on cointegration and tests of forward market unbiasedness

On Cointegration and Tests of Forward Market UnbiasednessAuthor(s): Dean Corbae, Kian-Guan Lim and Sam OuliarisSource: The Review of Economics and Statistics, Vol. 74, No. 4 (Nov., 1992), pp. 728-732Published by: The MIT PressStable URL: http://www.jstor.org/stable/2109389 .

Accessed: 28/06/2014 13:19

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review ofEconomics and Statistics.

http://www.jstor.org

This content downloaded from 91.213.220.163 on Sat, 28 Jun 2014 13:19:02 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mitpress

http://www.jstor.org/stable/2109389?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


728 THE REVIEW OF ECONOMICS AND STATISTICS

ON COINTEGRATION AND TESTS OF FORWARD MARKET UNBIASEDNESS

Dean Corbae, Kian-Guan Lim and Sam Ouliaris*

Abstract-This paper provides univariate and multivariate tests of the unbiasedness hypothesis in forward market efficiency studies using canonical regression procedures for cointegrated systems. The advantage of conducting inference on levels rather than differenced data include greater asymptotic efficiency in estimation, and overcoming the contemporaneous correlation problem among regressors and a stationary risk premium. We demonstrate that the procedure can isolate the time series properties of the forward market risk premium. Canonical regression procedures can also be used to identify which forward rates predict future spot rates in semi-strong form efficiency tests.

I. Introduction

The aim of this paper is to test the unbiasedness hypothesis in forward market efficiency studies. In con- trast to methodologies based on differenced data (see Cornell (1977), Bilson (1981), Cumby and Obstfeld (1981), Geweke and Feige (1979), and Hakkio (1981)), we formulate the forward efficiency hypothesis using the levels of spot and forward exchange rates as in, among others, Frenkel (1976, 1977, 1979, 1980).1 There are several advantages in using levels rather than differenced data to test the simple efficiency hypothesis. First, a differenced model yields estimates that con- verge to the true parameter estimates at the rate e

(where T is the sample size) rather than rate T for levels (see Stock (1987) and Park and Phillips (1988)). Second, while a stationary stochastic risk premium may exhibit stochastic correlation with differenced regressors, the levels approach is not affected by such correlation since the order of nonstationary regressors dominates the order of the stationary risk premium (see Phillips and Durlauf (1986, p. 486)).

In particular, the simple efficiency hypothesis2 is restated in terms of the Engle and Granger (1987)

theory of cointegrated processes. Canonical regression procedures for cointegrated systems are then used to test the null hypothesis of forward market unbiasedness of forward market efficiency. The procedures provide a simple test for the presence of a forward-market risk premium and are able to characterize its time series properties. Finally, the methodology is used to examine the issue of multi-market or semi-strong form efficiency and, in particular, identify the currencies in the traders' information sets that may help to predict future spot rates.

The rest of the paper is organized as follows. Section II discusses the forward exchange efficiency hypothesis in terms of a cointegrated system. Section III discusses the data and the empirical findings. Section IV con- cludes the paper.

II. Tests of Forward Exchange Prediction

Let s' denote the natural logarithm of the spot exchange of currency i in terms of a numeraire currency, the US$ at time t. Let fti k denote the natural logarithm of the forward exchange rate of currency i contracted at time t for delivery at time t + k. Allow- ing for unit root processes with deterministic drifts, we specify St+k and fti k to follow:

St+k = A + St+k-1 + Es,t+k (1)

and

ft, k = A + ft-1, k + E%t,k (2)

where E5,t+k and E%,t,k are mean zero stationary errors that may be drawn from the ARMA(p, q) family. By iterative substitution, we can show that St+k =1at +

St+k and fti k = A' t + ft,*k where St'*+k and tt*k are the stochastic components of st+k and ft k and comprise sums of the stationary errors as well as a lagged term in the levels.

The simple efficiency hypothesis imposes a specific cointegrating vector between spot and forward exchange rates. In general, suppose S+k and ft/k are stochastically cointegrated with cointegrating vector (1, -f3j). Note that this is equivalent to saying that

StI+k-I3iftk is a trend stationary ARMA(p, q) process. Then from (1) and (2), the deterministic part of the cointegrating vector is constrained, viz:

S I = ai + - + i3tf k + 1t+k (3)

where 7qti+k is a mean zero stationary error comprising terms in E' and E' -j k for some i 2 O, and a i is a constant term reflecting the initial values of s and f (possibly zero). Letting yi = A' - /3ji,i, a testable im-

Received for publication October 9, 1990. Revision accepted for publication April 2, 1991.

