Section 4c

4.4 Multiple Correlation Coefficient

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Problem 4c.1

Suppose that $\beta_{1} = \beta_{2} = \cdots = \beta_{p-1} = 0$ . Find the distribution of $R^{2}$ and hence prove that

\mathbb {E}\left[R^{2}\right] = \frac{p-1}{n-1}

Problem 4c.2

For the general linear full-rank regression model, prove that $R^{2}$ and the $F$ -statistic for testing $H: \beta_{j} = 0(j \neq 0)$ are independent of the units in which the $Y_{i}$ and the $x_{i j}$ are measured.

Problem 4c.3

(a)(b)

Given the full-rank model, suppose that we wish to test $H: \beta_{j} = 0$ , $j \neq 0$ . Let $R_{H}^{2}$ be the coefficient of determination for the model with $\beta_{j} = 0$ .

(a)

Prove that the $F$ -statistic for testing $H$ is given by

F = \frac{R^{2}-R_{H}^{2}}{1-R^{2}} \cdot \frac{n-p}{1}

(This result shows that $F$ is a test for a significant reduction in $R^{2}$ .)

(b)

Deduce that $R^{2}$ can never increase when a $\beta$ coefficient is set equal to zero.