198 The Basics of financial economeTrics
cointegration tests, the null hypothesis states that the variables lack
cointegration and the alternative claims that they are cointegrated.Step 3. To test for cointegration, test for the stationarity in zt. The most
often used stationarity test is the Dickey-Fuller test. That is, the fol-
lowing autoregression of the error term should be considered:Δzt = p zt− 1 + ut (10.3)
where zt is the estimated residual from equation (10.2). The Dickey-
Fuller test focuses on the significance of the estimated p. If the esti-
mate of p is statistically negative, we conclude that the residuals, zt,
are stationary and reject the hypothesis of no cointegration.
The residuals of equation (10.3) should be checked to ensure the
residuals are not autocorrelated. If they are, the augmented Dickey-
Fuller test should be employed. The augmented Dickey-Fuller test is
analogous to the Dickey-Fuller test but includes additional lags of Δzt
as shown in the following equation:Δzt = p zt− 1 + a 1 Δzt− 1 +... +anΔzt−n + ut (10.4)
The augmented Dickey-Fuller test for stationarity, like the Dickey-
Fuller test, tests the hypothesis of p = 0 against the alternative
hypothesis of p < 0 for equation (10.4).
Generally, the OLS-produced residuals tend to have as small a
sample variance as possible, thereby making residuals look as sta-
tionary as possible. Thus, the standard t-statistic or augmented
Dickey-Fuller test may reject the null hypothesis of nonstationarity
too often. Hence, it is important to have correct statistics. Fortu-
nately, Engle and Yoo provide the correct statistics.^10 Furthermore,
if it is believed that the variable under investigation has a long-run
growth component, it is appropriate to test the series for stationar-
ity around a deterministic time trend for both the Dickey-Fuller and
augmented Dickey-Fuller tests. This is accomplished by adding a
time trend to equations (10.3) or (10.4).Step 4. The final step for the Engle-Granger conintegration test involves
estimating the error-correction model. Engle and Granger showed
that if two variables are cointegrated, then these variables can be(^10) Robert Engle and Byung Yoo, “Forecasting and Testing in Co-integrated Systems,”
Journal of Econometrics 35 (1987): 143−159.