In a model I am trying to justify, a mortgage rate spread is estimated by regression on a swap spread using around 40 monthly data-points. The model fails the assumptions of heteroskedasticity and serial correlation but the coefficients are still significant after using white-corrected std. errors (using vcovHAC, sandwich package).
My questions are about how to go about testing the stationarity of the model. I have conducted ADF, KPSS and PP tests.
For dependent and independent variables, all three tests point towards non-stationarity.
For model residuals, KPSS and PP tests points towards non-stationarity while in ADF test, the null hypothesis of unit root is not rejected.
My first question is that can these tests be applied on residuals. Are same set of critical values be used for residuals as for X and Y?
My another question is that if residuals are stationary, can I ignore the non-stationarity of X any Y variables in this model? Is it possible only due to cointegration? What is the way forward if cointegration is not detected (I performed Johansen and Phillips-Ouliaris Cointegration Tests)?
I read in the R documentation for PP test that it is a non-parametric test and robust to heteroskedasticity. Given that PP test gives contradictory results to ADF test, shall I give it more weight in reaching the conclusion?