The **Ljung-Box test **is a statistical test that checks if autocorrelation exists in a time series.

It uses the following hypotheses:

**H _{0}:** The residuals are independently distributed.

**H _{A}:** The residuals are not independently distributed; they exhibit serial correlation.

Ideally, we would like to fail to reject the null hypothesis. That is, we would like to see the p-value of the test be greater than 0.05 because this means the residuals for our time series model are independent, which is often an assumption we make when creating a model.

This tutorial explains how to perform a Ljung-Box test in Python.

**Example: Ljung-Box Test in Python**

To perform the Ljung-Box test on a data series in Python, we can use the acorr_ljungbox() function from the **statsmodels **library which uses the following syntax:

**acorr_ljungbox(x, lags=None)**

where:

**x:**The data series**lags:**Number of lags to test

This function returns a test statistic and a corresponding p-value. If the p-value is less than some threshold (e.g. α = .05), you can reject the null hypothesis and conclude that the residuals are not independently distributed.

The following code shows how to use this function to perform the Ljung-Box test on the built-in statsmodels dataset called “SUNACTIVITY”:

import statsmodels.api as sm #load data series data = sm.datasets.sunspots.load_pandas().data #view first ten rows of data series data[:5] YEAR SUNACTIVITY 0 1700.0 5.0 1 1701.0 11.0 2 1702.0 16.0 3 1703.0 23.0 4 1704.0 36.0 #fit ARMA model to dataset res = sm.tsa.ARMA(data["SUNACTIVITY"], (1,1)).fit(disp=-1) #perform Ljung-Box test on residuals with lag=5 sm.stats.acorr_ljungbox(res.resid, lags=[5], return_df=True) lb_stat lb_pvalue 5 107.86488 1.157710e-21

The test statistic of the test is **107.86488** and the p-value of the test is **1.157710e-21**, which is much less than 0.05. Thus, we reject the null hypothesis of the test and conclude that the residuals are not independent.

Note that we chose to use a lag value of 5 in this example, but you can choose any value that you would like to use for the lag. For example, we could instead use a value of 20:

#perform Ljung-Box test on residuals with lag=20 sm.stats.acorr_ljungbox(res.resid, lags=[20], return_df=True) lb_stat lb_pvalue 20 343.634016 9.117477e-61

The test statistic of the test is **343.634016 **and the p-value of the test is **9.117477e-61**, which is much less than 0.05. Thus, we reject the null hypothesis of the test once again and conclude that the residuals are not independent.

Depending on your particular situation you may choose a lower or higher value to use for the lag.