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?

  • $\begingroup$ 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. $\endgroup$ – IrishStat Jun 3 '19 at 14:26

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.


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