Understanding convergence of OLS estimator

Question

From a linear regression with one explanatory variable, $ y = \beta_0 + \beta_1x+e$, the OLS estimator can be written as

\begin{equation} \hat{\beta}_1 = \frac{\widehat{cov(y,x)}}{\widehat{var(x)}}. \end{equation}

The text I am reading says that this equation

means $\hat{\beta}_1$ converges to:

\begin{equation} \hat{\beta}_1 \rightarrow \beta_1 + \frac{{cov(e,x)}}{{var(x)}}. \end{equation}

I understand how to arrive at the first equation, but I don't understand what 'convergence' means in the context of an OLS estimator. I know that if the estimator is unbiased, then $E(\hat{\beta_1}) = \beta_1$. Where does the $\frac{{cov(e,x)}}{{var(x)}}$ term come in?

For reference, the text is here and I am trying to understand equations (1) through (6).

Answer 1

The population parameter is equal to (equation $(4)$ in your note) \begin{equation} \beta_1 = \frac{cov(y,x)}{var(x)}-\frac{cov(e,x)}{var(x)}. \tag1 \end{equation}

whereas the OLS estimate of it is given by

\begin{equation} \hat{\beta}_1 = \frac{\widehat{cov(y,x)}}{\widehat{var(x)}}. \tag2 \end{equation}

As the sample size increases

\begin{equation} \frac{\widehat{cov(y,x)}}{\widehat{var(x)}} \rightarrow \frac{cov(y,x)}{var(x)}. \tag3 \end{equation}

This convergence is in probability, that is, the probability that

$$\left|\frac{\widehat{cov(y,x)}}{\widehat{var(x)}}-\frac{cov(y,x)}{var(x)}\right|$$

is larger than any $\epsilon>0$ goes to zero as the sample size increases.

Putting equations $(1)$, $(2)$, and $(3)$ together we arrive at

\begin{equation} \hat{\beta}_1 \rightarrow \beta_1 + \frac{{cov(e,x)}}{{var(x)}}. \end{equation}