Asymptotic Theory and Stochastic Regression
Chapter 13
Asymptotic Theory and Stochastic Regressors
The nature of explanatory variable is assumed to be
non-stochastic or fixed in repeated samples in any regression
analysis. Such an assumption is appropriate for those experiments
which are conducted inside the laboratories where the experimenter
can control the values of explanatory variables. Then the repeated
observations on study variable can be obtained for fixed values of
explanatory variables. In practice, such an assumption may not
always be satisfied. Sometimes, the explanatory variables in a
given model are the study variable in another model. Thus the study
variable depends on the explanatory variables that are stochastic
in nature. Under such situations, the statistical inferences drawn
from the linear regression model based on the assumption of fixed
explanatory variables may not remain valid.
We assume now that the explanatory variables are stochastic but
uncorrelated with the disturbance term. In case, they are
correlated then the issue is addressed through instrumental
variable estimation. Such a situation arises in the case of
measurement error models.
Stochastic regressors model
Consider the linear regression model
yX
be
=+
where
X
is a
(
)
nk
´
matrix of
n
observations on
k
explanatory variables
12
,,...,
k
XXX
which are stochastic in nature,
y
is a
(
)
1
n
´
vector of
n
observations on study variable,
b
is a
(
)
1
k
´
vector of regression coefficients and
e
is the
(
)
1
n
´
vector of disturbances. Under the assumption
(
)
2
()0,,
EVI
ees
==
the distribution of
,
i
e
conditional on
'
,
i
x
satisfy these properties for all all values of
X
where
'
i
x
denotes the
th
i
row of
X
. This is demonstrated as follows:
Let
(
)
'
|
ii
px
e
be the conditional probability density function of
i
e
given
'
i
x
and
(
)
i
p
e
is the unconditional probability density function of
i
e
. Then
(
)
(
)
(
)
(
)
''
||
0
iiiiii
iii
i
Expxd
pd
E
eeee
eee
e
=
=
=
=
ò
ò
(
)
(
)
(
)
(
)
2'2'
2
2
2
||
.
iiiiii
iii
i
Expxd
pd
E
eeee
eee
e
s
=
=
=
=
ò
ò
In case,
'
and
ii
x
e
are independent, then
(
)
(
)
'
|.
iii
pxp
ee
=
Least squares estimation of parameters
The additional assumption that the explanatory variables are
stochastic poses no problem in the ordinary least squares
estimation of
b
and
2
s
. The OLSE of
b
is obtained by minimizing
(
)
(
)
'
yXyX
bb
--
with respect
b
as
(
)
1
''
bXXXy
-
=
and estimator of
2
s
is obtained as
(
)
(
)
2
1
'
syXbyXb
nk
=--
-
.
Maximum likelihood estimation of parameters:
Assuming
(
)
2
~0,
NI
es
in the model
yX
be
=+
along with
X
is stochastic and independent of
e
, the joint probability density function
e
and
X
can be derived from the joint probability density function
of
y
and
X
as follows:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
'''
1212
'
11
''
11
''
1
'
1
'''
1212
,,,...,,,,...,
|
|
,
,,...,,,,...,
,.
nn
nn
ii
ii
nn
iii
ii
n
iii
i
n
ii
i
nn
fXfxxx
ffx
fyxfx
fyxfx
fyx
fyyyxxx
fyX
eeee
e
==
==
=
=
=
æöæö
=
ç÷ç÷
èøèø
æöæö
=
ç÷ç÷
èøèø
=
=
=
=
ÕÕ
ÕÕ
Õ
Õ
This implies that the maximum likelihood estimators of
b
and
2
s
will be based on
(
)
(
)
'
11
|
nn
iii
ii
fyxf
e
==
=
ÕÕ
so they will be same as based on the assumption that
',1,2,...,
i
sin
e
=
are distributed as
(
)
2
0,.
N
s
So the maximum likelihood estimators of
2
and
bs
when the explanatory variables are stochastic are obtained
as
(
)
(
)
(
)
1
2
''
1
.
