How many lags to use in the Ljung-Box test of a time series?

Question

After an ARMA model is fit to a time series, it is common to check the residuals via the Ljung-Box portmanteau test (among other tests). The Ljung-Box test returns a p value. It has a parameter, h, which is the number of lags to be tested. Some texts recommend using h=20; others recommend using h=ln(n); most do not say what h to use.

Rather than using a single value for h, suppose that I do the Ljung-Box test for all h<50, and then pick the h which gives the minimum p value. Is that approach reasonable? What are the advantages and disadvantages? (One obvious disadvantage is increased computation time, but that is not a problem here.) Is there literature on this?

To elaborate slightly.... If the test gives p>0.05 for all h, then obviously the time series (residuals) pass the test. My question concerns how to interpret the test if p<0.05 for some values of h and not for other values.

Answer 1

The answer definitely depends on: What are actually trying to use the $Q$ test for?

The common reason is: to be more or less confident about joint statistical significance of the null hypothesis of no autocorrelation up to lag $h$ (alternatively assuming that you have something close to a weak white noise) and to build a parsimonious model, having as little number of parameters as possible.

Usually time series data has natural seasonal pattern, so the practical rule-of-thumb would be to set $h$ to twice this value. Another one is the forecasting horizon, if you use the model for forecasting needs. Finally if you find some significant departures at latter lags try to think about the corrections (could this be due to some seasonal effects, or the data was not corrected for outliers).

Rather than using a single value for h, suppose that I do the Ljung-Box test for all h<50, and then pick the h which gives the minimum p value.

It's a joint significance test, so if the choice of $h$ is data-driven, then why should I care about some small (occasional?) departures at any lag less than $h$, supposing that it is much less than $n$ of course (the power of the test you mentioned). Seeking to find a simple yet relevant model I suggest the information criteria as described below.

My question concerns how to interpret the test if $p<0.05$ for some values of $h$ and not for other values.

So it will depend on how far from the present it happens. Disadvantages of far departures: more parameters to estimate, less degrees of freedom, worse predictive power of the model.

Try to estimate the model including the MA and\or AR parts at the lag where the departure occurs AND additionally look at one of information criteria (either AIC or BIC depending on the sample size) this would bring you more insights on what model is more parsimonious. Any out-of-sample prediction exercises are also welcome here.

Answer 2

Before you zero-in on the "right" h (which appears to be more of an opinion than a hard rule), make sure the "lag" is correctly defined.

http://www.stat.pitt.edu/stoffer/tsa2/Rissues.htm

Quoting the section below Issue 4 in the above link:

"....The p-values shown for the Ljung-Box statistic plot are incorrect because the degrees of freedom used to calculate the p-values are lag instead of lag - (p+q). That is, the procedure being used does NOT take into account the fact that the residuals are from a fitted model. And YES, at least one R core developer knows this...."

Edit (01/23/2011): Here's an article by Burns that might help:

http://lib.stat.cmu.edu/S/Spoetry/Working/ljungbox.pdf

Answer 3

Escanciano and Lobato constructed a portmanteau test with automatic, data-driven lag selection based on the Pierce-Box test and its refinements (which include the Ljung-Box test).

The gist of their approach is to combine the AIC and BIC criteria --- common in the identification and estimation of ARMA models --- to select the optimal number of lags to be used. In the introduction of they suggest that, intuitively, ``test conducted using the BIC criterion are able to properly control for type I error and are more powerful when serial correlation is present in the first order''. Instead, tests based on AIC are more powerful against high order serial correlation. Their procedure thus choses a BIC-type lag selection in the case that autocorrelations seem to be small and present only at low order, and an AIC-type lag section otherwise.

The test is implemented in the R package vrtest (see function Auto.Q).

Answer 4

... h should be as small as possible to preserve whatever power the LB test may have under the circumstances. As h increases the power drops. The LB test is a dreadfully weak test; you must have a lot of samples; n must be ~> 100 to be meaningful. Unfortunately I have never seen a better test. But perhaps one exists. Anyone know of one ?

Paul3nt