* The University of Iowa, the National University of Singa- pore, and the National University of Singapore, respectively.

We are grateful to two referees for helpful comments on an earlier draft of this paper. Remaining errors are the authors' sole responsibility.

1 Early studies using levels typically neglected the nonsta- tionarity of the data. The purposes of this paper is to address such data.

2As defined in Hansen and Hodrick (1980), "simple efficiency" is a joint hypothesis of rational expectations and risk neutral arbitrage yielding a zero risk premium. Here, we do not address the information processing aspect of the joint hypothesis, only the unbiasedness condition. Rejection of the unbiasedness condition is suggestive of risk aversion on the part of traders, where the premium can be characterized by the conditional covariance between the marginal rate of substitution in consumption and the future spot exchange rate (see Baillie and McMahon. 1989).

Copyright C) 1992



NOTES 729

plication of the "simple efficiency hypothesis" is

Ho: ai = ? Yi = O, 8i = 1, (4)

provided that r7ti+k is uncorrelated with information at time t, including ftik as well as its own lags 7-q i 2 0. Obviously, the test is based on the joint hypotheses of market rationality and a zero risk premium.

Hansen and Hodrick (1980) have used sampling data where the sampling interval is finer than the forward contract interval k. In this more general setup, the forward forecast error t+k would be serially correlated up to lag k. We also use overlapping forward contract intervals. The cointegration regression frame- work that we use allows 7qti+k to be ARMA(p, q) in general, and so accommodates the error serial correlation induced by the overlapping data.

The presence of a risk premium suggests one or more of the three constraints may be violated: ai = 0, yi = 0, f3i = 1. In addition, even under market rationality, the disturbance ti+k and 1tik may be correlated. For this alternative hypothesis, the risk premium as defined in Fama (1984) is given by

Pt,k =ft,k - Et(St+k)

=ft,k -[a + Yt + Ift,k + Et(-qt+k)]

using (3)

= (1 - 18M, k + Pt*, k (5)

where

Pt,k = - [a + yt + Et(t+k)]

and a, y and Et(7t+k) are in general non-zero. Under the simple efficiency hypothesis the risk premium Pt k

is zero. Interestingly, under the alternative hypothesis, the risk premium Pt k can be characterized explicitly. In particular, when ft k is an I(1) process, f3 = 1, and

Ptk is trend stationary (with a and y not equal to zero), the components in (5) represent a permanent and a transitory component that are embodied in the risk premium. However, if I8 = 1, but a and y are not equal to zero, the risk premium is Pt* k, which contains a linear time trend as well as a non-zero stochastic component Et(7t +k)*

If 7't +k in (3) is stationary, canonical regression techniques should be u'sed to estimate (3) since the estimates of ai, yi and f3i will be consistent in the limit. Since the risk premium (if it exists) is not inde- pendent of the contemporaneous forward rate, -t+k may be correlated with the innovations of 17 k. How- ever, canonical regression procedures for cointegrated systems explicitly allow for such contemporaneous correlation. It follows that the procedures provide a test of forward rate prediction under the more general conditions that a risk premium may be present.

We test the unbiasedness condition using the null hypothesis that 8i = 1 and ai = 0 and yi = 0. This null hypothesis may be viewed as unbiased prediction

(j3i = 1), together with a zero risk premium (ai = 0 and yi = 0). Park's (1992) estimator for cointegrated systems yields a test statistic for the model which is asymptotically distributed as x2 where q is the number of restrictions on the parameters.

In the test of the model, the alternative hypothesis allows the slope term i3i to take values other than one. In this case, there would be evidence of forward rate prediction even though the unbiasedness hypothesis may not be true. However, using the conventional tests where the dependent variable is s+k - ftik, it is not convenient to test other hypotheses, e.g., fi E (0, 1). This is due to the fact that if s'+k and fti k are cointegrated with cointegrating vector (1, -/3i), then the forward forecast error becomes an integrated process. The additional possibilities of estimation and inference using the cointegration regression make it a more useful methodology than the conventional tests. In the next section, model (3) is tested for each of the currencies under study.

