Instrumental Variable - Interpretation As Two-stage Least Squares

Interpretation As Two-stage Least Squares

One computational method which can be used to calculate IV estimates is two-stage least-squares (2SLS or TSLS). In the first stage, each endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model, including both exogenous covariates in the equation of interest and the excluded instruments. The predicted values from these regressions are obtained.

Stage 1: Regress each column of X on Z,

and save the predicted values:

In the second stage, the regression of interest is estimated as usual, except that in this stage each endogenous covariate is replaced with the predicted values from its first stage model from the first stage.

Stage 2: Regress Y on the predicted values from the first stage:

The resulting estimator of is numerically identical to the expression displayed above. A small correction must be made to the sum-of-squared residuals in the second-stage fitted model in order that the covariance matrix of is calculated correctly.