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The bias of the 2SLS variance estimator

a b ,

_*

Jan F. Kiviet , Garry D.A. Phillips

a

Tinbergen Institute and Faculty of Economics and Econometrics, University of Amsterdam, Amsterdam, The Netherlands

b

School of Business and Economics, University of Exeter, Streatham Court, Exeter EX4 4PU, UK Accepted 8 April 1999

Abstract

In simultaneous equation models the two stage least squares (2SLS) estimator of the coefficients, though consistent, is biased in general and the nature of this bias has given rise to a good deal of research. However, little if any attention has been given to the bias that arises when an estimate of the asymptotic variance is used to approximate the small sample variance. In this paper we use asymptotic expansions to show that, in general, the asymptotic variance estimator has an upwards bias.  2000 Elsevier Science S.A. All rights reserved. Keywords: 2SLS estimation; Nagar expansions; Asymptotic variance; Variance estimation bias

JEL classification: C30

1. Introduction

The seminal paper of Nagar (1959) presented approximations for the bias of the 2SLS estimator to

21 22

order T and for the mean squared error to the order of T , where T is the sample size. By subtracting the square of the bias approximation from the mean squared error approximation we

22

obtain an estimator for the variance to the order of T . Little use seems to have been made of this particular approximation; indeed, the problem of bias in the estimation of the coefficient estimator variance seems generally to have been neglected. However, this approximation can be used to explore the bias of the estimated asymptotic variance as an estimator of the small sample variance. By finding

22 an approximation for the expectation of the asymptotic variance estimator to the order of T , and comparing it with the variance approximation of the same order, we may deduce immediately the

22

approximate bias in the asymptotic variance estimator. This bias, which is of order T , is found to be non-negative for all coefficients in the 2SLS estimator showing that, in general, the traditional

*Corresponding author. Tel.: 144-1392-263-241; fax: 144-1392-263-242. E-mail address: G.D.A.Phillips@exeter.ac.uk (G.D.A. Phillips)

0165-1765 / 00 / $ – see front matter  2000 Elsevier Science S.A. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 ( 9 9 ) 0 0 2 3 3 - 5

estimator is upwards biased. This is the main theoretical result in the paper. Given that an explicit expression for the bias approximation is obtained, a bias correction can routinely be applied.

2. Model and notation

Consider a general static simultaneous equation model containing G equations which may be written as

A9y_t1B9x_t5e_t, t51, . . . ,T, (1)

where y is a G_t 31 vector of endogenous variables, x is a K_t 31 vector of strongly exogenous variables which we shall treat as non-stochastic, and e_t is a G31 vector of structural disturbances.

A9and B9 are respectively, G3G and G3K matrices of structural coefficients. With T observations

on the above system, we may write

YA1XB5E (2)

where Y is a T3G matrix of observations on the endogenous variables, X is a T3K matrix of

observations on the exogenous variables, and E is a T3G matrix of structural disturbances. We shall

be particularly concerned with that part of the system (2) which relates to the first equation. The reduced form of the system includes

Y₁5XP₁1V₁ (3)

where Y 5( y :Y ), X5(X :X ), P 5(p :P ) and V 5(v :V ).P is a K3( g11) matrix of reduced

1 1 2 1 2 1 1 2 1 1 2 1

form parameters and V is a T1 3( g11)matrix of reduced form disturbances. In addition, the following assumptions are made:

• The rows of V are independently and normally distributed with mean vector 0₁ 9 and non-singular covariance matrix V5

h j

v_ij.

• The T3K matrix X is of rank K (,T ), and the elements of the K3K matrix X9X are O(T ).

• The first equation of system (1) is overidentified with the order of overidentification, L, being at least 2. This ensures that the first two moments of 2SLS exist; see Kinal (1980).

3. Asymptotic approximations

The first equation of (2) may be written as

y₁5Y₂b1X₁g1e₁ (4)

where y and Y are, respectively, a T_{1 2} 31 vector and a T3g matrix of observations on g endogenous

variables and X is a T1 3k matrix of observations on k non-stochastic exogenous variables. The

2 of independently and identically distributed normal random variables with mean zero and variances . The 2SLS estimators of the unknown parameters of (4) are given by

21

ˆ ˆ ˆ

₉

₉

ˆ

₉

b* Y Y Y X_{2 2 2 1} Y₂

5 y (5)

S D S

₉

_ˆ

₉

D S D

₉

1.

g* X Y X X_{1 2 1 1} X₁

21

ˆ ˆ

where Y₂5XP₂5X(X9X ) X9Y is the T₂ 3g matrix of fitted values obtained in the regression of Y₂

on X. From (5) we may write the estimation error as

21

ˆ ˆ ˆ

₉

ˆ

₉

b* b Y Y Y X_{2 2 2 1} Y₂

2 5 e . (6)

S D S D S

₉

_ˆ

₉

D S D

₉

1

g* g X Y X X_{1 2 1 1} X₁

In what follows it will be convenient to re-write (4) in the form

y₁5Z₁a1e₁ (7)

where Z₁5(Y :X ) and_{2 1} a5(b9,g9)9. The 2SLS estimator may then be written as

21

ˆ ˆ ˆ

9

a*5(Z Z )1 1 Z y1 1 (8)

ˆ ˆ

where Z₁5(Y :X ) is a T_{2 1} 3( g1k) matrix of regressors at the second stage of the 2SLS procedure.

