highly significant at any level in the corresponding asymptotic x22 distribution.’ As can be seen from Tables 2-1 and 2-2, the estimated values for a+fi are close to 1" for some currencies, indicating the probable existence of an integrated GARCH, or IGARCH process; see Bollerslev (1989), Engle and Bollerslev (1986). The autoregressive term 1.3“. the coefficients of hf,1 are highly significant, which tells us that changes in volatility of exchange rates have a high degree of persistence. It is also interesting to note that the implied estimate of the conditional kurtosis’, 3(9-2 )(i‘z-h)’1 is in close accordance with the sample analogue for 2211? (which is k in Table 3) for most currencies (see Tables 51 didn't show all test results in the Table for other exchange rates, but obviously they have the same results; see each Table. “The GARCH (1,1) process is wide-sense stationary iff a 4- fl < 1. See Bollerslev, T (1986) for the proof. The time series (Kt, teZ}, with index set Z-(0,_+.l,i2,---) is said to be wide-sense stationary or covariance stationary if (1) E|x,|2 < on for all teZ, (ii) Ext - m for all tez, and (iii) 7,(r,s) - 1,(r+t, s+t) for all r,s,te2, where 7x(r,s) - Cov(X,,X,). If a + fl 2 1, then it blows up and we have an explosive ARMA model (see Bollerslev (1989) for discussions about IGARCH.). 7From Kendall and Stuart (1969),, the fourth moment is equal to an; I a“) - 3(y-2)(y-4)‘1h§,,_1,u>a. 16 2-1 and 2-2). This means that even in the weekly data, the t-distributed GARCH (1,1) model works well“. This estimate of the conditional kurtosis differs significantly from the normal value of three, as seen by the LRuy-o test for the GARCH (1,1) model with conditional normal errors with xi distribution (see Table 4) . The estimated value of each u‘1 is the inverse of the degree of freedom parameter (see Tables 2-1 and 2-2). In conclusion, as expected, GARCH models worked very well for my purposes and this model is used to test the EMS currency volatility. 3.2 Tests for EMS Currency Volatility 3.2.1 Test Method Because EMS implemented the Exchange Rate Mechanism (ERM)° in March 13, 1979, to test the volatility, I will designate the time period before March 13, 1979 as pre-ERM and after March 13, 1979 as post-ERM and see whether there is a difference in volatility in both periods. To test volatility we simply could test the following null hypothesis: Ho : Pre-ERM 85,31“, :8“ are same as those of Post-ERM in our Eq. (3-4) hu- “’1 + ali‘i-l+filiht-l (i - 1,2). However, if It, is rejected, does this imply increasing ‘volatility, decreasing volatility, or neither? We can find no distinction between 8Baillie and.Bollerslev (1987, 1989) have found.that.with weekly data the assumption of normality is generally appropriate. 9Presently, Belgium, Luxembourg, Denmark, France, the Federal Republic of Germany, Ireland, Italy, and the Netherlands participate in the Exchange Rate Mechanism. Great Britain, Spain, Portugal and Greece are not in the ERM, but in the EMS. Hereafter, the term ERM is used to indicate these countries or their currencies. l7 them. One possible way to structure our test would be to test 3),, fin, and 211- but, 3's and 2's have really nothing to do with volatility levels. Therefore, $13 will be used to test the change in volatility, i.e., the differences in 85' s. y; - bo + 0(L)ec (3.1) 9a)., - e, + o,.,-, + 92%-, (3-2) ‘t I own - D(0,ht) (3-3) he. " “a + “’11): + ai‘zt-i +filht-l (340' r - 1 if post-ERM where D 1 t - 0 if pre-ERM 3.2.2 Test Results of ERM Currency Volatility First, the nominal exchange rate volatilities were estimated against intra-ERM (D-mark against Lira, D-mark against French franc, and D-mark against the Netherlands guilder). The existence of the ERM since 1979 has coincided with a marked reduction in the volatility of exchange rates within the ERM. This was one major goal of the system, and to this end, the intervention arrangement and other elements of the exchange rate mechanism were established (see Table 2-1, Column 1 to 3). However, in terms of the nominal volatility against the US dollar, the ERM currency volatility increased during the ERM period. It is statistically significant at the 5% level for the cases which I have studied with the D-mark, Danish krone, the Netherlands guilder and Belgian franc against 18 the US dollar (see Table 2-2). To compare the volatility level change between ERM currencies with that betwaen non—ERM currencies, I have estimated the volatility of the Canadian dollar, Swiss franc, and Pound sterling against the US dollar, which showed an increase in nominal volatility during the ERM period in each case (see Table 2-1, last three columns and Table 5 for the summary of t statistics). Figures 1, 3 and a confirm these changes, showing the residual movements in our model. Figure 1 shows that after the ERM system there was a decrease in the volatility in the case of intra-ERM currencies, while Figure 3 shows an increase in ERM currency volatilities against the US dollar. Figure 4 reveals an increase in volatility'between.non-ERM currencies after March, 1979. In.the previous case we checked the volatility level changes against two key currencies, the US dollar and the D-mark, which are both major reserve currencies and transaction currencies. In addition, we used the D-mark, because West Germany is the leading country in the ERM. However, due to those factors, the measure of the exchange rate volatility might be distorted. To eliminate this problem we used the Pound sterling instead of the US dollar and the Italian lira instead of the D-mark as base currencies and tested the significance of the change in the level of the volatility again. The results are shown in Table 6. As expected, in the case of intra-ERM currencies (the French franc and the Netherlands guilder against the Italian lira), there were significant decreases in the volatility after March, 1979 (see the first two columns in Table 6). But in the case of the ERM currencies, (Italian lira and Netherlands guilder) against the non-ERM currencies (Pound 19 sterling), the Netherlands guilder, which showed an increase in volatility after March, 1979 when it is measured against the US dollar, showed at least constant volatility after March, 1979 (see Table 6, fourth column). Also, between non-ERM currency volatility, the Swiss franc, which showed an increase in 'volatility against the 'US dollar, accepts the null hypothesis that there was no change in the volatility even after March, 1979 (see Table 6 column 5). These results are confirmed in Figures 2, 5, and 6. Figures 5a and 6b imply the constant volatility movements. As we suspected that United Kingdom might try to stabilize her currency volatility against the other ERM currencies, as they are her neighbors, we tested the non-ERM currency volatility against the Japanese yen and Canadian dollar as base currencies. Table 7 shows that the Yen/guilder and Yen/Sfr had at least constant volatilities again, and we can confirm our results. The clear diminution of exchange-rate volatility in the case of intra-ERM is certainly consistent with the view that the system has been successful in contributing to exchange-rate stability among participating countries. However, as is shown in Tables 2-1 and 2-2, in the exchange rate volatilities against the US dollar the volatility of the ERM currencies showed different patterns under their exchange-rate mechanisms from those of the non-ERM currencies. Hence, we can say that decreasing volatility of the intra-ERM does closely follow the increasing volatility against the US dollar. This was already noted by Cohen (1981), who said that "--- effort to maintain.the joint float could increase the volatility of fluctuations between participating and non-participating 20 currencies---" (see p. 14). It appears that such effort may do so at the cost of increased instability of exchange rates against the US dollar.10 Even though there was a significant reduction in volatility after joining the ERM and although the study as a whole suggest fairly distinct patterns to the results, no strong conclusions as to cause or effect can be drawn. For example, it is impossible to say how far the reduced volatility among ERM currencies is due to the operations of the ERM itself. In addition, even if the coincident fall in volatility among ERM versus non-ERM currencies and the constant volatility among the non-ERM currencies are not a reliable indication of the way in which the ERM rates would have behaved in the absence of the system, it does nevertheless somewhat weaken the claim that the reduction in the volatility of inter- ERM rates is due to the creation of the system alone. 4. The Multivariate Generalized ARCH Approach In previous sections we estimated the univariate GARCH models, and they offered good statistical descriptions of exchange rate movements. However, they are not satisfying compared to a multivariate model because the multivariate approach gives some advantages for the following reasons: 1) Nonzero covariances among exchange rate innovations require joint estimation.of sets of regression.if efficient.estimation in parameters is to be achieved. 1° Also Marston (1980) says that the volatility of the dollar exchange rate of that member country disturbs economic relationships between the two members of the union by changing cross-exchange rates between member currencies. 21 2) Exchange rates are bilateral rates, and if new information comes to the foreign exchange market (e. g., the US money supply, the US budget deficit, the German trade surplus, etc.), it should affect all rates as market dealers change their demands of' specific currency' and. it affects their portfolios. The multivariate ARCH (q) model was originally introduced in Engle and Kraft (1982), and then used by Diebold and Nerlove (1986). Later it was generalized by Bollerslev, Engle and Wooldridge (1988). Baillie and Bollerslev (1987) modelled risk premia in forward exchange-rate markets with a multivariate GARCH approach. 4.1 Estimation of Multivariate GARCH Models In order to deal with the multivariate GARCH (1,1) model, the following SUR system is estimated for the set of N currencies: y, - bo + 6: (4-1) ‘elnvq ” N(0:Ht) (4'2) Vech(Ht) - Co + CID“ + A1Vech(et-1e 1-1) + BIVech(H,’-1), (4-3) where y, is vector of first-differenced N currencies and.b° and.et are le vector of constants and innovation vector. The Vech (.) denotes the column-stacking operator of the lower portion of a symmetric matrix. Co and C1 are N(N+1)/2 x 1 vector, and A1 and 81 are N(N+1)/2 x N(N+1)/2 matrices. The conditional log likelihood function for (4-1)-(4-3) for the single time period t can be expressed as 1.,(0) - -N/2 10321: - 1/2 log|Hc(0)| - 1/2 e,(o)'a,j1(o)e,(o), 22 where all the parameters have been combined into 9' - (b'°,C'°, '1,Vec(A1) ' ,Vec(B1) ' ) . Thus, conditional on the initial values, the log likelihood function for the sample 1,2, ------ ,T is given by T L(0) -t§1Lt(9). As is obvious from univariate case, the log likelihood function L(0) depends on the parameter 0 in a nonlinear form, and the maximization of L(0) requires iterative methods. While the multivariate GARCH (1,1) of manageable size is considered here, a natural simplification is to assume that each covariance depends only on its own.past values. We restrict our attention.to two currencies, the Italian lira and the Netherlands guilder, first against the Deutsche mark and then against other cross currencies. Weekly data from the FRB tape are used, as in the univariate GARCH models. The model considered here becomes the bivariate GARCH model Y1: ' bi + ‘1: (4°1)' Culnru ” N(0,H.) (4’2)' hth " CoiJ + (311,191: I“ “13‘: t-l‘Jt-l "’ fliJhth-l 1:3 " 1'2 (“JV where subscript i refers to the ith elements of the corresponding vector and ij to the ijth element of the corresponding matrix. 23 4.2 Model Estimates The maximum likelihood estimates of the model obtained by the BHHH (1974) algorithm for the case of the Netherlands guilder (yn) and the Italian lira (yap) against the Deutsche mark are:11 I 1 F T [ 1 y -0.001 c It (0.06) 1t - + (4-4a) y 0.61** e 1 2: < + (0 22) { L 2: J l hilt 0.31** 0.05** a: c-1 1 [ 0.93** hll t 1 (0.08) (0.01) (0.05) ' h - 3.15** -+ 0.104** e e + 0.73** h 21c (1 17) (0 02) 1 c-1 2 c-1 (0 05) 12 c-1 2 h 16.49** 0.31** e O.61** h L 22‘} L (2.03) J L (0.03) 2 c'1 j 50'03) 22 “'1 i -0.29** l (0.08) + -2.9l** D (4-4b) 1: (1.13) -11.59** L (1.79) J log likelihood function - ~3970.3668. In the case of the Netherlands guilder against the D-mark, in the conditional covariance equation of the hrun 0.31 is the intercept, a - 0.05, and fl - 0.93, with -0.29 as the intercept change after March 1979. The 1‘21: is for the conditional covariance of the Netherlands guilder against the Italian lira, and the significant value of the coefficient of Dunne“ -2.9l) shows the decreasing volatility after ERM as already 1‘ Hereafter, * indicates significance at the 5% level and ** at the 1% level. Asymptotic standard errors are in parentheses under corresponding parameter estimates. 