Full text of DOI:10.1111/1468-0262.00171

Ž
.
Econometrica, Vol. 68, No. 6 November, 2000 , 1517᎐1527
COVARIANCE MATRIX ESTIMATION AND THE POWER OF THE
OVERIDENTIFYING RESTRICTIONS TEST
1
BY ALASTAIR R. HALL
1. INTRODUCTION
GENERALIZED METHOD OF MOMENTS ŽGMM. Ž
Ž
..
Hansen 1982 has been widely applied to
estimate the parameters of economic and statistical models based on time series data.2 In
many cases of interest, the parameter vector is overidentified and so inference is based
Ž
.
on the two-step estimator proposed by Hansen 1982 . The second step of this procedure
requires the construction of a positive semidefinite consistent estimator of the long run
covariance matrix of the sample moment. Unless the economic model implies a certain
form for this covariance matrix, it is desirable to use an estimator that is consistent under
the weakest possible conditions. This requirement has motivated the development of the
class of heteroscedasticity and autocorrelation consistent covariance HACC matrix
estimators that are now routinely used by practitioners in the calculation of the two-step
GMM estimator. In applications, it is customary to assume the model is correctly
specified during the estimation, and only to assess the specification after the second-step
using a statistic such as the overidentifying restrictions test. One important consequence
of this methodology is that the HACC estimator is calculated under the assumption that
the model is correctly specified.
Ž
.
In this paper, we examine the implications of model misspecification for the HACC
covariance matrix estimator and the overidentifying restrictions test. It is shown that the
HACC estimator is asymptotically equivalent to the sum of two matrices: one of these
Ž .
Ž
.
matrices is nonsingular and O 1 ; the other is of rank one and O bT where bT is the
bandwidth used in the HACC estimator. It is shown that this structure implies the
inverse of the HACC estimator converges to a singular matrix as bT ªϱ with T, and this
limiting matrix has rank qy1 where q is the dimension of the moment condition. It is
shown that this limiting behavior translates into an overidentifying restrictions test that is
Ž
.
Op TrbT . In contrast, it is shown that the overidentifying restrictions test is consistent
Ž
.
and Op T if the covariance matrix estimator is consistent under both null and alterna-
tive. This can be achieved by constructing the HACC estimator using the sample moment
in mean deviation form.3
1I am grateful for useful conversations about this work with Peter Burridge, Atsushi Inoue, and
Ada Wossink, and for the comments from seminar participants at the Departments of Economics at
Cornell University, North Carolina State University, Pennsylvania State University, and the Univer-
sity of Arizona. This paper has also greatly benefited from the comments of Don Andrews and three
anonymous referees. Part of this work was undertaken while the author was a visiting research
fellow at the Department of Economics at the University of Birmingham, UK, and this support is
gratefully acknowledged.
2
Ž
.
Ž
.
For example, see the citations in the review articles by Hall 1993 and Ogaki 1993 .
3Since this research was undertaken, Don Andrews has drawn my attention to footnote 3 in his
Ž
Ž
..
paper Andrews 1999 on moment selection in which it is recommended that variances be
calculated using the data in mean deviation form. However, as he also pointed out to me, he
provides neither a citation nor analysis to support the recommendation in his paper.
1517

1518
ALASTAIR R. HALL
An overview of the paper is as follows. Section 2 presents a brief review of the two-step
GMM estimator and HACC estimators. Section 3 examines the properties of these
estimators and the overidentifying restrictions test when the model is misspecified. All
proofs and certain other technical details are relegated to an Appendix.
2. TWO-STEP GMM ESTIMATION IN CORRECTLY SPECIFIED MODELS
Consider the case in which a model implies the p=1 vector of parameters, ␪0, and a
set of observed variables, ¨t, satisfy the q=1 population moment condition
Ž .
1
w Ž
.x
0
E f ¨t , ␪ s0.
Ž .
The GMM estimator of ␪ based on 1 is defined to be
0
ˆ
Ž .
2
Ž . Ž .
␪T sArgmin␪ g⌰ gT ␪ ЈWT gT ␪
y
1
T
Ž .
Ž
.
where gT ␪ sT Ýts1 f ¨t , ␪ and WT is a positive semidefinite weighting matrix that
converges in probability to a positive definite matrix of constants. If q)p, then the
choice of WT becomes important because it determines the asymptotic covariance matrix
ˆ
Ž
.
of ␪T. In these circumstances, Hansen 1982 proves that the optimal choice of weighting
matrix converges in probability to Sy where
1
1
Ss lim var T r2gT
Ž .
