1.5 Prediction
We estimate the mean, \(x_{+}^{T}\beta\), for \(Y\) at values of the explanatory variables given by \(x_+^T=\begin{pmatrix} 1&x_{+1}&\ldots&x_{+p}\end{pmatrix},\) which may or may not match a set of values observed in the data, using \[ \hat Y_{+} = x_{+}^{T}\hat{\beta}. \] Then \[ \hat Y_{+}\sim N(x_{+}^{T}\beta, \sigma^{2}h_{++}) \] where \(h_{++} = x_{+}^{T}(X^{T}X)^{-1}x_{+}\). Hence confidence intervals for predictive means can be derived using \[ \frac{\hat Y_+-x_+^T\beta}{sh_{++}^{1/2}}\sim t_{n-p-1}. \]
For predicting the actual value \(Y_+=x_{+}^{T}\beta+\epsilon_+\), the predictor \(\hat Y_{+}\) is also sensible, as \(E(\hat Y_{+} - Y_{+}) =0\). Now \[ \hat Y_{+} - Y_{+}\sim N(0,\sigma^{2}(1+h_{++})). \] as \(\hat Y_+\) and \(Y_+\) are independent. Hence predictive confidence intervals can be derived using \[ \frac{\hat Y_+-Y_+}{s(1+h_{++})^{1/2}}\sim t_{n-p-1}. \]