You can calculate residuals that have auto regression in them. Residuals are simply the difference between what you predict with the model and the observed results. There is no way that having autoregression could make it impossible to generate residuals, residuals are how you test for errors in the assumptions usually (specialized tests using lags is how you test for autoregression, some tests are better than others). You might look up the use of HAC to deal with autoregression (or the use of ARMA or ARIMA models).