Economics Letters 66 2000 7–15
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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.Phillipsexeter.ac.uk G.D.A. Phillips 0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved.
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8 J
.F. Kiviet, G.D.A. Phillips Economics Letters 66 2000 7 –15
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
A9 y 1 B9x 5 e , t 5 1, . . . ,T,
1
t t
t
where y is a G 3 1 vector of endogenous variables, x is a K 3 1 vector of strongly exogenous
t t
variables which we shall treat as non-stochastic, and e is a G 3 1 vector of structural disturbances.
t
A9and B9 are respectively, G 3 G and G 3 K matrices of structural coefficients. With T observations on the above system, we may write
YA 1 XB 5 E 2
where Y is a T 3 G matrix of observations on the endogenous variables, X is a T 3 K matrix of observations on the exogenous variables, and E is a T 3 G 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
Y 5 XP 1 V 3
1 1
1
where Y 5 y :Y , X 5 X :X , P 5p :P and V 5 v :V .P is a K 3 g 1 1 matrix of reduced
1 1
2 1
2 1
1 2
1 1
2 1
form parameters and V is a T 3 g 1 1matrix of reduced form disturbances. In addition, the
1
following assumptions are made: •
The rows of V are independently and normally distributed with mean vector 09 and non-singular
1
covariance matrix V 5 v .
h j
ij
• The T 3 K matrix X is of rank K , T , and the elements of the K 3 K matrix X9X are OT .
• 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
