55
2.
There is no effect on the coefficients of the other variables.
For the dummy variable coefficients, with the
full set of
n
dummy variables, each coefficient is
y
i
*
= mean residual for the
i
th group in the regression of
y
on the
x
s omitting the dummy variables.
(We use the partitioned regression results of Chapter 6.)
If an overall constant term and
n
1 dummy variables
(say the last
n
1) are used, instead, the coefficient on the
i
th dummy variable is simply
y
i
*

y
1
*
while the
constant term is still
y
1
*
For a full proof of these results, see the solution to Exercise 5 of Chapter 8 earlier in
this book.
3.
(a)
The pooled OLS estimator will be
1
11
nn
ii
i
i
−
==
′
′
⎡
⎤⎡
⎤
=Σ
Σ
⎣
⎦⎣
⎦
bX
X
X
y
where X
i
and y
i
have T
i
observations.
It remains true that
y
i
=
X
i
β
+
ε
i
+
u
i
i
, where Var[
ε
i
+
u
i
i
X
i
] = Var[
w
i

X
i
] =
σ
ε
2
I
+
σ
u
2
ii
′
and,
maintaining the assumptions, both
ε
i
and u
i
are uncorrelated with X
i
.
Substituting the expression for y
i
into
that of b and collecting terms, we have
1
i
i
−
′′
⎡⎤
=+Σ
Σ
⎣⎦
X
X
w
β
.
Unbiasedness follows immediately as long as E[w
i
X
i
] equals zero, which it does by assumption.
Consistency,
as mentioned in Section 9.3.2, is covered in the discussion of Chapter 4.
We would need for the matrix
Q
=
1
i
n
i
nT
=
′
Σ
XX
to converge to a matrix of constants, or not to degenerate to a matrix of zeros.
The
requirements for the large sample behavior of the vector in the second set of brackets is quite the same as in
our earlier discussions of consistency.
The vector
(
1
/)
(
1
i
i
i
′
Σ=
Σ
Xw
v
has mean zero.
We would
require the conditions of the LindebergFeller version of the central theorem to apply, which could be
expected.
(b)
We seek to establish consistency, not unbiasedness.
As such, we will ignore the degrees of freedom
correction, K, in (937).
Use n(T1) as the denominator.
Thus, the question is whether
plim
2
2
.
()
(1
)
it i
t i
ee
ε
ΣΣ
−
= σ
−
If so, then the estimator in (937) will be consistent. Using (933) and
e
it

i
i
ey
a
′
=
−−
xb
, it follows that
(
)
it
i
it
i
it
i
−=
ε−
ε
− −
−
xxb
β
. Summing the squares in (937), we find that the estimator in (937)
2
2
.
1
.
1
ˆ
(
)
(
)
(
)
)
 2(
)
(
)(
)
T
it
i
it
i
t
it
i
it
i
it
i
n
n
T
=
−
=σ
+
−
−
⎢⎥
−
ε
−
ε
∑∑
∑
bx
x
x
x
b
x
ββ
β
The second term will converge to zero as the center matrix converges to a constant Q and the vectors converge
to zero as b converges to
β
. (We use the Slutsky theorem.)
The third term will converge to zero as both the
leading vector converges to zero and the covariance vector between the regressors and the disturbances
converges to zero.
That leaves the first term, which is the average of the estimators in (934).
The terms in
the average are independent.
Each has expected value exactly equal to
σ
ε
2
.
So, if each estimator has finite
variance, then the average will converge to its expectation.
Appendix D discusses various different conditions
underwhich a sample average will converge to its expectation.