# Should I keep a non-significant intercept in a GARCH model?

I've estimated an ARMA-GARCH model. All the parameters except the intercept ($\omega$) in the GARCH part are significant. So I tried to estimate the model again imposing that $\omega = 0$. I compared the information criteria of the two models and found out that keeping omega in the model gives the model a better fit.

Questions:

1. Does it make sense?
2. Can I keep $\omega$ in the model, although it appears non significant at 5% level, just because of better information criteria?
3. Is it rigorous from a statistical point of view?
• Lot of similar questions on this site, start with stats.stackexchange.com/questions/11064/… Feb 12, 2017 at 15:51
• @kjetilbhalvorsen, indeed, though the GARCH case is a little special as discussed in the first two bullet points of my answer, for example. Mar 29, 2018 at 9:41

The intercept of a GARCH model should be kept in the model for the following reasons.

1. If you force the intercept to be zero AND the sum of ARCH and GARCH coefficients is less than one (which will happen by the design of the estimation procedure that restricts the parameters to a stationary region defined by their sum being less than one), then your model implies the conditional variance is decreasing over time, which is generally undesirable.
2. If you force the intercept to be zero AND the sum of ARCH and GARCH coefficients equals one, then you end up with an EWMA estimator of the conditional variance, and the conditional variance is a random walk (which again might be undesirable).
3. Also, note that testing for $\text{H}_0\colon \text{intercept}=0$ is testing a hypothesis that the parameter is on the boundary of the space (intercept cannot be negative). This might have implications on the null distribution of the test statistic, making the regular $p$-value associated with the $t$-statistic inappropriate. There might be some relevant information in Francq & Zakoian "Testing the nullity of GARCH coefficients: correction of the standard tests and relative efficiency comparisons" (2009), but I am not entirely sure.

Also note that statistical significance need not be a good basis for variable selection. Rob J. Hyndman explains in his blog post "Statistical tests for variable selection" why statistical significance testing should not be used for variable selection and suggests using cross validation or information criteria instead.

Statistical significance is not usually a good basis for determining whether a variable should be included in a model <...>. Statistical tests were designed to test hypotheses, not select variables. Tests on coefficients are answering a different question from whether the variable is useful in forecasting. It is possible to have an insignificant coefficient associated with a variable that is useful for forecasting. It is also possible to have a significant variable associated with a variable that is better omitted when forecasting.

The post is so good and clear that you better read the entire piece.

Hence, the answers to your questions 1., 2. and 3. are "YES", "YES" and "YES".

• Hi, Richard again. May I know how you exactly interpret both the intercepts in the return and volatility equations within GARCH?
– Eric
Mar 29, 2018 at 9:17
• If the conditional mean equation only has an intercept and no other terms, then this is just the unconditional mean -- the mean return. If it has autoregressive terms, then this is a little different and I am not sure if there is a nice interpretation for that case. The intercept in the GARCH equation is basically a lower bound for the conditional variance. Mar 29, 2018 at 9:39
• Thank you so much, Richard. So if my GARCH model has all significant coefficients including the intercept, shall I say that the lower bounds for the conditional mean and variance are both significant? (my model has significant intercept both on the return and volatility equations).
– Eric
Mar 29, 2018 at 9:41