Residuals plot for Detection of Autocorrelation

The existence and pattern of autocorrelation may be detected using a graphical representation of residuals obtained from ordinary least square regression. One can draw the following residuals plot for detection of autocorrelation:

• A plot of the residual plot against time.
• A plot of the $\hat{u}_t$ against $\hat{u}_{t-1}$
• A plot of standardized residuals against time.

Note that the population disturbances $u_t$ are not directly observable, we use their proxies, the residuals $\hat{u}_t$.

• A random pattern of residuals indicates the non-presence of autocorrelation.
• A plot of residuals for detection of residuals used for visual examination of $\hat{u}_t$ or  $\hat{u}_t^2$ can provide useful information not only about the presence of autocorrelation but also about the presence of heteroscedasticity. Similarly, the examination of $\hat{u}_t$ and $\hat{u}_t^2$ provides useful information about model inadequacy or specification bias too.
• The standardized residuals are computed as $\frac{u_t}{\hat{\sigma}}$ where $\hat{\sigma}$ is standard error of regression.

Note: The plot of residuals against time is called the sequence plot. For a time-series data, researcher can plot (graphically draw) the residuals verses time (called a time sequence plot), he may expect to observe some random pattern in the time series data, indicating that the data is not autocorrelated. However, if researcher observes some pattern (other than the random) in the graphical representation o f the data, then it means that the data is autocorrelated.

See more on: Autocorrelation