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 is
The estimator for 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):
and is a vector of 's.
Fixed effects estimator:
where and are time-demeaned matrices:
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 is .
I'm hoping to add more useful equations from Wooldridge later.