We now turn to the issue of multi-market efficiency. Geweke and Feige (1979) and Hansen and Hodrick (1980) have tested for multi-market efficiency, which may be considered a semi-strong form test. Their tests indicated evidence that the forward forecast error of some currencies might be correlated with lagged forecast errors of other currencies as well. Geweke and Feige suggested that risk aversion rather than transac- tions costs might explain the significance of lagged forecasts. Since we allow for risk premiums to be incorporated in the error term and the intercept, our models would provide less ambiguous tests of market efficiency. We run a second set of cointegrated regressions as follows:

m

St+k = ai + yit + iftj,k + vt+k, (6) j=1

where m is the number of currencies excluding the numeraire currency and vt+k is the residual error. If there is weak-form efficiency, then the coefficients of terms f/J k for j 0 i should be zero. On the contrary, if there is an inefficiency, then the same coefficients would not be trivial. This may be explained as follows. If the forward rates are integrated I(1) processes, then lagged rates are the best forecasts of present rates. Therefore, if ft, k5, i j i, have explanatory power on St +k, then the lagged forward rates, f/uk, i 2 0, would provide information on s'+k when ft!k' I , i, are excluded in the regression. Such an observation would be consistent with multi-market inefficiency reported by Geweke and Feige and others.

The test of (6) is based on the following null hypothesis: Ho: ai = 0, Yi = ?, .8i = 1, 8jIi = ?. Park's (1992) CCR is applied to (6). A joint Wald-test of the restrictions is provided by a test statistic that is asymptotically distributed as x2 where q is the number of restrictions.




III. Data and Empirical Results

Weekly data on exchange rates are used to test the simple efficiency hypothesis. They are the Wednesday twelve o'clock Canadian dollar, Deutschemark, Swiss franc, French franc, Japanese yen, and U.K. pound-U.S. dollar spot bid rates, and the midpoint of the twelve o'clock bid-ask rates on the three-month forward contracts. These data, collected by the Federal Reserve System, are for the period beginning 2 January 1976 and ending 2 January 1985.

We first pre-test (see Engle and Granger (1987)) the time series data si+k and ftik for unit root or I(1) behavior. Table 1 presents the results of applying the Said and Dickey (1984) Augmented Dickey-Fuller (ADF) procedure and Phillips' (1987) Zt statistic to the levels of the exchange rate data. In applying these two statistics, we included a time trend in the fitted regression. Including the time trend in the fitted regression purges the drift term from the exchange data and thus helps to remove deterministic non-stationarity. Table 1 also reports the Park and Choi (1988) G(j, k) statistic for j = 0 and k = 4.3 This statistic uses a null hypothesis of stochastic stationarity, and possesses an asymptotic X2-j distribution. Note that, like the Zt and ADF statistics, the G(j, k)-statistic can allow for deterministic drift in the null hypothesis by setting j = 0. The statistical results imply that the hypothesis of unit root behavior with deterministic drift cannot be rejected at the 5% significance level.

Since the exchange data are consistent with the I(1) hypotheses, we can test whether (3) is a cointegrated system. Table 2 reports the results of applying Park's (1992) CCR to (3). The H(], k) statistic for j = 1 and k = 3 is also reported. The H(1, 3) statistic tests the null hypothesis of cointegration. We use k = 1 to allow for the possibility that yi is non-zero. The Wald statistic for the null hypothesis for ai = 0, yi = 0, and f3i = 1 is also reported in table 2.

The H(1, 3) statistics imply that the null hypothesis of cointegration cannot be rejected at the 10% level of significance for any of the countries in the data set. For the levels regression, with the exception of the U.K. pound, all the estimated values of the intercepts are negative. In the case -of the Japanese yen, the t-statistics for the ai and /3i estimators reject unbiasedness and a zero intercept at the 5% level (two-tail test). Based on X3 tests on the joint restrictions in (4), the null hypothesis of unbiasedness is rejected at the 5% significance level for the Swiss franc and the Japanese yen. In the case of the deutschemark, the unbiasedness hypothesis can be rejected using a 10% level of significance.

When i3i deviates from one, as in the case of the Japanese yen, (5) implies that the risk premium is non-stationary, comprising a permanent I(1) component and a transitory error. This characterization of the risk premium is more general than the usual stationary risk premium assumption used in existing studies. For example, if ft k and E,(- +k) were stationary, then the risk premium would be covariance stationary.