Before stating the approximations that are the focus of interest, we shall define the following:

¯ ¯ ¯

Z₁5(Y :X ) is a T_{2 1} 3( g1k) non-stochastic matrix where Y₂5E(Y ),₂

21

¯ ¯

₉

¯

₉

Y Y_{2 2} Y X_{2 1 21}

¯ ¯

₉

Q5

S

₉

_¯

₉

D

5(Z Z )1 1 ,

X Y_{1 2} X X_{1 1}

1 1 _˘ 2 2

]_TE Zf

9

1 1e g5]_TE V

f

2 1

9

e

g

5s tf 9,09 5g s c

˘

where V₂5(V :0) has the last k columns zero and₂ c5ft9,09gis ( g1k)31 with the last k elements zero,

1 2

1 _{˘ ˘ s tt9 0 ]E(W9W ) 0}

]

9

C5_TE(V V ), C2 2 15

3

4

and C25

3

T

4

0 0 0 0

where W5V₂2e t₁ 9 with W distributed independently of e₁, see Nagar (1959), and

C5C₁1C .₂

With the above definitions we may state the following:

21

2 21

E(a*2a)5s (L21)Qc1o(T ) (9)

22

• 2SLS mean squared error to order T : Nagar (1959, p. 579).

2 2

E (ha*2a)(a*2a)9 5j s Q1s

f

tr(CQ )22(L21)tr(C Q ) Q₁

g

2 2 22

1s

f

(L 23L14)QC Q1 2(L22)QCQ

g

1o(T ) (10)

21

• Bias of the residual variance estimator to order T : Nagar (1961, p. 240).

2 2 2 21

E(s_*2s )5 2s

f

2(L21)tr(QC )₁ 2tr(QC )

g

1o(T ) (11)

e9*e*

2 _]]]]

where s*5_{T2( g1k)} and e*5y12Z1a* is a T31 vector of 2SLS residuals.

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

Var(a*)5s Q1s h[tr(CQ )22(L21)tr(C Q )]Q1 2(L23)QC Q1 2(L22)QCQj1o(T ). (12)

2 21 2

ˆ ˆ ˆ

In practice the estimated asymptotic variance, Var(a*)5s_*(Z9Z ) , 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 Var(a*)5s (Z9Z ) 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 Var(a*)5s (Z9Z ) can

be approximated as

2 2 22

˜

E Var(

f

a*)

g

5s Q1s ftr(CQ )Q2(L22)QCQg1o(T ). (13)

The result of this lemma, combined with (12), leads to the following.

22

˜

Theorem 1. The bias of the non-operational variance estimator Var(a*), to order T , is given by

2 22

˜

Notice that tr(C Q )Q and QC Q are both positive semi-definite matrices where tr(C Q )Q1 1 1 $QC Q;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 Var(a*)5s (Z Z ) to the * 1 1

22

order of T is given by

2 2 22

ˆ

E

f

(Var(a*)

g

5s Q1s

f

2(L21)QCQ14QC Q₁ 22(L21)tr(QC )Q₁ 12tr(QC )Q

g

1o(T ). (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 Var(a*)5s (Z9Z ) , to the order of *

22

T , is given by

2 22

ˆ

E

f

Var(a*)2Var(a*)

g

5s

f

tr(QC )Q1(L11)QC Q₁

g

1o(T ). (16)

Noting that both tr(QC )Q and QC Q are positive semi-definite, it is clear that the estimated1 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 Var(a*) depends on

˜

the matrix C whereas the bias of Var(₂ a*) does not.

ˆ

Given that an explicit expression for the bias of the asymptotic variance estimator Var(a*) 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

This paper shows that the traditional variance estimator for the 2SLS coefficient estimator is, in

22

general, biased upwards to order T . This is a surprisingly strong result for it applies whatever the data and parameter set. The magnitude and the effects of this bias have not been studied. We can, of course, speculate that 2SLS confidence intervals will be conservative in small samples but this is not a clear cut matter since there is also a coefficient estimator bias. In fact, given this bias, it is possible that the upward bias in the variance might lead to improved confidence interval coverage. This issue, the option to correct for the coefficient bias, and the effects on hypothesis tests etc, requires further study.

