# Checking for White Noise Errors After ARIMA Model

Jan 24, 2024

Fit an Autoregressive Integrated Moving Average (ARIMA) model is one of the widely used methods for time series forecasting. Once you have developed an ARIMA model, checking the residuals or errors for white noise is an essential process to ensure that the model is appropriately capturing the underlying patterns in the data. White noise refers to the random errors that are independent and identically distributed, which means the model has successfully accounted for the data's underlying structure.

Here is a step-by-step guide to check if the errors are white noise after fitting the ARIMA model:

1. Calculate residuals: First, fit the ARIMA model to the time series and then calculate the residuals. These residuals represent the differences between the actual values and the predicted values provided by the ARIMA model. You can simply subtract the predicted values from the actual values to obtain the residuals.

2. Check the mean and variance of residuals: One critical aspect of white noise errors is that they should have zero mean and constant variance. To verify this, calculate the mean and variance of the residuals. If the mean is close to zero and the variance is constant, it is an initial indication of white noise.

3. Autocorrelation check: Plot the autocorrelation function (ACF) of the residuals. ACF measures the correlation between the residuals at different time lags. If the errors are white noise, there should be no significant autocorrelation between them. Observe the ACF plot and look for any significant peaks beyond the confidence intervals.

4. Ljung-Box (Q) Test: This statistical test checks for the absence of autocorrelation in residuals. Perform Ljung-Box test on the residuals by providing the appropriate lag length (often, the same as the maximum lag order used in the ARIMA model). If the p-values are not significant, it indicates a lack of autocorrelation and supports the white noise hypothesis of errors.

5. Normality Check: White noise errors should be normally distributed. To check for normality in the residuals, consider plotting a histogram of the residuals, along with a Q-Q plot to visualize any discrepancies from the normal distribution. Also, apply a statistical test like the Shapiro-Wilk or Jarque-Bera test to test for normality. Non-significant p-values indicate that the errors follow a normal distribution.

If your residual analysis passes these tests and suggests white noise errors in your ARIMA model, it indicates a well-fitted model that has accounted for the series' underlying structure. After identifying any issues in the initial model, you can fine-tune the model further to improve its performance.