The t-statistics for the yi estimators imply that there is no significant drift at the 5% significance level (two- tail test). Therefore, the transitory term has no trend, and may be characterized as a stationary ARMA(p, q) error. In the cases when the forward rates are unbiased predictors of the future spot rates, the risk premiums for the currencies involved would then be stationary and generally negative (with negative ai's) in the period of study. The negative premiums may be reflecting the cost of hedging against US$ appreciation in the early eighties.

The results for the multi-market tests are reported in table 3. Again, a test of stochastic cointegration is provided by the H(1, 4) statistic. The Wald statistic reported in table 3 tests the restriction that ai, yi, and f3j i are jointly zero. The own forward rate coefficient f31 in this case is one.

The H(1, 4) tests do not reject the null hypotheses of cointegration at the 5% significance level. The Wald statistic rejects the null hypothesis that Ho: ai = 0,

Yi = 0, p3i = 1, f3 i = 0, at the 5% level of significance for all the countries. This result implies that the own forward rate forecast is biased for the Canadian dollar, the yen, and the U.K. pound, and that other forward rates can explain spot rate movements. In the case of

TABLE 1.-UNIT ROOT STATISTICS FOR THE FOREIGN

EXCHANGE DATAa

Currency ADF(6 t)b Zt(6, t) G(0, 4)c

A. Spot Exchange Rates

Canadian $ -2.5474 - 2.4265 38.115 DM -1.2989 -1.1995 34.704 Swiss franc -1.1443 -1.1487 31.546 French franc -0.9166 -0.9079 37.909 Yen -2.0628 -1.9319 26.350 U.K. pound -0.1521 - 0.0235 36.852

B. Forward Exchange Rates

Canadian $ - 2.2988 - 2.2781 37.189 DM -1.1447 -1.1528 34.969 Swiss franc -1.1201 -1.1151 32.186 French franc -0.6855 -0.6588 38.052 Yen -1.8317 - 1.8030 27.833 U.K. pound - 0.3167 - 0.2869 36.807

a Data are expressed in logarithms. bThe ADF and the Zt statistics are based on a null hypothesis of unit root

process with trend t. 5% critical values: ADF(O, t) = Z,(O, t) = - 3.4500. c The G(0, 4) statistic is based on a null hypothesis of stationary process.

5% critical value: G(0, 4) = X = 9.49.

3Our qualitative findings are not sensitive to the choice of k.



NOTES 731

TABLE 2.-UNIVARIATE TESTS OF FORWARD EXCHANGE PREDIcTION

True Model: St+k = fti k + Et +k

Fitted Model:a S,+k = ai + y,t + Pi,k + Et+k

Currency a1i Y, i H(1, 3)b

Canadian $ - 0.1278E-02 - 0.6307E-02 0.9777 3.7686 0.4345 t-stat/p-valued (0.2904) (0.3357) (0.3135) (0.1519) (0.9330) DM - 0.2679E-01 - 0.2557E-01 0.9660 1.1378 7.6676 t-stat/p-value (0.9177) (1.3366) (0.9063) (0.5661) (0.0534) Swiss franc -0.5071E-01 -0.1851E-01 0.9386 1.7631 10.4410 t-stat/p-value (1.4904) (0.7915) (1.4362) (0.4141) (0.0160)e French franc -0.7100E-01 -0.5245E-01 0.9417 2.4816 2.9835 t-stat/p-value (1.3558) (1.6490) (1.5597) (0.2891) (0.3942) Yen -0.7362 -0.1122E-01 0.8670 2.9425 11.652 t-stat/p-value (2.7129)f (0.0521) (2.7129)f (0.2296) (0.009)f UK/pound 0.3207E-01 - 0.3858E-01 0.9782 2.4635 2.8378 t-stat/p-value (0.9910) (1.6785) (0.5151) (0.2918) (0.4173)

a The equation is estimated using a Parzen window with 30 lags. (See Park (1992).) b2 Based on a null of cointegration, this statistic is distributed as X2. c This statistic is based on the null of a, = 0, y, = 0, and 13, = 1, and is distributed as X3- dAbsolute t-statistics are given in the parentheses for ?4, , and 13. For 13, the t-statistic is computed for the

null hypothesis that the coefficient is equal to unity. P-values are provided for the H(1, 3) and the W-statistics. eRejection at the 5% level.