Appendix A

2 21 22

˜ ˆ ˆ

9

Here we derive the expected value of Var(a*)5s (Z Z )1 1 to the order T . We need only

2 _{ˆ ˆ}

consider the inverse matrix since s is constant. First we examine the matrix Z₁5(Y :X ) where_{2 1}

21 21

ˆ

Y25X(X9X ) X9Y25XP21X(X9X ) X9V . Noting that this last term is stochastic, it is seen that we2

ˆ

may write the matrix Z as the sum of two matrices, one stochastic and one non-stochastic, as follows:₁

21 21

ˆ ˘ ¯

Z₁5(X(X9X ) X9V :O )₂ 1(XP₂:X )₁ 5X(X9X ) X9V₂1Z₁

21

¯ ¯ ¯

˘

where we have put V₂5(V :O ) and Z_{2 1}5(XP₂:X ). Noting that X(X₁ 9X ) X9Z₁5Z we may write:₁

21 21

ˆ ˆ

9

¯ ¯

9

˘

9

¯ ¯

9

˘ ˘

9

˘ ˘

9

¯ ¯

9

˘ ˘

9

¯ ˘

Z Z_{1 1}5Z Z_{1 1}1V Z_{2 1}1Z V_{1 2}1V X(X₂ 9X ) X9V₂5Q 1V Z_{2 1}1Z V_{1 2}1V MV_{2 2}

21

¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

˘

5

f

I1(V Z_{2 1}1Z V )Q_{1 2} 1V MV Q Q_{2 2}

g

21 _{¯ ¯ ¯} 21

9

where Q 5Z Z and M_{1 1} 5X(X9X ) X9. Inverting both sides yields:

21 21

ˆ ˆ

9

˘

₉

¯ ¯

₉

˘ ˘ ¯ ˘

(Z Z )_{1 1} 5Q I

f

1(V Z_{2 1}1Z V )Q_{1 2} 1V MV Q_{2 2}

g

¯ ¯ ¯ ¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

˘ ˘

₉

₉

˘ ˘

₉

₉

˘

5Q I

f

2(V Z2 11Z V )Q1 2 2V MV Q2 2

g

1Q (V Z

f

2 11Z V )Q(V Z1 2 2 11Z V )Q1 2

g

22

1o (T_p ) (A.1)

1

]

2 21

¯ ¯ 2 ¯

˘

₉

₉

˘ ˘

₉

˘

where (V Z_{2 1}1Z V )Q is O (T_{1 2 p} ) and V MV Q is O (T_{2 2 p} ). Here the inverse matrix has been

21

ˆ ˆ

9

expanded in terms of decreasing orders of smallness. To obtain E (Z Z )

f

1 1

g

we shall take expectations of each of the relevant terms in (A1) as follows:

¯ ¯ ¯

˘

₉

₉

˘ ˘

• E Q(V Z

f

2 11Z V )Q1 2

g

50 since Q and Z are fixed and E(V )1 2 50.

¯ ¯ ¯ ¯

˘

₉

˘ ˘

₉

˘

• E QV MV Q

f

_{2 2}

g

5QE(V MV )Q_{2 2} 5tr(M )QCQ5K QCQ where tr(M )5K, see Kiviet and Phillips

(1996, p. 166),

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

₉

˘ ˘

₉

˘

₉

˘

₉

₉

˘

• E Q(V Z

f

_{2 1}1Z V )Q(V Z_{1 2 2 1}1Z V )Q_{1 2}

g

5E Q(V Z )Q(V Z )Q

f

_{2 1 2 1} 1Q(V Z )Q(Z V )Q_{2 1 1 2}

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ¯

₉

˘

1Q(Z V )Q(V Z )Q_{1 2 2 1} 1Q(Z V )Q(Z V )Q_{1 2 1 2}

g

˘

On noting that we may write V₂5(W1e t₁ 9:0) where W ande₁ are independent, see Nagar (1959), we may use the results in Mikhail (1972) or Kiviet and Phillips (1996, p. 166) to show directly that:

¯ ¯ ¯ ¯

˘

₉

˘

₉

˘

₉

₉

˘

E Q(V Z )Q(V Z )Q

f

2 1 2 1

g

5QCQ, E Q(V Z )Q(Z V )Q

f

2 1 1 2

g

5( g1k)QCQ

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ¯

₉

˘

E Q(Z V )Q(V Z )Q

f

1 2 2 1

g

5tr(QC )Q, E Q(Z V )Q(Z V )Q

f

1 2 1 2

g

5QCQ.

Adding terms we find that in (iii)

¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

_{9 9}

˘

E Q(V Z

f

_{2 1}1Z V )Q(V Z_{1 2 2 1}1Z V )Q_{1 2}

g

5( g1k12)QCQ1tr(QC )Q. (A.2)

2

Finally, using (i)–(iii) above, we have found that the expected value ofs times (A1) is given by:

2 _{ˆ ˆ}

₉

2 22

Appendix B

2 21

ˆ ˆ ˆ

9

In this Appendix we prove Lemma 2. Thus we derive the expected value of Var(a*)5s*(Z Z )1 1

22 2

to o(T ) where s_* is given in (11). To find the required expansion we first note that the 2SLS residual vector is given by

*

e₁ 5y₁2Z₁a*5e₁2Z (₁ a*2a) and the sum of squared residuals is then

9

*

9

9

9

e *e1 1 5e e1 122(a*2a)9Z1 1e 1(a*2a)9Z Z (1 1 a*2a). The disturbance variance estimator is

9

*

9

9

9

e *e1 1 e e1 1 (a*2a)9Z1 1e (a*2a)9Z Z (1 1 a*2a)

2 _{]]]] ]]]] ]]]]] ]]]]]]]]}

s_*5 5 22 1 (B.1)

T2( g1k) T2( g1k) T2( g1k) T2( g1k)

21 / 2 21

where the first term is O (1), the second is O (T_{p p} ) and the last is O (T_p ).