3
w
Ž
.x
␪ .
0
Tªϱ
y
1
ˆ
ˆ
This limit can be achieved by setting WT sST where ST is a positive semidefinite
consistent estimator of S. Since S typically depends on ␪0, the use of this optimal
weighting matrix necessitates a two step estimation procedure. In this paper, we consider
ˆ
Ž .
the case in which the first step estimator, ␪ 1 , is used to construct the following HACC
T
estimator:
Ty1
ˆ
ST s
ˆ
i
Ž .
4
Ž
.
␻ irbT ⌫
Ý
isyTq1
where
T
y
1
1
ˆ
ˆ
ˆ
Ž .. Ž Ž ..
f ¨tyi , ␪ 1
T
Ž
⌫isT
f ¨t , ␪
1
Ј
for iG0,
for i-0,
Ý
T
tsiq1
T
sTy
f ¨tqi , ␪
1
Ј
ˆ
ˆ
Ž ..
1
T
Ž
Ž .. Ž
f ¨t , ␪
Ý
T
tsyiq1
Ž .
and the kernel, ␻ и , and bandwidth, bT , satisfy certain restrictions given below. The
y
ˆ
1
ˆ
Ž
.
Ž .
Ž .
second step or optimal estimator, ␪ 2 , is then calculated using WT sST in 2 .
Up to this point, the model is assTumed to be correctly specified. It is only after the
second step estimation that credence is given to the possibility of misspecification, and

COVARIANCE MATRIX ESTIMATION
1519
Ž .
the overidentifying restrictions test is applied to assess the validity of 1 . This statistic is
ˆ
Ž .
conveniently calculated as T times the second-step GMM minimand evaluated at ␪ 2 ,
namely
T
y
JT sTgT
␪
ЈST 1gT
ˆ
ˆ
ˆ
Ž ..
␪ 2 .
T
Ž .
5
Ž
Ž ..
2
Ž
T
2
Ž
.
Ž .
Hansen 1982 shows that if 1 holds, then JT converges to a ␹
distribution. This
q
sequence of events has the important consequence that in many apyplpications the HACC
estimator used in the second step estimation, and hence the overidentifying restrictions
Ž .
test, is calculated under the assumption that 1 holds.
3. TWO STEP GMM ESTIMATION IN MISSPECIFIED MODELS
In this section, we consider the case where the original population moment condition
Ž .
in 1 is invalid. Certain assumptions are required to facilitate the analysis. However, for
brevity, we only highlight in the text those assumptions that are relevant to the exposition
and relegate the rest to the mathematical Appendix.
Ž
.
Following Hansen 1982 , we impose the following condition on ¨t.
ꢀ
4
ASSUMPTION 1: ¨tgV , ts1, 2, . . . is a sequence of strictly stationary and ergodic
random ¨ectors where V:ᑬs.
Ž .
If the original population moment condition in 1 is invalid, then there must be no
value of ␪ at which the population moment condition is satisfied. So, if we define
Ž .
6
w Ž
.x
Ž .
E f ¨t , ␪ s␮ ␪ ,
then model misspecification is captured by the following assumption.
ASSUMPTION 2: ␮: ⌰ªᑬq such that ␮ ␪ )0 for all ␪g⌰.
5
Ž .5
To deduce the impact of this misspecification on the HACC estimator and the
overidentifying restrictions test, it is necessary to begin with the first step estimator. Let
WT denote the weighting matrix for the first step estimation. For completeness, we
formally impose the conditions described in the previous section.
ASSUMPTION 3: WT is a positi¨e semidefinite matrix that con¨erges in probability to the
positi¨e definite matrix of constants W.
So, the first step estimator is defined as
ˆ
Ž .
7
Ž . Ž .
Ž .
␪
1 sArgmin␪ g⌰ gT ␪ ЈWT gT ␪
T
p
ˆ
Ž .
where ⌰;ᑬ . To establish that ␪ 1 converges to a probability limit, it is necessary to
T
impose an identification condition.4
4
Assumption 4 is similar to the ‘‘identifiable uniqueness’’ condition that underpins Gallant and
Ž
.
White’s analysis of misspecified models; see Gallant and White 1988, p. 19 .