24 verified in the univariate case (see Table 6). The hug is for the conditional covariance of the Italian lira against the D-mark, and the significant value of the coefficient D1,.(i.e., -ll.59) also shows the decreasing volatility after joining the ERM. The estimates for the model are appealing, and the estimated value for each coefficient is reasonable and highly significant, lending some support for the arguments that time series for exchange rates works well under the GARCH model and that the intra-ERM currencies show decreasing volatilities after participating in the ERM system” However, the likelihood ratio between the univariate GARCH (-2758.883 for DM-LIRA and -2017.937 for DM-Netherlands guilder) and the multivariate GARCH (-3970.3668) implies that the multivariate GARCH is more efficient than the univariate model. Compared with the univariate model, significant coefficients of a and 8 are achieved in the case of Guilder/Lira. I estimated the Lira and the Netherlands guilder against the Japanese yen as one test of the change in volatility of the ERM against the non-ERMJ In.the univariate GARCH model, the decreasing volatility was shown at the 5% significant level in the case of the Yen-Lira and at least no change in volatility in the case of the Yen-Netherlands guilder (see Table 7). Even in the case of the ERM currency against the non-ERM currency, there was at least constant volatility after March 13, 1979 and demonstrated that the reduction in the volatility of ERM currency could result even in ERM vs. non-ERM cases. The following multivariate GARCH estimates assured such a claim. The volatility of the Yen-Lira (yap) is decreased after'March, 1979 with a 5% significant level and the volatility of the Yen-Guilder (yap) is shown to be at least constant. Here, again, 25 the multivariate model becomes more efficient than univariate models when we consider their likelihood functions. Furthermore, the constant term in the case of Yen/Guilder [see Eq. (4-5a)] shows significance at the 5% level, which was not significant in the univariate case. y“ ‘ I1. 389“? 81;] (0.403) - + (4-5a) Lth 0.604* 62% 60.367)J € \ I ‘ " 2 \I 4' W 1111: 30.63” 0.205“ £1 0']. 0.67“ hi]. t'l (2.82) (0.02) (0.02) 1112‘ - 19.89“ + 0.196 61 $-16: t‘l + 0.67“ hi: 3'1 (2.65) (0.02) (0 02) K1122.7} 21.4w: O.l86** .3 ,-, 0.701“ hzz ,-, L(3.71)j L(0.02) ‘ ) ,(0.025) j (~5.65*‘ (2.89) + 2.72 0,, (4-5b) (3.14) -0.45 ((3.43)/( log likelihood function - -5170.6051 Lastly, the volatilities of the Canadian dollar and the Swiss franc against the Japanese yen were estimated to see whether among non-ERM currencies they have increased volatilities after March, 1979. The following,multivariate estimates show’that, in the case of the Swiss franc against the Yen, there was a decreasing volatility after March, 1979 at the 5% significant level. In the case of univariate estimates, it showed at least constant volatility (see Table 7). In Eqs. (4-6a) and (4-6b), 26 y“ denotes the first-differenced Canadian dollar/Yen, and Y2: denotes the first-differenced Yen/Swiss franc. Y1: r'0°91* 7 ‘18 (0.47) YZB 0.61 (2‘ (0.41) J )L (hm f9.62**\ (0.08» .2, ,-, ”0.86” hu m) (1.89) (0.01) (0 01) 1112‘ '- “8.05“ + 0.09“ El t’l £2 t’z + 0.80“ 1112 t'l (2.52) (0.02) (0.04) hm l6.01** 0.15“ 5220-1 0.77“ 11,, H \ \(3-97) ) ((0.03) J (0.03) 1 J ’ 5.30*4‘ (1.43) + 1.79 (1.88) 0,, (4-6b) .s.24* (2.84) 1 J log likelihood function - -5844.3448 As we have seen, the estimates of multivariate GARCH models are efficient relative to univariate GARCH estimates, and it is important to have simultaneous multivariate estimation for the reasons mentioned. However, the magnitude and sign of the coefficients which were used to test for the change in the volatility after joining in the ERM did not vary with the multivariate estimates. 27 5. Conclusion The European Monetary System was established on Murch 13, 1979. After ten.years, the time is ripe to evaluate this scheme and.consider its possible future contribution to European and world-wide monetary relations as well as to European integration. I have empirically studied the after-EMS currency volatilities and demonstrated them. with multivariate GARCH models as well as with univariate GARCH.models. Although the intra-EMS showed stable volatility after March, 1979, one can not say that these stable exchange volatilities result from the system itself, because we have found.that even in some ERM vs. non-ERM cases, as well as among non-ERM currency cases, there existed at least constant volatilities. Furthermore, decreasing volatility of intra-ERM closely follows the increasing volatility against the US dollar, and an effort to maintain the joint float increases the volatility of fluctuations between participating currencies and the US dollar. Proposals for policy coordination among the major industrial economies have been discussed in recent years. But if such proposals utilize the successful EMS-member coordination for stable exchange rates, they should be considered carefully, because our experience indicates it is not used without cost. Table l Phillips-Perron Unit Root Tests on Exchange Rate Series y-u+pumn)+w t mg) -1.3388 -l.4455 -l.2731 -l.3907 -0.8800 -l.2l38 -l.0358 -l.9050 -l.7106 -0.7591 -l.9090 -2.1850 -0.1924 -1.0746 t-l t * * “ y ' n T ay 0 y ' “y + t t-l C C C-l 2(02) :3-05-03-1 2(03) :3-0de-1,Z(c;) :a-l 2(83) : 3 -1 , 2(01):4*- 0 and a*- 1, Lag - 10 in Newey and West(1987) Yfifi’iiflii‘fis) “'3’ 2(02) “'3’ Canadian 8 0.8985 1.1039 ~0.4762+ Pound Sterling 1.1994 0.9803 -1.3970 Irish Pound 0.9044 1.0334 -l.0969 Italian lira 0.9932 1.9399 -0.8229 French franc 1.0638 0.7877 -l.4448 Belgian franc 1.1551 0.7876 -1.4757 Danish krone 1.0127 0.6799 -1.4258 Deutsche mark 1.8170 1.5909 -l.8384 Dutch guilder 1.4877 1.2487 -l.684l Swedish krona 1.0141 0.8579 -l.4159 Aust.schilling 1.8367 1.6774 -l.8514 Swiss franc 2.6288 2.5395 -2.1850 Japanese yen 1.3073 1.7751 -1.3598 Norweg.krone 3.1458 2.1024 -2.3415 weighted- US dollar 0.8523 0.5976 -l.2980 O CHUNOHNOOONHHH -l.2577 Key: * indicates + indicates Note: Under the tively, and values for 2(t3) are -3.96 and -3.441 at 1% and 5%. significance at the .01 percentile and ** at .05 significance at the .95 percentile and ++ at .99 percentile percentile null hypothesis, the 95% and 99% critical values of Z(t;), 2(03) and 2(02) are -.94 and -.33, 4.68 and 6.09, and 6.25 and 8.27,respect- Also, at the 95% and 99% level the critical values of Z(ta,),2(t:) and Z(Ol)are 1.28 For 2(t3 ) and 2(tz), values and 2.00, -0.07 and 0.6, and 4.59 and 6.43. -l.95, are -2.58 and Phillips and Perron (1986)]. -3.43 and -2.86 at 1% and 5%, ,respectively [ see 29 Table 2-1 Estimation of GARCH Models with D-mark and Us dollar as Base Currency yc - b° + uc ; ut- ¢t+ 'l‘t-l+'2‘t-2 ; ‘t' 0t.1- D(0,ht,v); 2 hc - ”o + wch+ a1¢c_1+ fllht-l Parameters 1 6: Diagnostic ALL—— Statistics uss-cns ass-spa ass-0x5 Log L -2452 528 3482.335 3297 485 ............................................. b0 0.607** 0.047 0.816* -0.209 0.641 -O.67l (0.238) (0.062) (0.382) (0.169) (0.608) (0.567) 91 --- --- 0.219** 0.068* 0.069* 0.009 (0.048) (0.037). (0.035) (0.037) 02 ~-- --- 0 147** o.117** 0.106** 0.078* (0.035) 0.036) (0.026) (0.037) 40 19.397** 0.222** 66.990** 2.263** 156.463** 15.239** (2.182) (0.005) (2.799) ((0.902) (21.779) (3.481) 41 -l4.580** -0.209** -40.981** 3.385** l40.164** l4.626** (1.722) 44(9.058) (2.739) (1.430) (39.797) (5.437) 61 0.266** 0.048** 0.274** 0.169** 0.311** 0.109** (0.033) (0,005)!) (0 047) (0.045) (0.080) (0.023) pl O.63l** 0.947** --- 0.694** --- O.768** (0.033), (0.004) (0.079) (0.047) u'I normal normal normal 0.201** 0.212** normal (0.007) (0.056) m3 1.996 0.597 4.754 -O.868 0.240 -0.292 m“ 13.109 8.941 51.352 8.788 5.583 6.561 Q(10) 12.404 10.296 17.116 12.631 6.209 12.666 02(10) 18.750 4.571 0.382 8.582 141.254 3.714 3(y 2)/(u 4) [ N.A. N.A. N.A. 9.0 11.36 N.A. Note: 1. Asymptotic standard errors are in parentheses under corresponding parameter estimates. 2. * indicates significance at the 5 % level and ** at the l % level. 3. NGL stands for the Netherlands guilder,CN$ stands for the Canadian dollar, SFR for the Swiss franc, and FFR for the French franc. 4. U'denotes the degree of freedom with student t density. 30 Table 2-2 Estimation of GARCH Models with EMS Currencies against the US Dollar Parameters WI & Diagnostics USS-DH USS-DKR USS-NGL US$-BFR Statistics 1 Log L 1 -3316.977 -3309.l70 ~3287.396 -3287.438 -------------- )ooo-------o------o------------------o--------- - -{ bo 0.543 -18 033** O 422 0.216 (0451517 (0 513) (0.452) (0.536) 01 0.080* 0.081* 0.078* 0.084** ((0.036) ((0.038) (0.037) (0.034)“ 02 O.152** 0.119** 0.150** O.l40** (0.035) (0.035) (0.035) (0.035) wo 7.066** 12.020** 6.124** 5.773* (0.168) (4.883) (2.819 2.551 “1 12.572* 18.715* 10.754* 12.079* ((5.742) (8.348) (4.996) (5.546) ml 0.168** 0.185** 0.156** 0.153** (0.039) (0.045) (0.034) (0.036) 81 0f782** 0.726** 0.793** 0T7985; (0.044) (0.064) (0.042) (0.040) 0'1 0.159** 0.169** 0.123** 0.152** (9.032) (0.031) (0.020)( (0 013)? 1113 0.314 -0.067 0.349 0.321 ma 6.093 4.190 6.257 5.947 Q(10) 6.129 4.190 4.903 5.861 03(10) 3.484 4.453 4.903 3.290 3(u-2)/(v-4) 5.650 6.115 4.464 5.319 ............................................................. “-1 Note DKR stands for the Danish krone and BFR stands for the Belgian franc. 31 Table 3 Summary Statistics with the Implication of GARCH(l,l) Model y: . bo GMRCH(1,1)-t Q(IO) 02(10) k Q(IO) Q’(10) k ass-Du 8.52 21.21 11.72 6.13 3.48 6.09 ass-0x2 8.45 34.52 10.77 4.19 4.46 5.21 USS-NGL 9.93 33.43 13.01 6.26 4.90 5.12 USS-BER 7.64 29.45 12.39 5.86 3.29 5.95 USS-CNS 18.84 27.98 17.28 12.63 8.58 8.79 ass-spa 3.03 145.52 6.20 6.21 141.25 5.58 art-ass 10.94 38.43 6.07 12.67 3.71 6.56 Note: Q(IO) and 03(10) denote the Ljung-Box (1978) portmanteau tests for up to tenth-order serial correlation in the levels and the squares which are standardized, respectively. They have x3 distributions with a degree of freedom of 10, which has values of 15.987 and 18.307 with p- 0.10 and 0.05, respectively. k is the usual measure of kurtosis given by the fourth sample moment divided by the square of the 2nd moment. Table 4 Likelihood Ratio Tests USS-DH USS-NGL USS-DKR USS-BFR US$~CN$ USS-SFR USS-UKL LR 46.334 39.112 28.578 41.994 77.694 56.966 --- l/u-O LRa-fl-O 88.410 68.994 93.428 97.472 81.936 105.728 104.758 Note : For our reference, x3 - 6.638 (with P-0.01) and x3 - 9.210 ( with P-0.01). 32 Table 5 Summary of T-Statistica for Shift in Volatility after March.1979 yc - b o+ut ; ut- ¢ c+ al‘t-1+02‘t-2; ‘t' 0t_1-D(O,hc,v); 2: . hc - we + «81060-010c 1+ plht-l .whete Dt- 1 if Poet-ERM - 0 if Pro-ER“ Ho ”1 - 0 “1 8.3. t-statistic DM-LIRA -14.580 1.722 -8.467 DH-NGL -O.209 0.058 -3.603 Du-FFR -40.981 2.739 -14.962 US$-CN$ 3.385 1.430 2.367 USS-SFR 140.164 39.797 3.522 USS-UK; 14.626 5.437 2.690 mm US$-DM 12.572 5.742 2.190 USS-DKR 18.715 8.348 2.242 US$-NGL 10.754 4.996 2.153 USS-BFR 12.079 5.546 2.178 Note: to 05- 1.645, 70.025 - 1.960, and 70.01-2'326' 33 Table 6 Estimation of GARCH Models with the Italian Lira and the Pound Sterling as Base Currency yt- bo+ u t 3 “t' ‘c+'1‘t-1+'2‘t-2‘ ‘t' ot-l' D 0; st and (p-p'),‘ are then said to be cointegrated of order d,b. Here )9 is a cointegration vector. Testing the PPP hypothesis on such a cointegrating vector is conventionally done with regressions of the spot exchange rates on relative prices, with prices given exogenously. In this section, multivariate tests developed by Johansen (1988) are implemented in testing 54 for cointegrating vectors between exchange rates (st) and relative prices (p-p')t simultaneously by specifying a model with a vector autoregressive process (VAR). The VAR model is: Ax... " rims-firzAXn-z "" +Pk-1Axt-k4-1 ‘ nt‘k+ 5r. (3‘1) where xtx- (st, pt - p2)’ and £1 --- e, are iid N(0,A). This is expressed as a traditional first-differenced VAR-model except for the term xxt*. If rank (n) is not full rank, then the coefficient matrix (u) may convey information about the long-run structure of our chosen data. Johansen's method of hypothesis testing of cointegrating vectors formulates the hypothesis of reduced rank (-r) in z, or one which implies that there are matrices a and B of order v(-# of variables)x r, such that a - afi' where ,B'xt ~ 1(0). The properties of the Johansen's test are as follows; 1. The maximum likelihood estimator of the space spanned by 8 is the space spanned by r canonical variates corresponding to the r largest squared canonical correlations between the residuals of xbk and Axt, corrected for the effect of the lagged differences of the x process. 2. The likelihood ratio test statistic for the hypothesis that there are, at most, r cointegration vectors is given by V A -21n Q - - T E ln(l-Ai) i-r+l where ATHJ ----, Av.are the v-r smallest squared canonical correlations. 3. Under the hypothesis that there are r cointegrating vectors, the estimate of the cointegration space as well as x and A are consistent, and the likelihood ratio test statistic of this hypothesis is asymptotically distributed as an f3 BdB'[ f}, B(u)B(u)’du]’1 f1, dBB'} 55 where- B is a v-r dimensional Brownian motion with covariance matrix I. The proofs of these results are presented in Johansen (1988). Given these properties we test for cointegration using maximum likelihood estimates. First, to Eq.(3-l) we add constant terms and seasonal dumies, since they turn out to be very significant. Hence, our model will be: 11 Ax,-r1Ax,-,+r2Ax,-z- -+l‘k-1Axt-k+1-1rxt-k+ constant + E thufls (3-2) 1 . where the 6th term in RHS denotes seasonal dummies. Before testing the cointegrating vectors in x - afl' we must decide how many legs are needed to get uncorrelated residuals in Eq.(3-2). With the VAR model, the 'k' is determined when the residuals in the data clearly passes the test for no autocorrelation. The estimates F, s, and A are given in Table 10 for the case of USArGermany exchange rates and prices. The hypothesis of cointegrating vectors in the variables (3 and p-p') according to Johansen's test method is H°z 1r - afi' , where a and ,8 are 2 x r matrices with rank (1!) - r s v (-2) . If r - 0, we can say that there is no cointegrating vector between s and (p-p') [Johansen (1988) and Johansen and Juselius (1988)]. First we tested for cointegration between 0.3. variables and four other country variables: W. Germany, Canada, Switzerland, and Japan. Next, to eliminate the effect of the USA variables, we tested for cointegration among W. German variables and three other country variables: Denmark, Japan, and Switzerland. (For reference, Frankel (1981) says that departures from PPP are a USA phenomenon.). Table 11 reports the results of calculating the likelihood ratio tests of cointegrating vectors in each case. In Table ll we find that the null hypothesis of no S6 cointegrating vector is rejected at the 95% or greater quantile in every case except that of Canada-USA. In cases where we reject the hypotheses of no cointegrating vector, the null hypotheses of at most one cointegrating vectors are accepted, and we can say that there is one cointegrating vector between the exchange rates and relative prices, except in the Canada—USA case. With these test results, we estimated the cointegrating relations, 8, in the second column in V from Table 12. In this case it seemed natural to normalize 5 by the coefficient of s--l. The normalized coefficients of (p-p') are reported on Table 13 as (-¢/a). This made it straightforward to interpret the cointegrating vector in terms of an error-correction mechanism measuring the excessive movement of exchange rates, where the equilibrium relation is given by s - 1(p-p') + constant. Similarly, 3’s can be found in the second column in the matrix of estimated Alphas in Table 12. a is discussed further in Section 4. Wald tests for the significance of each coefficient in the cointegrating vector were done with the hypothesis krfl - (l,0)(§1)- 0 or - (O,l)(§l)- 0 , where 51 is the coefficient of 2 z the exchange rate(s), and 82 is that of relative price index (p-p')’. .As can be seen in Table 13, the elements of cointegrating vectors are significant at any reasonable level. We also formulate a linear restriction on the cointegrating vectors and use Wald tests to see whether the traditional PPP theory that the coefficients of s and (p-p') are equal 7 The test statistic is T1’2K'fi / { (Ail - l) (K'f-f’K) )“2 with xf. X1 is the maximal eigenvalue, and B is the corresponding eigenvector, and the remaining eigenvector forms i, [see Johansen and JuSelius (1988) Corollary 3.17.]. 57 with opposite signs is acceptable. The restriction in a matrix formulation can be expressed as: k' fil a - (1, -1) 5 - o. 2 'For example, in the case of Germany - USA, the Wald test statistic is 0.7346, which is less than xi(p-O.99) - 6.635 and the null hypothesis of identical coefficients with opposite sign in s and (p-p') is not rejected. The other cases can be seen in Table 13. In the case of Sfr/US, which has a large coefficient, the null hypothesis of identical coefficients with opposite signs is not rejected at xfijfl (with 1 degree of freedom), but it is rejected at x338. The Sfr/Mark also has a large coefficient on the relative price indexes, but the null hypothesis is not rejected. Only in the Yen/Mark case is the null hypothesis of identical coefficients with opposite signs rejected. From these results, we are unable to reject the hypothesis of the simple version of PPP Theory; it generally holds under our multivariate approach. Therefore, we see that the alleged failure of PPP after 1973 is due to imprecise parameter estimates and improper specification of error structures. This result is opposite to Frenkel's (1981), where he has imprecise parameter estimates and concludes that departures from PPP are a USA phenomenon, because we have cases where the PPP theory holds when it involves the US dollar and the US price level. This result also is similar to that of Hakkio (1984), who is unable to reject PPP theory when it is viewed in a multivariate context. However, this approach is different from his; first, we recognize the cointegration problem which is not treated in his paper, and second, his paper uses instrumental 58 variables for relative prices, which did not need to use when using the simultaneous approach due to Johansen. To this point, we use cointegration tests in a multivariate context. Now we will see which results one can expect with conventional univariate cointegration tests. We applied the conventional univariate unit root tests to the deviations from PPP, defined as 2‘, with Zt'- constant + a, - (p - 33,, using Said and Dickey (1984, 1985), Phillips and Perron (1986, 1988) procedures and Dickey and Fuller's likelihood ratio tests (1979, 1981). First, in ADF tests [Table 14], only in the Denmark-Germany and Swiss-Germany cases are the null hypotheses of’a unit root rejected at the 0.10 and 0.025 significance levels, respectively. In other cases which showed long-run equilibrium relations (see Table 13), the null hypotheses can not be rejected. Next, we used Phillips and Perron tests, and surprisingly, in the two cases cited, in which we reject the null hypotheses of a unit root by ADF tests, the null hypotheses of a unit root can not be rejected using Z(t;) statistics.a Lastly, the low power of univariate unit root tests is examined again with the Dickey and Fuller likelihood ratio tests (1981), by comparing them with the Phillips and Perron tests (1986, 1988). Dickey and Fuller (1981) assume that the time series is adequately represented by the model a The ADF test is HO: 82 - 1 with P Yr. ‘ 59731: + flZYt-l + 2 “Ye-1 ‘ Yt-l-i) + 5t.- i-l The 2(tu') in the Phillips and Perron tests is Ho: 52 - 1, with yy- 8dt51(t-T/2) + fizde + uq. See Section 2 for the conditions on {ut). 59 P Zt - BC + filt + azt_1 + jfl ¢j(Zt-j - zt-j-l) + 5t (3-4) where Zt are deviations from PPP and at are independent identically - distributed (0,02) random variables. The hypotheses are: [11:50-81-0, a-l H2: [31-0, a-land Ho: [90 - 0, a - l with Zr. - ,90 + oz“ + at. The test statistics of O2, 03, and 01 for each of above hypotheses are given in Dickey and Fuller (1981, P. 1063). The test of these hypotheses is basically the same as that of Phillips and Perron. However, Table 14 reports that even when we test the same hypothesis, most of the Phillips and Perron tests (1986, 1988) accept the hypotheses of Ho and H1, which are rejected.by the Dickey and Fuller tests( in the case of'wa, the results are mixed). For example, in the case of Denmark-Germany, to test the hypothesis that fio-fil-O and a-l against the general alternative of Eq. (3- 4) we first compute (RSS1 - RSSz)/m .01878 - .01461 <12 - - - 17.81, ass2 / (T-k) 3(.01461/ 171) where RSS1 denotes the restricted residual sum of squares, and R832 denotes unrestricted residual sum of squares, m denotes number of restrictions, T denotes total number of observations, and k denotes number of coefficients. As there were 179 observations in the regression, the 97.5% point of the distribution of O2, as given in Dickey and Fuller (1981, P. 1063), was 5.40. Therefore, the hypothesis 8° - 81 - 0 and a --11 is rejected at the 2.5 percent level. On the other hand, under the Phillips 60 and Perron test statistics, the hypothesis of 8° - 81 - 0 and a - l in Z, - 50 + 191(t-T/2) +aZt.1 + ut is tested with the 2(02) statistic, which is basically identical to Oz in the Dickey and Fuller tests (1981) . However, the 2(02), 4.07 is less than 6.25 at the .95 percentile, and the null hypothesis of a unit root is accepted. That one test accepts the null hypothesis of a unit root, and the other test rejects the hypothesis can be seen in other cases in Table 14. 4. SHORT-RUN DYNAMICS Based on the estimated coefficient matrix of 9r, which conveys information about the long-run PPP as discussed above, we can explore short-run movements between exchange rates and relative prices in the context of the error-correction model. Error-correction models are usually interpreted by the partial adjustment approach of Engle and Granger (1987), but another interpretation is the rational expectations approach of Campbell and Shiller (1988). The former says that, most of the time, the economy system is out of equilibrium, but there is a tendency for the system to return to equilibrium [see Engle and Granger (1987) and Johansen and Juselius (1988) for this interpretation.]. On the other hand, Campbell and Shiller have an alternative interpretation for cointegrated models. They say that the error-correction model may also arise because one variable forecasts another. Engle and.Granger believe that the motivation for cointegration is that equilibrium error causes changes in the variables of the model. However, Campbell and Shiller emphasize the possibility that the equilibrium error results from agents' forecasts of 61 these changes. According to Campbell and Shiller, cointegration.can.arise even in a well—organized market with no adjustment costs, where there is no true causal role for the equilibrium error. It can arise when agents forecast and have rational expectations. In this study, the exchange rates and consumer price indexes have very different characteristics; it is generally accepted that the exchange rate, which is the relative price of two durable assets (monies), can be best treated by within an analysis of asset prices, which strongly depends on expectations concerning the future. 0n the other hand, aggregate price indexes reflect the prices of goods and services and are less sensitive to news. This distinction between aggregate price indexes and asset prices results in short-run deviations from PPP, and the stickiness exhibited by the aggregate price indexes may reflect the cost of price adjustment, which results in finite nominal contracts. Given such differences between exchange rates and consumer price indexes, we cannot avoid the partial adjustment approach in our analysis of short—run movements. In our VAR model, A A 11 A AXt - Plet'l + ' ' + Pk-1AXt-k-1-%{t-k+ conétant + Z k iqlb (4‘ 1) A 1-]. where Axt - (Ast, A(p-p*)g)', and F and % are matrices of order 2x2, and where seasonal dummy Q“ is llxl [see Table 10 for an example of USA- Germany]. Changes in exchange rates and relative prices at time t are expressed with lagged changes in the variables and error-correction terms as well as constants and seasonal dummies. If we replace the a with 08' to Eq.