XXXy
yXyX
n
b
sbb
-
=
¢
=--
%
%%
%
Alternative approach for deriving the maximum likelihood
estimates
Alternatively, the maximum likelihood estimators of
b
and
2
s
can also be derived using the joint probability density function
of
and.
yX
Note: Note that the vector
'
x
is represented by an underscore in this section to denote that
it ‘s order is
(
)
11
k
´-
éù
ëû
which excludes the intercept term.
Let
'
,1,2,...,
i
xin
=
are from a multivariate normal distribution with mean vector
x
m
and covariance matrix
xx
S
, i.e.,
(
)
'
~,
ixxx
xN
m
S
and the joint distribution of
y
and
'
i
x
is
'
~,.
y
yyyx
x
xyxx
i
y
N
x
m
s
m
éù
S
æö
æö
æö
êú
ç÷
ç÷
ç÷
SS
êú
èø
èø
èø
ëû
Let the linear regression model is
'
01
,1,2,...,
iii
yxin
bbe
=++=
where
'
i
x
is a
(
)
11
k
´-
éù
ëû
vector of observation of random vector
0
,
x
b
is the intercept term and
1
b
is the
(
)
11
k
-´
éù
ëû
vector of regression coefficients. Further
i
e
is disturbance term with
(
)
2
~0,
i
N
es
and is independent of
'
x
.
Suppose
~,.
y
yyyx
x
xyxx
y
N
x
m
s
m
éù
S
æö
æö
æö
êú
ç÷
ç÷
ç÷
SS
êú
èø
èø
èø
ëû
The joint probability density function of
(
)
'
,
i
yx
based on random sample of size
n
is
(
)
(
)
'
1
1
2
2
11
,'exp.
2
2
yy
k
xx
yy
fyx
xx
mm
mm
p
-
éù
--
æöæö
êú
=-S
ç÷ç÷
--
êú
èøèø
S
ëû
Now using the following result, we find
1
:
-
S
Result: Let
A
be a nonsingular matrix which in partitioned suitably as
,
BC
A
DE
æö
=
ç÷
èø
where
E
and
1
FBCED
-
=-
are nonsingular matrices, then
111
1
111111
.
FFCE
A
EDFEEDFCE
---
-
------
æö
-
=
ç÷
-+
èø
Note that
11
.
AAAAI
--
==
Thus
1
1
12111
2
1
1
,
yxxx
xxxyxxxxxyyxxx
s
s
-
-
----
æö
-SS
S=
ç÷
ç÷
-SSS+SSSS
èø
where
21
.
yyyxxxxy
ss
-
=-SSS
Then
(
)
(
)
(
)
(
)
(
)
{
}
2
121
1
2
2
2
11
,'exp''.
2
2
yxxxxyxxxx
k
fyxyxxx
mmsmm
s
p
--
éù
éù
=----SS+-S-
êú
ëû
ëû
S
The marginal distribution of
'
x
is obtained by integrating
(
)
,'
fyx
over
y
and the resulting distribution is
(
)
1
k
-
variate multivariate normal distribution as
(
)
(
)
(
)
(
)
1
1
1
2
2
11
'exp'.
2
2
xxxx
k
xx
gxxx
mm
p
-
-
éù
=--S-
êú
ëû
S
The conditional probability density function of
y
given
'
x
is
(
)
(
)
(
)
(
)
(
)
{
}
2
'
1
2
2
,'
|'
'
11
exp
2
2
yxxxxy
fyx
fyx
gx
yx
mm
s
ps
-
=
éù
=----SS
êú
ëû
which is the probability density function of normal distribution
with
· conditional mean
(
)
1
(|')'
yxxxxy
Eyxx
mm
-
=+-SS
and
· conditional variance
(
)
(
)
2
|'1
yy
Varyx
sr
=-
where
1
2
yxxxxy
yy
r
s
-
SSS
=
is the population multiple correlation coefficient.
In the model
01
',
yx
bbe
=++
the conditional mean is
(
)
(
)
'
01
01
|'|
'.
ii
EyxxEx
x
bbe
bb
=++
=+
Comparing this conditional mean with the conditional mean of
normal distribution, we obtain the relationship with
0
b
and
1
b
as follows:
1
1
'
01
.
xxxy
yx
b
bmmb
-
=SS
=-
The likelihood function of
(
)
,'
yx
based on a sample of size
n
is
(
)
'
1
1
2
2
11
exp.
