8.4 Dummy variables
A dummy variable takes the value \(1\) for observations in a category and \(0\) otherwise: female, urban, after privatisation, December. Dummies let a regression shift between groups without splitting the sample.
Intercept and slope dummies
With \(D_i = 1\) for observations in the second period, \[ Y_i = \alpha + \beta X_i + \gamma D_i + \delta D_iX_i + u_i \] gives the line \(\alpha + \beta X\) in the first period and \((\alpha + \gamma ) + (\beta + \delta )X\) in the second. \(\gamma \) is a shift in the intercept and \(\delta \) a change in the slope; \(t\) tests on them, or an \(F\) test on both, say which part of the relation changed.
Theorem 8.4.1. In the regression of \(Y\) on \(X\), \(D\) and every interaction \(D \times X_j\) (including \(D\) itself as the interaction with the constant), the residual sum of squares is \(\text {RSS}_1 + \text {RSS}_2\), and the \(F\) test that all the dummy coefficients are zero is exactly the Chow test of Theorem 8.3.1.
Proof. The fully interacted model has the same column space as the block-diagonal model in the proof of Theorem 8.3.1: in the first set the regressors are \(X_{(1)}\) and zeros, in the second \(X_{(2)}\) and \(X_{(2)}\), and \([X_{(2)}\ \ X_{(2)}]\) spans the same space as \([0\ \ X_{(2)}]\) together with \([X_{(2)}\ \ 0]\). Equal column spaces give equal fitted values and residuals (Section 6.6). Setting the \(k\) dummy coefficients to zero gives the pooled regression. □
Setting only some dummy coefficients to zero gives the more specific tests. “Did only the intercept change?” compares the model with \(D\) alone to the fully interacted model. When the Chow test rejects, this is how the source of the instability is found.
Several categories and the dummy variable trap
A variable with \(m\) categories, such as the four quarters, is represented by \(m - 1\) dummies, with one category as the base. Including all \(m\) together with an intercept makes \(X\) rank-deficient, because the dummies sum to the constant column. This is the dummy variable trap, a case of perfect multicollinearity. Each dummy coefficient measures the difference from the base category. Seasonal dummies used in this way remove fixed seasonal patterns, and by the Frisch–Waugh–Lovell theorem including them is equivalent to deseasonalising every variable by subtracting its seasonal means.
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