I'm doing a regression analysis with non stationary time series. If I run the regression the residuals are auto correlated and non stationary. If i add a lag of the dependent variable (the estimated coefficient is about 0.75 so the dynamic is stable), residuals become well behaved and i have a really high R^2. It's ok to proceed in this way or it's still a spourious regression?Standard errors are still valid?
There are different types of nonstationarity. Have you performed stationarity tests (e.g. Augmented Dickey-Fuller or KPSS) that reveal a unit root? If there is a unit root, estimate the model in first-differences. If there the series is $I(0)$ and there is a deterministic trend, then include $t$ as a regressor. I am assuming, based on the way you have written your question, that you are working with a univariate time series.