8.5 Recursive residuals and the CUSUM test
The Chow tests need the break date to be known. When it is not, recursive estimation tracks the coefficients through the sample. For \(t = k + 1, \dots , n\), let \(b_{t-1}\) be the least squares estimate from the first \(t - 1\) observations and \(X_{t-1}\) their regressor matrix. The recursive residual is the standardised one-step-ahead prediction error, \[ w_t = \frac {Y_t - x_t'b_{t-1}}{\sqrt {1 + x_t'(X_{t-1}'X_{t-1})^{-1}x_t}} . \]
Theorem 8.5.1. Under the classical normal model, the \(n - k\) recursive residuals \(w_{k+1}, \dots , w_n\) are independent \(N(0, \sigma ^2)\).
Proof. By Theorem 7.6.1, \(Y_t - x_t'b_{t-1}\) is normal with mean zero and variance \(\sigma ^2\bigl (1 + x_t'(X_{t-1}'X_{t-1})^{-1}x_t\bigr )\), so \(w_t \sim N(0, \sigma ^2)\). For independence, take \(s < t\). Then \(w_s\) is a linear function of \(Y_1, \dots , Y_s\), all of which are used in \(b_{t-1}\), and the prediction error at \(t\) is uncorrelated with any linear function \(a'Y_{1:t-1}\) that has mean zero for all \(\beta \). Indeed, \(\Cov \bigl (a'U_{1:t-1},\ u_t - x_t'(b_{t-1} - \beta )\bigr ) = -\sigma ^2a'X_{t-1}(X_{t-1}'X_{t-1})^{-1}x_t\), and \(a'X_{t-1} = 0\) for such a function, since its mean \(a'X_{t-1}\beta \) is zero for every \(\beta \). Since \(w_s\) has mean zero for every \(\beta \), the two are uncorrelated, and jointly normal, so they are independent. □
Under correct specification the recursive residuals are therefore white noise. If the coefficients change at some point, the predictions after the change are systematically wrong, and the \(w_t\) start to have a non-zero mean. The CUSUM statistic of Brown, Durbin and Evans plots the cumulative sum \[ W_t = \frac 1{\hat \sigma }\sum _{j=k+1}^t w_j, \qquad t = k + 1, \dots , n , \] which under the null wanders around zero like a random walk. Crossing the lines \(\pm a\bigl (\sqrt {n - k} + 2(t - k)/\sqrt {n - k}\bigr )\), with \(a = 0.948\) for a \(5\%\) test, signals instability. The companion CUSUMSQ test uses cumulative sums of \(w_t^2\) and is sensitive to changes in the variance.
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