(4-1) which we derived in the previous section, fl'xt can be used as an instrument or an error-correction term to estimate an error-correcting 62 model, and a can be interpreted as the weight with which deviations from PPP enter the equations of our system. In this case a can be given an economic interpretation as the speed of adjustment towards the estimated equilibrium state. The MLEs of the error-correction model, in the case of USA-Germany, is as follows: ( Ast 1 { .29** .80 1 ( Ast_1 1 L.015 .95 1 { Ast_2 1 (.079) (.75) (.079) (.73) - + Arpt .008 .22** Arpt-1 .003 .11** Arpt-2 .008 .08 .008 .08 [JL‘H’H JL‘)()HJ 023 s - 014 ' t-3 ' ll 6 (l, -.68) (.013) It - -.004 rpt.3 + -.004** + 1ElkiQit + ‘zt (.001) In this equation, s denotes the logged exchange rates, rp denotes the logged relative prices, i.e., ln(p / p'), and As and Arp denote the first differenced. term. of each ‘variable. The fifth term represents the coefficients of seasonal dummies, which are not reported for reasons of space. With two lagged first-differenced terms, the residuals clearly pass the test for being uncorrelated ( the diagnostic tests are done with Box-Pierce statistics.). In the parentheses are standard errors, and ** denotes significance at the .05 level. The third term in the RHS includes the long-run equilibrium relationship discussed in the previous section; the first vector is a: and the second one is ,9' in 1r - 018'. The significance of 1r, which is shown as (28' in the above equation, is reported in Table 10. The vector, 8, i.e., (1, —.68)' has significant elements, and it was not possible to reject the null hypothesis of identical elements with opposite signs as (l, -l)', which implies the 63 long-run PPP [Table 13]. Under such restrictions on p, we construct a likelihood ratio test9 for the hypothesis that the second element of a is zero, for this element is small when we compare it with first element. This comparison implies that the cointegration relation enters only the first equation. However, this hypothesis is rejected at the 99% level. The first equation with Ast as a dependent variable is expressed as: ** As: ' (2339) Asc-l+ (2??)ArPc-1 ' (:8i3)A3c-2 + (:3§)ArPc-2 11 ** ** -.02(1.0 St-3 - .68 rpt_3) - (.813) + 131 ki Qit + e t . This equation can be interpreted as demonstrating how the change in exchange rates at time t is related to lagged changes in exchange rates and prices, with deviations from long-run PPP. Only the coefficients of the changes in exchange rate at lag one and the departures from long-run PPP are significant at the .05 level, which explain the changes in exchange rate at time t. The deviations from the long-run PPP are entered into the parentheses in the 5th term in the RHS, and the coefficient of 0.02 indicates the speed of adjustment towards the estimated equilibrium states. For Denmark, during the sample period, the estimated error- correction model against Germany is as follows: 9 The test statistic for this hypothesis of a is -2ln(Q) - T{ ln( 1 - I1) - ln(l - A;)} with xf, where 11 is eigenvalues under a and fl restrictions, and A; is eigenvalues under [9 restrictions. The a restriction is a - (3) (a1,0), and the 5 restriction is 81 - -fiz [see Johansen and Joselius (1988), Theorem 2.4 and its proof.]. J J J .19** .03 J J J As As c (.08) (.10) :4 ' .16* -.02 “Pt (.06) (.08) Arpc-1 J J J J J -.02 .06 J J J (.08) (.09) A“Hz-3 + .11 .05 A " (.06) (.08) rpc-s J J J J 11 e + E k.Q. + 1t 1-1 1 1t 6 2t 64 .067 .038' [ -.015 (.08) .05 (.06) (1. .09 J J l (.90) As c-2 -.05 (.08) At'I’c-2 J [ J [ .079**J sc-a (.027) -.90) + _.033 rpm. (.02) J J The fourth term in the RHS includes the long-run relationship shown in Table eliminate correlated residuals. l3. ** denotes significance at the Three lagged first-differenced terms in the RHS are chosen to .005 level, * denotes significance at the .025, and + denotes significance at the . 05 level . The fourth term comes from the vector autoregressive estimates of n - a5' and the x has significance with the matrix of [ -.06** (.02) .03 (.02) In the vector 8, i.e., .06** J (.02) .03* (.02) (1. -.90)', the null hypothesis of that both elements are zero is rejected and the null hypothesis of identical elements with opposite signs (1, -l)' is accepted. This implies the long- run PPP [see the previous section.]. With Ast as a dependent variable, the equation becomes 65 ASc'(:03)XSc-1 I (:98) Arpc-1 ' (903) Asc-2 + (:93) Arpt_2 ' (:83) Ass-3 ** ** ** 11 2.89)Arpt-3 - .07 (1.0 st-4 - .90 rpt-4) + (:83? + 131 kiQit + et' The first, seventh, and eighth coefficients are significant at the .05 level. Therefore, like the USArGermany case, the changes in exchange rates at time t are primarily explained by the changes in exchange rates at lag one and departures from long-run PPP as well as the constant terms. The speed of adjustment is approximately 0.07 and it is marginally higher than that in the USA-Germany case. We have examined the short-run dynamics and speeds of adjustment with vector autoregressive regressions and derived the long-run relationship from x - afi' as proposed by Johansen (1988). We conclude that PPP theory holds in the long-run. However, we observe short-run deviations from PPP. 5. CONCLUSIONS We have reviewed some of the evidence on prices and exchange rates, with the intention of testing the validity of the PPP. Using univariate unit root tests, we find that most CPI's and some of relative price indexes are I(2). Therefore, we cannot expect a stationary result with conventional cointegration tests if we use these price series. Cointegration between exchange rates and relative prices is tested in a multivariate context, using MLE estimates of vector autoregressive processes developed by Johansen (1988). In most cases tests of cointegrating vectors reject the null hypotheses of no cointegration in 66 relative prices and exchange rates. The null hypotheses of identical coefficients with opposite signs between exchange rates and relative price indexes are usually not rejected. From these results, we are unable to reject the hypothesis of PPP theory in the post-1973 data. This result is contrary to that of Frenkel (1981), who has imprecise parameter estimates and rejects PPP in the USA data. The result resembles that of Hakkio (1984), who finds support for PPP in a multivariate context. It differs from his, because we recognize the cointegration problem that he did not address. The results suggest that second unit root tests must be done for price series, and that a multivariate approach to testing PPP theory is needed for more precise parameter estimates. 67 Table'l Logged Monthly Spot Exchange Rates A. Sample Autocorrelations ooooooooooooooooooooooooooooooooooooooo goo-cc.---a-coo-ooooouoooooocococooo Level First differenced Yen/Us DM/US Yen/DM DK/DM Yen/US DM/Us Yen/DH DK/DM l 97 97 .98 .98 .35 31 37 .22 2 94 94 .95 .97 -.00 10 06 .08 3 92 91 .90 .96 .06 08 08 .04 4 89 88 .87 .94 .15 05 09 -.00 5 79 85 .84 .95 .08 06 - 02 .14 6 76 81 .81 .94 -.04 04 - 02 .03 7 68 77 .78 .92 -.00 04 05 -.02 8 65 73 .76 .89 .05 07 00 .02 9 61 69 .73 .88 -.02 04 - 02 .01 10 57 64 .70 .86 - 03 06 . 05 -.01 11 S4 59 .67 .85 03 02 - 08 -.01 12 51 55 .65 .83 02 - 01 - 09 -.02 Note: DK denotes Danish krone. B. Diagrams of ACF (Autocorrelation Function) and PACE ( Partial Autocorrelation Function) in the case of Yen / U$ LEEEL W act m 1r acr Pact J: J: . [I J .. g], I iiJlL'. , [in EE'EEiIJUJHu. . . . .. 5 1.. I unit .1 . h . . 11 1.1 w :04 w v w ‘ 9 PM w W“ 1 .‘. 1’01": [“11 I 68 Table 2 Unit Root Tests on Logged Monthly Exchange Rate Series Phillips-Perron test ADF Currencies 2(03) 2(02) 2(t3) 2(61) Z(t:) Z(t;) (Efége§(1984)) Against US Canada 1.372 1.219 -0.082 1.752 -1.574 -0.052 -1.787(2) Italy 1.091 1.259 -0.844 1.823 -1.454 -1.141 -1.571(2) France 0.926 0.652 -1.333 0.682 -1.122 0.134 -l.359(2) Denmark 0.794 0.529 -1.215 0.658 -1.144 -0.131 -1.302(2) Germany 1.124 1.051 -1.487 1.484 -1.435 -1.184 -1.587(2) Holland 1.098 0.940 -1.464 1.389 -1.464 -1.013 -1.542(2) Switzerland 2.135 2.161 -1.997 2.874 -1.901 -1.899 -2.114(2) Japan 1.424 1.900 -1.254 1.280 0.008 -1.591 -2.005(3) Against D-mark Canada 1.694 1 671 -1.786 2.119 -1.638 -1.656 -1.431(1) Italy 4.458 6 982** -1.732 8.234** -2.665 -3.225++ -2.705(2) France 2.653 4 603 -2.282 3.365 -0.756 -2.192+ -2.715(3) Denmark 0.877 3 712 -1.143 4.265 -0.863 2.593** -1.551(2) Holland 1.985 2 065 -1.888 1.565 -1.155 0.812 -l.752(2) Switzerland 4.412 3 775 -2.452 4.316 -2.555 -0.841 -2.805(3) Japan 5.218* 3 841 -2.989 0.577 -0.544 -0.920 -3.207(3) Note: 1.Phillips-Perron tests are with truncated lag - 10 in Newey and West (1987). '*' indicates significance at the .01 percentile and '**' at .05. '+" indicates significance at the .95 percentile and ”++” at .99. [See Table 6 for critical values] y - u + fi(t-n/2) + my + u t t-l t *+ * + * yt “ O‘ytrl “t ' A y: ' “yr-1 + “t 2(02) : Z - 0 B - 0 a -1 , 2(03) : z - 0 and a -1 , 2(c;) : a - 1 z - “ 1 z 0 - * 0 d * 1 z ' * 1 (c3) . a - , ( 1).u - an a - , (ca*) . a - 2. For Said and Dickey test, test statistics are from Fuller(l976,p.373). "(1)" represents 'Z,statistic, "(2)" represents 774 and "(3)", 7'6 . 69 Table 3 Logged Consumer Price Indexes A. Sample Autocorrelations for Level Lag Italy Germany Nether- Canada Denmark Swiss Japan U.S. France lands 1 98 98 .98 98 .98 98 98 97 98 2 97 97 .97 96 .97 97 97 95 97 3 95 95 .95 94 .95 95 95 93 95 4 94 94 .94 93 .94 94 94 91 94 5 92 92 .93 91 .92 92 92 89 92 6 91 91 .91 89 .90 91 90 87 91 7 89 89 .90 87 .89 89 89 85 89 8 87 88 .88 86 .87 87 87 83 88 9 86 86 .87 84 .86 86 85 81 86 10 85 85 .85 82 .84 84 84 79 85 11 83 83 84 80 .83 83 83 77 83 12 82 82 82 79 .81 81 81 75 82 Lag Italy Germany Nether- Canada Denmark Swiss Japan U.S. - France lands 1 .98 .98 98 .98 .98 .98 .98 .97 .98 2 -.01 .00 - 01 -.01 -.01 -.01 -.03 -.29 -.01 3 -.01 -.02 - 01 .00 -.01 -.01 -.01 .03 -.00 4 -.01 -.01 - 01 -.09 -.01 .00 -.04 -.05 -.01 5 -.01 -.01 - 01 .01 -.00 -.00 .03 -.01 -.00 6 -.01 -.00 - 01 -.02 -.00 -.00 -.02 -.02 -.01 7 -.01 -.00 - 01 -.02 -.01 -.01 -.01 -.04 -.01 8 -.00 -.01 - 01 -.01 -.01 -.01 .00 -.01 -.00 9 -.00 -.01 -.01 .01 -.01 -.00 .02 -.01 -.00 10 -.00 -.00 -.01 .01 -.01 -.00 .00 -.03 -.01 11 -.01 -.00 -.01 -.00 -.01 - 00 .02 -.01 -.01 12 —.01 -.01 -.01 -.02 -.01 - 01 -.01 .01 -.01 70 Table 4 A. Sample Autocorrelations for First Differenced Logged Consumer Price Indexes lands 1 52 38 -.10 64 .20 07 26 - 12 63 2 37 23 .12 23 .27 05 25 06 49 3 27 26 -.05 11 .29 - 05 11 10 40 4 25 20 .11 - 00 .21 03 14 04 34 5 23 17 .04 16 .24 - 21 - 03 10 .32 6 32 11 .19 15 .18 13 - 11 08 31 7 25 13 .03 14 .28 03 - 10 - 05 33 8 21 20 .09 - 05 .22 08 01 07 33 9 14 18 -.04 05 .16 03 01 11 40 10 .13 .21 .01 .02 .24 -.04 .12 -.01 .38 11 .09 .25 .05 -.03 .16 -.10 .13 .19 .34 12 .03 .09 .26 -.02 .09 .06 .16 .03 .22 lands 1 - 06 - 02 .01 - 14 -.05 - 01 - 03 02 - 11 2 07 02 .11 08 .19 05 15 01 08 3 03 15 -.02 13 .22 - 06 10 18 04 4 07 09 .ll - 04 .11 05 13 06 04 5 - 00 09 .08 09 .17 - 21 - 09 04 07 6 22 01 .20 10 .09 15 19 03 02 7 06 04 .06 19 .22 02 - 08 - 02 09 8 08 12 .09 03 .15 08 04 01 - 00 9 00 05 -.03 11 .07 03 02 01 19 10 05 11 01 11 .19 - 03 09 07 11 Table 5A Phillips-PerronfiUhit Root Tests on Logged Censumer Price Indexes ot‘west.Germany y - u + fi(t-n/2) + «7 +u t t-l t * * * ‘ ‘ y - p + ay +u , y - ay + u t t-l t t t-l t 2(02) : 3 - 0 B - 0 a -1 2(03) : 3 - 0 and 3 -l , Z(t;) : a - 1 2(c3) : a -1 , 2(01):u*- 0 and a*- 1, 2(ta*) : 0* - 1 Lags in Newey and West(1987) Lags 2(03) 2(02) 2(t3) 2(01) Z(t:) 2(ta) lags - 0 19.651++ 77.677++ 1.148++ 109.696++ -5.633** 2.261++ lags - 2 10.515++ 32.768++ 0.660++ 46.760++ -4.239** 7.966++ lags - 4 7.851++ 21.648++ 0.475++ 30.927++ -3.724** 6.449++ lags - 6 6.696++ 16.689++ 0.419++ 23.7OS++ -3.473** S.622++ lags - 8 5.973+ 13.818++ 0.401++ 19 402++ -3.302* 5.064++ lags - 10 5.334+ 11.900++ 0.357++ 16.421++ -3.140* 4.638++ lags - 12 4.716+ 10.527++ 0.259++ 14.186++ -2.971* 4.289++ lags - 14 4.212 9.548++ 0.145++ 12.495++ -2.820 4.006++ lags - 20 3.470 8.033+ -0.032++ 9.471++ —2.558 3.435++ lags - 25 3.173 7.513+ -0.116++ 8.025++ -2.428 3.123++ lags - 30 2.988 7.308+ -0.195++ 7.033++ -2.329 2.887++ lags - 35 2.989 7.285+ -0.214++ 6.336+ -2.269 2.706++ lags - 40 2.840 7.373+ -0.231++ 5.804+ -2.223 2.558++ lags - 45 2.812 7.528+ -0.210++ 5.398+ -2.196 2.436++ lags - 50 2.794 7.731+ -0.178++ 5.072+ -2.177 2.333++ lags - 55 2.784 7.965+ -0.129++ 4.807+ -2.164 2.243++ lags - 60 2.779 8.221+ -0.060++ 4.588 -2.157 2.165++ Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 Note: See Table 6 for critical values. 