清华经管计量经济学Lecture7

Shengjie Hong SEM, Tsinghua) hongshj@ Lecture 7 5 / 52
p
d
−→ VS −→
d
p
Relation between convergence in distribution and convergence in probability: If Xn −→ X , then this implies that Xn −→ X . The inverse is NOT true. Xn −→ X does NOT necessarily imply p Xn −→ X . However, there is one special case:If Xn converges to a CONSTANT p a, then Xn −→ a.
1
The Weak Law of Large Numbers (WLLN): If we have an i.i.d. n sample of {xi }i =1 (which is typically the case in our course), then the sample average p ¯n −→ E (x ) . x The Continuous Mapping Theorem (CMT): If we can show Xn −→ X and Yn −→ Y , then we also know that aXn −→ aX , Xn p (d ) X p (d ) p (d ) Xn + Yn −→ X + Y , Xn Yn −→ XY , and −→ . Yn Y
I NTRODUCTORY E CONOMETRICS
Lecture 7
Spring 2014, Tsinghua University
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
1 / 52
Asymptotic Properties of the OLS Estimators
p (d ) p (d ) p (d )
2
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
9 / 52
The WLLN is often essential for proving convergence in probability. The CMT usually comes in handy, when we Know the main components of a sequences convergence in probability/distribution, respectively; And want to prove the sequence itself converges in probability/distribution. We have a third tool, which is essential for proving convergence in distribution. It is called the Central Limit Theorem (CLT).
hongshj@ Lecture 7 7 / 52
2
Shengjie Hong (SEM, Tsinghua)
The estimator as a function of the sample size n ˆ1 θ ˆ2 θ ˆn θ ˆn+1 θ = = ··· = = ··· As n goes to infinity, we obtain a sequence of r.v. ˆn θ
1
Given a sample {(xi , yi )}N i =1 , we can interpret it as N repeated observation on the same pair of random variables (x , y ). Alternatively, we can interpret the sample as a single observation on N pairs of random variables (x1 , y1 ), (x2 , y2 ), ..., (xN , yN ).
∞
g ((x1 , y1 )) ; g ((x1 , y1 ) , (x2 , y2 )) ; g ((x1 , y1 ) , . . . (xn , yn )) ; g ((x1 , y1 ) , . . . (xn , yn ) , (xn+1 , yn+1 )) ; .
n =1
“What would happen when n goes to infinity” is asking, literally:
1 2 3
ˆn Is there any trend/tendency of the sequence of θ
∞
as n increases?
n =1
If yes, what is the pattern of this trend/tendency? Does this trend/tendency converge to a limit in some sense?
d p d
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
6 / 52
W HY TO BOTHER WITH THESE NEW TYPES OF CONVERGENCE ? We want to know what would happen to our estimators as the sample size n increases. Understanding an (i.i.d.) sample:
x1 = x2 = 1, (0.5 × 0.5 = 0.25) .
Shengjie Hong (SEM, Tsinghua)
hongshj@
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
11 / 52
¯n T HE F INITE S AMPLE D ISTRIBUTION OF x
For example, suppose xi = −1, (0.5) ; 1, (0.5).
−1 1
with prob. 0.5 . Then with prob. 0.5
¯1 x
=
x1 =
¯2 x
=
−1, x1 + x2 = 0, 2 1,
x1 = x2 = −1, (0.5 × 0.5 = 0.25) ; x1 = 1, x2 = −1, or, x1 = −1, x2 = 1, 0.52 + 0.52 = 0.5 ;
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
10 / 52
¯n T HE D ISTRIBUTION OF x
¯n ? What is the probability distribution of x Let’s first consider the finite sample distribution. For any fixed sample size: ¯n is UNKNOWN unless we know the exact The exact distribution of x distribution of thesexi ’s. ¯n Even if we know the distribution of xi ’s, the exact distribution of x can be VERY COMPLICATED.
∞
ˆn These questions are all about θ
, which is a sequence of r.v.. So it
n=1
makes sense to answer these questions in terms of convergence in probability and/or convergence in distribution.
Shengjie Hong (SEM, Tsinghua) hongshj@ Lecture 7 8 / 52
U SEFUL T OOLS
Proving a sequence converges in probability/distribution can be very hard, especially if you want to prove it directly by definitions. We have several effective tools to help us. We’ve learned two of them:
Shengjie Hong (SEM, Tsinghua)
hongshj@
Lecture 7
3 / 52
Last time, we introduce two different types of convergence. Convergence in Probability Definition: Xn converges in probability to X if, for any distance ε > 0, Prob (| Xn − X | > ε) −→ 0 as n −→ ∞. Notation: We can write “Xn converges in probability to X ” as Xn −→ X . Convergence in Distribution Definition: Xn converges in distribution to X if the CDF of Xn , Fn (·) converges pointwise to the CDF of X , F (·). Notation: We can write “Xn converges in distribution to X ” as Xn −→ X .