Root n Consistency
How ‘fast’ does b β? Asy.Var[b] =σ2/n Q-1 is O(1/n)
Convergence is at the rate of 1/n n b has variance of O(1)
Is there any other kind of convergence?
Suppose zn is not exactly normally distributed, but (1) E[zn] = , (2)
Var[zn] = 2, (3) the limiting distribution of zn is normal. Then
Asymptotic Distribution
b
1 n
X'X
1
1 n
n i1
x
i
i
The lim iting behavior of b is the sam e as
that of the statistic that results when the
m om ent m atrix is replaced by its lim it. W e
plim
1 n
'M
plim
1 n
'
1 n
'X
(
X
'X
)
-
1
X
'
plim 1 ' n
plim
1 n
'X
plim
1 n
(X'X
)-1
p
lim
1 n
X'
plim 1 ' 0'Q-1 0 n
W hat m ust be assum ed to claim plim 1 '= E[2 ] 2 ? n
C o n c lu d e : n (b ) d N[0 , 2 Q -1 ]
A p p ro x im a te ly : b a N[ 0 , ( 2/ n ) Q -1 ]
Asymptotic Properties
Probability Limit and Consistency Asymptotic Variance Asymptotic Distribution
Limiting Normality
n 1 X'ε n 1
n
n
n i1
xii
n1 n
w n
i1 i
1
n
w n
i1 i
Mean
of
a
sample.
Independent observations.
nw
Mean converges to zero (plim (1/n)X'ε = 0 already assumed
Applied Econometrics
William Greene Department of Economics Stern School of Business
Applied Econometrics
12. Asymptotics for the Least Squares Estimator in the Classical Regression Model
x1,…,xn = a sample from exponential population; min has variance O(1/n2). This is ‘n – convergent’
Certain nonparametric estimators have variances that are O(1/n2/3). Less than root n convergent.
and identical distributions.
(2) i = a random variable with a constant distribution with finite mean and variance and E[i]=0 (3) xi and i statistically independent. Then, zi = xii = an observation in a random sample, with constant variance matrix and mean vector 0.
Mean Square Convergence
E[b|X]=β for any X. Var[b|X]0 for any specific X b converges in mean square to β
Probability Limit
b
1 n
X'X
1
1 n
n i1
x
ii
b
-
Slutsky theorem and the delta method apply to functions of b.
Test Statistics
We have established the asymptotic distribution of b. We now turn to the construction of test statistics. In particular, we based tests on the Wald statistic
Wald Statistics
General approach to the derivation based on a univariate distribution (just to get started).
A. Core result: Square of a standard normal variable chi-squared with 1 degree of freedom.
1
n
z n
i1 i
converges
to
its
expectation
by
the
law
of
large
numbers.
What is wrong with this approach?
Consistency of s2
s2 1 e'e 1 'M n 1 'M
nK
nK
nK n
n 1 nK
p lim s2
distribution?
Has no ‘limiting’ distribution
Variance 0; it is O(1/n) Stabilize the variance? Var[n b] ~ σ2Q-1 is O(1) But, E[n b]= n β which diverges
n (b - β) = n (X’X)-1X’ε = n (X’X/n)-1(X’ε/n)
Limiting behavior is the same as that of n Q-1(X’ε/n)
Q is a fixed matrix. Behavior depends on the random vector n (X’ε/n)
n w = a candidate for the Lindberg-Feller Central Limit Theorem. Variance of each term (|xi) is 2xixi '. Variance of w 2Q/n. Var n w 2Q
Based on the CLT, n w d N[0, 2Q]
n (b - β) a random variable with finite mean and variance. (stabilizing transformation)
b apx. β +1/ n times that random variable
Limiting Distribution
exam ine the behavior of the m odified sum
Q
1
1 n
n i1
x
i
i
Asymptotics
Q
1
1 n
n i1
x
i
i
What is the mean of this random vector?
What is its variance?
Do they 'converge' to something? We use
regressors.
What
must
be
assumed
to
get
p lim
1 n
X
'
0?
Crucial Assumption of the Model
What
must
be
assumed
to
get
plim
1 n
X'
0
?
(1) x i = a random vector w ith finite m eans and variance
Suppose z ~ N[0,2], i.e., variance not 1. Then (z/)2 satisfies A. Now, suppose z~N[,2].
Then [(z - )/]2 is chi-squared with 1 degree of freedom. This is the normalized distance between z and , where distance is measured in standard deviation units.
this method to find the probability limit.