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

\beta=[E(x'x)]^{-1}E(x'y).

The estimator for \beta is

\hat{\beta}=\left(N^{-1}\sum_{i=1}^{N}x_i'x_i\right)^{-1}\left(N^{-1}\sum_{i=1}^{N}x_i'y_i\right).


The bias in the equation is given by:

\hat{\beta}=\beta+\left(N^{-1}\sum_{i=1}^{N}x_i'x_i\right)^{-1}\left(N^{-1}\sum_{i=1}^{N}x_i'u_i\right)


Homoskedastic standard errors, variance matrix $V$:

\hat{V}=\hat{\sigma}^2(X'X)^{-1}


Heteroskedastic standard errors, variance matrix $V$:

\hat{V}=(X'X)^{-1}\left(\sum_{i=1}^{N}\hat{u}_i^2x_i'x_i\right)(X'X)^{-1}

Random effects estimator (from Chapter 10 in Wooldridge):

\hat{\beta}=\left(\sum_{i=1}^NX_i'\hat{\Omega}^{-1}X_i\right)^{-1}\left(\sum_{i=1}^NX_i'\hat{\Omega}^{-1}y_i\right)


where

\Omega=\sigma_u^2I_T+\sigma_c^2j_Tj_T'


and

\hat{\Omega}=\hat{\sigma_u}^2I_T+\hat{\sigma_c}^2j_Tj_T'


and j_T is a vector of 1's.

Fixed effects estimator:

\hat{\beta}=\left(N^{-1}\sum_{i=1}^{N}{F}_i'{F}_i\right)^{-1}\left(N^{-1}\sum_{i=1}^{N}{F}_i'{g}_i\right)


where F and g are time-demeaned matrices:

F=Q_TX_i


and

g=Q_ty_i


where

Q_T=I_T-j_T(j_T'j_T)j_T'


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

\hat{V}=(F'F)^{-1}\left(\sum_{i=1}^NF_i'\hat{w}_i\hat{w}_i'F_i\right)(F'F)^{-1}


where w_i is Q_Tu_i.

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

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