Iterative Estimation in Partitioned Regression Models. This is based on Fiebig (1995). Consider the partitioned regression model given in (7.8) and let X2 be a single regressor, call it x2 of dimension n × 1 so that β2 is a scalar. Consider the following strategy for estimating β2: Estimate β1 from the shortened regression of y on X1. Regress the residuals from this regression on x2 to yield b(1)

2 .

(a) Prove that b(1)

2 is biased.

Now consider the following iterative strategy for re-estimating β2:

Re-estimate β1 by regressing y − x2b(1)

2 on X1 to yield b(1)

1 . Next iterate according to the following scheme:

b(j)

1 = (X



1X1)−1X



1(y − x2b(j)

2 )

b(j+1)

2 = (x



2x2)−1x



2(y − X1b(j)

1 ), j= 1, 2, ...

(b) Determine the behavior of the bias of b(j+1)

2 as j increases.

(c) Show that as j increases b(j+1)

2 converges to the estimator of β2 obtained by running OLS on (7.8).