1520
ALASTAIR R. HALL
Ž
.
Ž .
ꢀ
4
)
AŽ S.SUMPwTIOŽN 4: There exists ␪ g⌰ such that Q
.x
w Ž
.x )
␪
-Q0 ␪ , ᭙␪g⌰_ ␪ , where
0
)
Q0 ␪ sE f ¨t , ␪ ЈWE f ¨t, ␪ .
Notice that in general ␪ depends on W, although we suppress this dependence for
ease of notation. Once th)is identification condition is imposed, we can appeal to a
Ž
.
Ž
.
combination of Newey and McFadden’s 1994 Theorem 2.1 and Wooldridge’s 1994
Theorem 7.1 to deduce the following result.5
LEMMA 1: If Assumptions 1᎐4, and Assumptions A.1᎐A.4 in the Appendix hold, then
p
ˆ
Ž .
1 ª ␪) .
␪
T
After the first step estimation, it is the behavior of the population moment condition at
that becomes important. Consequently, we introduce the notation
␪
)
Ž .
8
w Ž .x
E f ¨t , ␪ s␮) .
)
Notice that Assumption 2 implies ␮)/0.
We now turn to the impact on misspecification on both the HACC estimator and its
inverse because it is the latter that appears in the overidentifying restrictions test. For
this analysis, it is necessary to impose certain additional restrictions. First we limit
attention to HACC estimators whose kernels and bandwidths satisfy the following
properties.
Ž .
<
Ž . <
Ž
.
Ž . Ž .
Ž .
ASSUMPTION 5:
i For all xgᑬ, ␻ x F1, ␻ yx s␻ x , ␻ 0 s1, ␻ x is continu-
y
i x␭
Ž .2
ous at zero and for almost all xgᑬ, Hᑬ ␻ x dx-ϱ, Hᑬ ␻ x e
Ž .
and bT ªϱ; iii Hᑬ ␻ x dxsc where 0-c-ϱ.
dxG0 for all ␭gᑬ;
1
r2
Ž .
Ž
.
Ž
.
Ž .
ii bT so T
Ž
It can be verified that Assumption 5 is satisfied by the Bartlett Newey and West
Ž
..
Ž
Ž
..
Ž
Ž
..
1987 , Parzen Gallant 1987 , or the quadratic spectral Andrews 1991 kernels.
It is also necessary to assume the following.
y
1r2
T
ts1
w
Ž Ž
.
)
.x
ASSUMPTION 6: VslimT ªϱ var T
of constants.
Ý
f ¨t, ␪ y␮# is a positi¨e definite matrix
Ž
.
We can now appeal to Gallant and White’s 1988 Theorem 6.8 to deduce the
following lemma.6
LEMMA 2: If Assumptions 1᎐6 and A.1᎐A.8 in the mathematical Appendix hold, then
X
Ty1
ˆ
Ž .
Ž
.
ST sVqMT qop 1 where MT sBT ␮ ␮) , BT sÝisyTq1 ␻ irbT , and hence BTrbT sc
)
Ž .
Ž .
qo 1 for c defined in Assumption 5 iii .
5Our identification condition is different from Wooldridge’s but his proof is easily adapted to take
Ž
.
account of this difference. Note that Wooldridge’s result is based on Newey and McFadden’s 1994
Theorem 2.1 and these authors employ an analogous identification context to Assumption 4 albeit in
the context of correctly specified models.
6Strictly, Gallant and White 1988 only prove their result for bT so T
1r4
Ž
.
Ž
.
and kernels of the
1r2
Ž .
Ž
.
and
form ␻ x s0 for x)1. However, it is straightforward to extend the result to bT so T
Ž .
Ž
.
kernels satisfying Assumption 5 i using Andrews’s 1991 Theorem 1.

COVARIANCE MATRIX ESTIMATION
1521
ˆ
Lemma 1 indicates that the large sample behavior of ST is identical to the behavior of
Ž .
VqMT. The matrix V is positive definite and O 1 ; the matrix MT has rank equal to one
y
ˆ
1
and increases at rate bT. We now consider the implications of Lemma 2 for ST
.
THEOREM 1: If Assumptions 1᎐6 and A.1᎐A.8 hold then:
Ž .
i
p
1
X
STy1 ª S) sVy1
y
Vy1␮ ␮)Vy1
;
ˆ
X
)
␮)Vy1
␮
)
Ž .
ii S) is a positi¨e semidefinite matrix with rank equal to qy1 and a nullspace spanned
by ␮).
