The **Ljung (pronounced Young) Box test** (sometimes called the modified Box-Pierce, or just the

*Box test*) is a way to test for the

*absence*of serial autocorrelation, up to a specified lag

*k*.

The test determines whether or not errors are iid (i.e. white noise) or whether there is something more behind them; whether or not the autocorrelations for the errors or residuals are non zero. Essentially, it is a test of *lack *of fit: if the autocorrelations of the residuals are very small, we say that the model doesn’t show ‘significant lack of fit’.

## Ljung Box Test Hypotheses

The null hypothesis of the Box Ljung Test, H_{0}, is that our model *does not* show lack of fit (or in simple terms—the model is just fine). The alternate hypothesis, H_{a}, is just that the model *does *show a lack of fit.

A significant p-value in this test rejects the null hypothesis that the time series *isn’t *autocorrelated.

## Calculating the Ljung Box Test Statistic

Most statistical packages can run a Ljung Box test. For example, in R, you can implement the test with the **Box.test** function.

To run the Ljung Box test by hand, you must calculate the statistic Q. For a time series Y of length n:

=

Where:

- r
_{j}= the accumulated sample autocorrelations, *m*= the time lag.

We reject the null hypothesis and say that the model shows lack of fit if

- χ
^{2}_{1-α,h}= the value found on the chi-square distribution table for significance level α and h degrees of freedom.

When the Ljung Box test is applied to the residuals of an ARIMA model, the degrees of freedom h must be equal to m-p-q, where p and q are the number of parameters in the ARIMA(p,q) model.

## Alternative ways to test for autocorrelation

The Durbin-Watson test is a popular way to test for autocorrelation. However, it can’t be used if you have lagged dependent variables; If you have those, used the Breusch-Pagan-Godfrey test instead.

The Ljung Box test is a **modification of the Box Pierce Test**. The difference is in how the test statistic is calculated (Kleiber & Zeileis, 2008). Both approximate a chi-squared statistic, based on autocorrelations up to order *p*.

**Box Pierce test statistic**: n * sum of squared autocorrelations,**Ljung Box test statistic**: the squared autocorrelations are weighted at lag*j*by (*n*+ 2)/(*n*–*j*) (*j*= 1…,*p*).

## References

NIST. Box-Ljung Test. Engineering Statistics Handbook 6.4.4.8.1. Retrieved from https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4481.htm on July 28, 2014.

Hyndman, Rob J. Thoughts on the Ljung-Box Test. Hyndsight, published online January 24 2014. Retrieved from https://robjhyndman.com/hyndsight/ljung-box-test/ on July 28, 2014.

Kleiber, C. & Zeileis, 2008. Applied Econometrics with R. Springer Science & Business Media, Dec 10, 2008

PennState Statistics. Lesson 3.2: Diagnostics. Stat 510 Applied Time Series Analysis. Retrieved from https://onlinecourses.science.psu.edu/stat510/node/65/ on July 28, 2014.