[SOLVED] Statistics 215b assignment 4

For this assignment you will use simulation to study the performance of OLS and IVLS coefficient
estimators, as well as two different estimators of the error variance. Consider the model
Yi D Xiˇ C i
; (1)
Xi D Ui C 2Vi C ıi
:
Everything in sight is scalar. The vectors
.Ui
; Vi
; i
; ıi/; i D 1; : : : ; n;
are iid across i. Each vector is normal with mean zero. For each i we take the three random objects
(1) Ui
, (2) Vi
, and (3) .i
; ıi/ to be mutually independent; furthermore,
Var.Ui/ D Var.Vi/ D 1;
Var.i/ D Var.ıi/ D 
2
; and
Cov.i
; ıi/ D :
In (1), the variable Xi
is endogenous (explain why), and .Ui
; Vi/ are instruments (explain why).
Suppose ˇ D 3, 
2 D 1, and  D 3=4. For each of 1,000 simulation runs, generate n D 100
independent realizations of the vector .Ui
; Vi
; i
; ıi
; Xi
; Yi/ according to (1) and (2). Use OLS to
obtain the estimate ˇO
OLS, and IVLS to obtain ˇO
IVLS. Plot the histogram for each estimator; report
the mean, SD, and RMSE in each case. What are the relative merits of OLS versus IVLS? For
each simulation, estimate the error variance 
2
in two ways: first using the residuals obtained from
plugging ˇO
IVLS into (1), then using the residuals from the transformed equation
.Z0Z/1=2Z
0Y D .Z0Z/1=2Z
0Xˇ C  :
Here Z is the n 2 matrix of instruments, X is the n 1 design matrix, and Y is the n 1 vector
of responses. What is the appropriate denominator in each case? Plot the histograms for the two
estimators and report sample means and SDs. Comment briefl