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problem

I figured out part a pretty easily, but I'm having trouble figuring out how to do b and c.

I know the formula for a normal estimator of beta: $\frac{\sum(X_i-\bar{X_1}_i)(Y_i-\bar{Y_1}_i)}{\sum(X_i-\bar{X_1}_i)^2}$

But what exactly can I divide by (${X_2}_i$) to get their vesion of this estimator? And also how can I use this to find the variance of the estimator?

Thank you for your help, much appreciated!

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1 Answer 1

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Note that you are interested in the estimator of $\beta$ for model without an intercept, i.e., $$ y_i = \beta x_i + \epsilon_i, \quad, i=1,...,n $$ by minimizing $$ S(\beta) = \sum_{i=1}^n(y_i - \beta x_{1i})^2, $$ you can show that $$ \hat{\beta} = \frac{\sum x_{1i}y_i}{\sum x_{1i}^2}\, $$ is the OLS estimator of $\beta$. And its variance is given by $$ var(\hat{\beta})= \frac{\sum x^2_{1i}var(y_i)}{(\sum x_{1i}^2)^2} = \frac{\sigma^2}{\sum x_{1i}^2} . $$ Now, just divide by $x_{i2}$ every $y_i$ and $x_{1i}$.

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