10 J
.F. Kiviet, G.D.A. Phillips Economics Letters 66 2000 7 –15
2 21
Ea 2 a 5 s L 2 1Qc 1 oT 9
22
• 2SLS mean squared error to order T
: Nagar 1959, p. 579.
2 2
E a 2 aa 2 a9 5 s Q 1 s trCQ 2 2L 2 1trC Q Q h
j
f g
1 2
2 22
1 s L 2 3L 1 4QC Q 2 L 2 2QCQ 1 oT
10
f g
1 21
• Bias of the residual variance estimator to order T
: Nagar 1961, p. 240.
2 2
2 21
Es 2 s 5 2 s 2L 2 1trQC 2 trQC 1 oT 11
f g
1
e9e
2
]]]] where s 5
and e 5 y 2 Z a is a T 3 1 vector of 2SLS residuals.
1 1
T 2 g 1 k These are slight adaptations of the published results which we shall use later in the paper. In fact
Nagar 1961 deflates the sum of squared residuals by T and, as a result, the estimator is biased to
21
order T . We prefer to use the less biased version: see also Kiviet and Phillips 1998.
4. The bias of the asymptotic variance estimator
Subtracting the outer product of 9 from 10, we may deduce an approximation to the variance of 2SLS as follows:
2 2
22
Vara 5 s Q 1 s h[trCQ 2 2L 2 1trC Q]Q 2 L 2 3QC Q 2 L 2 2QCQj 1 oT .
1 1
12
2 21
2
ˆ ˆ ˆ
In practice the estimated asymptotic variance, Vara 5s Z 9Z , where s is defined in 11, is
used to estimate the variance in finite samples and it is the bias of this estimator which is the main focus of interest in this paper. However, we shall first consider the bias of a non-operational estimator
2 21
2
˜ ˆ ˆ
of the variance given by Vara 5 s Z 9Z where s is known. Since none of the resulting bias
2
can be attributed to the estimator of s , a consideration of this case will be helpful in analysing the source of the bias in the estimated asymptotic variance. In Appendix 1 the following result is proved:
2 21
˜ ˆ ˆ
Lemma 1. The expected value of the non-operational variance estimator Vara 5 s Z 9Z can
be approximated as
2 2
22
˜ E Vara 5 s Q 1 s trCQ Q 2 L 2 2QCQ 1 oT
. 13
f g
f g
The result of this lemma, combined with 12, leads to the following.
22
˜
Theorem 1. The bias of the non-operational variance estimator Vara , to order T , is given by
2 22
˜ E Vara 2 Vara 5 s 2L 2 1trC Q Q 1 L 2 3QC Q 1 oT
. 14
f g
f g
1 1
J .F. Kiviet, G.D.A. Phillips Economics Letters 66 2000 7 –15
11
Notice that trC Q Q and QC Q are both positive semi-definite matrices where trC Q Q QC Q;
1 1
1 1
see Kadane 1971, p. 728. Hence the bias matrix above is positive semi-definite for L 2. However, L 2 is a requirement for the variances to exist so that, in general, the non-operational variance
22
estimator is biased upwards to order T . Next we examine the bias of the asymptotic variance
estimator. In Appendix 2 we show the following result.
2 21
ˆ ˆ ˆ
9
Lemma 2. The expected value of the asymptotic variance estimator Vara 5 s Z Z to the
1 1
22
order of T is given by
2 2
22
ˆ E
f
Vara
g
5 s Q 1 s 2 L 2 1QCQ 1 4QC Q 2 2L 2 1trQC Q 1 2trQCQ 1 oT
.
f g
1 1
15 An approximation for the bias can now be readily obtained. Combining the result in Lemma 2 with
the approximation in 12 gives the next theorem.
2 21
ˆ ˆ ˆ
Theorem 2. The bias of the asymptotic variance estimator Vara 5 s Z 9Z , to the order of
22
T , is given by
2 22
ˆ E
f
Vara 2 Vara
g
5 s trQCQ 1 L 1 1QC Q 1 oT
. 16
f g
1
Noting that both trQC Q and QC Q are positive semi-definite, it is clear that the estimated
1
asymptotic variance is, in general, biased upwards to the order of the approximation. Comparing
2
Theorems 1 and 2, it is seen that the direction of the bias is unaltered by the need to estimate s although the bias expression itself changes. It does not appear possible to make general statements
ˆ about the relative magnitudes of the two biases. This is partly because the bias of Vara depends on
˜ the matrix C whereas the bias of Vara does not.
2
ˆ Given that an explicit expression for the bias of the asymptotic variance estimator Vara has been
found, a bias corrected estimator can be obtained straightforwardly. Estimates are available for the relevant terms in the bias approximation so that an estimate of the bias can be obtained which is then
subtracted from the original estimator. We shall not pursue the matter further in this paper however.
5. Conclusion
