THREE_STAGE_REGRESSION

Estimate a simultaneous equation system with Three-Stage Least Squares (3SLS).

Purpose

The THREE_STAGE_REGRESSION command estimates a system of simultaneous linear equations using Three-Stage Least Squares. The method combines instrumental variables estimation with Seemingly Unrelated Regression, making it useful when explanatory variables are endogenous and equation errors are correlated across the system.

Syntax

THREE_STAGE_REGRESSION(simultaneousRegression, instruments)
Parameter Description
simultaneousRegression A SIMU_REGRESSION object containing the system of regression equations.
instruments A LIST(...) of exogenous variables used to instrument endogenous regressors.

Example

r1 = REGRESSION(LOGDIFF(LIST(CPIAUCSL,PPIACO,M2SL)))

r2 = REGRESSION(LOGDIFF(LIST(PCE,CPIAUCSL,PAYEMS)))

r3 = REGRESSION(LOGDIFF(LIST(REALLN,PAYEMS,M2SL)))

system = SIMU_REGRESSION(r1,r2,r3)

t = THREE_STAGE_REGRESSION(system, LIST(PPIACO,PAYEMS,M2SL))

SLIDESHOW(t)

How 3SLS Works

Stage 1 — Instrument Endogenous Variables

Each endogenous right-hand-side variable is regressed on the supplied instrument set. The fitted values from these first-stage regressions replace the original endogenous variables.

Stage 2 — Estimate Each Equation

Each equation is estimated independently using the instrumented variables. This is the two-stage least squares portion of the procedure and addresses simultaneity bias.

Stage 3 — Estimate the System Jointly

The residual covariance matrix from the instrumented equations is estimated, and the full system is then estimated with SUR/FGLS. This step uses cross-equation residual information to improve efficiency.

Slideshow Output

Panel Description
Interpretation Explains the simultaneous system, endogenous variables, instruments, and estimation procedure.
Diagnostics Shows residual dependence, average residual correlation, system SSR, and efficiency diagnostics.
First Stage Lists endogenous explanatory variables and the fitted variables created during instrumentation.
Coefficient Comparison Compares equation-by-equation estimates with final system-adjusted 3SLS estimates.

When to Use

Use THREE_STAGE_REGRESSION when variables are jointly determined across equations.

Note: Instruments should be correlated with the endogenous variables but uncorrelated with the structural equation errors. Weak instruments can produce unreliable estimates.

Mathematical Background

Three-Stage Least Squares combines instrumental variables estimation with SUR. In practical terms: 

3SLS = Instrumented Equations + SUR/FGLS

If no endogenous right-hand-side variables are present, ordinary SUR is usually sufficient. If equation residuals are not correlated, equation-by-equation 2SLS may be adequate. 3SLS is most useful when both simultaneity and cross-equation residual correlation are present.

Historical Note

Three-Stage Least Squares was introduced by Arnold Zellner and Henri Theil as an extension of Seemingly Unrelated Regression. It remains a classic method in structural econometrics and is commonly discussed in graduate texts such as Jan Kmenta's Elements of Econometrics and Johnston's Econometric Methods.

Related Commands

Returns

The command returns a 3SLS system object containing: