I am trying to forecast using independent economic variables such as GDP, unemployment, etc. I made a model that looked like a good fit, as it had a high R-square value, but the Durbin-Watson statistic was very low (0.30). I'm guessing it wasn't actually a "good" fit, it appears to be a spurious relationship lm (NPL ~ MortgageRate + Unemployment + 3monthTres)
Anyways, to correct the autocorrelation, I decided to difference the y and x variables (which worked, my new Durbin-Watson value is 1.8). I followed the logic:
$$\Rightarrow \Delta Y_t=B\Delta X_t+\Delta e_t$$
So my questions are this:
My coefficients for the differenced model are slightly different than the original linear model I made, which confuses me because theoretically, isn't the beta supposed to not change? I assumed it was because I lost an observation with the lagged term - does that make sense?
If the beta doesn't change, how does this change the model?
Since the new model doesn't have an intercept, can I forecast using the predicted changes in Y values and just add it to the previous Y value? Or should I be using a different method to forecast?
Is this a good method to use? I am hesitant to do any ARIMA type of models because my sample size is small (20-30 observations) so I don't want to lose observations through multiple lags. I also tried doing linear regression with just one lagged term, but it had a dominant effect on my model (lagged term coefficient was 0.80, but the coefficients for my other terms were 0.0001 and under).
Any help is much appreciated!