COINTEGRATION TESTS WITH DAILY EXCHANGE RATE DATA

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<ul><li><p>OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 53,2(1991)0305-9049 $3.00</p><p>COINTEGRATION TESTS WITH DAILYEXCHANGE RAIE DATA</p><p>Laurence S. Copeland*</p><p>INTRODUCTION</p><p>The efficiency of foreign currency markets must by now be one of the mostintensively researched topics in economics.' By far the majority of the workhas focused on the relationship between spot and forward rates, for the mostpart using monthly data (e.g. Frenkel (1979), Edwards (1983)), though oftenalso using weekly data (Baillie, Lippens and McMahon (1983)). Relativelylittle research effort has been directed at examining the evidence in dailydata,2 a state of affairs which means that potentially useful information isbeing neglected. By and large, the conclusion which appears to have emergedis that the forward exchange rate is almost certainly not an unbiased predictorof the future spot, and attention has increasingly turned to explanations forthis apparent bias in terms of a risk premium (Hodrick and Srivastava (1984),Domowitz and Hakkio(1985)).</p><p>Before addressing the question of forward market efficiency, this paperstarts by examining a related issue: the question of whether or not the foreignexchange markets are cross-sectionally efficient. In other words, we testwhether it is possible to profit by trading across currency markets, exploitingmovements in exchange rate A to predict movements in exchange rate B.There are a number of reasons why this type of test is of interest. First, withlow transactions costs and increasing ease of access to information in foreignexchange markets, cross-currency arbitrage must be very easy for large scaletraders. Second, unlike the situation in the forward markets, the issue in spotmarkets is not clouded to the same extent by the existence of a risk premium.3Thirdly, we use daily rather than monthly data. This fact makes the tests moresearching, both in the econometric sense that far more information is being</p><p>*The author wishes to express his gratitude to the Nuffield Foundation for providing supportfor the research reported here, and also to seminar groups at the Universities of Keele andLancaster for a number of helpful comments.</p><p>'Seethe survey by Levich(1985).2 see Baillie and Bollerslev (1989).</p><p>a risk premium cannot be entirely ruled out, it seems likely to be negligibly small,given that the market is dominated by banks, multinationals etc., with no obvious preferredhabitats.</p><p>185</p></li><li><p>186 BULLETIN</p><p>utilized and in the straightforward economic sense that it requires anyinefficiencies to be eliminated within one day, rather than one quarter, monthor week.</p><p>The tests rely on the results given in a series of papers by Granger, Engleand others on the concept of cointegration in economic variables.4 Forpurposes of the present paper, there are two crucial results in that literature.First, the proof that cointegrated variables can be represented by an errorcorrection mechanism (ECM) implies immediately that cointegration isinconsistent with market efficiency (Section I below). Secondly, thanks toadvances in the study of time series with unit roots (e.g. Dickey and Fuller(1979)), a number of techniques have been developed to test formally forcointegration. The original two-stage approach proposed by Engle andGranger (1989) (see also Engle and Yoo (1989)) suffers from a number ofshortcomings, however. The present paper follows the method outlined byJohansen (1988) for testing for the presence of a cointegration relationshipand for extracting the cointegrating vectors, a technique which has theadvantage of being robust with respect to different possible processes in theunderlying time series. Section III discusses the results for pairings of fivespot exchange rates (against the US dollar) for the period from 1976 to 1989,and subperiods thereof, while Section IV focuses on the respective spot andforward rates.5</p><p>I. COINTEGRATION AND EFFICIENCY</p><p>Granger (1981) has shown that, if the n-dimensional vector time series x iscointegrated of order one, it can be represented both by a Vector Autoregres-sion (VAR), of the form:</p><p>A(B)x1v, (1)and by an Error Correction Mechanism (ECM):</p><p>A*(B)(1 B)x= - (2)where B is the backshift operator, and A(0) = A*(0) = I,.</p><p>The implication is easiest seen in the special case where n =2. For twonon-stationary series X and Y,, cointegration implies the existence of aunique representation:</p><p>Y,=b0+b1X,+u, (3)such that u, is a stationary error term.</p><p>See particularly Granger (1981), (1986), and Hendry (1986).recent paper by Hakkio and Rush (1989) looked at similar tests for only one currency</p><p>pairing (the Pound and Deutschemark), and estimated over monthly data ending in mid-i 986,ending too soon therefore to reflect the policy of shadowing the DM pursued by the UKauthorities from about 1986 to late 1989. Their results indicate no cointegration. Similarly,MacDonald and Taylor (1989) performed two of the tests used here on monthly data over theperiod 1973 to 1985, and also found no cointegration between spot rates for differentcurrencies.