1.2 KiB
#stats #econ
review from earlier:
$\beta_0 = \mu_y-\beta_1\mu_x$
\beta_1=p*\frac{\sigma_y}{\sigma_x}=\frac{\sigma_{xy}}{\sigma_x^2}
$\hat \beta_0 = \bar Y-\hat \beta_1\bar X$
$\hat \beta_1=r*\frac{S_y}{S_x}=\frac{S_{xy}}{S_x^2}$
sample variance:
total variations squared divided by one minus the sample size:
$S_x^2=\frac{1}{n-1}\sum^n{(x_i-\bar x)^2}$
sample covariance:
$S_{xy}=\frac{1}{n-1}\sum^n{(X_i-\bar X)(Y_i-\bar Y)}$
sample r value:
\rho = \frac{\sigma_{xy}}{\sigma_x\sigma_y} so the sample rho is $r = \frac{S{xy}}{S_x S_y}$
The only difference between these equations and those we used earlier in class to find the population regression is the \frac{1}{n-1} instead of $\frac{1}{n}$
Average sample error term \hat \varepsilon = 0. That was kinda the whole point of least squared error (Least Squares Regression).
since variance of the error term is \sigma_\epsilon^2 = E[(\varepsilon_i-\mu_\varepsilon)^2] and E(\mu_\varepsilon) = 0 then the variance of the error term is S\epsilon^2 = \frac{1}{n-2}\sum_{i=1}^n[\varepsilon_i^2] and that = S_{\varepsilon\varepsilon} and that's a chi-square distribution.
- TODO: make flashcards based off of these notes.