Simple linear regression

Activity 11.1   Find the mean and variance of the estimator of slope for the model of a regression line through the origin.
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Answer 11.1   For a regression line through the origin,

$$B= \sum_1^nw_i\left(Y_i/x_i\right)$$

where $\sum_1^nw_i=1$. First the expected value:

$$E[B] = E[\sum_1^nw_i\left(Y_i/x_i\right)]$$

$$=\sum_1^nw_iE\left(Y_i/x_i\right)$$

$$=\sum_1^nw_i\beta =\beta.$$

So, the estimator of slope is unbiased. Now for the variance:

$$\operatorname{var}B = \operatorname{var}\sum_1^nw_i\left(Y_i/x_i\right)$$

$$=\sum_1^nw_i^2\operatorname{var}\left(Y_i/x_i\right)$$

$$=\sum_1^nw_i^2\left(\sigma^2/x_i^2\right)$$

$$=\sigma^2\sum_1^nw_i^2/x_i^2$$

$$=\sigma^2/\left(\sum_1^n x_i^2\right).$$

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Activity 11.2   Why are the two equations the same when $x=0$?
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Answer 11.2   When $x=0$ the fitted value become the intercept $\alpha$, so the results should be the same.
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Activity 11.3   Find the variance of $\hat{Y}$ for regression through the origin.
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Answer 11.3   Here $\hat{Y}=Bx$, so $\operatorname{var}\hat{Y}=x^2\operatorname{var} B=x^2/\left(\sum_1^n x_i^2\right)$ from Activity 11.1
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Activity 11.4   Which assumptions go wrong in each of the cases just mentioned?
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Answer 11.4   For the first case, when the regression is quadratic, the model that says the regression is linear is clearly wrong.

If there is some increase in variation around the regression line with $x$, then the assumption that the error variance is the same for each observation must be wrong.

If there s a wild observation, then that observation does not follow the model at all, so the model does not hold for all the observations.
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