21

ˆ ˆ

9

The inverse matrix (Z Z )1 1 is similarly expanded in terms of decreasing stochastic order of magnitude in (A.1) as

21

ˆ ˆ

9

¯

₉

˘ ˘

₉

¯ ˘

₉

¯ ˘ ¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ˘

₉

¯

(Z Z )_{1 1} 5Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2} 1Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

22

1o (Tp ) (B.2)

21 23 / 2 22

where the successive terms are O(T ), O (T_p ) and O (T_p ) respectively. 2 _{ˆ ˆ}

₉

21

To find the appropriate expansion for s*(Z Z )1 1 we combine (B.1) and (B.2) to yield

9

e e_{1 1}

2 _{ˆ ˆ}

₉

21 _{¯ ˘ ˘ ¯ ˘ ¯ ˘}

]]]]

9

9

9

s_*(Z Z )_{1 1} 5

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2} T2( g1k)

2

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ˘

₉

¯ ]]]]

9

1Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

g

2 (a*2a)9Z_{1 1}e

T2( g1k)

9

(a*2a)9Z Z (_{1 1} a*2a) ₂₂

¯

₉

˘ ˘

₉

¯ ]]]]]]]]

3

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1}

g

1 Q1o (T_p ). (B.3)

T2( g1k)

We shall take expectations of each of these terms. However the analysis of the first is simplified by noting that

9

9

e e_{1 1} e e_{1 1 2 2}

]]]]5

S

]]]]2s

D

1s

T2( g1k) T2( g1k)

21 / 2

₉

where the first part on the right is O (T_p ). Hence when (e e_{1 1}) /(T2( g1k)) multiplies terms which

22 2

are O (Tp ), it can be replaced bys and we shall still get an approximation to the expectation of the desired order. We thus consider

9

e e_{1 1}

¯ ˘ ˘ ¯ ˘ ¯ ˘

]]]]

9

9

9

E

H

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2}

g

J

T2( g1k)

e9e1 _{¯ ˘ ˘ ¯ ¯ ˘ ˘ ¯}

]]]]

9

9

9

9

1E

H

f

Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

g

J

T2( g1k)

e9e_{1 2 2}

¯ ¯ ¯ ¯ ¯

˘ ˘ ˘ ˘ ˘ ˘

]]]]

9

9

9

9

5E

H

_{T2( g1k)}Q

J

1E

h

s

f

2QV MV Q2 2

g

j

1E

h

s

f

Q(Z9V21V Z )Q(Z V2 1 1 21V Z )Q2 1

g

j

where terms involving products of an odd number of zero mean normal random variables have been ignored. The first two expected values are;

9

e e1 1 2 ( g1k)

]]]] ]]]]

E

H

Q

J S

5s 11

D

Q

T2( g1k) T2( g1k)

2

₉

_¯ 2 _¯ 2

E

h

s

f

2QV MV Q_{2 2}

g

j

5 2s trM. QCQ5 2s K QCQ, (B.4)

see Kiviet and Phillips (1996, p. 166).

The third term can be readily evaluated as in (A.2) so that

2 _¯

₉

_{˘ ˘}

₉

_{¯ ¯}

₉

_{˘ ˘}

₉

_¯ 2 2

E

h

s

f

Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 2}

g

j

5s ( g1k12)QCQ1s tr(QC )Q (B.5) Gathering terms from (B.4) and (B.5), we have found the expectation of the first term of (B.3). Next we consider the second term in (B.3). To analyse this term we first note the expansion

21

¯

₉

˘

₉

¯ ¯

₉

˘ ¯

₉

˘

₉

¯ ¯

₉

(a*2a)5Q(Z_{1 1}e 1V M₂ e₁)2QZ V QZ_{1 2 1 1}e 2QV Z QZ_{2 1 1 1}e 1o (T_p ) (B.6)

¯ ˘

9

9

9

which is given in Nagar (1959, p. 582). Using this result, and noting that Z_{1 1}e 5Z_{1 1}e 1V_{2 1}e where 1 / 2

the right hand side terms are, respectively, O (T_p ) and O (T ), we may write_p

¯ ¯ ¯

˘

₉

₉

₉

₉

e9V QZ e e Z QZ e

1 _{¯ ˘ ˘ ¯} 2 1 1 1 1 1 1

]]]](a*2a)9Z_{1 1}

9

e

f

Q2Q(Z V

9

_{1 2}1V Z )Q₂

9

₁

g

5]]]]Q1]]]]Q

T2( g1k) T2( g1k) T2( g1k)

¯ ¯ ¯ ¯ ¯ ¯

˘ ˘ ˘ ˘ ˘ ˘ ˘

9

9

9

9

9

9

9

9

9

e₁V QV M_{2 2} e₁ e₁V QZ V QZ_{2 1 2} 9e₁ e₁V QV Z QZ_{2 2 1 1 1}e e₁V QZ_{2 1 1}e

¯ ˘

]]]] ]]]]]] ]]]]]] ]]]]

9

1 _{T2( g1k)} Q1 _{T2( g1k)} Q1 _{T2( g1k)} Q1_{T2( g1k)}Q(Z V1 2

22

¯ ˘

₉

1V Z )Q2 1 1o (Tp ). (B.7)

The first of these terms involves a product of an odd number of normal random variables with zero mean and so has expected value zero. Taking expected values for the remaining terms we have;

¯ ¯

9

9

e1Z QZ1 1 1e 2 ( g1k)

]]]] ]]]]

E

H

J

Q5s Q

T2( g1k) T2( g1k)

¯ ˘

₉

˘

₉

e9V QV M2 2 e1 2 TKtr(QC )1

]]]]] ]]]]

E

H

J

Q5s Q

¯ ¯

˘ ˘

9

9

9

e₁V QZ V QZ_{2 1 2 1 1}e ₂ T tr(QC )₁

]]]]]] ]]]]

E

H

2

J

Q5 2s Q

T2( g1k) T2( g1k)

¯ ¯

˘ ˘

9

9

e₁V QV Z QZ_{2 2 1} 9e_{1 2}T( g1k)tr(QC )₁

]]]]]] ]]]]]

E

H

2

J

Q5 2s Q

T2( g1k) T2( g1k)

2

¯ ˘

9

9

e1V QZ2 1 1e _{¯ ˘ ˘ ¯} 2s

]]]]

9

9

]]]]

E

H

2 Q(Z V_{1 2}1V Z )Q_{2 1}

J

5 2

f

(T12)QC Q₁ 1QC Q₂

g

T2( g1k) T2( g1k)

2

2s T 22

]]]]

5 2 QC Q₁ 1o (T_p ). (B.8)

T2( g1k)

Here the results for evaluating the second, third and fourth terms are given in Nagar (1961, p. 242).

˘

An evaluation of the last term proceeds from putting V₂5W1e t₁ 9 where W and e₁ are independent. The required analysis is straightforward but lengthy and so is not included here. The authors will provide details on request. Collecting the terms in (B.8) and multiplying by 22 yields the expectation of the second term of (B.3). To complete the analysis we need the expected value of the last term in (B.3). From Nagar (1961, p. 243) we find that

2

9

(a*2a)9Z Z (1 1 a*2a) s fg1k1Ttr(QC )g 21

]]]]]]]] ]]]]]]]

E

H

Q

J

5 Q1o(T ). (B.9)

T2( g1k) T2( g1k)

Collecting the various terms we have the result given in Lemma 2.

References

Kadane, J.B., 1971. Comparison of k-class estimators when the disturbances are small. Econometrica 39, 723–737. Kinal, T.W., 1980. The existence of moments of k-class estimators. Econometrica 48, 241–249.

Kiviet, J.F., Phillips, G.D.A., 1996. The bias of the ordinary least squares estimator in simultaneous equation models. Economics Letters 53, 161–167.

Kiviet, J.F., Phillips, G.D.A., 1998. Degrees of freedom adjustment for disturbance variance estimators in dynamic regression models. Econometrics Journal 1, 44–70.

Mikhail, W.M., 1972. The bias of the two stage least squares estimator. Journal of the American Statistical Association 67, 625–627.

Nagar, A.L., 1959. The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575–595.

2 21

E(a*2a)5s (L21)Qc1o(T ) (9)

22

• 2SLS mean squared error to order T : Nagar (1959, p. 579).

2 2

E (ha*2a)(a*2a)9 5j s Q1s

f

tr(CQ )22(L21)tr(C Q ) Q₁

g

2 2 22

1s

f

(L 23L14)QC Q1 2(L22)QCQ

g

1o(T ) (10) 21

• Bias of the residual variance estimator to order T : Nagar (1961, p. 240).

2 2 2 21

E(s_*2s )5 2s

f

2(L21)tr(QC )₁ 2tr(QC )

g

1o(T ) (11)

e9*e*

2 _]]]]

where s*5_{T2( g1k)} and e*5y12Z1a* is a T31 vector of 2SLS residuals.

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

Var(a*)5s Q1s h[tr(CQ )22(L21)tr(C Q )]Q1 2(L23)QC Q1 2(L22)QCQj1o(T ). (12)

2 21 2

ˆ ˆ ˆ

In practice the estimated asymptotic variance, Var(a*)5s_*(Z9Z ) , 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 Var(a*)5s (Z9Z ) 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 Var(a*)5s (Z9Z ) can

be approximated as

2 2 22

˜

E Var(

f

a*)

g

5s Q1s ftr(CQ )Q2(L22)QCQg1o(T ). (13)

The result of this lemma, combined with (12), leads to the following.

22

˜

Theorem 1. The bias of the non-operational variance estimator Var(a*), to order T , is given by

2 22

˜

Notice that tr(C Q )Q and QC Q are both positive semi-definite matrices where tr(C Q )Q1 1 1 $QC Q;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 Var(a*)5s (Z Z ) to the

* 1 1

22

order of T is given by

2 2 22

ˆ

E

f

(Var(a*)

g

5s Q1s

f

2(L21)QCQ14QC Q₁ 22(L21)tr(QC )Q₁ 12tr(QC )Q

g

1o(T ). (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 Var(a*)5s (Z9Z ) , to the order of

*

22

T , is given by

2 22

ˆ

E

f

Var(a*)2Var(a*)

g

5s

f

tr(QC )Q1(L11)QC Q₁

g

1o(T ). (16)

Noting that both tr(QC )Q and QC Q are positive semi-definite, it is clear that the estimated1

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 Var(a*) depends on

˜

the matrix C whereas the bias of Var(₂ a*) does not.

