Testing for Autocorrelation

Motivation

Sometimes, we’ll need to know if there is any autocorrelation of any form in our series at all. In other words, we will be interested in whether or not all autocorrelations up to some lag are 0. The most common case arises when evaluating the results of a model, such as an ARIMA model.

ARIMA models require that prediction errors (or residuals) are independent and identically distributed (IID) with a constant mean and variance, i.e., white noise. In this definition, independent means that the residuals are not serially correlated. If an ARIMA model is not producing this type of residual, it fails to capture all the underlying patterns in the data. In this lesson, we will go through some tests that will let us assess if our data is autocorrelated.

The Ljung-Box test

The Ljung-Box (LB) test is a commonly used test in econometrics and finance to assess the presence of autocorrelation in a time series. Its null hypothesis is that all autocorrelations up to lag $k$ are jointly 0:

The alternative hypothesis, $H_1$ of the LB test, is more loosely defined. It allows for any autocorrelation coefficient to be different from 0. This means that $H_0$ can be disproved no matter how many autocorrelations are not 0, as long as one isn’t.

The LB test has a cumulative nature, as it is a function of the sum of several autocorrelation coefficients. We can see this cumulative nature in the definition of the LB test statistic below:

In the equation above, ...