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I am using the function McLeod.Li.test in the TSA package. According to the help file in R, this function “Performs the McLeod-Li test for conditional heteroscedasticity (ARCH).”

However, it has been mentioned on page 50 of the book “Introduction to Time Series and Forecasting, Brockwell and Davis, 2nd edition” that the McLeod-Li test “can be used as a further test for the iid hypothesis”.

Question: Is it true to say that the McLeod-Li test can be used to check the “conditional heteroscedasticity” as well as the “iid assumption”? If not, which one is the correct null hypothesis?

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The McLeod-Li test can be used as a test of independence and it can also be used to test the hypothesis of no ARCH. In fact, the McLeod-Li test can be used as a general misspecification test against nonlinearity. See Teräsvirta, Tjøstheim and Granger (2010), (pages 140-1 and 178-9).

You may find it useful to know that it is stated in Patterson, and Ashley (2000), (page 40), that the null hypothesis is that the series is i.i.d.

I hope this information helps you, and rather than go into the particulars, I refer you to the aforementioned sources.

As a side note, it's worth mentioning that some tests can be used for a variety of purposes. For example, the BDS test can be used to test the i.i.d hypothesis, but it can also be used to test for the presence of non-linearity. I think this is a similar situation to the usage of the McLeod-Li test.

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