Download 計量經濟分析（一）

國立東華大學經濟系
94 學年度第二學期
計量經濟分析（二）
第一次作業
1.
2006 年 3 月 27 日繳交
Let y and x be scalars such that
E ( y | x)   0  1 ( x   )   2 ( x   ) 2
where   E (x ) .
a、 Find E ( y | x ) / x , and comment on how it depends on x .
b、 Show that 1 is equal to E ( y | x ) / x averaged across the distribution of
x.
c、 Supposed that x has a symmetric distribution, so that E[( x   ) 3 ]  0 .
Show that L( y | 1, x )   0  1 x for some  0 . Therefore, the coefficient
on x in the linear projection of y on (1, x ) measures something useful
in the nonlinear model for E ( x   ) : it is the partial effect E ( y | x ) / x
averaged across the distribution of x .
2.
Let ˆ be a N -asymptotically normal estimator for the scalar   0 . Let
ˆ  log( ˆ) be an estimator of   log(  ) .
a、 Why is ˆ a consistent estimator of  ?
b、 Find the asymptotic variance of
variance of N (ˆ   ) .
N (ˆ   ) in terms of the asymptotic
c、 Suppose that, for a sample of data, ˆ  4 and se(ˆ)  2 . What is ˆ and
its (asymptotic) standard error?
d、 Consider the null hypothesis H 0 :   1 . What is the asymptotic t statistic
for testing H 0 , given the numbers from part c?
e、 Now state H 0 from part (d) equivalently in terms of  , and use ˆ and
se (ˆ ) to test H 0 . What do you conclude?
3.
Let the probability density function of X is
f ( x) 
1

ex /   0 .
Let {xi : i  1,2, , N } be an independent, identically distributed sequence with
E ( xi2 )   . Let X  i 1 X i
N
國立東華大學經濟系
94 學年度第二學期
a、 Prove that X is a consistent estimator of E (x ) .
b、 Prove that X is asymptotically normal.
4.
Suppose that the Xi and Yi are independently distributed as B(n,p) and B(n,q). Let
pq
be the parameter of interest.
 2 2
p  q  pq
pˆ qˆ
a、 Prove that the estimate ˆ  2
is a consistent estimate.
pˆ  qˆ 2  pˆ qˆ
b、 Prove that ˆ is asymptotically normally distributed and find the
asymptotic variance. (Hint: Use the delta method!)
5.
Consider estimating the effect of personal computer ownership, as represented by
a binary variable, PC on college GPA, colGPA. With data on SAT scores and high
school GPA, hsGPA you postulate the model
colGPA   0  1hsGPA   2 SAT   3 PC  u
a、 Why might u and PC be positively correlated?
b、 If the given equation is estimated by OLS using a random sample of
col/lege students, is ˆ likely to have an upward or downward asymptotic
3
bias?
c、 What are some variables that might be good proxies for the unobservables
in u that are correlated with PC?