Residual Independence Verified through Model Diagnosis
Diagnostic Methods
Intuitive Pattern Recognition
Using standardized residual plots, we can check if the regression analysis was performed correctly. To confirm independence, there should be no distinct patterns appearing in the residual plots. Unfortunately, diagnosing independence can be very subjective compared to other assumptions of regression analysis.
A common example of lacking independence is seeing an unidentified straight line as shown above. It could be by chance, but usually, it indicates a misunderstanding of the data or missing crucial data.
At first glance, the case above may seem problem-free, but on closer inspection, the same pattern repeats every $9$ periods. If such an explicit regularity is observed, one can confidently say that independence is compromised. In such cases, the residuals exhibit Autocorrelation, and it is advisable to consider analysis including time series.
Statistical Testing
Such issues could easily be detected if the analyst thought they were not good at observing data and used the Durbin-Watson test. However, one should not entirely rely on the Durbin-Watson test because it only identifies autocorrelation and does not guarantee independence. The image below is deliberately created for easier understanding, but there are plenty of examples where independence is compromised even after passing the Durbin-Watson test.
In the figure, the residuals almost precisely fall on $0$ for the first time, and the next two times as well, but it is impossible to predict when the first $0$ will occur. In such cases, though there’s a serious issue with the independence of residuals, it’s too irregular to claim autocorrelation. While it might be a coincidence, the judgment is entirely up to the analyst.