Estimation of a Triangular, Seemingly Unrelated Regression System by OLS. This is based on Sentana (1997). Consider a system of three SUR equations in which the explanatory variables for the first equation are a subset of the explanatory variables for the second equation, which are in turn a subset of the explanatory variables for the third equation.

(a) Show that SUR applied to the first two equations is the same (for those equations) as SUR applied to all three equations. Hint: See Schmidt (1978).

(b) Using part

(a) show that SUR for the first equation is equivalent to OLS.

(c) Using parts

(a) and

(b) show that SUR for the second equation is equivalent to OLS on the second equation with one additional regressor. The extra regressor is the OLS residuals from the first equation. Hint: Use Telser’s (1964) results.

(d) Using parts (a),

(b) and

(c) show that SUR for the third equation is equivalent to OLS on the third equation with the residuals from the regressions in parts

(b) and

(c) as extra regressors.