7.6 Prediction
For a new observation with regressor row vector \(x_0'\), the point prediction is \(\hat Y_0 = x_0'b\).
Theorem 7.6.1. If \(Y_0 = x_0'\beta + u_0\), with \(u_0\) uncorrelated with \(U\), mean zero and variance \(\sigma ^2\), then under (A1)–(A3) \[ \Var \bigl (x_0'b - x_0'\beta \bigr ) = \sigma ^2x_0'(X'X)^{-1}x_0, \qquad \Var (Y_0 - \hat Y_0) = \sigma ^2\bigl (1 + x_0'(X'X)^{-1}x_0\bigr ) . \]
Proof. \(x_0'b - x_0'\beta = x_0'(b - \beta )\) has variance \(x_0'\Var (b)x_0\). The prediction error is \(Y_0 - \hat Y_0 = u_0 - x_0'(b - \beta )\), and \(u_0\) is uncorrelated with \(b\), which depends on \(U\) only, so the variances add. □
For the two-variable model \(x_0 = (1, X_0)'\), and \(x_0'(X'X)^{-1}x_0 = \frac 1n + (X_0 - \bar X)^2/\sum x_i^2\), which is Theorem 2.5.1. Under normality, \((Y_0 - \hat Y_0)\big /\bigl (s\sqrt {1 + x_0'(X'X)^{-1}x_0}\bigr ) \sim t_{n-k}\), which gives prediction intervals. When many predictions are made at once, the prediction errors are correlated through their common dependence on \(b\). Their joint distribution is the basis of the Chow forecast test of the next section.
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