# Lag length selection in ARCH-LM test

How to determine the lag length $q$ in the ARCH-LM test?

If I set $q=1$, the result is homoscedastic (failed to reject H0).
But if I set $q=4$ for example, the result is heteroskedastic (reject H0).

Which one should I use?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? – Richard Hardy Feb 20 '17 at 19:17

## 1 Answer

The ARCH-LM test is a portmanteau test. It tests a number of lags (lag 1 through lag $q$) at once as a group and tells you whether the average ARCH effect within the group is large. The maximum lag $q$ governs the tradeoff between power and generality:

• Small $q$
If you have a small group of lags, you will have good power against any individual lag within the group, because the average will be relatively sensitive to each individual. If there is a substantial ARCH effect among them, the test statistic will pick that up.
However, a small group means that you might miss a lag with substantial ARCH effect that is outside the group, unlike in the case of large $q$.

• Large $q$
If you have a large group, you will have low power against any individual lag within the group, because the average will be relatively insensitive to each individual. If there is a substantial ARCH effect in the group, the test statistic might not pick that up because all the other lags without ARCH effects would swamp the one with the ARCH effect. But at least you cover a large group and do not ignore the higher order lags altogether as in the small $q$ case.

In general, lag length selection can be motivated by subject-matter knowledge. You should check the lags where you could plausibly expect ARCH effects. You do not need to check too distant lags where ARCH effects in a given sample could mainly be due to chance.

You could try a few different lag lenghts to better elicit possible ARCH effects.

Diagnostic plots of autocorrelations and partial autocorrelations of squared residuals could also be useful.

• Thank you for your explanation. In Enders, on engle's research, first he set lag=1 but it failed to reject H0, then he set lag=4, and he succeed to reject H0. And he concluded that there was arch effect. – Fatma Sep 5 '16 at 9:08
• So, how far should i set the q if lag=4 was also failed to reject H0? – Fatma Sep 5 '16 at 9:16
• It depends on the data frequency (high frequency allows for longer lags) and subject-matter knowledge (probably you expect ex ante that some lag will be significant). Based on the limited information you have given us, advising you to rely on any particular lag lenght (like 4) would be pretty careless. So far there is no simple answer. – Richard Hardy Sep 5 '16 at 16:47
• Why partial test in mean and variance equation on garch model using z statistic? Not t statistic like in arima model? – Fatma Sep 5 '16 at 23:50
• What partial test? Since it is not the ARCH-LM test (if I understand correctly), please ask a separate question. Also, be aware that you may accept the answer by clicking on the tick mark to the left of it. – Richard Hardy Sep 6 '16 at 7:15