ˆ

Given that an explicit expression for the bias of the asymptotic variance estimator Var(a*) 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

This paper shows that the traditional variance estimator for the 2SLS coefficient estimator is, in 22

general, biased upwards to order T . This is a surprisingly strong result for it applies whatever the data and parameter set. The magnitude and the effects of this bias have not been studied. We can, of course, speculate that 2SLS confidence intervals will be conservative in small samples but this is not a clear cut matter since there is also a coefficient estimator bias. In fact, given this bias, it is possible that the upward bias in the variance might lead to improved confidence interval coverage. This issue, the option to correct for the coefficient bias, and the effects on hypothesis tests etc, requires further study.

Appendix A

2 21 22

˜ ˆ ˆ

9

Here we derive the expected value of Var(a*)5s (Z Z )1 1 to the order T . We need only

2 _{ˆ ˆ}

consider the inverse matrix since s is constant. First we examine the matrix Z₁5(Y :X ) where_{2 1}

21 21

ˆ

Y25X(X9X ) X9Y25XP21X(X9X ) X9V . Noting that this last term is stochastic, it is seen that we2

ˆ

may write the matrix Z as the sum of two matrices, one stochastic and one non-stochastic, as follows:₁

21 21

ˆ ˘ ¯

Z₁5(X(X9X ) X9V :O )₂ 1(XP₂:X )₁ 5X(X9X ) X9V₂1Z₁

21

¯ ¯ ¯

˘

where we have put V₂5(V :O ) and Z_{2 1}5(XP₂:X ). Noting that X(X₁ 9X ) X9Z₁5Z we may write:₁

21 21

ˆ ˆ

9

¯ ¯

9

˘

9

¯ ¯

9

˘ ˘

9

˘ ˘

9

¯ ¯

9

˘ ˘

9

¯ ˘

Z Z_{1 1}5Z Z_{1 1}1V Z_{2 1}1Z V_{1 2}1V X(X₂ 9X ) X9V₂5Q 1V Z_{2 1}1Z V_{1 2}1V MV_{2 2}

21

¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

˘

5

f

I1(V Z_{2 1}1Z V )Q_{1 2} 1V MV Q Q_{2 2}

g

21 _{¯ ¯ ¯} 21

9

where Q 5Z Z and M_{1 1} 5X(X9X ) X9. Inverting both sides yields:

21 21

ˆ ˆ

9

˘

₉

¯ ¯

₉

˘ ˘ ¯ ˘

(Z Z )_{1 1} 5Q I

f

1(V Z_{2 1}1Z V )Q_{1 2} 1V MV Q_{2 2}

g

¯ ¯ ¯ ¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

˘ ˘

₉

₉

˘ ˘

₉

₉

˘

5Q I

f

2(V Z2 11Z V )Q1 2 2V MV Q2 2

g

1Q (V Z

f

2 11Z V )Q(V Z1 2 2 11Z V )Q1 2

g

22

1o (T_p ) (A.1)

1 ]

2 21

¯ ¯ 2 ¯

˘

₉

₉

˘ ˘

₉

˘

where (V Z_{2 1}1Z V )Q is O (T_{1 2 p} ) and V MV Q is O (T_{2 2 p} ). Here the inverse matrix has been 21

ˆ ˆ

9

expanded in terms of decreasing orders of smallness. To obtain E (Z Z )

f

1 1

g

we shall take expectations of each of the relevant terms in (A1) as follows:

¯ ¯ ¯

˘

₉

₉

˘ ˘

• E Q(V Z

f

2 11Z V )Q1 2

g

50 since Q and Z are fixed and E(V )1 2 50.

¯ ¯ ¯ ¯

˘

₉

˘ ˘

₉

˘

• E QV MV Q

f

_{2 2}

g

5QE(V MV )Q_{2 2} 5tr(M )QCQ5K QCQ where tr(M )5K, see Kiviet and Phillips

(1996, p. 166),

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

₉

˘ ˘

₉

˘

₉

˘

₉

₉

˘

• E Q(V Z

f

_{2 1}1Z V )Q(V Z_{1 2 2 1}1Z V )Q_{1 2}

g

5E Q(V Z )Q(V Z )Q

f

_{2 1 2 1} 1Q(V Z )Q(Z V )Q_{2 1 1 2}

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ¯

₉

˘

1Q(Z V )Q(V Z )Q_{1 2 2 1} 1Q(Z V )Q(Z V )Q_{1 2 1 2}

g

˘

On noting that we may write V₂5(W1e t₁ 9:0) where W ande₁ are independent, see Nagar (1959), we may use the results in Mikhail (1972) or Kiviet and Phillips (1996, p. 166) to show directly that:

¯ ¯ ¯ ¯

˘

₉

˘

₉

˘

₉

₉

˘

E Q(V Z )Q(V Z )Q

f

2 1 2 1

g

5QCQ, E Q(V Z )Q(Z V )Q

f

2 1 1 2

g

5( g1k)QCQ

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ¯

₉

˘

E Q(Z V )Q(V Z )Q

f

1 2 2 1

g

5tr(QC )Q, E Q(Z V )Q(Z V )Q

f

1 2 1 2

g

5QCQ.

