## Linear Algebra Equations for Econometrics

The purpose of this post is to outline the linear algebra of popular regression strategies. It is essentially an extremely short summary of the parts of Wooldridge's authoritative econometrics textbook that I use most often.

Equations for the single equation model (Wooldridge, Chapter 4):
The theoretical value of $\beta$ is

The estimator for $\beta$ is

The bias in the equation is given by:

Homoskedastic standard errors, variance matrix $V$:

Heteroskedastic standard errors, variance matrix $V$:

Random effects estimator (from Chapter 10 in Wooldridge):

where

and

and $j_T$ is a vector of $1$'s.

Fixed effects estimator:

where $F$ and $g$ are time-demeaned matrices:

and

where

which (according to Wooldridge, and comically in my opinion) is "easily seen to be a TXT symmetric, idempotent matrix with rank T-1." The asymptotic covariance matrix for a fixed effects estimation is

where $w_i$ is $Q_Tu_i$.

I'm hoping to add more useful equations from Wooldridge later.