Correct notation when proving that $\hat{\beta_1}$ is linear

Question

When reviewing lecture slides for the proof that $\hat{\beta_1}$ is linear in OLS-regression my teacher posted the following on the lecture slides: $$\hat{\beta_1}=\frac{\sum(X_i-\bar{X})Y_i}{\sum(X_i-\bar{X})^2}= \sum a_iY_i \text{ where }a_i= \frac{(X_i-\bar{X})}{\sum(X_i-\bar{X})^2}$$

But when I look at the same proof in the textbook the teacher referenced for a further discussion of this proof (Introduction to Econometrics by Stock&Watson, appendix 5.2) the same proof have the following notation: $\hat{a_i}= \frac{(X_i-\bar{X})}{(X_i-\bar{X})^2}$ . Note that there is no $\sum$-sign in the denomination in the second case.

So my questions are:

1. Which is the correct notation?
2. If the correct notation is $\frac{(X_i-\bar{X})}{\sum(X_i-\bar{X})^2}$, I would be grateful for an explanation of why this is the case since $\frac{(X_i-\bar{X})}{(X_i-\bar{X})^2}$ intuitively feels correct.

Answer 1

I'm assuming we are talking about the simple linear regression model

$$ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i,\quad i=1,\ldots,n, $$ with $\epsilon_i\sim N(0,\,\sigma^2)$, $\sigma^2>0$, independently and identically. Doing the calculations yourself will notice that the OLS estimator for $\beta_1$ is

\begin{align*} \hat \beta_1 &= \frac{\sum_i (Y_i-\bar Y)(X_i-\bar X)}{\sum_i(X_i-\bar X)^2} = \frac{\sum_i Y_i(X_i-\bar X)-\sum_i \bar Y(X_i-\bar X)}{\sum_i(X_i-\bar X)^2}\tag{*}\\ &= \sum_i a_iY_i.\tag{**} \end{align*}

The first equality in (*) is just OLS, the second equality is algebra and equality in (**) is due to a certain property of the average. I'm leaving the rest of the details to you since this is a self-study question.