I have two I(1) time series and I regressed one against the other and found that it had low to moderate R-squared but my DW statistic is about 0.015. I know the literature says this is the case of spurious regression? Now, upon running co-integration tests on the residuals (I ran an ADF test using updated MacKinnon's p table, used Phillips Ouliaris test, Johansen test and Elliott Rothenberg and stock test). Now, all my tests pass except for Phillips Ouliaris and Johansen test. These are the only tests where I am not getting the residuals from the data. I believe the PO test automatically runs an regression of y against x and uses the Phillips Ouliaris distribution rather than the ADF distribution on residuals.
My main question is, which test do I trust and whether these tests even make any sense considering I have a spurious regression phenomena? I believe if the two series are co-integrated, then the residuals won't be spurious correct? So my main questions are the follows:
- Can you have co-integration even with spurious regression?
- which test do I ultimately have to chosen from? PO test, Johansen test (both these tests accept the series are not co-integrated). ERS test passes on residuals and so does the R functions
- My time series is from 1998 to 2015. Sometimes, daily gives me co-integration, but monthly doesn't. What time frame is most acceptable?