Cointegration 197
engle-granger Cointegration tests
The Engle-Granger conintegration test, developed by Engle and Granger,^7
involves the following four-step process:
Step 1. Determine whether the time series variables under investigation
are stationary. We may consider both informal and formal meth-
ods for investigating stationarity of a time series variable. Informal
methods entail an examination of a graph of the variable over time
and an examination of the autocorrelation function. The autocor-
relation function describes the autocorrelation of the series for vari-
ous lags. The correlation coefficient between xt and xt−i is called the
lag i autocorrelation. For nonstationary variables, the lag 1 auto-
correlation coefficient should be very close to one and decay slowly
as the lag length increases. Thus, examining the autocorrelation
function allows us to determine a variable’s stationarity.
Unfortunately, the informal method has its limitations. For station-
ary series that are very close to unit root processes, the autocorrelation
function may exhibit the slow-fading behavior as lag length increases.
If more formal methods are desired, the Dickey-Fuller statistic, the
Augmented Dickey-Fuller statistic,^8 or the Phillips-Perron statistic^9
can be employed. These statistics test the hypothesis that the variables
have a unit root, against the alternative that they do not. The Phillips-
Perron test makes weaker assumptions than the Dickey-Fuller and
augmented Dickey-Fuller statistics and is generally considered more
reliable. If it is determined that the variable is nonstationary and the
differenced variable is stationary, proceed to Step 2.Step 2. Estimate the following regression:yt = c + dxt + zt (10.2)To make this concrete, let yt represent some U.S. stock market
index, xt represents stock dividends on that stock market index, and
zt the error term. Let c and d represent regression parameters. For(^7) Robert Engle and Clive Granger, “Cointegration and Error-Correction: Representa-
tion, Estimation, and Testing,” Econometrica 55 (1987): 251−276.
(^8) David Dickey and Wayne Fuller, “Distribution of the Estimates for Autoregressive
Time Series with a Unit Root,” Journal of the American Statistical Association 74
(1979): 427−431.
(^9) Peter Phillips and Pierre Perron, “Testing for a Unit Root in Time Series Regres-
sion,” Biometrica 75 (1988): 335−346.