Multiple Regression and Interaction Terms

(20 points) Researchers study how a child’s reading score \(Y\) depends on home reading time \(x_{1}\) (hours/week) and classroom support level \(x_{2}\) (a small integer score). It is believed that extra home reading time is more effective when classroom support is higher, so the mean is modeled with an interaction between \(x_{1}\) and \(x_{2}\). To make the intercept interpretable, home reading time is centered at \(1\) hour/week.

Consider the linear model \[ Y_{i} = \beta _{0} + \beta _{1}(x_{1,i} - 1) + \beta _{2}(x_{1,i} \times x_{2,i}) + \epsilon _{i}, \qquad i = 1, \ldots , 5, \] where \(\epsilon _{i} \sim \mathcal {N}(0, \sigma ^{2})\) are independent. The observed data consist of: \[ (x_{1}, x_{2}, Y) = (0, 1, 4), (1, 0, 3), (2, 1, 7), (1, 2, 6), (3, 1, 9). \]

A least squares fit yields \(\hat {\beta } = (\hat {\beta }_{0}, \hat {\beta }_{1}, \hat {\beta }_{2})^{\top } = (2.10, 0.90, 1.20)^{\top }\) and \(RSS = 1.50\).

1. Write down the design matrix \(\mathbf {X}\) for this regression model and the parameter vector \(\beta = (\beta _{0}, \beta _{1}, \beta _{2})^{\top }\). Identify the estimated variance \(S^{2}\).
2. We wish to test the joint null hypothesis that there is no effect of the predictors beyond the baseline, \[ H_{0}: \beta _{1} = \beta _{2} = 0. \] Express this hypothesis in matrix form \(A\beta = 0\) for an appropriate matrix \(A\), and state \(q = \operatorname {rank}(A)\).
3. Write down the \(F\)-statistic for testing \(H_{0}\) and state the distribution of \(F\) under \(H_{0}\). (You do not need to evaluate the formula.)
4. Using Scheffé’s method, construct simultaneous 95% confidence intervals for \(\beta _{0}, \beta _{1}, \beta _{2}\). You may express these intervals in terms of \((\mathbf {X}^{\top }\mathbf {X})^{-1}\), but plug in all other quantities.

Derive, using only first principles, the least-squares estimators of the slope and the intercept of a linear regression problem. What can you say about the optimality properties of those estimators?