Question

10.13 Suppose that the full model is $y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \epsilon_i$, $i = 1, 2, ..., n$, where $x_{i1}$ and $x_{i2}$ have been coded so that $S_{11} = S_{22} = 1$. We will also consider fitting a subset model, say $y_i = \beta_0 + \beta_1 x_{i1} + \epsilon_i$. a. Let $\hat{\beta}^_1$ be the least-squares estimate of $\beta_1$ from the full model. Show that $\text{Var}(\hat{\beta}^_1) = \sigma^2 / (1 - r_{12}^2)$, where $r_{12}$ is the correlation between $x_1$ and $x_2$. b. Let $\hat{\beta}_1$ be the least-squares estimate of $\beta_1$ from the subset model. Show that $\text{Var}(\hat{\beta}_1) = \sigma^2$. Is $\beta_1$ estimated more precisely from the subset model or from the full model? c. Show that $E(\hat{\beta}_1) = \beta_1 + r_{12} \beta_2$. Under what circumstances is $\hat{\beta}_1$ an unbiased estimator of $\beta_1$? d. Find the mean square error for the subset estimator $\hat{\beta}_1$. Compare MSE($\hat{\beta}_1$) with Var($\hat{\beta}^*_1$). Under what circumstances is $\hat{\beta}_1$ a preferable estimator, with respect to MSE?

          10.13 Suppose that the full model is $y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \epsilon_i$, $i = 1, 2, ..., n$,
where $x_{i1}$ and $x_{i2}$ have been coded so that $S_{11} = S_{22} = 1$. We will also consider
fitting a subset model, say $y_i = \beta_0 + \beta_1 x_{i1} + \epsilon_i$.
a. Let $\hat{\beta}^*_1$ be the least-squares estimate of $\beta_1$ from the full model. Show that
$\text{Var}(\hat{\beta}^*_1) = \sigma^2 / (1 - r_{12}^2)$,
where $r_{12}$ is the correlation between $x_1$ and $x_2$.
b. Let $\hat{\beta}_1$ be the least-squares estimate of $\beta_1$ from the subset model. Show
that $\text{Var}(\hat{\beta}_1) = \sigma^2$. Is $\beta_1$ estimated more precisely from the subset model
or from the full model?
c. Show that $E(\hat{\beta}_1) = \beta_1 + r_{12} \beta_2$. Under what circumstances is $\hat{\beta}_1$ an unbiased
estimator of $\beta_1$?
d. Find the mean square error for the subset estimator $\hat{\beta}_1$. Compare MSE($\hat{\beta}_1$)
with Var($\hat{\beta}^*_1$). Under what circumstances is $\hat{\beta}_1$ a preferable estimator, with
respect to MSE?

10.13 Suppose that the full model is yi = β0 + β1 xi1 + β2 xi2 +, i = 1, 2, ..., n,
where xi1 and xi2 have been coded so that S11 = S22 = 1. We will also consider
fitting a subset model, say yi = β0 + β1 xi1 +.
a. Let β̂^*1 be the least-squares estimate of β1 from the full model. Show that
Var(β̂^*1) = σ^2 / (1 - r12^2),
where r12 is the correlation between x1 and x2.
b. Let β̂1 be the least-squares estimate of β1 from the subset model. Show
that Var(β̂1) = σ^2. Is β1 estimated more precisely from the subset model
or from the full model?
c. Show that E(β̂1) = β1 + r12β2. Under what circumstances is β̂1 an unbiased
estimator of β1?
d. Find the mean square error for the subset estimator β̂1. Compare MSE(β̂1)
with Var(β̂^*1). Under what circumstances is β̂1 a preferable estimator, with
respect to MSE?

Added by Michael B.

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