71 72 Table SB Phillips-Perron Unit Root Tests on Logged Consumer Price Indexes of Switzerland y - p + fi(t-n/2) + ay +u t t-l t e e * ‘ ‘ y - p + ay +u , y - ay + u t t-l t t t-l t 2(02):;-03-03-1 2(03) :5-0and3-1,Z(t;) :a-l - * * 2(ta) : a -1 , 2(01):u - 0 and a - l, Z(ta*) o* - l Lags in Newey and West(1987) L888 20%) 202) 2(t;) 2031) 20:3) 2(ta) lags - 0 1.734 26 810++ -0 562+ 40 165++ -1 833 8.604++ lags - 2 1.493 14 758++ -0 816+ 21 626++ -1 619 6.284++ lags - 4 1.458 10.802++ -0 995 15.074++ -1 500 5.222++ lags - 6 1.479 9.099++ -1 089 11.931++ -1 444 4.624++ lags - 8 1.499 8.232+ -1 129 10.09l++ -1 419 4.234++ lags - 10 1.522 7.725+ -1 168 8.796++ -1 398 3.936++ lags - 12 1.560 7.424+ -1 227 7.796++ -1 372 3.688++ lags - 14 1.605 7.273+ -1 288 7.022++ -1 347 3.483++ lags - 20 1.705 7.315+ -1 397 5.565+ -1.308 3.056++ lags - 25 1.758 7.6l9+ -1 447 4.834+ -1 291 2.813++ lags - 30 1.797 8.044+ -1 482 4.317 -1 281 2.625++ lags - 35 1.808 8.534++ -1 489 3.937 -1.278 2.475++ lags - 40 1.793 9.061++ -1 473 3.647 -1.282 2.351++ lags - 45 1.758 9.613++ -1 437 3.418 -1.291 2.246++ lags - 50 1.693 10.164++ .1 366 3.237 -1.312 2.157++ lags - 55 1.627 10.728++ -1 281 3.089 -1.337 2.079++ Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 Note: See Table 6 for critical values. 73 Table 5C Phillips-Perron Unit Root Tests on Logged Consumer Price Indexes of the United States :7 - 9 + awn/2) + ay +9 t t-l t * * e ‘ y - p + ay +u , y - ay + u t t-l t t t-l t 2002):,‘1-03-03-1 2(03):3-0and;-1,2(c;) :a-l - * * 2(t3) : a -1 , 2(01):p - 0 and a - 1, Z(ta*) 0* - 1 Lags in Newey and West(1987) Lags 2(03) 2(02) 2(t3) 2(01) Z(t:) 2(ta) lags - 0 26.016++178.503++ 0 663++ 258.906++ ~6.768** 18.125++ lags - 2 11.336++ 64.957++ 0 179++ 94.419++ -4.554** 10.916++ lags - 4 7.783++ 40.790++ -0.020++ 59.096++ -3.817** 8.6l3++ lags - 6 6.17S++ 30.304++ -0.140++ 43 556++ -3.424* 7.374++ lags - 8 5.212+ 24.442++ -0.232++ 34.705++ -3.158* 6.565++ lags - 10 4.530 20.701++ -0.319++ 28.921++ -2.949* 5.976++ lags - 12 4.040 18.144++ -0.398+ 24.851++ -2.783 5.524++ lags - 14 3.692 16.323++ -0.465+ 21 850++ -2.654 5.164++ lags - 20 3.116 l3.205++ -0.603+ 16 274++ -2.403 4.417++ lags - 25 2.890 11.941++ -0.669+ 13.596++ -2.278 4.007++ lags - 30 2.768 11.255++ -0.700+ 11.782++ -2.195 3.701++ lags - 35 2.701 10.898++ ~0.703+ 10.474++ «2.141 3.462++ lags - 40 2.662 10.746++ -0.687+ 9.481++ -2.102 3.268++ lags - 45 2.641 10.736++ -0.649+ 8.706++ -2.077 3.106++ lags - 50 2.628 10.806++ -0.596+ 8.084++ -2.061 2.969++ lags - 55 2.620 10.949++ -0.533+ 7.572++ -2.049 2.850++ lags - 60 2.615 11.139++ -0.459+ 7.146++ -2.042 2.746++ Key: * indicates significance at the .05 percentile and ** at .01 ‘+ indicates significance at the .95 percentile and ++ at .99 Note See Table 6 for critical values. 74 Table 5D Phillips-Perron Unit Root Tests on Logged Consumer Price Indexes of Italy y - p + 5(t-n/2) + ay +u t t-l t * s e ‘ ‘ y - u + ay +u , y - ay + u t t-l t t t-l 1: 2(02) : 3 - 0 3 - 0 a -1 2(03) : B - 0 and 3 -1 . 2(c;) : a - 1 a * * 2(ta) : a -1 , 2(01):u - 0 and a - 1, Z(ta*) : a* - l Lags in Newey and West(1987) lags - 0 44.544++285.750++ 2.463++ 389.278++ -8.247** 19.287++ lags - 2 20.149++103.165++ 1.593++ 140.986++ -5.665** 11.575++ lags - 4 14.146++ 64.535++ 1.319++ 87.946++ -4.812** 9.117++ lags - 6 11.055++ 47.534++ 1.136++ 64.248++ -4.301** 7.772++ lags - 8 9.147++ 38.091++ 0.987++ 50.807++ -3.946** 6.892++ lags - 10 7.938++ 32.219++ 0.885++ 42 223++ -3.699** 6.265++ lags - l2 7.046++ 28.234++ 0.790++ 36.204++ -3.502** 5.785++ lags - 14 6.383++ 25.404++ 0.712++ 31.759++ -3.346** 5.403++ lags - 20 5.210+ 20.584++ 0.595++ 23 443++ -3.034* 4.603++ lags - 25 4.636 18.655++ 0.541++ 19 368++ -2.856 4.152++ lags - 30 4.240 17 663++ 0.482++ 16 581++ -2.717 3.813++ lags - 35 3.981 17 223++ 0.440++ 14.573++ -2.613 3.547++ lags - 40 3.802 17 126++ 0.395++ 13 062++ -2.531 3.331++ lags - 45 3.688 17 253++ 0.378++ 11.893++ -2.471 3.153++ lags - 50 3.610 17 530++ 0.376++ 10.960++ o2.426 3.003++ lags - 55 3.559 17 910++ 0.400++ 10.202++ -2.393 2.874++ lags - 60 3.526 18 363++ 0.452++ 9 573++ -2.370 2.762++ Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 See Table 6 for critical values. 75 Table 6 Phillips-Perron Unit Root Tests on Logged Consumer Price Indexes y - u + 5(t-n/2) + or +9 t: t-l t e * * ‘ ‘ y - p + ay +u , y - ay + u t t-l I: t t-l 1: 2(02) : 3 - o z - 0 3 -1 2(03) : 3 - 0 and 3 -1 , 2(c;) : a - 1 - * * 2(c3) : a -1 , 2(01):p - 0 and m - 1, 2(ca*) : o* - 1 Lags are in Newey and West(1987) CPI 2(03) 2(02) Z(t;) 2(01) 2(tz) Trupsggion Germany 2.898 7.285+ -0.214++ 6.336+ -2.269 35 United States 2.641 10.730++ -0.649+ 8.706++ -2.077 45 Netherlands 5.162+ 6.979+ -1.100 6.426+ -3.011* 40 Japan 7.585++ 7.177+ -3.505* 6.598++ -3.449** 40 France 1.693 10.165++ 1.366++ 3.237 -1.312 50 Italy 3.802 17.126++ 0.395++ 13.062++ -2.531 40 Denmark 3.192 9.556++ 0.163++ 7.928++ -2.481 25 Swiss 1.705 7.315+ -1.397 5.565+ -1.308 20 Canada 2 10.452++ -2.256 40 .997 14.744++ -0.294++ Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 Note: Under the null hypothesis the 950 and 99‘ critical values of Z(t;), 2(03) and 2(02) are -.94 and -.33,4.68 and 6.09,and 6.25 and 8.27 respectively and for 2(t3) are -3.96 and -3.441 at 18 and 5§.A1so at the 958 and 990 level the critical values of 2(t3.),2(t:) and Z(§l)are 1.28 and 2.00,-0.07 and 0.6,and 4.59 and 5.43.90: 2(c3 ) and 2(62) -2.58 and -1.95,-3.43 and -2.86 at 18 and 58 respectively. [ See Phillips and Perron(1986)] 76 Table 7 Second Unit Root Tests on Logged Consumer Price Indexes A. Phillips-Perron Test (Ayt- yt- yy_1) Ay - p + fi(t-n/2) + aAy + u t 't-l t * * * ‘ * Ay - p + aAy +u , Ay - aAy + u t t-l t t t-l c 2(02) 5 - 0 3 - 0 a -1 , 2(03) : 3 - 0 and a -1 , 2(:;) : a - 1 .. * * 2(c3) : a -1 , 2(01):p - 0 and a - 1, 2(ca*) : a* - 1 CPI 2(93) 2(92) 2(t3) 2(01) 2(t3) 2(ta) Trupsgsion Germany 287.16++ l95.41++ -9.39** 181.75++ -6.82** -2.86** 35 United States 65.32++ 43.73++ -5.28** 32.92++ -3.3l** -1.22 45 Netherlands 859.13++ 579.63++ ~17.44** 433.24++ ~10.79** -5.56** 40 Japan .1880.04++ 1224.78++ -26.94** 1097.15++ -19.02**-12.60** 40 France 131.61++ 88.87++ -8.67** 37.98++ -3.58** -0.96 50 Italy 104.59++ 71.77++ -7.13** 43.ll++ -4.05** -l.05 40 Denmark 775.l9++ 520.6l++ ~17.41** 612.54++ -l4.8l** -5.l9** 25 Swiss 446.05++ 298.91++ ~12.75** 427.44++ ~12.48** -7.09** 20 Canada 621.80++ 421.79++ -15.79** 345.24++ -10.l7** -l.9l 40 Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 Note: See Table 6 for critical values. B. Dickey-Pantula tests for two unit roots in logged CPI Country . Germany USA Netherlands Japan France Test Results 2 - -l.l9 ‘zp- -2.11 I - -1.543 1- -2.09 2' - -.77 which imply 1(2) 1(2) 1(2) 1(1) I(2) Country Italy Denmark Switzerland Canada Test Results Z. - -.12 Ip- -3.81 '(p- -3.88 (y- -2.0 which imply 1(2) 1(1) 1(1) 1(2) Note: see Dickey and Pantula(l987) for test statistics. 77 Table 8 millipe-Pei'rm Tests for an Unit knot :11 Logged Relative Price Ratios Lags in Newey and West(1987) are in parenthesis. Pfl354v° 2(03) 2(02) 2(t3) 2(01) Z(t:) Z(t;) 1.Against USA CPI Germany (10) 1.439 7.475+ -1.175 9.726++ -1.438 -1.924 (40) 1.716 6.131 -1.495 3.853 -l.243 -1.677 Canada (10) 1.721 2.239 -1.849 1.748 -0.840 0.696 (40) 1.517 2.463 -l.735 1.735 -0.790 0.443 Netherlands(10)2.632 4.110 -1.928 3.187 0.468 -0.159 (40) 3.217 4.323 -2.308 1.386 0.054 -0.652 Japan (10) 7.103++ 5.947 -3.242* 2.246 0.741 0.847 (40) 6.499++ 5.527 -3.279* 0.845 0.117 -0.040 Swiss (10) 2.865 6.452+ -0.774 9.305++ -2.378 -3.042** (40) 2.075 4.027 -l.017 3.841 -1.857 -2.298* Italy (10) 3.002 10.995++ -0.128++ 13.792++ -2.404 -0.489 (40) 2.481 11.682++ -0.448+ 5.030+ -1.971 -0.805 France (10) 1.105 3.809 -0.664+ 5.085+ -1.421 0.270 (40) 1.643 4.102 -1.447 2.465 -1.294 -0.303 Denmark (10) 2.348 3.792 -2.065 3.014 -1.018 0.160 (40) 1.770 4.435 -1.682 2.228 -l.002 -0.l63 2.Against German CPI Canada (10) 2.214 12.369++ -1.110 12.939++ -1.827 -0.512 (40) 2.149 17.054++ -1.028 4.482 -1.669 -O.835 Netherlands(10)9.579++ 7.212+ -3.266 10.683++ -4.340** -4.569** (40) 6.669++ 5.044 -3.067 6.598++ -3.524** -3.663** Japan (10) 12.508++ 8.824++ -4.356** 12.659++ -4.858** -4.298** (40) 9.09l++ 6.376+ -3.920* 9.46l++ ~4.277** -3.579** Swiss (10) 3.224 2.247 -0.682+ 1.779 -l.782 -1.643 (40) 2.450 1.681 -O.418+ 1.949 -l.903 -1.627 Italy (10) 5.342+ 20.389++ 0.530++ 26.584++ -3.184* -0.992 (40) 3.212 16.529++ 0.707++ 8.508++ -2.399 -l.094 France (10) 4.912+ 16.577++ 1.260++ 21.162++ -2.924* -0.823 (40) 2.385 15.545++ 0.496++ 6.689++ -2.068 -0.991 Denmark (10) 2.163 9.246++ -1.024 9.698++ -1.836 -0.972 (40) 2.152 12.697++ -0.550+' 3.627 -1.819 -1.094 Key: * indicates significance at the .05 percentile and ** at .01 + indicates significance at the .95 percentile and ++ at .99 Note : Truncation legs are 10 and 40. 10 was chosen since this is normal size in usual cases, and 40 was chosen since this lag was needed in our cases. The choice of lags did not affect our conclusion that these series are at least 1(1). 78 Table 9 Second Unit Root Tests on Logged Relative Price Ratios Phillips and Perron Test Dickey and Pantula gglgggve Z(t;) Z(t:) 2(ta) Test l.Against USA CPI Germany (10) -6.063** -5.837** -3.628** E - -2.92** (40) -6.705** -6.397** -4.234** Canada (10) -13.144** -13.258** -12.718** 2- -2.34** (40) -21.891** -22.131** -21.139** Netherlands(10) -10.715** -10.596** -9.327**‘ 'Z,- -1.83+ (40) -15.025** -14.746** -12.803** Japan (10) ~16.429** -15.644** -14.602**’ '2 - -1.90+ (40) -26.707** -25.491** -23.457** Swiss (10) -10.424** -9.171** -6.043** ‘Zh - -3.30** (40) -l4.752** -12.563** -7.671** Italy (10) -5.141** -4.624** -2.605** 'zfi - -2.98* (40) -5.451** -4.756** -2.287* France (10) -7.474** -7.219** -5.800** Z- -1.362 (40) -9.498** -9.025** -6.98l** Denmark (10) -12.585*# ~12.678** -12.211** lab - -4.918** (40) -20.331** ~20.556** -19.712** 2.Against German CPI . Canada (10) -15.487** -15.065** -6.750** .Zfl - -3.921** (40) -24.789** -23.886** -9.531** Netherlands(10) -ll.129** ~10.224** -9.888** ‘ZL- -2.46** (40) . -16.208** -14.796** -l4.246** Japan (10) -l4.494** -12.995** -12 755** 7b - ~3.104** (40) -23.028** -20.779** -20.176** Swiss (10) -15.520** -14.718** -14.918** '2.- -2.42** (40) -24.334** -22.902** -23 228** - Italy (10) -7.404** -5.847** -2.l43* 'Zfi - -3.17** (40) -9.373** -6.793** -l.802 France (10) -7.474** -7.219** -5.800** ‘Z.- -0.92 (40) -10.319** -7.910** -2.518* Denmark (10) -13.488** -13.306** -8.739** ‘2; - -5.99** (40) -22 098** -21 721** -13 359** Keyzl. In Phillips and Perron test statistics, * 2. 3. the .05 percentile and ** at the .01 . the .95 percentile and ++ at the .99 + indicates significance at indicates significance at In Dickey and Pantula test statistics, * indicates significance at the 0.025, * at the 0.05 and + at the 0.10. Lags in Newey and West(1987) are in parentheses for Phillips and Perron tests. 79 Table 10 Vector Autoregressive Estimates for the USA-Germany 11 Axt - -P1 Axt_1 + P2 Axt-Z'th-3 + const +1§ifiiQit + ct ( 539) (:79) 2:873) (??3) (:093* (:81) P1 ' (:888) (:53)* F2 ' (:888) (308) ' ' (:88i)* (2881) Note: xt - ( logged Mark/US, logged CPIUSA-Germany )'. t-statistics are in the parentheses; Key: ** at the .005%, and * at the .05 %. Estimated Correlations and Variances of Regression Residuals .583828-03 P e -.6943lE-01 .30837E-02 Box-Pierce Q-statistics A(U$/Mark) Q(39) 48.32 37.71 Note: x§o(p - 0.01) - 50.892. 80 Table 11 Test Statistics for the Hypothesis for Various Values of Cointegration Vectors between Log of Exchange Rates and Log of Relative Prices 0 AGAINST THE U.S.A.(l974,3 ~ 1988,11) # of -2 ln(Q) cointe- gration W.Germany Switzerland Japan Canada vectors r = 0 13.003* 22.135** 13.062* 6.013 r S 1 0.325 2.056 0.725 1.002 Key: ** denotes significance at the 97.5% quantile and * at the 95% quantile. o AGAINST THE GERMANY(1974,3 ~ 1988,11) # of -2 ln(Q) cointe- gration Switzerland Japan Denmark vectors r = 0 12.152* 25.706** 14.277** r 5 1 1.489 1.315 3.088 81 Tahhelz 'Ihe Eigenvalues: and Eigenvectofs V and Estimated Alphas Eigenvalues 1 (0.0018, 0.0695) (0.0116, 0.108) (0.0042, 0.0688) A Eigenvectors V 3 -1.908 -6.363 3 2.949 -6.005) s 1.45 -7.14 rp{:-6.595 4.353 rp 3.332 9.068 rp -10.27 6.32 3 Alpha x 10 3 -0.914 -3.696 5 3.151 -1.396 s 1.66 -0.34 rp -0.053 0.722 rp 0 058 1.327 rp -0.02 1.70 Swiss--Germany Japan-Germany Denmark-Germany Eigenvalues 1 (0.0084, 0.0591) (0.0074, 0.129) (0.017, 0.0619) A Eigenvectors V 3 3.004 -13.107) s 5.236 -2.182] 3 [’6.856 -41.937] rp - rp -44.960 24.662 rp 10.755 21.557 0.994 37.831 Alpha x 103 3 -0.132 -3.105 8 1.841 0.960. 3 0.513 -1.602 rp -0.326 0.128 rp -0.033 2.462 rp 0.745 0.908 Note: s denotes logged spot exchange rates and rp denotes logged relative price indexes. 82 Table 13 Tests of PPP in a§t+ ¢(§ - 13*)t - ct, where e is stationary. 1'. Exchange Wald Tests Rates ¢ '¢/0 for 0 ' '¢ Mark/US -6.36** 4.35** 0.68 0.7436 (146 15) (5.72) Sfr /US -6.0** 9.06** 1.51 5.05 (88 21) (157.48) Yen/US -7.14** 6.32* 0.89 0.112 (310.4) (4.84) Sfr/Mark -13 1** 24.86+ 1.89 0.863 (209 26) (3.36) Yen/Mark -2.2* 21.5** 9.7 38.39** (4.54) (105.06) Kroner -41.94** 37.83** 0.9 5.67 /Mark (432.23) (16735.36) Note: 1. In the parentheses we have the Wald test result on the significance of each coefficient. ** indicates significance at xi(p-0.0l), * at xi(p-0.05), and + indicates significance at xi(p-0.l). 