We now consider how this behavior impacts the properties of the overidentifying
restrictions test.
Ž
.
.
THEOREM 2: If Assumptions 1᎐6 and A.1᎐A.8 hold, then JT sOp TrbT
This result states that the overidentifying restrictions test can not increase faster than
rate TrbT. By itself, Theorem 2 does not imply JT is a consistent test; however this is in
fact the case.7
Inspection of the proof reveals that this dependence on bT stems directly from the
invalid assumption that the model is correctly specified. Therefore it seems intuitively
reasonable that a more powerful test will be obtained if the HACC matrix is constructed
in such a way that it is consistent regardless of whether or not the population moment
condition is correct. This can be achieved by using the estimator
Ty1
ˆ
VT s
˜
⌫
i
Ž .
9
Ž
.
␻ irbT
Ý
isyTq1
where
T
y
1
˜
ˆ
ˆ
Ž Ž ..x
ygT ␪ 1
T
w Ž
Ž ..
1
⌫isT
f ¨t , ␪
Ý
T
tsiq1
ˆ
ˆ
Ž ..x
␪ 1
T
Ј
w Ž
Ž ..
1
Ž
= f ¨tyi , ␪
ygT
for iG0,
T
T
1
sTy
f ¨tqi , ␪
ygT
ˆ
ˆ
Ž ..x
␪ 1
T
w Ž
Ž ..
1
Ž
Ý
T
tsyiq1
ˆ
ˆ
Ј
w Ž
= f ¨t , ␪
Ž ..
1
Ž
Ž ..x
1
ygT
␪
for i-0.
T
T
For the rest of the paper, we consider the properties of the GMM estimator and
overidentifying restrictions test when this ‘‘centered’’ version of the HACC estimator is
ˆ
used instead of the ‘‘uncentered’’ version, ST. Therefore we now consider the second step
GMM estimator
y
2 sArgmin␪ g⌰ gT ␪ ЈVT 1gT
˜
ˆ
Ž
.
Ž .
Ž .
Ž .
␪
10
␪
T
p
7In an earlier version of this paper, it is shown that bTrT JT ª a where a is a finite positive
Ž
.
constant, and hence that the test is consistent. However, this analysis was dropped for brevity on the
advice of the editor and referees.

1522
ALASTAIR R. HALL
and its associated overidentifying restrictions test
y
11
␪
ЈVT 1gT
˜
JT sTgT
˜
ˆ
˜
Ž ..
␪ 2 .
T
Ž
.
Ž
Ž ..
2
Ž
T
ˆ ˜
˜
Ž .
The following theorem establishes the properties of VT , ␪ 2 , and JT. However, before
T
these results can be presented it is necessary to introduce the following identification
condition.
˜
˜
Ž .
Ž
.
ꢀ
)) 4
and
)) g⌰ such that Q0
␪
-Q0 ␪ , ᭙␪g⌰_ ␪
AŽ S.SUMPwTIOŽN 7: There exists ␪
.x
w Ž
.x
))
y
1
˜
Q0 ␪ sE f ¨t , ␪ ЈV E f ¨t, ␪ .
p
p
ˆ
˜
T
Ž .
Ž . Ž .
THEOREM 3: If Assumptions 1᎐7 and A.1᎐A.9 hold, then: i VT ª V; ii ␪ 2 ª ␪
;
))
p
y
1
˜
Ž
.
iii T JT ª ⑀ for some finite positi¨e constant ⑀.
˜
Ž
.
Theorem 3 iii implies that JT is consistent and increases at rate T, which, from
˜
Theorem 2, is faster than JT. Therefore JT is more powerful than JT in large samples.
To conclude this paper, we present the results from a simulation study designed to
illustrate the extent to which our asymptotic results manifest themselves in finite
samples.8 Data were generated from the following model:
ytsxtq␥ z1, t qut ,
xtsz1, t qz2, t qet ,
Ž
.
