Serial Correlation in Residuals

The correlogram can also be used to check whether the assumption that model errors have zero covariance-an important assumption in the proof of the Gauss-Markov theorem. The example that illustrates this is based on the Phillips curve that relates inflation and unemployment. The data used are from Australia and reside in the phillips-aus. gdt dataset.

The model to be estimated is

inf = ві + в2 Аи* + et (9.6)

The data are quarterly and begin in 1987:1. A time-series plot of both series is shown below in Figure 9.10. The graphs show some evidence of serial correlation in both series.

image304

Figure 9.10: This plot shows the relationship between inflation and the change in unemployment in Australia, 1987:1 – 2009:3.

The model is estimated by least squares and the residuals are plotted against time. These appear in Figure 9.11. A correlogram of the residuals that appears below seems to confirm this. To generate the regression and graphs is simple. The script to do so is:

ols inf const d_u

2 series ehat = $uhat

3 gnuplot ehat —time-series

4 corrgm ehat

Unfortuantely, gretl will not accept the accessor, $uhat, as an input into either gnuplot or corrgm. That means you have to create a series, ehat, first. Once this is created, both functions work as expected.

The GUI is even easier in this instance once the model is estimated. The model window offers a way to produce both sets of graphs. Simply choose Graphs>Residual plot>Against time to produce the first. The second is Graphs>Residual correlogram. The latter opens a dialog box allowing you to specify how many autocorrelations to compute. In this example, I set it to 12.

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