2. "k" denotes the lag term in the vector autoregressive regres- sion [see Eq.(342)]. 3. -a/¢ equals the coefficient of 1 in the equation of st- a + 7(p - p*)+ constant. 83 Table 14 Univariate Unit Root Tests to Deviations from PPP l. Dickey and Fuller(l981) F-Test : P Zt - 80 + filt + alt + ”812:1- Zt-j-l) + 6t 2. Phillips and Perron(1986,l988) Z - flo + 81(t-T/2) + ¢Zt-l + Ct t: 01 02 03 [ ADF 2(t5) Ho (51.0) (0 1) “1 (fl J31.a)-(0.0 1) H2 (J9 191.0) (.8 0 1) .......................................................... ,----------.------n Denmark (1) l7.8l** l6.29** 5 48+ 'er-3 20+ -3 07 -Germany(2) 4.07 4.22 4.92* Swiss (l) 7.91** ll.59** 9.49** ‘er-3 98** -3.17 -Germany (2) 3.55 4.58 5.68* Swiss (1) 5.09* 4.62+ 5.2 'Zp--l.74 -1.75 -USA (2) 1.74 1.19 1.57 Germany (1) 9.74** 7.51** 1.38 'Zp--l.50 -l.29 ~USA (2) 0.83 0.60 0.90 Japan (l) 8.41** 6.57** 1.33 .2p-'1'07 -l.27 -USA (2) 1.08 1.14 1.03 ........................................................... [.---------.------“ Note: l.The "(1)" indicates the likelihood ratio statistics under Dickey and Fuller(l981). The distributions are on page 1063 of that paper. ** indicates significance at the 99 % level, * at the 95 %, and + at the 90 % level. 2.The "(2)" indicates the Phillips and Perron test statistics of 2(01), 2(02) and 2(03) respectively. * indicates significance at the .95 quantile. The 2(t3 ) is under the null hypothesis of ¢ - l and all test statistics accept the null of a unit root. 3.The distribution of a is from Table 8.5.2 of Fuller(l976). + indicates significance at the .10 and ** at the .025. 84 LIST OF REFERENCES Adler, M. and B. Lehmann. "Deviations from Purchasing Power Parity in the Long Run", Inn Jgunnal of Finangg, 38, 5, pp. 1471-87. Baillie, R.T., "Commodity Prices and Aggregate Inflation: Would a Commodity Price Rule be Worthwhile7", W Eglicy anfgrenge Eaneg, Forthcoming, Fall 1989. Baillie, R.T. and D.D. Selover, "Cointegration and Models of Exchange Rate DeterminAtion". Iatsraati2ea1.2221n51_2£_fisrssastins. 1987. 3. PP- 43-51. Campbell, J.Y. and R.J. Shiller. "Interpreting Cointegrated Models", Jgugnal 9: Economic Dynamigg nnn Contzgl 12, 1988, Pp. 505-22. Corbae, D. and S. Ouliaris, "Cointegration and Tests of Purchasing Power Parity", I02 Revigw of Econonigg and Sgngisgigg, 1988, Pp. 508-11. Dickey, D.A. and W.A. Fuller, "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root", gnnnnmgnzinn, 49, 4, 1981, Pp. 1057-71. Dickey, D.A. and W.A. Fuller, "Distribution of the Estimates for Autoregressive Time Series with a Unit Root," gnurna], of the Amegican Statistical Assogiation 74, 366, 1979, Pp. 427-31. Dickey, D.A. and S.G. Pantula. "Determining the Order of Differencing in Autoregressive Processes," o o B o Sgntigticg, October 1987, Vol. 5, No. 4, Pp. 455-61. Dornbusch, R. "Exchange Rate Economics: Where Do We Stand?" annkings Pnnggs on Egnngmig Agtivity 1980, 1, Pp. 143-185. Dornbusch, R. Ennnnngg_3§;g_nnn_lnflnninn, The MIT Press, 1988. Edison, H.J. and E. Fisher. WA Long-Run View of the European Monetary System”, a ona cu on a e , No. 339, Federal Reserve System, January, 1989. Enders, W. "Arima and Cointegration Tests of PPP Under Fixed and Flexible Exchange Rate Regimes", Inn Bgview of Econonigs and Statistigs, Pp. 504-8, August, 1988. Engle, R.E. and C.W.J. Granger. "Cointegration and Error Correction: Representation, Estimation, and Testing", Egonometgicn, 55,2, 1987, Pp. 251-76. Engle, R.E. and B.S. Yoo. "Forecasting and Testing in Co-integrated Systems", gnuznal of Ecnnongtging, 35, 1987, Pp. 143-159. 85 Frankel, J.A. "The Collapse of Purchasing Power Parities During the 1970's", Eugnnean Egonomig Revigw 16, 1981, Pp. 145-65. Frankel, J.A. "Purchasing Power Parityw Doctrinal Perspective and Evidence from the 1920's". WWW. 8. 2, 1978, Pp. 169-91. Fuller, W.A. Intzoduction to §§atisgig§1 Ting Segigg, Wiley, New York, 1976. Hakkio, C.S. "A Reexamination of Purchasing Power Parity: A Multi-Country and Multi-Period Study". Wm 17. 1984, Pp. 265-277. Hasza, D.P. and W.A. Fuller. "Estimation.for.Autoregressive Processes with Unit Roots," The Annnlg of Stagisgigs, 1979, Vol 7, No. 5, Pp. 1106- 1120. Johansen, S. "Statistical Analysis of Cointegration Vectors", lnn;nnl_nfi WW 12. June-Sept- 1988. PP- 231-54- Johansen, S. and K. Juselius. ”Hypothesis Testing for Cointegration Vectors with an Application to the Demand for Money in Denmark and Finland", e ri a m a , University of Copenhagen, March, 1988. Katseli-Papaefstratiou, Louka T. "The Reemergence of the Purchasing Power Parity Doctrine in the 19703", W Eggnomigs, 13, Princeton University, Dec. 1979. Krugman, P.R. "Purchasing Power Parity and Exchange Rates: Another Look at the Evidence". W 8. 1978. PP- 397-407. Newey, W.N. and K.D. West "A Simple, Positive Semidefinite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Egnnnngnzing 55, 1987, Pp. 277-301. Officer. L. W. JAI Press. 1984. Phillips, P.C.B. "Understanding Spurious Regressions in Economics", W 33. 1986. PP-311-340- Phillips, P.C.B. and P. Perron. "Testing For a Unit Root in Time Series Regression", Binnggxika 75, 2, 1988, Pp. 335-46. Phillips, P.C.B. and P. Perron. "Testing for a Unit Root in Time Series Regression," Yale University, w ou da D scu a N2, 281, 1986. 86 Said, S.E. and D.A. Dickey "Testing for Unit Roots in Autoregressive- Moving Average Models of Unknown Order", Biometgika 71, 3, 1984, Pp. 599-607. Said, S.E. and D. A. Dickey. "Hypothesis testing in ARIMA.(p,1,q) Models", Jougnal of the Amegican fitngisgigal Assngintion 80, 1985, Pp. 369- 74. Schmidt, P. "Dickey-Fuller Tests With Drift", Michigan State University Eggnometgics and Economig Ingnzy Enng; No. 8717, June 1988. Schwert, G.W. "Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data", Jougnal of'Mongtnny Eggnomigg 20, 1987, Pp. 73- 103. Taylor, M.P. and M.J. Artis. "What has the European Monetary System Achieved?" Bank of Englann gesggggn Pang: Hg, 31, March, 1988.3 87 III. MULTIVARIATE COINTEGRATION TESTS FOR A SET OF FOREIGN EXCHANGE RATES AND A COMPARATIVE STUDY OF THE FORECASTING ACCURACY OF THE RANDOM-WALK AND THE ERROR-CORRECTION MODELS 88 1. INTRODUCTION This study begins with the work of Baillie and Bollerslev (1989) on the common stochastic trends in a system of exchange rates. In their paper, multivariate tests for unit roots showed the existence of one long- run relationship between a set of seven daily exchange rate series during l980:3:l through l985:l:28. This result indicates a.perceptible deviation from weak-form efficiency for each of the exchange rates because in the first order error-correction model, if two or more prices of different currencies are cointegrated, part of the changes will usually be predictable. Several questions may be raised by this result, but the discussion centers around two related questions. First, one cointegrating factor between seven exchange rates arises because any two, any three, or any four, etc. are cointegrated. Are any rates redundant to this relationship, or is there one. or more driving currency? Second, although each exchange rate series has a univariate representation as a martingale, which is similar to a 'random.‘walk [see Section. 3 for discussion], it is also true that a vector of the first differenced exchange rates should have a lagged error-correction term applied to it, since there is one cointegrating vector between the rates. This result implies that the daily exchange rate in each of the rates is partly determined by an I(0) equilibrium error and will in general be predictable. One interesting question is whether this representation can be used in forecasting, and if it outperforms the random walk. This question is related to the interpretation of the error-correction model; 89 the error-correction models for cointegrated economic variables are commonly interpreted by Engle and Granger (1987) as reflecting partial adjustment of one variable to another. The motivation for cointegration is that an equilibrium error causes changes in the variables of the model. However, Campbell and Shiller (1988) have an alternative approach, maintaining that the error-correction model may also arise because one variable forecasts another. Campbell and Shiller emphasize the possibility of these changes and that cointegration can arise even in a well-organized market with no adjustment costs. This study follows the general interpretation of error-correction models by Engle and Granger (1987) and analyzes the two questions. This paper is divided into 5 sections. Section 2 tests for cointegrated vectors on a set of exchange rates, by pair, threesome, and so forth, to find the redundant rates and driving currencies. After examining the random-walk representation of daily exchange rate series in Section 3, out-of-sample forecasts are performed in Section.4 to determine the accuracy of forecasting in the error-correction model compared to the random-walk model. Conclusions follow. I took the same daily spot exchange rate data which were used by Baillie and Bollerslev (1989), from the New York Foreign Exchange Market between March 1, 1980 and January 28, 1985, which constitutes a total of 1,245 observations. The data were originally provided by Data Resources Incorporated (DRI) and are opening bid prices for the UK pound, West German mark, French franc, Italian lira, Swiss franc, Japanese yen, and Canadian dollar vis-a-vis the US dollar. 90 2. MULTIVARIATE TESTS FOR UNIT ROOTS IN A SET OF EXCHANGE RATES With Baillie and Bollerslev's finding of one cointegrating vector, we can estimate the parameters in this long-run relationship between the set of seven exchange rates, and secondly, find redundant currencies in the set by Johansen's technique, which was used by the authors. By redundant currencies we mean currencies that have zero coefficients in the cointegrating vector. Before proceeding, we should briefly explain the Johansen test: Hypothesis testing of a cointegrating vector is originally done with regression estimates; however, the multivariate tests developed by Johansen (1988) are implemented in testing for cointegrating vectors between exchange rate series simultaneously by specifying a model with a vector autoregressive process (VAR). This approach gives more efficient estimates than the conventional regression estimates, since it not only takes into account the error structure of the underlying process, which the conventional regression estimates do not, but it also gives maximum likelihood estimates. The VAR form of our model looks like Ast - I‘ Asvl - as”; + 6,. (l) where st is a vector of logged daily exchange rates of 7 currencies; UK pound, German mark, Japanese yen, Canadian dollar, French franc, Italian lira, and Swiss franc vis-a-vis the US dollar. £1 --- e, are iid N(0,A). This model is the first differenced form of st - «15,...1 + «2st,; + e... (t-1,---,T) Using A - l-L, where L is the lag operator, we have the model (1), when P - -l + «1 and x - l- a} - «2. The model (1) is expressed as a traditional first differenced VAR model except for the term "Sc-2o If the rank (1r) is not full, then the 91 coefficient matrix (a) may convey information about the long-run.structure of the chosen data. Johansen’s hypothesis testing of cointegration vectors formulates the hypothesis of reduced rank (-r) in 1r, or the hypothesis which implies that there are matrices a and 6 of order v(- # of variables) x r such that s - afi', where fi'stFV’I(0). The properties of Johansen's test are as follows: 1. The maximum likelihood estimator of the space spanned by 8 is the space spanned by r canonical variates, corresponding to the r largest squared canonical correlations between the residuals of 3%-; and Ass, corrected for the effect of the lagged differences of the 3 process. 2. The likelihood ratio test statistic for the hypothesis that there are at most r cointegration vectors is V -21nQ - - T 2 ln(l-Ai) i-r+1 where Arfln ----, Av are the v-r smallest squared canonical correlations. 