Ž
.
for ts1, 2 . . . T where z1, t, z2, t, ut, et Ј;IN 0,⌺ and ⌺ is the symmetric matrix with
i-jth element ␴ whose nonzero upper triangular elements are given by ␴iis1 for
i j
is1, 2 . . . 4, ␴12s␴34 s0.5. GMM estimation of a scalar parameter ␪ is performed under
the assumption that the following population moment condition holds:
Ž
.
w Ž
.x
w
Ž
.x
12
E f ¨t , ␪ sE zt ytyxt␪ s0
Ž
.
Ž .
where zts z1, t , z2, t Ј. Notice that if ␥s0, then 12 holds at ␪s1; otherwise the model
is misspecified. Simulation results reported in an earlier version of this paper indicate
ˆ
ˆ
Ž .
5
Ž
Ž ..5
<
<
that both gT
␪
1
and ␪ 1 y1 increase monotonically with ␥ over the range of
values considereTd. On each rTeplication, both JT and JT are calculated using a HACC
˜
estimator with the Bartlett kernel and a bandwidth chosen via the data based method
Ž
.
proposed by Newey and West 1994 . To present this method, it is notationally conve-
nient to consider the generic case in which it is desired to estimate the long run variance
y
limT ªϱ var T 1r2 ÝTts1 dt for some random vector dt.
w
x
9
Ž
.
Newey and West’s 1994 bandwidth selection method
1. Construct the scalar random variable htswЈ)dt where w is a prespecified vector
of constants discussed below.
y
1
T
Ž
.
2. Construct ␴ s Ty1 Ýtsjq2 ht htyj for js0, 1, . . . n.
ˆ
3. Calculate sŽ␯j . s2Ýnjs1 j␴ and sŽ0. s␴ q2Ýn ␴ .
ˆ
ˆ
ˆ
ˆ0
js1 ˆ
j
j
Ž
r3
4. Calculate ␥s1.1447 s ␯ .rsŽ0.
.
wꢀ
42x1
ˆ
ˆ
ˆ
1
ꢀ
r34
.
5. Set b sint ␥T
ˆ
T
8All calculations are performed using MATLAB.
9
Ž
.
It should be noted that Newey and West 1994 actually recommend the use of a prewhitened
and recolored covariance matrix estimator. However, this step is omitted in our design to highlight
more clearly the impact of misspecification on HACC estimators.

COVARIANCE MATRIX ESTIMATION
TABLE I
1523
SUMMARY STATISTICS FOR BANDWIDTH AND OVERIDENTIFYING RESTRICTIONS TEST AT Ts300
ˆ
˜
˜
˜
Ž .
Pow JT
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
c
␥
Med bT
Med JT
Pow JT
Med bT
Med JT
4
0.000
0.125
0.250
0.375
0.500
10.00
0.00
0.125
0.250
0.375
0.500
10.00
4
4
4
5
6
10
11
11
12
15
18
24
0.476
1.310
4.736
9.474
13.744
21.661
0.501
1.341
4.300
7.389
9.228
11.370
0.051
0.186
0.600
0.923
0.996
1.000
0.046
0.171
0.563
0.905
0.993
1.000
4
4
4
4
4
0.481
1.353
5.296
12.016
20.721
110.900
0.518
1.473
5.702
12.867
22.059
116.227
0.061
0.213
0.631
0.934
0.997
1.000
0.078
0.241
0.651
0.940
0.997
1.000
4
12
12
12
12
11
11
12
Ž
.
Ž .
denotes the median of the statistic J; Pow J denotes the power of the test based on J with nominal
Notes: Med
size 0.05.
J
ˆ
Ž
Ž ..
1
If the HACC is calculated with uncentered data, then dtsf ¨t, ␪
and the selected
T
ˆ
bandwidth is denoted bT. If the HACC is calculated with centered data, then dts
ˆ
ˆ
˜
Ž
Ž ..
Ž
Ž ..
1 , and the selected bandwidth is denoted bT. To implement this
T
f ¨t, ␪ 1 ygT
␪
T
2
Ž
.
ꢀŽ
.
r94
approach, it is necessary to specify w and n. We set ws 1, y1 Ј, nscint Tr100
,
and consider two values for c, that is cs4, 12.10
For brevity, we only report results in Table I for ␥s0.0, 0.125,0.25,0.375,0.5,10.00 and
Ts300, but summarize corresponding results for Ts1000 in the text. All calculations
Ž
.
are based on 10,000 replications. First consider the size properties of the test i.e. ␥s0 .