2
2
n
iyiy
n
nk
ixix
i
yy
L
xx
mm
mm
p
-
=
éù
ìü
--
æöæö
ïï
êú
=-S
íý
ç÷ç÷
--
êú
èøèø
ïï
S
îþ
ëû
å
Maximizing the log likelihood function with respect to
,,and,
yxxxxy
mm
SS
the maximum likelihood estimates of respective parameters are
obtained as
1
23
1
'
1
1
1
1
(,,...,)
1
'
1
n
yi
i
n
xik
i
n
xxxxii
i
n
xyxyii
i
yy
n
xxxxx
n
Sxxnxx
n
Sxynxy
n
m
m
=
=
=
=
==
===
æö
S==-
ç÷
èø
æö
S==-
ç÷
èø
å
å
å
å
%
%
%
%
'
23
1
where (,,...,), is [(-1)(-1)] matrix w
ith elements ()() and is
1
[(-1)1] vector with elements ()().
iiiikxxtiitjjxy
t
tiii
t
xxxxSkkxxxxS
n
kxxyy
n
=´--
´--
å
å
Based on these estimates, the maximum likelihood estimators
of
10
and
bb
are obtained as
(
)
1
1
01
1
0
1
'
''.
xxxy
SS
yx
XXXy
b
bb
b
b
b
-
-
=
=-
æö
==
ç÷
ç÷
èø
%
%%
%
%
%
Properties of the estimators of least squares estimator:
The estimation error of OLSE
(
)
1
''
bXXXy
-
=
of
b
is
(
)
(
)
(
)
(
)
1
1
1
''
''
''.
bXXXy
XXXX
XXX
bb
beb
e
-
-
-
-=-
=+-
=
Then assuming that
(
)
1
''
EXXX
-
éù
ëû
exists, we have
(
)
(
)
{
}
(
)
(
)
1
1
1
()''
''
''
0
EbEXXX
EEXXXX
EXXXE
be
e
e
-
-
-
éù
-=
ëû
éù
=
ëû
éù
=
ëû
=
because
(
)
1
''
XXX
-
and
e
are independent. So
b
is an unbiased estimator of
b
.
The covariance matrix of
b
is obtained as
(
)
(
)
(
)
(
)
(
)
(
)
(
)
{
}
(
)
(
)
(
)
(
)
(
)
(
)
11
11
11
11
2
1
2
'
''''
''''
''''
'''
'.
VbEbb
EXXXXXX
EEXXXXXXX
EXXXEXXXX
EXXXXXX
EXX
bb
ee
ee
ee
s
s
--
--
--
--
-
=--
éù
=
ëû
éù
=
ëû
éù
=
ëû
éù
=
ëû
éù
=
ëû
Thus the covariance matrix involves a mathematical expectation.
The unknown
2
s
can be estimated by
(
)
(
)
2
'
ˆ
'
ee
nk
yXbyXb
nk
s
=
-
--
=
-
where
eyXb
=-
is the residual and
(
)
(
)
(
)
22
2
2
ˆˆ
'
.
EEEX
ee
EEX
nk
E
ss
s
s
éù
=
ëû
éù
æö
=
ç÷
êú
-
èø
ëû
=
=
Note that the OLSE
(
)
1
''
bXXXy
-
=
involves the stochastic matrix
X
and stochastic vector
y
, so
b
is not a linear estimator. It is also no more the best linear
unbiased estimator of
b
as in the case when X is nonstochastic. The estimator of
2
s
as being conditional on given
X
is an efficient estimator.
Asymptotic theory:
The asymptotic properties of an estimator concerns the
properties of the estimator when sample size
n
grows large.
For the need and understanding of asymptotic theory, we consider
an example. Consider the simple linear regression model with one
explanatory variable and
n
observations as
(
)
(
)
2
01
,0,,1,2,...,.
iiiii
yxEVarin
bbeees
=++===
The OLSE of
1
b
is
(
)
(
)
(
)
1
1
2
1
n
ii
i
n
i
i
xxyy
b
xx
=
=
--
=
-
å
å
and its variance is
(
)
2
1
.
