Question

1. In deriving the least-squares estimator $\hat{\beta_1}$, we first take the partial derivative of SSE with respect to $\hat{\beta_1}$. The partial derivative of the SSE is (select one answer) A. $-2 \left( \sum_{i=1}^n y_i - \hat{\beta_1} \sum_{i=1}^n x_i \right)$ B. $-2 \left( \sum_{i=1}^n x_iy_i - \hat{\beta_1} \sum_{i=1}^n x_i^2 \right)$ C. $2 \sum_{i=1}^n (\hat{\beta_1} x_i^2 - x_iy_i)$ D. $2 \sum_{i=1}^n (\hat{\beta_1} x_i^2 - x_i^2 y_i^2)$ 2. $\hat{\beta_1}$ is the least-squares estimator of the parameter $\beta_1$ and can be expressed as A. $\frac{\sum_{i=1}^n x_iy_i}{\sum_{i=1}^n x_i^2}$ B. $\frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$ C. $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$ D. $\sum_{i=1}^n x_iy_i - \frac{1}{n} \sum_{i=1}^n x_i \sum_{i=1}^n y_i$ 3. Is $\hat{\beta_1}$ an unbiased estimator of the parameter $\beta_1$? Which of the following statements is correct. A. $\hat{\beta_1}$ is an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n x_i E(Y)}{\sum_{i=1}^n x_i^2}$ B. $\hat{\beta_1}$ is an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n x_i^2 E(Y)}{\sum_{i=1}^n x_i^2}$ C. $\hat{\beta_1}$ is not an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n \beta_1 x_i^2}{\sum_{i=1}^n x_i^2}$ D. $\hat{\beta_1}$ is not an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n \beta_1 x_i^2}{\sum_{i=1}^n x_i^2}$

          1. In deriving the least-squares estimator $\hat{\beta_1}$, we first take the partial derivative of SSE with respect to $\hat{\beta_1}$. The partial derivative of the SSE is (select one answer)
A. $-2 \left( \sum_{i=1}^n y_i - \hat{\beta_1} \sum_{i=1}^n x_i \right)$
B. $-2 \left( \sum_{i=1}^n x_iy_i - \hat{\beta_1} \sum_{i=1}^n x_i^2 \right)$
C. $2 \sum_{i=1}^n (\hat{\beta_1} x_i^2 - x_iy_i)$
D. $2 \sum_{i=1}^n (\hat{\beta_1} x_i^2 - x_i^2 y_i^2)$

2. $\hat{\beta_1}$ is the least-squares estimator of the parameter $\beta_1$ and can be expressed as
A. $\frac{\sum_{i=1}^n x_iy_i}{\sum_{i=1}^n x_i^2}$
B. $\frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$
C. $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$
D. $\sum_{i=1}^n x_iy_i - \frac{1}{n} \sum_{i=1}^n x_i \sum_{i=1}^n y_i$

3. Is $\hat{\beta_1}$ an unbiased estimator of the parameter $\beta_1$? Which of the following statements is correct.
A. $\hat{\beta_1}$ is an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n x_i E(Y)}{\sum_{i=1}^n x_i^2}$
B. $\hat{\beta_1}$ is an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n x_i^2 E(Y)}{\sum_{i=1}^n x_i^2}$
C. $\hat{\beta_1}$ is not an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n \beta_1 x_i^2}{\sum_{i=1}^n x_i^2}$
D. $\hat{\beta_1}$ is not an unbiased estimator of the parameter $\beta_1$ and $E(\hat{\beta_1}) = \frac{\sum_{i=1}^n \beta_1 x_i^2}{\sum_{i=1}^n x_i^2}$

1 in deriving the least squares estimator hatbeta1 we first take the partial derivative of sse with respect to hatbeta1 the partial derivative of the sse is select one answer a 2 left sum 52744

Added by Tom-S N.

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