Two Stage Least Squares

Definition: Two stage least squares is an instrumental variables estimation technique. Extends the IV idea to a situation where one has more instruments than independent variables in the model. Suppose one has a model:

y = Xb + e

Here y is a T x 1 vector of dependent variables, X is a T x k matrix of independent variables, b is a k x 1 vector of parameters to estimate, and e is a k x 1 vector of errors. But the matrix of independent variables X may be correlated to the e's. Then using a matrix of independent variables Z, uncorrelated to the e's, that is T x r, where r>k:

Stage 1: By OLS, regress the X's on the Z's to get Xhat = (Z'X)^-1Z'y

Stage 2: By OLS, regress y on the Xhat's. This gives an unbiased estimate of b.

The stages can be combined into one for maximum speed:

b = (X'P_zX)^-1X'P_zy

where P_z, the projection matrix of Z, is defined to be:

P_z = Z(Z'Z)^-1Z'

(Econterms)

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