Three-Stage Least Squares (3SLS)
Also known as: 3SLS, system instrumental variables, Üç Aşamalı En Küçük Kareler (3SLS)
Three-Stage Least Squares is a system estimator for simultaneous-equation models that accounts for the correlation of error terms across equations. Introduced by Zellner and Theil in 1962, it combines two-stage least squares with the seemingly-unrelated-regression idea to estimate all equations jointly and more efficiently.
Key highlights
- More efficient than equation-by-equation two-stage least squares when the error terms are correlated across equations, because it uses that information jointly.
- Handles endogeneity in a full simultaneous-equation system through instrumental variables.
- Provides a coherent, system-wide estimate of structural parameters for interrelated economic relationships.
Intuition
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How it works
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When to use it
Use 3SLS when you have a system of simultaneous equations with continuous outcomes, each equation is identified (satisfying the order and rank conditions), valid and relevant instruments are available for the endogenous regressors, and the error terms are genuinely correlated across equations. A reasonably large sample is needed (roughly 100 observations or more) because the gains come from estimating the cross-equation covariance. It suits supply-demand systems, macroeconomic models, and structural equation systems on cross-sectional or time-series data.
Strengths & limitations
- More efficient than equation-by-equation two-stage least squares when the error terms are correlated across equations, because it uses that information jointly.
- Handles endogeneity in a full simultaneous-equation system through instrumental variables.
- Provides a coherent, system-wide estimate of structural parameters for interrelated economic relationships.
- Misspecification in one equation propagates through the whole system, so careful specification of every equation is essential.
- Requires each equation to be properly identified and valid, relevant instruments to be available.
- Offers little or no advantage over two-stage least squares when the cross-equation error correlation is weak, and needs a fairly large sample.
Common pitfalls
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Applications
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Frequently asked
How does 3SLS differ from 2SLS?
Two-stage least squares estimates each equation on its own. Three-stage least squares adds a final stage that uses the correlation between the equations' error terms, estimating the whole system jointly via generalised least squares. When those errors are correlated, 3SLS is more efficient.
When does 3SLS not help over 2SLS?
If the error terms across equations are uncorrelated, there is nothing extra for 3SLS to exploit, so it gives essentially the same result as 2SLS while being more sensitive to misspecification. In that case 2SLS is the safer choice.
What does identification mean here?
Each equation must be identified, satisfying the order and rank conditions, meaning there are enough exogenous variables excluded from the equation to pin down its parameters. An underidentified equation cannot be estimated, and it invalidates the system estimate.
Why can a single bad equation ruin the estimates?
Because 3SLS estimates all equations jointly, misspecification or invalid instruments in one equation propagate through the system and bias the other equations' estimates too. Every equation must be specified carefully.
Sources
- 1.Zellner, A. & Theil, H. (1962). Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations. Econometrica, 30(1), 54–78.
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ScholarGate. (2026, June 1). Three-Stage Least Squares. ScholarGate. https://scholargate.app/econometrics/three-stage-least-squares