I am trying to empirically estimate the coefficient for the Okun's law as a relationship between output growth and unemployment. I am using the simple gap version, where I regress real output growth (ind. variable) computed as a log difference of real GDP on change in unemployment from previous period (dep. variable). I use quarterly data with 59 observations. I want to estimate it as simple OLS, test for autocorrelation (I know there probably will be some) and other possible violations of the assumptions and then add proper lags of both dependent and independent variables to obtain dynamic model with higher explanatory power.
However, for real GDP growth (which is log differenced real GDP), the ADF and ADF-GLS tests didn't reject null hypothesis of unit root (p-values 0.28 and 0.11 for one lag, for more lags p-values are larger), KPSS rejected the null of stationarity (test statistic 0.858 vs critical value at 99% level of confidence 0.727). On the other hand, difference in unemployment seems to be (marginally) stationary (ADF and ADF-GLS p-values: 0.19 and 0.054), KPSS didn't reject null (test statistic 0.356).
I don't want to difference again, as I have a priori specification from economic theory, which would be ruined. I also tested for cointegration but residuals are I(1), so I cannot use it. It's kind of obvious there should be a strong relationship between unemployment and economic growth, but is it a sufficient argument against formal statistical tests to regress nonstationary on (almost) stationary variable?