# Is it valid to forecast using a differenced regression equation?

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:

$$Y_t−Y_{t−1}=BX_t+e_t−BX_{t−1}−e_{t−1}$$

$$\Rightarrow \Delta Y_t=B\Delta X_t+\Delta e_t$$

So my questions are this:

1. 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?

2. If the beta doesn't change, how does this change the model?

3. 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?

4. 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!

1. Of course it will change your beta since you change your X and since you do MCO I presume, it will change slightly the estimates.

2. When you differenciate the series, the purpose is for autocorrelation in residuals, not on the estimate, basically you added a mobile average component to take account of autocorrelation in residuals.

3. Basically, you just cumulatively add the diff-forecast to the last cumulative obs. Lets yd be the diff data and y the original then:

yt+1 = yt + ydt+1 yt+2 = yt+1 + ydt+2 = yt + ydt+2 + ydt+2 and you continue like that

1. First of all, you should do unit root test for each variable serie to see if one is I(1) or I(0) cause if you have at least one I(1) in a linear combination, you will have a I(1) target variable.

What I suggest you if you are motivated to learn and if you have a fair strong relationship between those series is the VAR modeling, especially the VECM (johansen cointegration) that can be pretty good.

• Hi: Along the lines of what josef_joestarr said ( but a little different ) , you're better off estimating an ECM. It's a myth that both variables need to be I(1) for this to be valid. If you view it as an ECM, you'll be keeping the levels information and also incorporating the differences. What you did by differencing throws out the level information if there is any. – mlofton Jul 3 '19 at 2:26
• I was looking for something that explains the myth and this was not it but it's not bad in terms of explaining how an ECM can be "valid" even if variables are not I(1). Note that the interpretation is somewhat different from the ECM interp. nuff.ox.ac.uk/politics/papers/2005/… – mlofton Jul 3 '19 at 2:35