From DDWiki

Two-stage least squares (2SLS) is a estimation method to utilize instrumental variables with least-squares for structural model as endogenous variables exist.

Suppose a model:

where

y is Tx1 vector of dependent variables (observations)

ε is kx1 vector of error components

X is Txk matrix of independent variables, which may be correlated to error components

Z is assumed a independent variable Txr matrix (r>=k) uncorrelated to error components

Stage 1: Endogenous variables X are regressed on all valid instruments Z, including the full set of exogenous variables. Since the instruments Z are exogenous, the approximations on the endogenous covariates will not be correlated with the error term. Thus,

Stage 2: A small correction need to be made to cover the sum-of-squared residuals in order to associate standard errors correctly.

reference

• Greene, W.H., 2003, Econometric analysis, Prentice Hall, Upper Saddle River, N.J.
• Gujarati, D.N., 2003, Basic econometrics, McGraw-Hill, New York.