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## Robust Two-Stage Least Squares

The Two-Stage Least Squares (2-SLS) is a well known econometric technique used to estimate the parameters of a multi-equation (or simultaneous equations) econometric model when errors across the equations are not correlated and the equation(s) concerned is (are) over-identified or exactly identified. It is one of the members of the family of k-class estimators. Unlike the Three-Stage Least Squares, it does not estimate the parameters of all the equations of the model in one go. The 2-SLS estimates the parameters of an econometric model equation by equation, that is, one equation at a time.

Presence of outliers in data affects the OLS (Ordinary Least Squares) estimates adversely. This effect is much more disasterous in case of the 2-SLS because the OLS estimates at the first stage enter into the estimation at the second stage. As a matter of fact, the effects of outliers pervade through all the equations and the estimated structural parameters in them. These effects are so intricately pervasive that it is very difficult to assess the influence of outliers on the estimated structural parameters. However, using the procedure developed by Norm Campbell modified by using the measure of robust median deviation suggested by Hampel et al., it is possible to robustify the 2-SLS estimation procedure. The estimation method based on the original Campbell procedure performs poorly, while the method based on the modified Campbell procedure shows appreciable robustness. Robustness of the proposed method is not much destabilized by the magnitude of outliers, but it is sensitive to the number of outliers/perturbations in the data matrix. The breakdown point of the method, is somewhere between 45 to 50 percent of the number of points in the data matrix. We present the Fortran program (source codes) of this estimation method. It has the data generator program also for conducting experiments and testing the effectiveness of the method.

References :
• Campbell, N. A. (1980) "Robust Procedures in Multivariate Analysis I: Robust Covariance Estimation", Applied Statistics, 29 (3): 231-237.
• Mishra, S.K. (2008) " Robust Two-Stage Least Squares: Some Monte Carlo Experiments", Journal of Applied Economic Sciences, III(4(6)): 434-443, 2008. Available at SSRN: Download.

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