There are several issues of model specification explored here. First, it is possible to omit relevant independent variables from your model. A relevant independent variable is one that affects the mean of the dependent variable. When you omit a relevant variable that happens to be correlated with any of the other included regressors, least squares suffers from omitted variable bias.
The other possibility is to include irrelevant variables in the model. In this case, you include extra regressors that either don’t affect y or, if they do, they are not correlated with any of the other regressors. Including irrelevant variables in the model makes least squares less precise than it otherwise would be-this increases standard errors, reduces the power of your hypothesis tests, and increases the size of your confidence intervals.
The example used in the text uses the dataset edu-inc. gdt. The first regression
famine = ві + в2 he + p3we + в4 kl6 + в5Х*5 + вбХіб + e* (6.7)
where famine is family income, he is husband’s years of schooling, we is woman’s years of schooling, and kl6 are the number of children in the household under age 6. Several variations of this model are estimated. The first includes only he, another only he and we, and one includes the two irrelevant variables, x5 and x6. The gretl script to estimate these models and test the implied hypothesis restrictions follows. If you type this in yourself, omit the line numbers.
1 list all_x = const he we kl6 xtra_x5 xtra_x6
2 ols famine all_x
3 modeltab add
4 omit xtra_x5 xtra_x6
5 modeltab add
6 omit kl6
7 modeltab add
8 omit we
9 modeltab add io modeltab show
The models can be estimated and saved as icons (File>Save to session as icon) within gretl. Once they’ve all been estimated and saved as icons, open a session window (Figure 1.12) and drag each model onto the model table icon. Click on the model table icon to reveal the output shown in Figure 6.13.
In the above script, we have used the modeltab function after each estimated model to add it to the model table. The final line tells gretl to display (show) the resulting model table.
One word of caution is in order about the given script and its interpretation. The omit statement tests the implied restriction (the coefficient on the omitted variable is zero) versus the estimated model that immediately precedes it. Thus, when we test that the coefficient on kl6 is zero in line 6, the alternative model is the restricted model from line 4, which already excludes xtra_x5, and xtra_x6. Thus, only one restriction is being tested. If your intention is to test all of the restrictions (omit xtra_x5, xtra_x6 and kl6) versus the the completely unrestricted model in line 2 that includes all of the variables, you’ll need to modify your code. I’ll leave this an an exercise.