2SLS

Two-stage least squares (2SLS) is an instrumental variables estimation method used to obtain consistent estimates when one or more regressors are endogenous. It is commonly applied in econometrics to identify causal relationships when regressor endogeneity arises from omitted variables, measurement error, or simultaneity.

In a typical linear model, y = β0 + β1 D + Xβx + ε, where D is endogenous and X contains exogenous controls, Z denotes instruments for D. The procedure has two stages. First, regress D on Z and X: D = π0 + ZΠ + XΓ + v, and obtain the fitted values D̂. Second, regress y on D̂ and X: y = β0 + β1 D̂ + Xβx + ε̃. The coefficient β1 from this second stage is the 2SLS estimate of the causal effect of D on y.

Key assumptions include instrument relevance (Z is correlated with D) and the exclusion restriction (Z affects y only through D), along with independence (Z is uncorrelated with ε). If the model is overidentified (more instruments than endogenous variables), overidentifying restrictions can be tested (e.g., Hansen J test). When the number of instruments equals the number of endogenous variables, the model is exactly identified and 2SLS coincides with the IV estimator using Z.

Properties and limitations: 2SLS provides consistent estimates under its assumptions, but its performance deteriorates with weak instruments, where the first-stage F-statistic is low and estimates become biased. Standard errors should reflect the two-stage procedure, and weak instruments require caution. Extensions and alternatives include limited information maximum likelihood (LIML) and various robust diagnostics.