Varb
n
s
=
If the sample size grows large, then the variance of
1
b
gets smaller. The shrinkage in variance implies that as sample
size
n
increases, the probability density of OLSE
b
collapses around its mean because
()
Varb
becomes zero.
Let there are three OLSEs
123
,and
bbb
which are based on sample sizes
123
,and
nnn
respectively such that
123
,
nnn
<<
say. If c and
d
are some arbitrarily chosen positive constants, then the
probability that the value of
b
lies within the interval
c
b
±
can be made to be greater than
(
)
1
d
-
for a large value of n. This property is the consistency of
b
which ensure that even if the sample is very large, then we can
be confident with high probability that
b
will yield an estimate that is close to
b
.
Probability in limit
Let
ˆ
n
b
be an estimator of
b
based on a sample of size
n
. Let
g
be any small positive constant. Then for large
n
, the requirement that
n
b
takes values with probability almost one in an arbitrary small
neighborhood of the true parameter value
b
is
ˆ
lim1
n
n
P
bbg
®¥
éù
-<=
ëû
which is denoted as
ˆ
plim
n
bb
=
and it is said that
ˆ
n
b
converges to
b
in probability. The estimator
ˆ
n
b
is said to be a consistent estimator of
b
.
A sufficient but not necessary condition for
ˆ
n
b
to be a consistent estimator of
b
is that
ˆ
lim
n
n
E
bb
®¥
éù
=
ëû
and
ˆ
lim0.
n
n
Var
b
®¥
éù
=
ëû
Consistency of estimators
Now we look at the consistency of the estimators of
b
and
2
.
s
(i) Consistency of b
Under the assumption that
'
lim
n
XX
n
®¥
æö
=D
ç÷
èø
exists as a nonstochastic and nonsingular matrix (with finite
elements), we have
1
2
21
1'
lim()lim
1
lim
0.
nn
n
XX
Vb
nn
n
s
s
-
®¥®¥
-
®¥
æö
=
ç÷
èø
=D
=
This implies that OLSE converges to
b
in quadratic mean. Thus OLSE is a consistent estimator of
b
. This also holds true for maximum likelihood estimators
also.
Same conclusion can also be proved using the concept of
convergence in probability.
The consistency of OLSE can be obtained under the weaker
assumption that
*
'
plim.
XX
n
æö
=D
ç÷
èø
exists and is a nonsingular and nonstochastic matrix and
'
plim0.
X
n
e
æö
=
ç÷
èø
Since
1
1
(')'
''
.
bXXX
XXX
nn
be
e
-
-
-=
æö
=
ç÷
èø
So
1
1
*
''
plim()plimplim
.0
0.
XXX
b
nn
e
b
-
-
æöæö
-=
ç÷ç÷
èøèø
=D
=
Thus
b
is a consistent estimator of
b
. The same is true for maximum likelihood estimators also.
(ii) Consistency of s2
Now we look at the consistency of
2
s
as an estimate of
2
s
. We have
2
1
1
11
1
'
1
'
1
1''(')'
''''
1.
see
nk
H
nk
k
XXXX
nn
kXXXX
nnnnn
ee
eeee
eeee
-
-
--
=
-
=
-
æö
éù
=--
ç÷
ëû
èø
éù
æöæö
=--
êú
ç÷ç÷
èøèø
êú
ëû
Note that
'
n
ee
consists of
22
i
1
1
and{,1,2,...,}
n
i
i
in
n
ee
=
=
å
is a sequence of independently and identically distributed
random variables with mean
2
s
. Using the law of large numbers
2
11
1
*
212
2
'
plim
''''X''
plimplimplimplim
n
0..0
0
plim()(10)0
.
n
XXXXXXX
nnnnn
s
ee
s
eeee
s
s
--
-
-
æö
=
ç÷
èø
éùéù
æöæöæöæö
=
êúêú
ç÷ç÷ç÷ç÷
èøèøèøèø
êúêú
ëûëû
=D
=
éù
Þ=--
ëû
=
Thus
2
s
is a consistent estimator of
2
s
. The same holds true for maximum likelihood estimates also.
Asymptotic distributions:
Suppose we have a sequence of random variables
{
}
n
a
with a corresponding sequence of cumulative density
functions
{
}
n
F
for a random variable
a
with cumulative density function
.
