# For autoregressive time series modeling, does the AR(p) regressors have to be in order despite insignificance?

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?

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.
• The first equation seems to be misssing coefficients, otherwise $\phi_1=\phi_{12}=-\phi_{13}=1$. – Richard Hardy Apr 23 at 17:53
• Still looks the same to me: isn't $B^k x_t=x_{t-k}$? – Richard Hardy Apr 23 at 19:43
• @RichardHardy, it's actually $\phi_kx_{t-k}$. You might be more used to notation $(1-B^d)\phi(B)x_t$ – Aksakal Apr 23 at 19:47