8.3 The Chow test of structural change
When both sets are large enough to estimate the model separately (\(n_1 > k\) and \(n_2 > k\)), a sharper question can be asked: are the coefficients the same in both? The unrestricted model allows different coefficients, \[ Y_{(1)} = X_{(1)}\beta _{(1)} + U_{(1)}, \qquad Y_{(2)} = X_{(2)}\beta _{(2)} + U_{(2)} , \] and \(H_0\colon \beta _{(1)} = \beta _{(2)}\) imposes \(k\) restrictions.
Theorem 8.3.1 (Chow test). If the disturbances are \(N(0, \sigma ^2I_n)\) in both sets, with the same \(\sigma ^2\), then under \(H_0\) \[ F = \frac {\bigl (\text {RSS} - (\text {RSS}_1 + \text {RSS}_2)\bigr )/k}{(\text {RSS}_1 + \text {RSS}_2)/(n - 2k)} \sim F_{k,\ n-2k} , \] where \(\text {RSS}_1\) and \(\text {RSS}_2\) come from the separate regressions and RSS from the pooled regression.
Proof. Stack the unrestricted model as one regression with the block-diagonal regressor matrix \(\begin {pmatrix} X_{(1)} & 0 \\ 0 & X_{(2)} \end {pmatrix}\) and \(2k\) coefficients. Its normal equations separate into those of the two subsamples, so its residual sum of squares is \(\text {RSS}_1 + \text {RSS}_2\) with \(n - 2k\) degrees of freedom. The restricted model, with \(\beta _{(1)} = \beta _{(2)}\), is the pooled regression. Apply Theorem 7.4.1 with \(q = k\). □
The equal-variance assumption is essential and easily overlooked: the denominator pools the two residual variances. If the disturbance variance also changed between the periods, the test’s size is wrong. A first check compares \(s_1^2 = \text {RSS}_1/(n_1 - k)\) with \(s_2^2 = \text {RSS}_2/(n_2 - k)\) by their ratio, which is \(F_{n_2-k,\,n_1-k}\) under equal variances (the Goldfeld–Quandt test of the section on heteroscedasticity).
Example 8.3.2. Per capita food consumption \(q\) was estimated by \[ \log q = \alpha + \beta _1\log p + \beta _2\log Y + u , \] using annual averages for two periods of fifteen years each, before and after a wave of privatisation of state-owned companies. The residual sums of squares are
| Period | \(n\) | RSS |
| Before privatisation | 15 | 0.1151 |
| After privatisation | 15 | 0.0544 |
| Both combined | 30 | 0.2866 |
Did the coefficients remain stable after privatisation? Use the \(5\%\) level.
Solution. With \(k = 3\) and \(n = 30\), \(\text {RSS}_1 + \text {RSS}_2 = 0.1695\) and \[ F = \frac {(0.2866 - 0.1695)/3}{0.1695/24} = \frac {0.03903}{0.007063} = 5.53 . \] The \(5\%\) point of \(F_{3,24}\) is \(3.01\), so stability is rejected (\(p = 0.005\)). The demand relation shifted between the two periods.
The equal-variance assumption is plausible here: \(s_1^2 = 0.1151/12 = 0.00959\) and \(s_2^2 = 0.0544/12 = 0.00453\), a ratio of \(2.12\), below the \(5\%\) point of \(F_{12,12}\), which is \(2.69\). The test cannot say which coefficients changed, the intercept, the price elasticity or the income elasticity. Dummy variables answer that.
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