Suppose that we have postulated the model \[ Y_{i}=\beta_{1} x_{i}+\epsilon_{i} \quad i=1,2, \ldots, n, \] where the \( \epsilon_{i} \) 's are independent and identically distributed random variables with \( E\left(\epsilon_{i}\right)=0 \) and \( \operatorname{Var}\left(\epsilon_{i}\right)=\sigma^{2} \). Then \( \hat{y_{i}}=\hat{\beta}_{1} x_{i} \) is the predicted value of \( y \) when \( x=x_{i} \) and \( \mathrm{SSE}= \) \( \sum_{i=1}^{n}\left[y_{i}-\hat{\beta}_{1} x_{i}\right]^{2} \) - 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 \( \hat{\beta}_{1} \).
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Step 1: Derive the least-squares estimator \(\hat{\beta}_{1}\) To find the least-squares estimator \(\hat{\beta}_{1}\), we need to minimize the sum of squared errors (SSE), which is given by: \[ \text{SSE} = \sum_{i=1}^{n} (y_i - \hat{\beta}_1 x_i)^2 \] To minimize Show more…
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