Lecture 27 – Cointegration:IITesting for CointegrationThere are two basic approaches that are commonly to test for cointegration.- Residual Based Tests H0: no CI vs. HA: CI Use single-equation regression residualsEngle-Granger; Phillips-Ouliaris- Likelihood-Ratio Based Tests H0: CI of rank r vs. HA: CI of rank r+1H0: CI of rank r vs. HA: CI of rank > rUse restricted and unrestricted VECMsJohansenLet yt be an n-dimensional I(1) process.ConsiderH0: yt is not CIHA: yt is CISo, under H0, α’yt ~ I(1) for all nonzero α in Rn.The Engle-Granger (EG) Test1. Regress y1t on 1,y2t,…,ynt by OLSto get the residual series, tuˆ.2. Fit tuˆ to an ADF regression (no intercept or trend). That is, regress tuˆ on 111ˆ,...,ˆ,ˆptttuuu3. Compute the t-statistic for H0: ρ=1.4. Use the appropriate asymptotic null distribution for this test statistic (which is NOT the DF distribution).The Phillips-Ouliaris (PO) Test1. Regress y1t on 1,y2t,…,ynt by OLSto get the residual series, tuˆ.2. Fit tuˆ to a DF regression (no intercept or trend). That is, regress tuˆ on 1ˆtu3. Compute the t-statistic for H0: ρ=1and modify it as in the PP procedure.4. Use the appropriate asymptotic nulldistribution for this test statistic (which is the same as the asymptotic distribution of the EG stat). Notes- The asymptotic distribution of the EG and PO test stat’s does not depend on thenormalization chosen, i.e., which element of y is place on the l.h.s. of the regression. But, the test will have low power against alternatives in which yt is CI but the element of each CI vector corresponding to y1 (or whichever variable is placed on the l.h.s.) is zero.(For example, if y1t,y2t,y3t are CI but only because y2t and y3t are CI, then a residual-based test with y1t on the l.h.s. will have very low power.)- Aside from the issue above, the normalization selected may affect the test result in finite samples.- Suppose n > 2 and we reject H0. What is the CI rank of
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