# AR1 term significant in ARMA(2,1) but not in ARMA(1,1) or ARMA(1,0)

I'm building an GARCH model using function garchFit from "fGarch" package in R. When specifying the mean equation I have some difficulties understanding what's happening.

If I specify the mean equation be an ARMA(1,1) process, there are no significant coefficients.

If I specify an ARMA(2,1) model, the ar1 coefficient suddenly is significant. However, if I then attempt to "validate" this by merely specifying an AR(1) model as the mean equation, the coefficient is not significant again...

Adding further coefficients and building an ARMA(5,5) model gives me a lot more significant coefficients, but none of them are validated when removing the insignificant terms from the system...

In a typical reduced form regression, this would confuse me a lot and I now do not quite know how to proceed.

• Take a simple example. In a multiple regression model, when you add or remove a variable, the statistical significance of estimates of coefficients on the other variables change (unless the added/removed variable is uncorrelated to any linear combination of the other variables). In ARMA models it is something similar. You add or remove a term, and significance of the other terms changes. This is nothing special. Instead of focusing on significance, better look at other diagnostics (of model residuals) and/or predictive performance. Jul 10 '16 at 10:17
• @Richard That strikes me as a reasonable answer to teh question Jul 10 '16 at 11:04
• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? Feb 16 '17 at 14:31

Also, statistical significance is not such an interesting property in ARMA models. Recall that interpretation of individual coefficients of ARMA models is not that meaningful; to better understand the time series development at hand one should rather check impulse-response functions. If you are going to use your model for description of the data or forecasting, better look at diagnostics of model residuals (ideally, they should closely resemble an $i.i.d.$ process) or predictive performance, respectively.