# Intuition of second order differencing dependent variable on non-differencing independent regressor regression?

I have two time series sequences. One is $$y_t$$, which is non-stationary, and the other is $$x_t$$, which is stationary. Suppose I would like to do a regression of $$y_t$$ on $$x_t$$ to forecast $$y_t$$. The second order differencing on $$y_t$$ which is $$y_t-2y_{t-1}+y_{t-2}$$ is stationary. Now the regression becomes $$y_t-2y_{t-1}+y_{t-2}=\alpha+\beta x_t+\epsilon_t$$. May I know the interpretation of $$\beta$$ now? Or technically, it is better also to make second order differencing on $$x_t$$ as the new regressor, which could give a sensible intuition of $$x_t$$ on $$y_t$$?Inspired by this question

### My attempt to interpret, please correct me if I am wrong

If $$x_t$$ is stationary, then I assume that $$x_t$$ is independent with $$x_{t-n}$$, so now my regression equation becomes $$y_t=\alpha+\beta x_t-2y_{t-1}+y_{t-2}+\epsilon_t$$, which means I fixed the coefficients of $$y_{t-1}$$ as $$-2$$ and I fixed the coefficients of $$y_{t-2}$$ as $$1$$ on $$y_t$$, and $$\beta$$ only explains the effect of $$x_t$$ on $$y_t$$, is this correct?

• You might want to read my apparently unappreciated answer to stats.stackexchange.com/questions/169838/… as it reflects on the evolved PDL or ADL to suggest the conditional impact of x (and it;s lags) on y. – IrishStat Jun 3 '19 at 14:26

## 1 Answer

The second difference is analogous to the second derivative of a function which, broadly, tells you how "spiky" a function is. The model you're specifying is basically regressing that function on covariates. I.e. how covariate influence the spikiness of the time series.