I am trying to estimate the long-run relationship between equity prices and bond default risk measures using a 250*250 panel of firms. There is strong theoretical evidence that there is a relationship between the two, so I expected cointegration to exist. I have used Hadri, Fisher and IPS unit root tests to check the order of integration of my dependent / independent variables; the Hadri test rejects the null that all panels are stationary very convincingly, but the other tests also reject the H0 of unit roots with a p value of 0. Therefore I conclude that some panels contain stationary variables and others unit root variables. How should I approach testing / estimating the cointegration? I have used Pedroni and Westerlund tests to check for cointegrating relationships, and conclude that cointegration exists based on this test. But I am concerned because two I(0) variables cannot cointegrate, and the order of integration of vars in each panel is questionable.
How should I proceed? I do not know which panels contain stationary/nonstationary series - or in what percentage of my panels the series are not of the same order of integration. Are the above cointegration tests valid in this context? Should I just assume, based on theory, that cointegrating relationships should exist and proceed with this assumption?
Thank you in advance for your generous help.