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I am trying to fit a time series model using data of auto sales (DAUTONSA from FRED) and noticed that there is evidence of serial correlation. I’ve tried fitting a model with 4 lags but noticed that the second lag was insignificant. When I dropped the second lag but kept the first, third, and fourth lag, the BIC of the model was better. Do I have to keep the second lag or can I drop it and ignore it?

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You do not need to use all lags up to the maximum lag, you can very well skip some. This does not invalidate the model. So yes, you can drop the second lag, and doing that will make sense if BIC is a relevant criterion for your problem.

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It can be easier intuitively instead of dropping to think about including the lags, then it'll become obvious that you do not need to include every lag between the lags that are included.

For instance, consider a typical seasonal time series specified with a baskshift (B) operator notation: $$(1-B^{12})(1-B)x_t=\varepsilon_t$$ $$x_t=\phi_1x_{t-1}+\phi_{12}x_{t-12}+\phi_{13}x_{t-13}+\varepsilon_t$$ You see that only three lags are present. The rest were not dropped, they were not included.

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  • $\begingroup$ The first equation seems to be misssing coefficients, otherwise $\phi_1=\phi_{12}=-\phi_{13}=1$. $\endgroup$ – Richard Hardy Apr 23 at 17:53
  • $\begingroup$ @RichardHardy, I clarified the answer, B is a backshift operator notation here $\endgroup$ – Aksakal Apr 23 at 17:55
  • $\begingroup$ Still looks the same to me: isn't $B^k x_t=x_{t-k}$? $\endgroup$ – Richard Hardy Apr 23 at 19:43
  • $\begingroup$ @RichardHardy, it's actually $\phi_kx_{t-k}$. You might be more used to notation $(1-B^d)\phi(B)x_t$ $\endgroup$ – Aksakal Apr 23 at 19:47
  • $\begingroup$ Indeed, I am quite used to the latter notation. I do not think I have seen the former type of notation in time series literature ever before. Could you point out any credible sources (e.g. textbooks) using it? $\endgroup$ – Richard Hardy Apr 24 at 9:29

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