I am trying to learn about bias in simple linear regression. Specifically, I want to see what happens when the $cov(e,x) = 0$ assumption of the simple regression is violated.
If this assumption is violated, I arrive at
\begin{equation} \hat{\beta}_1 \rightarrow \beta_1 + \frac{{cov(e,x)}}{{var(x)}}. \end{equation}
This derivation is from this web page (equations 1 through 6). The web page says
If cov(e,x) =\= 0, the OLS estimator is inconsistent, i.e. its value does not converge to the true value of the parameter with the sample size. Moreover, the OLS estimator is biased.
To me, it is clear that $\hat{\beta}_1$ converges to a value that is not the true value $\beta_1$, so that makes it biased. However, the web page seems to conclude that this makes it inconsistent. Somehow, they conclude that the estimator is biased, but I am not sure (they simply use "moreover").
So here are my questions:
- What is the difference between bias and inconsistency in this case? (When they conclude that it is inconsistent, I conclude that it is biased.)
- Does it ever make sense to say that $\beta_1$ is biased? Or, can only an estimator $\hat{\beta_1}$ be biased?
- If the $cov(e,x) = 0$ assumption is violated, how can I find out what happens to the variance of $\hat{\beta}_1$? Can I tell if it increases or decreases?
EDIT To clarify question 3, I am wondering if there is a proof/argument for:
When the second assumption ($cov(e,x) = 0$) of Ordinary Least Squares is violated, the variance of $\hat{\beta_1}$ changes.
The only answer I can think of is using the result from omitted-variable bias. That is, comparing $var(\hat{\beta_1})$ and $var(\tilde{\beta_1})$ using the equations
\begin{align} var(\hat{\beta_1}) = \sigma^2/[SST_1(1-R_1^2)] \end{align}
\begin{align} var(\tilde{\beta_1}) = \sigma^2/SST_1. \end{align}
The full argument for the omitted-variable case comes from Wooldridge's text.
Since having an omitted variable is sufficient to violate $cov(e,x) = 0$, is the argument given by Wooldridge sufficient to prove that the variance is less than it would be if $cov(e,x) = 0$ held true?
(If my understanding is correct, I think that $\tilde{\beta_1}$ is the assumption-violating case. $\hat{\beta_1}$ is the 'true' case.)