Question

a) Define what is meant by cointegration. Describe the difference between Engle-Granger and Johansen cointegration tests.

b) Explain the design and the purpose of the Chow test for structural break.

Answer 1

Cointegration:

Cointegration is nothing but if two or more series is integrated individually but some linear combination of them has lower order of integration, then the series is said to be cointegrated. Formally if (X,Y,Z) are integrated in order of d, and there exists coefficients a,b,c such that Ax+BY+CZ is integrated of order 0, then x,y,z are cointegrated. Cointegration is considered to be an important property in time series analysis. In a stock market index & the price of its futures contract move in a random walk, testing the hypothesis that there is significant connection between futures price & spot price could be done by testing for existence of cointegrated combination of the two series.

Eagle granger two step method:

If xt & yt are non stationary & cointegrated, then a linear combination of them must be stationary. Where

Yt – bxt = ut

Ut can be used to test for stationary, but since the value of ut is not known, it must be estimated first using least squares method & then run the statinarity test on ut. A second regression is then run on differenced variables from the first regression & lagged residuals of ut-1 are included as regressor.

Johansen test:

This allows for more than one cointegrating relationship, unlike the eagle granger method. But the test is applicable for large sample. The results won’t be reliable for small samples & one should make use of auto regressive distributed lags (ARDL).

Chow test for structural breaks:

A chow test is used to determine whether there exists a structural break in the time series. A structural break in one series can give useful clues whether a change has been propagated across other variables. The chow test is mainly useful when it comes to analyzing structural breaks across time series that are normally stationary.

The structural test is considered to be useful because it can help in identifying the timing & statistical significance of program effects even if the timing is uncertain & can give difference inference from methods used usually.