Adding terms we find that in (iii)

¯ ¯ ¯ ¯

˘

₉

₉

˘ ˘

₉

_{9 9}

˘

E Q(V Z

f

_{2 1}1Z V )Q(V Z_{1 2 2 1}1Z V )Q_{1 2}

g

5( g1k12)QCQ1tr(QC )Q. (A.2)

2

Finally, using (i)–(iii) above, we have found that the expected value ofs times (A1) is given by:

2 _{ˆ ˆ}

₉

2 22

Appendix B

2 21

ˆ ˆ ˆ

9

In this Appendix we prove Lemma 2. Thus we derive the expected value of Var(a*)5s*(Z Z )1 1

22 2

to o(T ) where s_* is given in (11). To find the required expansion we first note that the 2SLS residual vector is given by

*

e₁ 5y₁2Z₁a*5e₁2Z (₁ a*2a) and the sum of squared residuals is then

9

*

9

9

9

e *e1 1 5e e1 122(a*2a)9Z1 1e 1(a*2a)9Z Z (1 1 a*2a). The disturbance variance estimator is

9

*

9

9

9

e *e1 1 e e1 1 (a*2a)9Z1 1e (a*2a)9Z Z (1 1 a*2a)

2 _{]]]] ]]]] ]]]]] ]]]]]]]]}

s_*5 5 22 1 (B.1)

T2( g1k) T2( g1k) T2( g1k) T2( g1k)

21 / 2 21

where the first term is O (1), the second is O (T_{p p} ) and the last is O (T_p ). 21

ˆ ˆ

9

The inverse matrix (Z Z )1 1 is similarly expanded in terms of decreasing stochastic order of magnitude in (A.1) as

21

ˆ ˆ

9

¯

₉

˘ ˘

₉

¯ ˘

₉

¯ ˘ ¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ˘

₉

¯

(Z Z )_{1 1} 5Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2} 1Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

22

1o (Tp ) (B.2)

21 23 / 2 22

where the successive terms are O(T ), O (T_p ) and O (T_p ) respectively.

2 _{ˆ ˆ}

₉

21

To find the appropriate expansion for s*(Z Z )1 1 we combine (B.1) and (B.2) to yield

9

e e_{1 1}

2 _{ˆ ˆ}

₉

21 _{¯ ˘ ˘ ¯ ˘ ¯ ˘}

]]]]

9

9

9

s_*(Z Z )_{1 1} 5

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2} T2( g1k)

2

¯

₉

˘ ˘

₉

¯ ¯

₉

˘ ˘

₉

¯ ]]]]

9

1Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

g

2 (a*2a)9Z_{1 1}e

T2( g1k)

9

(a*2a)9Z Z (_{1 1} a*2a) ₂₂

¯

₉

˘ ˘

₉

¯ ]]]]]]]]

3

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1}

g

1 Q1o (T_p ). (B.3)

T2( g1k)

We shall take expectations of each of these terms. However the analysis of the first is simplified by noting that

9

9

e e_{1 1} e e_{1 1 2 2}

]]]]5

S

]]]]2s

D

1s

T2( g1k) T2( g1k)

21 / 2

₉

where the first part on the right is O (T_p ). Hence when (e e_{1 1}) /(T2( g1k)) multiplies terms which

22 2

are O (Tp ), it can be replaced bys and we shall still get an approximation to the expectation of the desired order. We thus consider

9

e e_{1 1}

¯ ˘ ˘ ¯ ˘ ¯ ˘

]]]]

9

9

9

E

H

f

Q2Q(Z V_{1 2}1V Z )Q_{2 1} 2QV MV Q_{2 2}

g

J

T2( g1k)

e9e1 _{¯ ˘ ˘ ¯ ¯ ˘ ˘ ¯}

]]]]

9

9

9

9

1E

H

f

Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 1}

g

J

T2( g1k)

e9e_{1 2 2}

¯ ¯ ¯ ¯ ¯

˘ ˘ ˘ ˘ ˘ ˘

]]]]

9

9

9

9

5E

H

_{T2( g1k)}Q

J

1E

h

s

f

2QV MV Q2 2

g

j

1E

h

s

f

Q(Z9V21V Z )Q(Z V2 1 1 21V Z )Q2 1

g

j

where terms involving products of an odd number of zero mean normal random variables have been ignored. The first two expected values are;

9

e e1 1 2 ( g1k)

]]]] ]]]]

E

H

Q

J S

5s 11

D

Q

T2( g1k) T2( g1k)

2

₉

_¯ 2 _¯ 2

E

h

s

f

2QV MV Q_{2 2}

g

j

5 2s trM. QCQ5 2s K QCQ, (B.4)

see Kiviet and Phillips (1996, p. 166).