Rejection at the 1% level.

TABLE 3.-MULTI-MARKET TESTS OF FORWARD RATE PREDICTION USING PARK'S (1992) CCRa

m

Fitted Model: S'+k = ai + y1t + E Ptjk + Vi+k j=l

Currency aEi Yi yil 182 18,3 f4 13i5 13i6 W2 H(1, 4)C

Canadian $ -0.4464 -0.0377 0.6134 -0.1586 0.0447 0.1064 -0.0893 0.0022 23.632 0.5959

(2.5195)d (0.9378) (4.0337)e (2.1448)e (0.8862) (1.4118) (2.7459)e (0.0529) (0.003)e (0.8973)

DM -0.5011 -0.0990 -0.1850 0.8549 0.2095 0.0857 -0.1446 -0.1774 25.108 1.0910

(1.0039) (0.8808) (0.6896) (0.7224) (1.4809) (0.4138) (1.5922)f (1.4993) (0.001)e (0.7792)

Swiss franc - 0.2149 - 0.1554 - 0.4154 0.2078 0.8115 - 0.0572 - 0.0363 - 0.0628 21.654 3.0761

(0.3932) (1.3026) (1.4312) (0.9374) (1.2077) (0.2542) (0.3604) (0.4901) (0.005)e (0.2535)

French franc 0.5114 -0.1230 -0.1474 0.2388 0.0791 0.9057 -0.1366 -0.1947 22.862 0.9700

(1.0357) (1.0971) (0.5602) (1.2136) (0.5693) (0.4598) (1.5379) (1.6702)f (0.003)e (0.8085) Yen - 1.1827 - 0.1014 0.0715 0.4886 - 0.1235 - 0.2943 0.8107 0.0241 43.292 0.4432

(2.9133)e (1.1202) (0.3221) (2.9759)e (1.0565) (1.7544)f (2.5066)e (0.2499) (0.000)e (0.9312)

U.K. pound 0.0576 0.0118 0.0607 0.3274 0.1098 0.1348 -0.1396 0.5580 89.575 4.2650

(0.1383) (0.1255) (0.2701) (1.9104)f (0.9263) (0.7694) (1.8287) (4.4546)e (0.000)e (0.2342)

a The CCR estimator is implemented using a Parzen window and 30 lags. The H-statistic for the null hypothesis that the forward rates are cointegrated is 12.590, with a p-value of 0.0134.

b 2 W2 is the Wald statistic is distributed as X8 cH(1, 4) tests the null of cointegration and is distributed as 23. dAbsolute t-statistics are given in the parentheses for 13, = 1, other coefficients zero. P-values are provided for the H(1, 4) and W-statistics. e Rejection at the 1% level.

Rejection at the 10% level.

the spot Canadian dollar, the forward deutschemark and yen had significant effects at the 1% level (two- tail test). For the yen, the deutschemark had a sim- ilar impact. Generally, the impact of the forward deutschemark and yen appear to be stronger than the other forward rates. Own forward rates, however, had more influence on spot rates. There is evidence that the coefficients are significantly less then one: consider the significant deviation from unity (1%, two-tail test) of the Canadian dollar, the yen and the U.K. pound.

The multivariate results suggests that the bloc of European countries (Germany, United Kingdom,

France) appear to act jointly to offset Japanese movements in its forward exchange rate. That is, fi5 < 0, at the 10% level for Germany and the United Kingdom, and nearly so for France.

The Wald statistic strongly rejects the joint unbiasedness and the zero risk premium hypotheses at the 1% significance level. The drift term $Yi is insignificant at the 10% level (two-tail test) for all the countries. This confirms our earlier finding that the transitory component of the risk premium has no time trend.

The results also substantiate the findings of Baillie and Bollerslev (1989) of a common stochastic trend in




the forward exchange market. Not only do own forward rates have predictive power (although this is somewhat less powerful than the simple efficiency hypothesis would suggest), other currency forward rates, particularly those of strong currencies, e.g., deutschemark and yen, also affect future spot rate movements. It appears that the linear combination of forward rates is a better predictor of the future spot rate than the own forward rate itself. A test of the forward rates shows that there is no evidence of cointegration amongst them. Therefore, there is no long-run equilib- rium relationship among the forward rates that would help to explain why other forward rates seemed to be able to help predict own spot rates.