</p></li><li><p>COINTEGRATION TESTS 187</p><p>The relevance of the cointegration concept to financial markets becomesclear from the (ECM), which in this simple case takes the form:</p><p>1a(Ylbb1X,)+X,+ &gt;1 iXX,,- ykAY ,</p></li><li><p>188 BULLETIN</p><p>which is precisely the starting point for most so-called efficiency tests.8 Avoluminous literature exists examining whether the data are consistent withthe hypothesis that b0 is zero and b1 unity (see the survey in (Levich (1985)).Much less attention has been focused on the properties of the error term in(5), with sometimes inadequate inspection for autocorrelation patterns.Unbiasedness plainly requires that u1 have no discernible structure whatever,otherwise the forward rate cannot be said to embody all the information inthe past history of the spot rate. Cointegration merely requires that any auto-correlation pattern in the residuals have no unit root, a much weakercondition (Hakkio and Rush (1989)).</p><p>It is worth considering in somewhat greater detail the implications ofdeviations from unbiasedness. In the first place, a non-zero intercept in (5) isneither a necessary nor sufficient condition for the existence of a riskpremium. On the one hand, it could be indicative of a systematic bias inmarket expectations. On the other hand, a zero intercept could simply be theoutcome of a risk premium which is uncorrelated both with the forward rateand its own past, but which fluctuates randomly around a zero mean - adistinct possibility in the light of the results typically reported in the literature(see in particular Fama (1984)). As far as the coefficient on the laggedforward rate is concerned, finding that b1 is significantly different from unityis most obviously consistent with expectations bias, but could also be theresult of a risk premium correlated in some way with the level of the forwardrate itself, implausible as that may seem prima facie (though see e.g. Frankeland Froot (1987)). Likewise, autocorrelated residuals could be evidence of anARJVIA process in the risk premium9 or equally the result of marketirrationality in failing to embody in the forward rate a systematic time seriescomponent in the spot rate.</p><p>II. THE DATA</p><p>The data used for the tests were taken from the DATASTREAM databank,and covered five currencies: the Deutschemark, Pound, French Franc, SwissFranc and Yen. Spot rates are daily closing rates against the US Dollar, andare averages of buying and selling rates. Forward rates are for one monthcontracts and, for purposes of the tests in Section IV below, were matched</p><p>So-called because different authors appear to use the term efficiency to mean differentthings. By and large, economists seem to use the term to preclude a risk premium (i.e. unbiased-ness of the forward rate), while the finance literature, from which the concept originates, allowsfor a risk premium, provided it is consistent with the maintained hypothesis about marketpreferences, the distribution of returns etc. Following the convention in Copeland (1989), thehypothesis that the forward rate equals the expected future spot will be called unbiasedness.Deviations from unbiasedness may or may not be consistent with efficiency, depending on thefeatures of the market structure assumed.</p><p>example, a serially correlated risk premium is consistent with a number of possibletheoretical models, as well as with empirical specifications along ARCH lines (e.g. Domowitzand Hakkio(1985)).</p></li><li><p>COINTEGRATION TESTS 189with spot rates according to the conventions of the foreign currency markets(Chrystal and Thornton (1989), Riehl and Rodriguez (1977)). In each case,the data were converted into natural logs to avoid the well-known difficultypresented by Jensen's inequality in the context of bilateral exchange rates.</p><p>All daily data were carefully checked for errors, with special attentiongiven to maxima and minima. Bank holidays were excluded from the dataset,' a fact which, in combination with the need to accommodate long lags inmany regressions, means that the number of degrees of freedom is oftensubstantially shorter than would appear from the data periods involved.</p><p>The order of integration of the variables was established by runningregressions of first differences on lagged levels, second differences on laggedfirst differences, and so on. In each case, care was taken to include sufficientlagged dependent variables to yield white noise residuals, a requirementwhich for some variables entailed using as many as 50 lags. On the standardDickey-Fuller tests, all were found to be unambiguously 1(1), as might havebeen expected in the light of the results reported by other researchers.'1</p><p>III. CROSS-CURRENCY TESTS</p><p>The evidence on cointegration in cross rates is presented first.Table 1 gives values of the likelihood ratio test statistic for the hypothesis</p><p>that the number of cointegration vectors is no greater than r:</p><p>LR=-T ln(1-) (6)1r+1</p><p>where Ar +i... are the N - r smallest squared canonical correlation coef-ficients between the residuals of x</p><p>- k and AX1 in regressions on lags of AX1.