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 residual plot for the detection of autocorrelation:
Detection of Autocorrelation from Residual Plots
- 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 time-series data, the researcher can plot (graphically draw) the residuals versus 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 the researcher observes some pattern (other than random) in the graphical representation of the data, then it means that the data is autocorrelated. The existence of some patterns shown in the above Figure can be used for the detection of autocorrelation.
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