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ECOM032 ECONOMETRICS B 20/21
发布时间:2022-05-11
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Main Examination Period 20/21
ECOM032 ECONOMETRICS B
Question 1
a) Suppose that 红0 is true.
i) Show that, with xy being
xi yi ,
)←n =
and that)←n =)0 + ←x2 | xZ ∶ Ω ┌ ┐ 〉←x2 | xZ ∶ Ω ┌
┐ .
ii) Show that E ←)←n |xn | zn ∶ =)0 and that Var /)←n |xn | zn 、≥ n2礻x2 |
How can you interpret (1)?
[10 marks]
(1)
Hint: (1) can be proved without any computation by invoking a result about ordinary least squares estimation. [5 marks]
b) Suppose that 红0 is true and that Ω = ┌ 0(1) 0(0) ┐ . Compute)←n and show that (1) becomes an equality for this choice of Ω.
c) Suppose that 红0 is true. Explain why
n = ┌
┐ -1
[5 marks]
(2)
is a good choice for Ω, and compute the corresponding)←n . Show that (1) becomes an equality when Ω = n .
[10 marks]
d) Discuss the relative merits of)←n compared to the OLS estimator of).
[5 marks]
e) It is assumed here that Ω = n . Suppose that 红1 is true. Show that)←n
)0 + ∆ where ∆
0.
[10 marks]
Question 2
For n as in (2) and)←n = arg minβ
n ())Y
n
n ()), define
n =
n /)←n、Y
n
n /)←n 、 where
n(2) =
i
1 /yi -
xi、2 |
Recall that 红0 and 红1 are the two hypotheses.
红0 : (xi | Zi ) is independent of –i vs 红1 : only Zi is independent of –i and E [xi –i] 0.
a) Give conditions ensuring that n(2)
礻2 under 红0 and 红1 .
[10 marks]
b) Recall that z,n ()) =
Zi (yi - xi)). Show that
n =
n(2)
/x2、 /,nz,n /)←n、、2 |
and interpret z,n /)←n、.
[10 marks]
c) Show that z,n /)←n 、 =
/Zi -
xi、–i , and that, under
红0 , n converges in distribution to a Chi Square distribution with one degree of freedom.
[15 marks]
d) Does n diverge to +o under 红1 ?
[10 marks]
e) Propose an asymptotic ( level test of the null hypothesis that xi is exogeneous. Explain when the proposed test is of level ( asymptotically and when it is consistent.
[10 marks]