F
Then
n
a
converges in distribution to
a
if
n
F
converges to
F
point wise. In this case,
F
is called the asymptotic distribution of
.
n
a
Note that since convergence in probability implies the
convergence in distribution, so
plim
D
nn
aaaa
=Þ¾¾®
(
n
a
tend to
a
in distribution), i.e., the asymptotic distribution of
n
a
is
F
which is the distribution of
a
.
Note that
(
)
:
E
a
Mean of asymptotic distribution
()
Var
a
: Variance of asymptotic distribution
lim()
n
n
E
a
®¥
: Asymptotic mean
2
limlim()
nn
nn
EE
aa
®¥®¥
éù
-
ëû
: Asymptotic variance.
Asymptotic distribution of sample mean and least squares
estimation
Let
1
1
n
nni
i
YY
n
a
=
==
å
be the sample mean based on a sample of size
n
. Since sample mean is a consistent estimator of population
mean
Y
, so
plim
n
YY
=
which is constant. Thus the asymptotic distribution of
n
Y
is the distribution of a constant. This is not a regular
distribution as all the probability mass is concentrated at one
point. Thus as sample size increases, the distribution of
n
Y
collapses.
Suppose consider only the one third observations in the sample
and find sample mean as
3
*
1
3
.
n
ni
i
YY
n
=
=
å
Then
(
)
*
n
EYY
=
(
)
(
)
3
*
2
1
2
2
2
9
and
9
3
3
0as.
n
ni
i
VarYVarY
n
n
n
n
n
s
s
=
=
=
=
®®¥
å
Thus
*
plim
n
YY
=
and
*
n
Y
has the same degenerate distribution as
.
n
Y
Since
(
)
(
)
*
nn
VarYVarY
>
, so
*
n
Y
is preferred over
.
n
Y
Now we observe the asymptotic behaviour of
n
Y
and
*
.
n
Y
Consider a sequence of random variables
{}.
n
a
Thus for all
n
, we have
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
**
**
2
2
2
2
2
**2
0
0
3
3.
nn
nn
nn
nn
nn
nn
nYY
nYY
EnEYY
EnEYY
VarnEYYn
n
VarnEYYn
n
a
a
a
a
s
as
s
as
=-
=-
=-=
=-=
=-==
=-==
Assuming the population to be normal, the asymptotic
distribution of
·
(
)
2
is0,
n
YN
s
·
(
)
*2
is0,3
n
YN
s
.
So now
n
Y
is preferable over
*
n
Y
. The central limit theorem can be used to show that
n
a
will have an asymptotically normal distribution even if the
population is not normally distributed.
Also, since
(
)
(
)
(
)
(
)
2
~0,
~0,1
n
n
nYYN
nYY
ZN
s
s
-
-
Þ=
and this statement holds true in finite sample as well as
asymptotic distributions.
Consider the ordinary least squares estimate
(
)
1
''of
bXXXy
b
-
=
in linear regression model
.
yX
be
=+
If
X
is nonstochastic then the finite covariance matrix of b is
21
()(').
VbXX
s
-
=
The asymptotic covariance matrix of b under the assumption
that
'
lim
xx
n
XX
n
®¥
=S
exists and is nonsingular. It is given by
(
)
1
22
21
1'
lim'limlim
.0.
0
nnn
xx
XX
XX
nn
ss
s
-
®¥®¥®¥
-
æöæö
=
ç÷ç÷
èøèø
=S
=
which is a null matrix.
Consider the asymptotic distribution of
(
)
nb
b
-
. Then even if
e
is not necessarily normally distributed, then asymptotically
(
)
(
)
(
)
(
)
21
2
2
~0,
'
~.
xx
xx
k
nbN
nbb
bs
bb
c
s
-
-S
-S-
If
'
XX
n
is considered as an estimator of
,
xx
S
then
(
)
(
)
(
)
(
)
22
'
'
''
XX
nbb
bXXb
n
bb
bb
ss
--
--
=
is the usual test statistic as is in the case of finite samples
with
(
)
(
)
1
2
~,'.
bNXX
bs
-
Econometrics | Chapter 13 | Asymptotic Theory and Stochastic
Regressors | Shalabh, IIT Kanpur
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