# Least Squares, Three-Stage

# Least Squares, Three-Stage

The term *three-stage least squares* (3SLS) refers to a method of estimation that combines system equation, sometimes known as *seemingly unrelated regression* (SUR), with two-stage least squares estimation. It is a form of instrumental variables estimation that permits correlations of the unobserved disturbances across several equations, as well as restrictions among coefficients of different equations, and improves upon the efficiency of equation-by-equation estimation by taking into account such correlations across equations. Unlike the two-stage least squares (2SLS) approach for a system of equations, which would estimate the coefficients of each structural equation separately, the three-stage least squares estimates all coefficients simultaneously. It is assumed that each equation of the system is at least just-identified. Equations that are underidentified are disregarded in the 3SLS estimation.

Three-stage least squares originated in a paper by Arnold Zellner and Henri Theil (1962). In the classical specification, although the structural disturbances may be correlated across equations (*contemporaneous correlation* ), it is assumed that within each structural equation the disturbances are both homoskedastic and serially uncorrelated. The classical specification thus implies that the disturbance covariance matrix within each equation is diagonal, whereas the entire system’s covariance matrix is nondiagonal.

The Zellner-Theil proposal for efficient estimation of this system is in three stages, wherein the first stage involves obtaining estimates of the residuals of the structural equations by two-stage least squares of all identified equations; the second stage involves computation of the optimal instrument, or weighting matrix, using the estimated residuals to construct the disturbance variance-covariance matrix; and the third stage is joint estimation of the system of equations using the optimal instrument. Although 3SLS is generally asymptotically more efficient than 2SLS, if even a single equation of the system is mis-specified, 3SLS estimates of coefficients of all equations are generally inconsistent.

The Zellner-Theil 3SLS estimator for the coefficient of each equation is shown to be asymptotically at least as efficient as the corresponding 2SLS estimator of that equation. However, Zellner and Theil also discuss a number of interesting conditions under which 3SLS and 2SLS estimators are equivalent. First, if the structural disturbances have no mutual correlations across equations (the variance-covariance matrix of the system disturbances is diagonal), then 3SLS estimates are identical to the 2SLS estimates equation by equation. Second, if all equations in the system are just-identified, then 3SLS is also equivalent to 2SLS equation by equation. Third, if a subset of *m* equations is overidentified while the remaining equations are just-identified, then 3SLS estimation of the *m* over-identified equations is equivalent to 2SLS of these *m* equations.

The 3SLS estimator has been extended to estimation of a nonlinear system of simultaneous equations by Takeshi Amemiya (1977) and Dale Jorgenson and Jean-Jacques Laffont (1975). An excellent discussion of 3SLS estimation, including a formal derivation of its analytical and asymptotic properties, and its comparison with full-information maximum likelihood (FIML), is given in Jerry Hausman (1983).

**SEE ALSO** *Instrumental Variables Regression; Least Squares, Two-Stage; Regression; Seemingly Unrelated Regressions*

## BIBLIOGRAPHY

Amemiya, Takeshi. 1977. The Maximum Likelihood and the Nonlinear Three-stage Least Squares Estimator in the General Nonlinear Simultaneous Equation Model. *Econometrica* 45 (4): 955–968.

Dhrymes, Phoebus J. 1973. Small Sample and Asymptotic Relations Between Maximum Likelihood and Three Stage Least Squares Estimators. *Econometrica* 41 (2): 357–364.

Gallant, A. Ronald, and Dale W. Jorgenson. 1979. Statistical Inference for a System of Simultaneous, Non-linear, Implicit Equations in the Context of Instrumental Variable Estimation. *Journal of Econometrics* 11: 275–302.

Robinson, Peter M. 1991. Best Nonlinear Three-stage Least Squares Estimation of Certain Econometric Models. *Econometrica* 59 (3): 755–786.

Sargan, J. D. 1964. Three-stage Least-Squares and Full Maximum Likelihood Estimates. *Econometrica* 32: 77–81.

Zellner, Arnold, and Henri Theil. 1962. Three-stage Least Squares: Simultaneous Estimation of Simultaneous Equations. *Econometrica* 30 (1): 54–78.

*Mitali Das*

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