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Identifying White Noise in VAR Model Residuals

Jan 23, 2024

One of the key assumptions in a vector autoregression (VAR) model concerns the residuals or error term of the model. Ideally, these residuals should be white noise—random and uncorrelated over time. But how can you determine if your residuals meet this criterion? This article outlines three common methods for verifying the presence of white noise in VAR residuals: plotting the residuals, performing the Ljung-Box test, and analyzing the autocorrelation function (ACF) and partial autocorrelation function (PACF).

  1. Plotting the Residuals: Visual inspection of the residuals can provide an initial indication of whether they might be white noise. Ideally, a plot of the residuals should reveal randomness, with no trend or seasonality. However, this method is subjective, and further statistical tests should be conducted for a more rigorous assessment.

  2. Ljung-Box Test: The Ljung-Box test is a statistical test that examines the autocorrelations in a time series dataset. In the context of VAR residuals, the null hypothesis of this test is that the residuals are independent (i.e., white noise). A low p-value (< 0.05) would indicate that the residuals are not white noise, while a high p-value (≥ 0.05) would suggest that the residuals are likely to be white noise. Note that the Ljung-Box test should be conducted for different lag lengths to ensure robustness.

  3. Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF): Analyzing the ACF and PACF plots can provide additional insight into whether the residuals are white noise. For a time series dataset that exhibits white noise, both the ACF and PACF should exhibit insignificant autocorrelations (i.e., within the confidence intervals) at all lags. If significant autocorrelations are present, the residuals are not white noise.

In summary, determining if your VAR model residuals are white noise involves a combination of visual inspection, statistical tests, and analyzing the autocorrelation properties of the residuals. By ensuring that your residuals meet this crucial assumption, you can increase the reliability and accuracy of your VAR model.

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