3. Under the hypothesis that there are r cointegrating vectors, the estimate of cointegration space as well as s and.A are consistent, and the likelihood ratio test statistic of this hypothesis is asymptotically distributed as trt 1'3888' [ f3 B(u)B(u)'du]-1 f3 dBB') where B is a v-r dimensional Brownian motion with covariance matrix I. The proofs and derivation of each parameter are presented in Johansen (1988). Given such properties, in order to find currencies redundant to the existence of one long-run relationship between a set of seven daily 92 exchange rates, I first performed Wald tests to examine the significance of each coefficient in the cointegrating vector among seven currencies with the hypothesis k'fi - ( 0, 0, 1, 0, ---, 0) 52 - 0 I”) where )6, is the coefficient to be tested for significance. The test statistic is Imk'fi / 1 ( ’3," - 1)( 1:494: ))1/2 with xf. Here 3, is the maximal eigenvalue, 5 the corresponding eigenvector, and the remaining eigenvectors form 9 [ see Johansen and Juselius (1988) Corollary 3.17 ]. As can be seen in Table 2, most elements of cointegrating vector are significant at the 99% level, except those of the UK pound and Japanese yen against the US dollar. With this result, we can expect that the Pound and the Yen may be redundant currencies to the existence of a long-run relationship. This turns out to be true, and without the U$/pound or U$/yenj a set of six currencies still have at least one long-run relationship [ Table l ]. Also, without both the Pound and Yen, one long-run relationship still exists with the remaining five currencies (German mark, French franc, Italian lira, Canadian dollar, and Swiss franc against the US dollar) [see Table 1 column 4]. This result implies that, during the sample period, both the Japanese yen and the UK pound played no important role for the existence of a long-run relationship between industrial-country currencies. Even if each of the five remaining exchange rates had a significant coefficient in the long-run relation, I continued to test cointegrating 93 vectors by reducing specific currencies one by one, so as to examine whether there are other redundant currencies, or to find some driving currencies [see Table 1]. First, without the Swiss franc/US rate, I found one cointegrating vector in the remaining six exchange rates. Second, I tested whether the Canadian dollar is redundant by testing cointegrating vectors without the Pound, Yen, Swiss franc and Canadian dollar. The remaining currencies, Mark, Lira and French franc, showed one cointegrating vector at the 95% level. Further reduction of a specific exchange rate can not reject the null hypothesis of no cointrgrating vector [see Table 1, Part 2]. We can conclusively say that the redundant currencies in the long-run relations between the seven currencies are the UK pound, Japanese yen, Swiss franc, and Canadian dollar. The remaining currencies, Mark, Lira and French franc, are major European Monetary System (EMS) currencies 1 and contribute to a long-run relation during our sample period. This result is consistent with the general view that the EMS has been successful in contributing to exchange rate stability among participating countries. However, this relative stability of internal EMS currencies is coincident with a lack of external tension in the system due to a strong and rising US dollar during the sample period. Here we can raise a question whether there are any causality relations between EMS stability and the US dollar strength. During the sample period, the strong and rising dollar was influenced by the monetary policy in the United States that led to high nominal and real US interest rates, both 1The EMS currencies include the German mark, French franc, Italian lira, Belgian franc, Netherlands guilder, Irish pound, and Danish krone. Among them the Mark acts as a leading currency and the French franc and Lira are from major industrial countries. ' 94 in absolute terms and relative to other countries. The value of the European Currency Unit (ECU) in terms of dollars, which had been as high as US 1.44 at the end of 1979, had fallen to a little less than a dollar by the turn of 1982, and was less than 0.7 at the beginning of 1985 [see Charts]. Even if the US dollar gradually appreciated relative to European currencies during our sample period, giving Germany a favorable current account and making a continuous difference in the participating countries' external positions, our cointegration test shows that the EMS experienced relative internal stability. This may be explained as follows: first, the EMS-participating members made more cooperative efforts for exchange rate stability between them; the EMS had five realignments from March 23, 1981 to July 22, 1985. Secondly, a more reasonable explanation is that the Mark, the leading EMS currency, did not come under upward pressure within the EMS, largely because of strong capital flows to the United States due to the strong US dollar. With cointegrating vector tests without the Mark, we accept the null hypothesis of no cointegration vector with six remaining currencies. This may imply that the Mark played a key role as a driving currency, as did three other EMS currencies together, to have a long-run relationship during our sample period [see Table 1, Part 2]. Since we found redundant currencies in the long-run relationships, we explored short-run movements of exchange rates and the comparison the forecasting accuracy of the random-walk and the error- correction models from the set of remaining currencies, Mark, Lira and French franc, from that point on. 95 In the VAR model, the vector of changes in the three exchange rates at time t are expressed with lagged changes in the exchange rates and error-correction terms as well as a constant term. Ast - constant + I‘ Ast-1 - «3..-; + at where Ast is a vector of changes of logged exchange rates of the Mark, French franc, and Lira against US dollar, and 1‘ and 1r are matrices of order 3x3, respectively. The estimated error-correction model is reported in Table 4; with one lagged first differenced term, the residuals for the exchange rate data clearly passed the test for no autocorrelation. Based on the estimated coefficient matrix of %, which conveys information about the long-run relationship between a set of three exchange rates, we can explore short-run movements between these exchange rates with Johansen's method. Maximum likelihood estimates of a and 6 in s - afi' are derived, and Table 3 reports the estimates of cointegrating vector 6 as the third column in V'and a as the third column in the matrix of Alphas. Tests for the significance of each element of the cointegrating vector are reported, and the results show that all elements are significant at the 99% level by Wald tests. Here we can raise a question: Each exchange rate series is said to have a univariate representation of being a martingale. But we found that a vector of the first differenced exchange rates should have a lagged error-correction term applied to it, since there is one cointegrating vector between them. Can this representation be used in forecasting? Or, does it outperform the random-walk forecasts? In Section 3, the univariate representation of a random walk is examined and we will compare the forecasting accuracy of an.error-correctioanodel with that of a random-walk model in Section 4. 96 3. ARIMA MODELS OF EXCHANGE RATE SERIES Univariate autoregressive moving average models for the endogenous variables of a dynamic simultaneous equations system can be interpreted as a form of solution to the system [see Zellner and Palm (1974) and Wallis (1977)]. Under this methodology, if the log of bilateral exchange rate is generally approximated by a random-walk model, then the stochastic processes generating the exogenous variables should also be random-walk models. For example, consider the following monetary model ( Baillie and Selover (1987)): St ' 31(‘1: ‘ mitt) + 320%. " )'.t) + 33(rt ' 1"'t.) + 343t(pt+1'p.t+1) + ‘t- where st is the logarithm of the nominal exchange rates, m,y and r represent the logarithms of domestic money supply, real output and short term interest rates, Egpud is the expected domestic rate of inflation; asterisks denote foreign quantities and ‘t is a stochastic disturbance term. If the exchange rate is a random-walk model, then mt - m'h, yt - y't, rt - r’t, etc should also be random-walk models. In this section we will examine the ARIMA model of exchange rate series to see its implication of random-walk models. Our empirical analysis begins with fitting univariate ARIMA models to the individual exchange rate series. When. we plot the sample autocorrelation functions (ACF) for a sample of 1245 observations, it can be seen from Table 5 that all of the autocorrelations of the seven exchange rate series lie outside of the bounds i 1.96 rf5, which implies significance different from zero at the 5% level. The partial autocorrelation functions (PACF) strongly suggest that the appropriate 97 models for this data are AR(l) processes. After first differencing, the autocorrelations of the five currencies excluding the Canadian dollar and Italian lira, lie between the bounds i 1.96 n", and we can not reject the hypothesis that the first differenced one is a white process i.e., (0,1,0). In the case of the Canadian dollar against the US dollar, however, inspection of the graph of ACF and PACF of the first differenced suggests that the appropriate model for this series may be an ARIMA (p,l,0) process [see Table S-B]. I estimated AR(p) models for p - 1,2,. .,12 and checked the significance of each AR coefficient; none of the coefficients have significant values at any reasonable levels. From the PACF graph, we may spot an autoregressive seasonal at lag nine; however, it is not significant at the 95% level. The implication of ARIMA (p,l,0) is applied to the Italian lira against the US dollar, also [see Table 5- C]. As can be seen from the PACF, the log of first differenced Lira may have significant autoregressive seasonals at lag 13 and 23. However, they turn out to be insignificant in each coefficient at the 95% level. Overall, a random-walk model appears to describe the stochastic process of each daily spot exchange rate series adequately as Ast - ct. This result imposes strong n_n;12;1 restrictions in any exchange rate model. The error-correction model, where the vector of first differenced exchange rates have a lagged error-correction term is: As,’ - As...1 - «3,,1 + et, in the VAR form. The result for a specific rate 1 from the VAR showed as fl U AS“. - 2‘03 ASJt-l ' 5.§"333t-2 '0' fit, With 1 - 1,2,. .,N. 3 If the matrix of 1r has significant elements [see Table 4], does this error-correction model outperform the.random-wa1k model? We will examine 98 the forecasting accuracy of the error-correction model relative to the random-walk model in the next section. 4. A COMPARATIVE STUDY OF THE FORECASTING ACCURACY OF THE ERROR- CORRECTION AND THE RANDOM-WALK MODELS This section compares the out-of-sample forecasting accuracy of the error-correction model (hereafter, ECM) and the random-walk model for the French franc/US rates. The selected currency is an EMS currency for two reasons: first, after eliminating the redundant currencies from a set of seven currencies, the three currencies (Lira, Mark and French franc) 21; n_21§ the US dollar show one long-run relationship in the cointegrating vector tests and three currencies make the computing work easier; and, secondly, among three currencies, the Franc/US rate has significant coefficients in the error-correction term compared to the other two currencies [see Table 4]. In our experiment, five models are set to compare their forecasting accuracies; in addition to the random-walk and the error-correction models, two modified versions of error-correction models and an unrestricted VAR model are included. The specific form of each model is given below. The parameters of each model are estimated on the basis of the most up-to-date information available at the time of a given forecast. This is accomplished by using rolling regressions to re- estimate the parameters of each model every forecast period. First, the random-walk model uses the current spot rate as a predictor of all future spot rates. I estimated a random-walk model with a drift term, which is very significant. Ast - constant + ‘t t (l) 99 The second model, the ECM, is: As“ - a +:§¢,As,,-,- was”) + a, 1 - 1,2,. .,N. (2) where fi'st-2, with )6 and spa as vectors, is the deviation from the long- run relation which we obtained in Section 2. Since the coefficient of Ask; for Lira rates is insignificant in our study [see Table 4] , I modify the ECM to have the third model, ECM-l, where insignificant coefficients are excluded: Asu- a: + §2¢3Asjt-1- 1903's,”) + 6,. i - 1,2,. .,N. (3) Also, I estimate the ECM with an error-correction term in the RHS as the fourth model, ECM-2: Asu- a - ¢(fi'st-1) + at (4) Finally, I use the VAR without any restrictions as our fifth model; Ast - ¢Ast-1 - «3%-; + at (5), where st is a vector of logged exchange rates of the Franc/US, Mark/U$ and Lira/US. These five models are estimated by OLS over a daily data series starting in March 1, 1980 and extending through April 30, 1984. Data ranging over May 1, 1984 to January 28, 1985 have been retained for ex- post out-of-sample forecasting exercises. Forecasts are generated at horizons of one through thirty days. Out-of-sample accuracy for each model is measured by three statistics: mean error (ME); mean absolute error (MAE); and the principle criterion, root mean square error (RMSE) [see Meese and Rogoff (l983,a) for their definitions]. Table 6 lists the forecast errors for the Franc/US rates at specific horizons. Each parenthesis contains a rank for each model. The striking feature of Table 6-A is that the random-walk model doesn't achieve lower RMSE than our ECM; 100 Although the differences in the RMSEs are small, the ECM still outperforms the random-walk model. The modified versions of ECM, i.e., Eqs. (3) and (4), and the unrestricted VAR do not outperform the random-walk model or our ECM. Overall, with RMSEs, the random--walk forecasts are not more accurate than our ECM forecasts. The MAE, which is less sensitive to outlier observations, shows a slightly different pattern [see Table 6-B]; our ECM outperforms the random-walk model up to horizon 12, and from horizon 18, the random-walk forecasts outperform our ECM forecasts. However, the difference in forecasting errors of MAEs is very small compared to that of the RMSEs. In this case, also, the modified versions of ECM and the unrestricted.VAR do not outperform our model or the random- walk model. The mean errors of the various models are listed in Table 6-C. They are smaller relative to the corresponding MAE, indicating that the models do not systematically over- or under-predict. Overall, with our examination of forecasting accuracy, we may conclude that the random-walk forecast is less accurate than our error- correction model, in which a vector of first differenced exchange rates has a lagged error-correction term. Although the forecasting errors (especially, RMSEs and MAEs ) are not significantly different from each other (the difference between the error-correcton model and the random- walk model is 0.5 8, on average), an obvious conclusion is that the random-walk model can not outperform the error-correction model. 5. CONCLUSIONS This study found that the EMS currencies contributed to the stability of a set of seven currencies, and that the stabilities of EMS 101 currencies were coincident to the strong US dollar during our sample period, raising the question of whether we can find a causal relationship between them. We made a comparative study of the forecasting accuracy of the error-correction and the random-walk models. Our error-correction forecasts showed a little improvement in accuracy compared to the random- walk forecasts. This result reminds us of the Meese and Rogoff demonstration of the superiority of the random-walk model not only to asset market models but also to all economic time series models, generally; Meese and. Rogoff (l983,a,b) found that the VAR. models' forecasts did not improve on their structural models, both being no better than a random-walk model. However, their VAR model with lagged explantory variables did not consider cointegration, i.e., whether the exchange rate and a given set of explanatory variables are cointegrated. Cointegration was rejected by Baillie and Selover (1987), for example. As shown by Engle and Granger (1987), cointegration is a necessary and sufficient condition for a vector of variables to bear an equilibrium relationship. Considering the cointegration, I applied the VAR.methodology to our model while introducing the error-correction term, which proves to be significantly different from zero, as an independant variable. The result was that the random-walk model did not outperform the forecastinng performance of the VAR model with an error-correction term applied to it. With root mean square error statistics, the random-walk. model was marginally less than the error-correction model. This study did not compare the forecasting accuracy of our error-correction model with that of asset-market models. In Woo's paper (1985), a monetary model with lagged dependent variables outperforms the random-walk in forecasting the 102 Mark/US rates at one- to twelve-month horizons. However, before comparing the forecasting accuracies, it is useful to check whether the exchange rate and. a given set of' explanatory ‘variables are cointegrated. as mentioned above. Then it is necessary to compare the stochastic processes of structural and time-series exchange-rate models according to the methodology developed by Zellner and Palm (1974) [See Ahking and Miller (1987) for its application to the exchange rate series. They rejected the univariate representation of the asset-market models.]. 103 Table 1 Dates: 1980:3:1 through 1985:1z28 WratesmSagainstdcnmsticamency): UKpound, Germanmark( BM ), yen, Canadian dollar, French franc( Ffr ), Italian Lira and Japanese Swiss franc( Sfr) Multivariate Tests for Cointegration Vectors in the Logarittms of Daily Spot Mange Ratesa 7 rates 6 rates 6 rates 5 rates 6 rates Quantiles w/o Pound w/o Yen w/o Yen w/o Sfr r r r r & Pound r 95% 9996 6 0.97 5 1.57 5 1.57 4 0.08 5 1.52 4.2 5.2 5 5.41 4 5.76 4 6.32 3 4.17 4 5.87 12.0 15.6 4 11.31 3 12.57 3 14.23 2 11.64 3 11.92 23.8 28.5 3 23.14 2 27.71 2 35.29 1 30.81 2 30.45 38.6 44.5 2 46.21 1 47.51 1 60.91* 0 61.38* 1 53.92 57.2 63.9 1 77.36 0 79.61* 0 93.27** 0 83.37* 78.1 86.6 0 124.64** 103.1 112.7 4rates 6rates Brates 2rates Zrates 2rates w/o Yen, w/o m w/ mash w/ m w/ m w/ lira r Pounds.Sfr r r r &Ffr r &L1ra° r&Ffr 3 1.39 5 0.07 2 1.25 l 0.04 1 0.12 1 0.10 2 6.55 4 4.66 l 8.32 O 3.91 0 7.01 O 9.75 1 17.41 3 11.25 0 27.23* 0 39.87* 2 25.12 1 49.09 0 77.76 a. Tests for r cointegration vectors in a VAR(1) . This is a likelihood ratio test, -21n(Qr) , for there being at most r cointegrating vectors with r=0,1,2,-—-,(p—1),M1erepdenotesthemnnberof variables [ See Johansen ( 1988) for the details of the tests. ]. b. DBincurTabledenotesthecurrencies of EM, FfrandLira. * Denotes significance at the 95 8 percentile. ** Denotes significance at the 99 % percentile. 104 Table 2 Eigenvalues, Eigenvectors and Estimated Alpha Coefficients with 7 Daily Exchange Ratesa EIGENVALUES: (0.00078, 0.00357, 0.00473, 0.00947, 0.01839, 0.02474, 0.03733) EIGENVECTORS(V): 14.9559 3.8858 3.18280 -15.6548 11.2340 2.0151 -2.2797 -4.7153 ~12.136 13.5431 -4l.8736 -3.0814 35.958 -87.952 -7.9889 -3.4311 -15.809 -5.43268 2.96194 2.1741 2.3036 18.2713 -21.944 ~21.109 -11.5129 -52.381 -23.485 -45.857 ~9.9867 13.769 -0.2498 10.64385 11.1770 -2.5308 -70.601 -5.2988 ~4.9388 ~3.2733 13.86174 -19.861 -16.256 111.360 11.6183 3.2015 -6.6825 34.53311 12.8260 -0 3785 51.6742 ESTIMATE OF ALPHA *1000: F 0.05409 0.01680 -0.05443 -0.46629 0.44189 -0.23807 -0.31666 0.01663 0.07516 -O.20700 -0.43194 0.41452 0.45515 -0.14963 ~0.02860 -0.01200 —0.36097 -0.12794 0.51288 -0.08337 -0.05450 0.03004 ~0.04313 -0.92046 -0.04543 -0.02583 0.08178 -0 28720 0.01949 0.20188 o0.178l4 -0.29764 0.51495 0.39532 -0.52211 0.04275 0.18192 -0.23069 -0.33l69 0.24361 0.23297 -0.07205 0 0.07593 0.02410 -0.20922 -0.30047 0.65882 0.47703 0.01148 b TEST FOR SIGNIFICANCE OF ESTIMATED BETA : BETA' - (-2.28, -37.95**, 2.30, -4S.86**, -70.62**, 111.3544, Sl.675** ) (0.40) (109.37) (0.69) (21.62) (449.87) (653.90) (83.19) a. The exchange rates (U$ against domestic currency) are as follows in order: UK pound; German mark; Japanese yen; Canadian dollar; French franc; Italian lira; and Swiss franc. b. The parentheses have the results of Wald tests with xi(p= 99.0%) - 6.63. ** Denotes significance at the 99 % level. 105 Table 3 Eigenvalues, Eigenvectors and Estimated Alpha Coefficients With 3 Daily Exchange Rates (D-mark, French franc and Italian lira) EIGENVALUES: (0.00100627, 0.00567086, EIGENVECTORS(;): -27.36069 -6.7l682 14.72755 ESTIMATE OF ALPHA *1000: -0.089277 -0.088014 -0.086243 TEST FOR SIGNIFICANCE OF ESTIMATED BETA: BETA' - ( ~52.69**, ~115.37**, (10.15) (187.96) 0.015118) 66.85266 -52.68962 -36.13997 -115.37290 -5.6497l 152.96214 0.079985 -0.055994 0.018223 -0.205984 -0.015824 -0.027654 152.96** ) (1794.21) Note: The pharentheses have the results of Wald tests with xi(p -99.0 % ) - 6 63. ** Denotes significance at the 99 % level. 106 Table 4 Vector Autoregressive Estimates with Lag 1 for 3 Daily Exchange Rates: D-mark, French franc and Italian lira 0 Ast - constant + PAst-l -«sc-2 + (t where at - ( D-mark(DM), French franc(FFR), Italian lira(LIRA) )' AD": (8:875) (8:07) (8:83) (8:89) ADMt-l AFFR: ' (8:073§* + (8:0§§* (8:38§** (8:88) AFFRc-l ALIRA: (8:83) (8:39)**(8:86§ (8:37§** ALIRAc-1 28:86) (8:81)* (8:8l§* FFRc-2 + ‘2.c \(8:887) (8:889) (8:881) LIRAt-Z ‘3,t * Denotes significance at the 9S 8 level, ** at the 97.5 4 level and *** at the 99.5 8 level. ((8:807) (8:89? 78:81) ”“62 ‘1,: o VARIANCE AND CORRELATION MATRIX ADM AFFR ALIRA ADM .48983E-04 .89 .91 AFFR .55737E-04 .89 ALIRA .39647E-04 o BOX-PIERCE Q-STATISTICS Q(105) ADM 105.245 AFFR 113.096 ALIRA 143.060 W no a 111111111111 31,1: ‘ 1111111111 1.; ,____ I We: 8 "I! II I! E E5~E AR Coefficients —.039 .011 016 .027 .034 .028 .011 -.014 .087 -.003 .036 ~.031 Ratio of AR Coefficients to (1. 96*etandarde u) .704 -.200 .298 .495 .620 .518 .197 -.262 1. 568 -.588 .649 -.573 3 HI? I 1 l E II HIT W 1L ” 1'” AR Coeftieienta . .053 -.000 .041.018 .016 .008-.004 .014 .038 .020 .013 -.032 - .070 .045 .040 - .034 .052 .017 .008 -.001 .015 -.031 .079 -.050 Ratio of AR Coetticienta to (1. 96*Itandard error) -.950 -.006 .736 .331 .297 .160 -.082 .264 .703 .367 .246 -.579 -1.27 .822 .734 -.622 -.949 .319 .141 -.024 .276 -.568 1.432 -.912 1‘ 1 1 108 Table 6 Initial Estimate Period: l980:3:3 ~ 1984:4z30 Forecasing Period: 1984 5:1 ~ 1984:12 7 A.Forecasting Percentage RMSE (Root Mean Square Error) ECM-l Horizon 3 6 9 12 15 18 21 24 27 30 Random Walk .7519(2) .7552(2) .7521(2) .7495(2) .7478(2) .7529(2) .7593(2) .7622(2) .7703(2) .7744(2) ECM .748l(l) .7510(1) .7484(1) .7458(1) .7439(1) .7496(1) .7554(1) .7583(1) .7666(1) .7703(1) .7625(5) .7650(5) .7623(5) .7592(5) .7577(5) .7631(5) .7695(5) .7724(5) .7807(5) .7850(5) ECM-2 .7543(3) .7572(3) .7544(3) .7520(3) .7503(3) .7556(3) .7620(4) .7648(4) .7730(4) .7770(4) B.Forecasting Percentage MAE (Mean Absolute Error) Horizon 3 6 9 12 15 18 21 24 27 30 Random Walk .5361(2) .5423(2) .5381(2) .5339(2) .5302(1) .5328(1) .5373(1) .5362(1) .5457(1) .5464(1) ECM .5354(1) .5409(1) .5375(1) .5338(1) .5302(1) .5337(2) .5378(2) .5367(2) .5471(2) .5473(2) ECM-l .5454(4) .5513(5) .5478(5) .5426(4) .5392(4) .5417(5) .5464(5) .5455(5) .5556(5) .5564(5) C.Forecasting Percentage ME (Mean Error) Horizon 12 15 18 21 24 27 30 Random Walk .0277(3) .0293(4) .0163(4) .0209(4) .0121(4) .0073(3) .0057(1) .0099(4) .0078(3) .0100(4) ECM .0125(1) .0136(1) .0000(1) .0044(1) .0045(3) .0088(4) .0105(4) .007l(3) .0088(4) .0069(3) ECM-1 .0155(2) .0170(2) .0034(2) .0082(3) ECM-2 VAR .7588(4) .7582(4) .7593(4) .7537(4) .7531(4) .7541(4) .7607(3) .7646(3) .7704(3) .7768(3) VAR .5376(3) .5431(3) .5399(3) .5361(3) .5330(3) .5353(3) .5404(3) .5392(3) .5488(4) .5494(3) ECM-2 .0180(4) .0185(3) .0042(3) .0080(2) .5517(5) .5503(4) .5521(4) .5433(5) .5412(5) .5382(4) .5433(4) .5438(4) .5477(3) .5539(4) VAR -.1022(5) -.0968(5) -.0966(5) -.09l8(5) .0008(1) .0049(1) .0062(2) .0030(1) .0045(1) .0024(1) .0014(2) .0063(2) .0077(3) .0037(2) .0058(2) .0035(2) .O922(5) .0861(5) .0856(5) .0913(5) .0852(5) .0825(5) 109 Charts Movements of the European Currency Unit (ECU) Against the US Dollar A. US Dollar per ECU, Monthly Average L5F__————'"“ ' 1.4 ”I’m \ B. ECU per US Dollar. Quarterly L3 . Average \ l.‘ 4." 12 L" 1 l.)- . hill) [.1 1.! b )..‘ 1 1.0 f .3 \U‘ ‘ 0., 1 \J\ J a '7 ' fl 0.0 j ()9 \ / 1 I.. I ,_L-.,-.--.- . -.--.., / “'5wszume‘pu-nvsuuafiséii V H l ‘ 0.8 \ 07 06 1070 1980 1081 19$: :93) 10x4 [985 IONA Source: International Monelarv Fund. lnu'rnmumul I’nmmml Slummus‘. various issues. 110 LIST OF REFERENCES Ahking, F.W. and S.M. Miller. ”A Comparison of the Stochastic Processes of Structural and Time-Series Exchange-Rate Models." Review 9: Esengaiss_and_§£a£istiss. 69. August. 1987. PP- 496-502- Baillie, R.T. and T. Bollerslev. ”Common Stochastic Trends in a System of Exchange Rates." gggzna1_gf_£1n§ngg, 44, March, 1989. Baillie, R.T. and.David,D. Selover. "Cointegration.and.Models of Exchange Rate Determination."1atsrnati2na1.123rna1_2£_figresas£ins. 3. 1987. Pp. 43-51 Campbell, J.Y. and R.J. Shiller. 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