Question

Suppose that we have postulated the model $Y_i = \beta_1 x_i + \epsilon_i$ $i = 1, 2, \dots, n$, where the $\epsilon_i$'s are independent and identically distributed random variables with $E(\epsilon_i) = 0$ and $Var(\epsilon_i) = \sigma^2$. Then $\hat{y}_i = \hat{\beta}_1 x_i$ is the predicted value of $y$ when $x = x_i$ and $SSE = \sum_{i=1}^n [y_i - \beta_1 x_i]^2$. • Derive $\hat{\beta}_1$, the least-squares estimator of the parameter $\beta_1$. • Show if $\hat{\beta}_1$ is or is not an unbiased estimator of the parameter $\beta_1$. • Find the variance of $\hat{\beta}_1$.

          Suppose that we have postulated the model
$Y_i = \beta_1 x_i + \epsilon_i$
$i = 1, 2, \dots, n$,
where the $\epsilon_i$'s are independent and identically distributed random variables with $E(\epsilon_i) = 0$
and $Var(\epsilon_i) = \sigma^2$. Then $\hat{y}_i = \hat{\beta}_1 x_i$ is the predicted value of $y$ when $x = x_i$ and $SSE = \sum_{i=1}^n [y_i - \beta_1 x_i]^2$.
• Derive $\hat{\beta}_1$, the least-squares estimator of the parameter $\beta_1$.
• Show if $\hat{\beta}_1$ is or is not an unbiased estimator of the parameter $\beta_1$.
• Find the variance of $\hat{\beta}_1$.

Suppose that we have postulated the model
Yi = β1 xi +
i = 1, 2, …, n,
where the 's are independent and identically distributed random variables with E() = 0
and Var() = σ^2. Then ŷi = β̂1 xi is the predicted value of y when x = xi and SSE = ∑i=1^n [yi - β1 xi]^2.
• Derive β̂1, the least-squares estimator of the parameter β1.
• Show if β̂1 is or is not an unbiased estimator of the parameter β1.
• Find the variance of β̂1.

Added by Emily H.

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