Hausmans Specification Test: 2SLS Versus 3SLS. This is based on Holly (1988). Consider the two-equations model, y1 = y2 + 1x1 + 2x2 + u1

Hausman’s Specification Test: 2SLS Versus 3SLS. This is based on Holly (1988). Consider the two-equations model, y1 = αy2 + β1x1 + β2x2 + u1 y2 = γy1 + β3x3 + u2 where y1 and y2 are endogenous; x1, x2 and x3 are exogenous (the y’s and the x’s are n × 1 vectors). The standard assumptions are made on the disturbance vectors u1 and u2. With the usual notation, the model can also be written as y1 = Z1δ1 + u1 y2 = Z2δ2 + u2 The following notation will be used: δ = 2SLS,δ = 3SLS, and the corresponding residuals will be denoted as u and u, respectively.

(a) Assume that αγ = 1. Show that the 3SLS estimating equations reduce to σ11Xu1 + σ12Xu2 = 0 σ12Z
2PX u1 + σ22Z
2PX u2 = 0 where X = (x1, x2, x3), Σ = [σij ] is the structural form covariance matrix, and Σ−1 = [σij ]
for i, j = 1, 2.

(b) Deduce that δ 2 = δ 2 and δ1 = δ1 − (σ12/σ22)(Z
1PXZ1)−1Z
1PXu2. This proves that the 3SLS estimator of the over-identified second equation is equal to its 2SLS counterpart. Also, the 3SLS estimator of the just-identified first equation differs from its 2SLS (or indirect least squares) counterpart by a linear combination of the 2SLS (or 3SLS) residuals of the over-identified equation, see Theil (1971).

(c) How would you interpret a Hausman-type test where you compareδ 1 andδ1? Show that it is nR2 where R2 is the R-squared of the regression of u2 on the set of second stage regressors of both equations  Z1 and  Z2. Hint: See the solution by Baltagi (1989).