(See Johansen (1988), Hall (1989)). The results are overwhelmingly con-sistent with the null hypothesis of no cointegration for every currency pair.Furthermore, tests on three sub-periods 1976-79, 1980-8 5 and 1986-90produced broadly similar results.'2 In no case could the hypothesis of zerocointegration be rejected at the standard 5% level.</p><p>number of tests were carried out using dummy variables to pick up possible day-of-the-week and pre-bankholiday or post-bankholiday effects. None were unambiguously significant,suggesting that seasonality (in other words, market anomalies') is not a problem with thisparticular data set. The need to check for these factors was indicated by, on the one hand, theevidence of seasonality in daily data presented by McFarland et al. (1982), Baillie andBollersiev (1989) and others, and on the other hand by the argument of Birchenall et aL (1987)that ignoring these effects can seriously distort the dynamic structure of the model.</p><p>See e.g. Baillie and Bollerslev (1989). For the variables in the data set used here, runningthe regression:</p><p>2S,c0 +clAS,+dk2Sk+u,typically resulted in a t-ratio on c1 in the region of - 250 to - 400.</p><p>12 results for the first subperiod are presented in Table 1</p></li><li><p>NotesFor each currency pair, the first (second) number is the value of the likelihood ratio given in</p><p>Johansen (1988) to test the hypothesis that the maximum number of cointegrating vectors is OThe appropriate 95 percent critical values are 38.6 and 23.8 respectively.</p><p>There are, however, two noteworthy features of the results. First, the testvalues are larger for the 1970's than for the later periods, which may suggestspeedier adjustment of cross rates as currency markets became moresophisticated in the later years. Second, this trend is particularly noticeablefor the relationship between the Yen, on the one hand, and both the Frenchand Swiss Francs, perhaps reflecting the increasing liberalization of Japanesecurrency markets.</p><p>190 BULLETINTABLE 1</p><p>Tests For Spot Rate cointegration</p><p>1976-90</p><p> DM YEN SF</p><p>DM</p><p>YEN</p><p>SF</p><p>FR</p><p>8.733.07</p><p>3.11 1.490.51 0.326.14 2.070.75 0.63</p><p>13.22 4.991.29 0.26</p><p>4.010.181.930.02</p><p>3.710.43</p><p>1976-79</p><p> DM YEN SF</p><p>DM 4.760.06</p><p>YEN 7.61 7.752.3 0.67</p><p>SF 3.44 8.2 10.390.22 1.55 0.43</p><p>FR 18.69 8.32 21.29 9.31.32 0.66 1.33 1.39</p></li><li><p>COINTEGRATION TESTS 191TABLE 2</p><p>Tests For Forward Rate Cointegration</p><p>1976-90 1976-79 1980-85 1986-90</p><p>NotesFor each currency, the first (second) number is the value of the likelihood ratio given in</p><p>Johansen (1988) to test the hypothesis that the maximum number of cointegrating vectors is O(1).</p><p>The appropriate 95 percent critical values are 38.6 and 23.8 respectively.The number in the third row is in each case the estimated cointegrating coefficient.</p><p>IV. SPOT AND FORWARD RATE TESTS</p><p>Table 2 shows the likelihood ratio test for cointegration of each spot rate withits own lagged forward rate. Plainly, the support for the existence of at leastone cointegration vector is extremely strong.</p><p>Although significance tests are not immediately available from theJohansen procedure, it is also notable that the point estimates of the slopecoefficient (i.e. b1 in equation (5)) are very close to unity.</p><p>However, as was pointed out in Section I, unbiasedness requires thesatisfaction of two other conditions on the cointegration regression. While itis difficult to summarize the evidence in a small number of statistics, all theindications pointed to the existence of highly significant but unstable lagpatterns in the residuals of the cointegrating regressions. In each case, theBox-Pierce Q-statistic was a long way above its critical value, often into the0.1 percent tail of the distribution, and the autocorrelation coefficients were</p><p> 1,798.10 2,144.60 1,677.80 2,118.000.93 0.29 2.51 27.170.999 0.981 0.999 0.998</p><p>DM 1,847.90 1,891.60 1,794.80 1,953.902.44 0.05 21.70 16.500.998 0.997 1.000 1.003</p><p>YEN 904.40 2,523.20 1,485.20 657.101.59 9.10 5.50 45.300.997 0.995 0.995 1.005</p><p>SF 1,733.10 1,626.50 1,762.70 1,946.204.50 1.04 19.50 12.701.001 1.000 1.000 1.004</p><p>FR 1,014.00 1,120.70 845.80 1,832.301.32 0.72 18.70 3.910.996 0.979 0.997 1.005</p></li><li><p>192 BULLETIN</p><p>significantly different from zero on the standard rule-of-thumb test13 for upto 30 lags and more.</p><p>On the other hand, no clear pattern was apparent. The maximum laglength is difficult to establish, and appears in any case to vary across thesample periods. This is illustrated in Tables 3 and 4, where the results ofattempting to fit an ECM to the data for the Pound and DM are shown.</p><p>To put these results in context, consider the general ECM (equation (4))applied in this case to spot and forward rates:</p><p>is,=au,, +0Af, kAfk Ykt-k +e (7)</p><p>where u,_, is the (lagged) residual from the cointegrating regression - theunbiasedness test given in equation (5).</p><p>Two points should be made about equation (7). First, as already pointedout, the residual ought to be serially uncorrelated. However, in practice, withthe data sets involved here, no reasona...</p></li></ul>