The third term can be readily evaluated as in (A.2) so that

2 _¯

₉

_{˘ ˘}

₉

_{¯ ¯}

₉

_{˘ ˘}

₉

_¯ 2 2

E

h

s

f

Q(Z V_{1 2}1V Z )Q(Z V_{2 1 1 2}1V Z )Q_{2 2}

g

j

5s ( g1k12)QCQ1s tr(QC )Q (B.5) Gathering terms from (B.4) and (B.5), we have found the expectation of the first term of (B.3). Next we consider the second term in (B.3). To analyse this term we first note the expansion

21

¯

₉

˘

₉

¯ ¯

₉

˘ ¯

₉

˘

₉

¯ ¯

₉

(a*2a)5Q(Z_{1 1}e 1V M₂ e₁)2QZ V QZ_{1 2 1 1}e 2QV Z QZ_{2 1 1 1}e 1o (T_p ) (B.6)

¯ ˘

9

9

9

which is given in Nagar (1959, p. 582). Using this result, and noting that Z_{1 1}e 5Z_{1 1}e 1V_{2 1}e where

1 / 2

the right hand side terms are, respectively, O (T_p ) and O (T ), we may write_p

¯ ¯ ¯

˘

₉

₉

₉

₉

e9V QZ e e Z QZ e

1 _{¯ ˘ ˘ ¯} 2 1 1 1 1 1 1

]]]](a*2a)9Z_{1 1}

9

e

f

Q2Q(Z V

9

_{1 2}1V Z )Q₂

9

₁

g

5]]]]Q1]]]]Q

T2( g1k) T2( g1k) T2( g1k)

¯ ¯ ¯ ¯ ¯ ¯

˘ ˘ ˘ ˘ ˘ ˘ ˘

9

9

9

9

9

9

9

9

9

e₁V QV M_{2 2} e₁ e₁V QZ V QZ_{2 1 2} 9e₁ e₁V QV Z QZ_{2 2 1 1 1}e e₁V QZ_{2 1 1}e

¯ ˘

]]]] ]]]]]] ]]]]]] ]]]]

9

1 _{T2( g1k)} Q1 _{T2( g1k)} Q1 _{T2( g1k)} Q1_{T2( g1k)}Q(Z V1 2

22

¯ ˘

₉

1V Z )Q2 1 1o (Tp ). (B.7)

The first of these terms involves a product of an odd number of normal random variables with zero mean and so has expected value zero. Taking expected values for the remaining terms we have;

¯ ¯

9

9

e1Z QZ1 1 1e 2 ( g1k)

]]]] ]]]]

E

H

J

Q5s Q

T2( g1k) T2( g1k)

¯ ˘

₉

˘

₉

e9V QV M2 2 e1 2 TKtr(QC )1

]]]]] ]]]]

E

H

J

Q5s Q

¯ ¯

˘ ˘

9

9

9

e₁V QZ V QZ_{2 1 2 1 1}e ₂ T tr(QC )₁

]]]]]] ]]]]

E

H

2

J

Q5 2s Q

T2( g1k) T2( g1k)

¯ ¯

˘ ˘

9

9

e₁V QV Z QZ_{2 2 1} 9e_{1 2}T( g1k)tr(QC )₁

]]]]]] ]]]]]

E

H

2

J

Q5 2s Q

T2( g1k) T2( g1k)

2

¯ ˘

9

9

e1V QZ2 1 1e _{¯ ˘ ˘ ¯} 2s

]]]]

9

9

]]]]

E

H

2 Q(Z V_{1 2}1V Z )Q_{2 1}

J

5 2

f

(T12)QC Q₁ 1QC Q₂

g

T2( g1k) T2( g1k)

2

2s T 22

]]]]

5 2 QC Q₁ 1o (T_p ). (B.8)

T2( g1k)

Here the results for evaluating the second, third and fourth terms are given in Nagar (1961, p. 242).

˘

An evaluation of the last term proceeds from putting V₂5W1e t₁ 9 where W and e₁ are independent. The required analysis is straightforward but lengthy and so is not included here. The authors will provide details on request. Collecting the terms in (B.8) and multiplying by 22 yields the expectation of the second term of (B.3). To complete the analysis we need the expected value of the last term in (B.3). From Nagar (1961, p. 243) we find that

2

9

(a*2a)9Z Z (1 1 a*2a) s fg1k1Ttr(QC )g 21

]]]]]]]] ]]]]]]]

E

H

Q

J

5 Q1o(T ). (B.9)

T2( g1k) T2( g1k)

Collecting the various terms we have the result given in Lemma 2.

References

Kadane, J.B., 1971. Comparison of k-class estimators when the disturbances are small. Econometrica 39, 723–737. Kinal, T.W., 1980. The existence of moments of k-class estimators. Econometrica 48, 241–249.

Kiviet, J.F., Phillips, G.D.A., 1996. The bias of the ordinary least squares estimator in simultaneous equation models. Economics Letters 53, 161–167.

Kiviet, J.F., Phillips, G.D.A., 1998. Degrees of freedom adjustment for disturbance variance estimators in dynamic regression models. Econometrics Journal 1, 44–70.

Mikhail, W.M., 1972. The bias of the two stage least squares estimator. Journal of the American Statistical Association 67, 625–627.

Nagar, A.L., 1959. The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575–595.