IV. Conclusion

The issue of whether forward exchange rates are unbiased forecasts of future spot rates has been the focus of numerous studies. A more recent issue has been the nature of the risk premium in the forward market. While the conventional regression approach typically assumes the risk premium is stationary, the cointegration approach is able to identify the nature of the risk premium explicitly. Our results indicate that when the forward rates are not unbiased predictors of the future spot rates, the risk premium is non-stationary, comprising an I(1) permanent deviation and a transitory component. The latter is stationary and does not carry a trend. In cases when the forward rates may be unbiased, the risk premium is then stationary and is on average negative for the currencies in the period of study.

Using multiple regression cointegration, we also find that other currency forward rates, particularly those of the deutschemark and the Japanese yen, have significant effects on most future spot rates in the period of study. However, these effects are not nearly as strong as those by the currency's own forward rate.

REFERENCES

Baillie, R. T., and T. Bollerslev, "Common Stochastic Trends in a System of Exchange Rates," Journal of Finance 44 (1) (1989), 167-181.

Baillie, R. T., and P. McMahon, The Foreign Exchange Mar- ket: Theory and Econometric Evidence (Cambridge: Cambridge University Press, 1989).

Bilson, J. F. 0., "The Speculative Efficiency Hypothesis," Journal of Business 54 (1981), 435-451.

Cornell, B., "Spot Rates, Forward Rates and Exchange Mar- ket Efficiency," Journal of Financial Economics 5 (1977), 55-65.

Cumby, R. E., and M. Obstfeld, "Exchange Rate Expecta- tions and Nominal Interest Differentials: A Test of the Fisher Hypothesis," Journal of Finance 36 (1981), 697-703.

Domowitz, I., and C. S. Hakkio, "Conditional Variance and the Risk Premium in the Foreign Exchange Market," Journal of International Economics (1985), 47-66.

Engle, R. F., and C. W. J. Granger, "Cointegration and Error Correction: Representation, Estimation and Testing," Econometrica 55 (1987), 251-276.

Fama, E. F., "Forward and Spot Exchange Rates," Journal of Monetary Economics 14 (1984), 319-338.

Frenkel, Jacob A., "A Monetary Approach to the Exchange Rate: Doctrinal Aspects and Empirical Evidence," Scandinavian Journal of Economics 78 (1976), 200-224. (Reprinted in Frenkel and Johnson (eds.)).

"The Forward Exchange Rate, Expectations and the Demand for Money: The German Hyperinflation," American Economic Review 67 (1977), 653-670. ,"Further Evidence on Expectations and the Demand for Money During the German Hyper-inflation," Jour- nal of Monetary Economics 5 (1979), 81-96.

______,"Flexible Exchange Rates in the 1970's," in Stabiliza- tion Policy: Lessons from the 1970's and Implications for the 1980's, Center for the Study of American Business at Washington University, St. Louis, Missouri (1980).

Geweke, J., and E. L. Feige, "Some Joint Tests of the Efficiency of Markets for Forward Foreign Exchange," this REVIEW 61 (1979), 334-341.

Hakkio, C., "The Term Structure of the Forward Premium," Journal of Monetary Economics 8 (1981), 41-58.

Hansen, L. P., and R. J. Hodrick, "Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis," Journal of Political Economy 88 (1980), 829-853.

Park, J. Y., "Canonical Cointegrating Regressions," Econo- metrica 60 (1992), 119-144.

Park, J. Y. and B. Choi, "A New Approach to Testing for a Unit Root," CAE Working Paper No. 88-23, Cornell University (1988).

Park, J. Y. and P. C. B. Phillips, "Statistical Inferences in Regressions with Integrated Processes: Part 1," Econo- metric Theory 4 (1988), 468-497.

Phillips, P. C. B., "Time Series Regression with a Unit Root," Econometrica 55 (1987), 277-301.

Phillips, P. C. B., and S. N. Durlauf, "Multiple Time Series Regression with Integrated Processes," Review of Eco- nomic Studies 53 (1986), 473-497.

Said, S. E., and D. A. Dickey, "Testing for Unit Roots in Autoregressive-moving Average Models of Unknown Order," Biometrika 71 (1984), 599-607.

Stock, J. H., "Asymptotic Properties of Least Squares Estima- tors of Cointegrating Vectors," Econometrica 55 (1987), 1035-1056.



on cointegration and tests of forward market unbiasedness

Documents