the significance levels commonly used for these tests suggest that the dice are loaded in favor of unit roots and no cointegration.

We can reverse the null and alternative hypotheses for the cointegration test if we reverse the null and alternative hypotheses for the unit root tests. That is, we adopt the null hypothesis and alternative in unit root tests as:

Hq. X, is stationary

H,: X, is a unit root process

with similar hypotheses for y,. Then for the cointegration test we have

(,: X, and y, are cointegrated ,: X, and y, are not cointegrated

We mentioned in Section 14.5 some unit root tests that use the null hypothesis of stationarity.

14.11 Cointegration and Testing of the REH and MEH

During recent years, cointegration theory has been used for testing the rational expectations hypothesis (REH) and the market efficiency hypothesis (MEH). In Section 10.11 we described some tests for the rationality of yf, where yf is the expectation of y, (obtained from survey data or other sources). The tests described there are, however, not valid ify, and/or y are 1(1). For the vaUdity of REH, y, - yf has to be 1(0) or else y, and yf will be drifting further apart over time, in which case the rationality of yf is violated. However, it is not sufficient that the forecast error y, - yf be 1(0). The forecast error has to be free of serial correlation or be white noise. Hence for the rationality of expectations we require the following three conditions:

1. y, and yf must be cointegrated.

2. The cointegrating factor must be I.

3. The difference (y, - yf) must be a white noise process.

Since the cointegrating factor is specified to be 1, we use what is known as the restricted cointegration test. That is, we first test, using unit root tests, whether y, and yf are both 1(1). We next consider p., = y, - yf and apply unit root tests to If the null hypothesis of a unit root (a null hypothesis of no cointegration) is rejected, y, and yf are cointegrated with a cointegrated factor of 1. We next test by using the g-statistics described in Section 13.5 for the presence of serial correlation in ,."

"Examples of such restricted cointegration tests of the REH appear in R. W. Hafer and S. E. Hein, "Comparing Futures and Survey Forecasts of Near Term Treasury Bill Rates," Federal Reserve Bank of St. Louis Review, May-June 1989, pp. 33-42, and Peter C. Liu and G. S. Maddala, "Using Survey Data to Test Market Efficiency in the Foreign Exchange Markets," Empirical Economics (1992), forthcoming.

"Craig Hakkio and Mark Rush, "Market Efficiency and Cointegration: An Apphcation to the Sterling and Deutsche Mark Exchange Markets," Journal of International Money and Finance, March 1989, pp. 75-88.

"Thus, if F, - S, is stationary, there is no point in further testing the MEH. If F, - S, is nonstationary, we have a regression of a stationary variable on a nonstationary variable. Hence plim p = 0 and the MEH is rejected.

As with the regression tests of the REH, the regression tests of the MEH are also not valid if the variables under consideration have unit roots. In this case, the cointegration tests should be appUed. The exact form of the MEH differs, depending on the markets being considered. For instance, in the case of the gold and silver markets, it should not be possible to forecast one price from the other. Thus gold and silver prices should not be cointegrated. In the case of foreign exchange rates, if the currency markets are efficient, spot exchange rates should embody all relevant information, and it should not be possible to forecast one spot exchange rate as a function of another. That is, the spot exchange rates across currencies should not be cointegrated. The same should be the case with forward rates. On the other hand, the future spot rate and the forward rate should be cointegrated because the forward rate is a predictor of the future spot rate.* The test of the last hypothesis in the regression context starts by estimating the equation

5,+ , = a + PF, + e,

The MEH states that a = 0 and p = 1. However, if both 5,+ , and F, are 1(1) series, the error e, is not a stationary white noise process unless the two variables are cointegrated with a = 0 and (3=1. Some economists try to avoid the nonstationary problem by considering the following equation.

5,+ , - S, = a+ P(F, - S,) + e,

Then they test whether or not a = 0 and = 1. However, the model is useful only if (F, - S,) is nonstationary. For simpUcity, assume that both variables are pure random walk processes. Then the left-hand side of the equation is stationary because (5,+, - S,) is a white noise process. But there is no guarantee that the variable on the right-hand side, F, - S„ is stationary. Note that (F, - S,) can be decomposed into (F, - F, ,) + (F, , - S,). While the first component is stationary, the second will have the same property only if the MEH holds. One can think of regressing the first difference of 5,+, on the first difference of F,. But this model is misspecified and gives inconsistent estimates of the parameters because the correct model to be used, if the two variables are cointegrated, is the error correction model,

5,,, - S, = a + P(F, - F,,) + -y(F, , - S,) + e,

The regression in first differences omits the last term, thus causing an omitted variable bias. Thus if the exchange rate is nonstationary, most of the models on testing the MEH are inappropriate. If 5,+, and F, are both 1(1), the propef procedure is to use the restricted cointegration test using 5,+, - F,.

14 2 A Summary Assessment of Cointegration

Cointegration tests have been used for a wide variety of problems, such as testing the permanent income hypothesis, testing rationaUty of expectations, testing market efficiency in different markets, and testing purchasing power parity. There are, however, many problems with the use of these tests and their interpretation. In a way, in the case of both unit roots and cointegration, there is too much emphasis on testing and too Uttle on estimation.

We discussed earlier the way the null and alternative hypothesis for the unit root tests and cointegration tests are formulated, and we have also discussed the arbitrariness in the universal use of the 5% and 1% significance levels. Conclusions may be reversed if the null and alternative are reversed. For instance, when the test is conducted with the null hypothesis as a unit root, the null hypothesis is not usually rejected if the conventional significance levels are used, and if the null hypothesis is that the time series is stationary (with the alternative that it has a unit root), again the null hypothesis is not rejected when the conventional significance levels are used. The same is the case with cointegration tests when the null hypothesis is of no cointegration, or of cointegration. There is also the problem of the power of these tests, as discussed in Section 14.4.

Another important issue is that of bivariate versus multivariate cointegrating regressions. The issue is similar to simple versus multiple regression. For instance, and JC, may not be cointegrated, but y, x,, and x, may be cointegrated. If y, JC, and Xl are all 1(1) and there exists a linear combination of these that is 1(0), so that = + fiiXi + e, where e is 1(0), then (y, x,, x are cointegrated. But when we consider

= px, +

since p, = fiiXi + e is 1(1), we will find and x, not to be cointegrated. This is the usual omitted-variable problem. In this case it is wrong to make inferences just because the hypothesis of no cointegration has not been rejected. For instance, if and X refer to prices in two related markets, it is tempting, if the hypothesis of no cointegration is not rejected, to conclude immediately that the two markets are efficient. This is indeed incorrect.

The analogy with the omitted-variable case in regression analysis cannot be pushed too far. In the case of the simple versus multiple regression that we have, if X and Xj are uncorrelated, the coefficient ofx, will be the same in both the regressions (with and without x,). In the case of cointegration, this condition is not sufficient. When will the bivariate and multivariate tests of cointegration give the same results? The answer is: when there is only one unit root process driving all the variables. Suppose that we are considering four series on exchange rates. Then if the matrix in the VAR representation (discussed