5
votes
What does it mean for $\hat\beta_1$ and $\hat\beta_0$ to have a variance?
The data you are feeding into your OLS model are random draws from some underlying population. You could either be drawing from the joint distribution of the predictors and the outcome, or from the ...
- 110k
5
votes
What does it mean for $\hat\beta_1$ and $\hat\beta_0$ to have a variance?
The result that we obtain from linear regression is a function of random variables, so the parameters are random variables. You can calculate variance for any random variable. The variance tells us ...
Tim♦
- 128k
5
votes
Accepted
Show that $E [u] = 0$ and $cov(u,x_j)=0$ does not imply $E[u|x]=0$
One way to contrive counterexamples is to let $X$ follow some (non-degenerate) distribution symmetric about $0$ and choose $c \in \mathbb R_{>0}$ s.t. $U \mathrel{:=} X^b - c$ has expectation $0$ ...
- 4,190
4
votes
What does it mean for $\hat\beta_1$ and $\hat\beta_0$ to have a variance?
The other answers are correct, but I think it might be helpful to simulate what is happening.
...
- 47.1k
2
votes
Why would bootstrap OLS standard errors differ from ML estimate?
You might just as well ask, "Why is the sample correlation between two independent variables always non-zero?" and the answer is the same: samples never conserve the properties of the ...
- 57.2k
2
votes
Accepted
Omitted variable problem
Start with
$$y=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \gamma q +\epsilon.$$
Say that the mean value of $q$ is $\bar q$. Then centering $q$ around its mean gives $q_c=q-\bar q$. Substitute into the ...
- 77.9k
2
votes
Performing 3 multivariate linear regressions at once
Assume $X,Y,Z$ are centered and organized as follows:
$$A=\begin{pmatrix}
\mid &\mid &\mid \\
X &Y &Z \\
\mid &\mid &\mid
\end{pmatrix}$$
And
$$A = A B + E$$
Where
$$B =
\...
- 17.1k
2
votes
What is R^2 if SSE = SST = 0
You should just continue to think of R2 as undefined in this situation. Thinking of it as 1 or zero obscures what's really go on here.
R2 answers the question of "what percent of the variance in ...
- 2,504
1
vote
Performing 3 multivariate linear regressions at once
There is another way to obtain these coefficients in a single regression.
It consists in having six variables, one for each coefficient we want to estimate.
If all you want are the OLS coefficients, ...
- 17.1k
1
vote
What is R^2 if SSE = SST = 0
$R^2$ in this situation is UNINTERESTING. (This is not to say, however, that the question is not worth asking, so this statement is not a criticism of the OP.)
The point of regression is that we want ...
- 47.1k
1
vote
Why is multiple R squared artificially inflated?
Assuming a linear model that is estimated using ordinary least squares (this seems to be the setting you mean), $R^2$ will not decrease, since the model always has the option to set the coefficient on ...
- 47.1k
1
vote
Is high multicollinearity always an issue in OLS?
There are competing factors.
One the one hand, multicollinearity inflates standard errors. On the other hand, removing a variable to remove the multicollinearity can lead to omitted-variable bias, and ...
- 47.1k
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