8.2 The Chow forecast test
An estimated relationship is of little use if it changes from one period to the next. The first test of stability asks whether a model estimated on part of the data predicts the rest. Divide the \(n\) observations into a first set of \(n_1\), used for estimation, and a second set of \(n_2\), used for prediction: \[ Y_{(1)} = X_{(1)}\beta + U_{(1)}, \qquad Y_{(2)} = X_{(2)}\beta + U_{(2)} , \] with \(n_1 > k\) and \(X_{(1)}\) of rank \(k\). Let \(b_{(1)} = (X_{(1)}'X_{(1)})^{-1}X_{(1)}'Y_{(1)}\), and define the vector of prediction errors \[ d = Y_{(2)} - X_{(2)}b_{(1)} . \]
Theorem 8.2.1. If the same model with \(E(U) = 0\) and \(\Var (U) = \sigma ^2I_n\) holds for both sets, then \[ E(d) = 0, \qquad \Var (d) = \sigma ^2\bigl [I_{n_2} + X_{(2)}(X_{(1)}'X_{(1)})^{-1}X_{(2)}'\bigr ] . \] Under normality, with \(e_{(1)}\) the residuals from the first set, \[ F = \frac {d'\bigl [I_{n_2} + X_{(2)}(X_{(1)}'X_{(1)})^{-1}X_{(2)}'\bigr ]^{-1}d\,/\,n_2}{e_{(1)}'e_{(1)}/(n_1 - k)} \sim F_{n_2,\ n_1-k} . \]
Proof. \(b_{(1)} = \beta + (X_{(1)}'X_{(1)})^{-1}X_{(1)}'U_{(1)}\), so \[ d = X_{(2)}\beta + U_{(2)} - X_{(2)}b_{(1)} = U_{(2)} - X_{(2)}(X_{(1)}'X_{(1)})^{-1}X_{(1)}'U_{(1)} . \] Both terms have mean zero, so \(E(d) = 0\). \(U_{(1)}\) and \(U_{(2)}\) are uncorrelated, so the variance is the sum of the two variances: \begin {align*} \Var (d) &= \sigma ^2I_{n_2} + X_{(2)}(X_{(1)}'X_{(1)})^{-1}X_{(1)}'(\sigma ^2I)X_{(1)}(X_{(1)}'X_{(1)})^{-1}X_{(2)}'\\ &= \sigma ^2\bigl [I + X_{(2)}(X_{(1)}'X_{(1)})^{-1}X_{(2)}'\bigr ] . \end {align*}
Under normality, \(d\) is normal and the numerator of \(F\), multiplied by \(n_2/\sigma ^2\), is \(\chi ^2_{n_2}\) by Theorem 5.4.2(ii). The denominator is \(\sigma ^2\chi ^2_{n_1-k}/(n_1 - k)\) by Theorem 5.5.1, applied to the first set. They are independent: \(e_{(1)} = M_{(1)}U_{(1)}\) is independent of \((X_{(1)}'X_{(1)})^{-1}X_{(1)}'U_{(1)}\) by Theorem 5.5.1(vi), and of \(U_{(2)}\) because the two sets are independent. □
The variance of \(d\) has two parts: the variance of the new disturbances, and the uncertainty in \(b_{(1)}\) carried into the prediction. It is the matrix version of Theorem 7.6.1. The statistic has a much simpler form, which is how it is computed in practice.
Theorem 8.2.2. Let \(\text {RSS}\) be the residual sum of squares from the regression on all \(n\) observations and \(\text {RSS}_1 = e_{(1)}'e_{(1)}\) that from the first \(n_1\). Then the statistic of Theorem 8.2.1 equals \[ F = \frac {(\text {RSS} - \text {RSS}_1)/n_2}{\text {RSS}_1/(n_1 - k)} . \]
Proof. Add to the full-sample regression one dummy variable for each observation of the second set: an \(n \times n_2\) matrix \(D = \binom {0}{I_{n_2}}\), so the model is \(Y = X\beta + D\gamma + U\). Each dummy fits its own observation exactly, so this augmented regression has zero residuals in the second set, and in the first set it is the regression on \(X_{(1)}\) alone. Its residual sum of squares is \(\text {RSS}_1\), its coefficients on \(X\) are \(b_{(1)}\), and its dummy coefficients are \(\hat \gamma = Y_{(2)} - X_{(2)}b_{(1)} = d\). The hypothesis \(\gamma = 0\) imposes \(n_2\) restrictions, and the restricted model is the full-sample regression, with residual sum of squares RSS. Theorem 7.4.1 gives \(F = \bigl ((\text {RSS} - \text {RSS}_1)/n_2\bigr )\big /\bigl (\text {RSS}_1/(n - k - n_2)\bigr )\), with \(n - k - n_2 = n_1 - k\). The Wald form of the same test uses \(\hat \gamma = d\) and its estimated variance, which is the expression in Theorem 8.2.1. By Theorem 7.4.1 the two forms are equal. □
The forecast test needs only \(n_2 \geq 1\); in particular it works when the second period is too short to estimate the model on its own. A large \(F\) says that the first period’s relationship does not predict the second period: something has changed.
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