# Difference between Ljung Box and McLeod Li Test?

I'm a little confused. What is the difference between the Ljung-Box test and the McLeod-Li test?

The Ljung-Box test is a test for linearity in the mean and the McLeod-Li is a test for nonlinearity on the mean and/or in variance?

Ljung-Box test is a test for autocorrelation in either raw data or model residuals. The test is described quite clearly in a Wikipedia article. I will add that alternatives to Ljung-Box test exist and in some situations are preferred; see this thread comparing Ljung-Box test with Breusch-Godfrey test.

McLeod-Li test is a test for autoregressive conditional heteroskedasticity in either raw data or residuals from a conditional mean model (but not for residuals from a GARCH model; there Li-Mak test should be used instead).

The similarity between the two tests is that McLeod-Li test takes the form of a Ljung-Box test applied on squared raw data or squared model residuals. Despite the similarity in their shapes, the purposes of the tests are different, as explained above.

Also, what you state in your second paragraph is incorrect.

"Li-Mak test should be used instead". I'm doing a Time Series Financial Course and we won't see this test. Is [this/there] a better test?

When applied on standardized residuals from a GARCH model, McLeod-Li test does not have the usual null distribution, thus you do not have a benchmark to compare the test statistic to. Consequently, you cannot obtain a p-value or tell whether the test statistic is above or below a 5% (or whatever %) critical value. Meanwhile, Li-Mak test is especially suited for this situation. See Li, W. K. and Mak, T. K. (1994) "On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627–36.

• Thanks Richards! Interesting your observation: "Li-Mak test should be used instead". Im doing a Time Series Financial Course and we wont see this test. Is a better test? – digsydinner Oct 2 '15 at 19:09
• When applied on standardized residuals from a GARCH model, McLeod-Li test does not have the usual null distribution, thus you do not have a benchmark to compare the test statistic to, and consequently you cannot obtain a $p$-value or tell whether the test statistic is above or below a 5% (or whatever %) critical value. Meanwhile, Li-Mak test is especially suited for this situation. See Li, W. K. and Mak, T. K. (1994) On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627–36. – Richard Hardy Oct 2 '15 at 19:23