˜
At Ts300, the empirical sizes of JT and JT tend to be closer to their nominal value of
0.05 with cs4 than with cs12. Of the two statistics, JT exhibits empirical size closer to
the nominal value as would be expected since it is based on the HACC estimator, which
Ž .
exploits the information in 1 . By Ts1000, the empirical sizes of JT with cs4, 12 and
˜
JT with cs4 are within two simulation standard errors of the nominal value; the
˜
corresponding quantity for JT with cs12 is slightly more than three simulation standard
Ž
.
errors from the nominal value. Now consider the power properties of the test i.e. ␥)0 .
˜
Three features stand out. First, JT is more powerful than JT and sometimes by as much
as approximately 10%. Second, the median of bT increases with the degree of misspeci-
fication i.e. ␥ but the median of bT is unaffected. The difference is even more striking
at Ts1000: when ␥s10, the medians of bT with cs4 and cs12 are respectively 16
and 36, whereas the corresponding medians of bT are 4 and 12 respectively. Third, the
median of JT increases far more rapidly than the median of JT. Again, the contrast is
more striking at Ts1000. For ␥s10, the median of JT is 359.637 cs4 and 363.018
ˆ
˜
Ž
.
ˆ
˜
˜
˜
Ž
.
10
Ž
.
The values of n are chosen to mimic those employed in Newey and West’s 1994 simulation
study. The choice of w requires a little more explanation. It turns out that within our simulation
Ž
.
Ž
.
design the moment condition approximately takes the form k,yk Ј. Therefore, if we set ws 1, 1 Ј
ᎏwhich is similar to the choice used by Newey and West 1994 in their simulationsᎏthen ht is
Ž
.
essentially the same process regardless of whether the data are centered or not. Since this is just an
Ž
.
artifact of the simulation design, we set ws 1,y1 in order to illustrate the potential impact of
misspecification on Newey and West’s method procedure for calculating the bandwidth.

1524
ALASTAIR R. HALL
Ž
.
cs12 . The corresponding figures for JT are 51.346 and 25.755. This evidence suggests
the use of Newey and West’s bandwidth selection method without the mean correction
can lead to relatively large values of the bandwidth, and hence exacerbate the problems
caused by the failure to account for misspecification in the construction of the HACC
estimator.
Dept. of Economics, North Carolina State Uni¨ersity, Box 8110, Raleigh, NC 27695-8110,
U.S.A.; alastair hall@ncsu.edu
᎐
Manuscript recei¨ed April, 1998; final re¨ision recei¨ed August, 1999.
MATHEMATICAL APPENDIX
We first list the additional assumptions for the Lemmas and Theorems presented in the paper.
ASSUMPTION A.1: f: V =⌰ªᑬq.
ASSUMPTION A.2: ⌰ is a compact set.
Ž
.
Ž
.
ASSUMPTION A.3: f и, ␪ is measurable for each ␪g⌰ and f ¨,и is continuous on ⌰ for e¨ery
¨ gV .
Ž
.
ASSUMPTION A.4: f ¨t, ␪ satisfies the Uniform Weak Law of Large Numbers on ⌰.
Ž
.
Ž .
ASSUMPTION A.5: f ¨, ␪ is continuously differentiable with respect to ␪ on int ⌰ and this
Ž
.
Ž .
differential Ѩ f и, ␪ rѨ␪ Ј is measurable on V for each ␪gint ⌰ .
Ž .
<
Ž
.<
-b ¨ for all i, js1, 2 . . . q and E b ¨ -D, a finite constant; there exist constants D, ␦)0 and
wꢀ
) .<44Ž rq␦ .
Ž . <
Ž
.
<
ASSUMPTION A.6: There exists a measurable function b ¨ such that fi ¨, ␪ -b ¨ , Ѩ fi ¨, ␪ rѨ␪
j
Ž .
rG1 such that E supi fi ¨, ␪
w Ž .2 x
<
Ž
x
-D.
Ž
.
ASSUMPTION A.7: ¨t is an ␣-mixing sequence with size y3rr ry1 , r)1.
ASSUMPTION A.8: ␪ is an interior point of ⌰.
)
ASSUMPTION A.9: ␪
is an interior point of ⌰.
))
More primitive conditions for the Central Limit Theorem and the Uniform Weak Law of Large
Ž
.
Numbers can be found in inter alia Woolridge 1994 .
Ž .
i is broken down into two parts. We first show that
PROOF OF THEOREM 1: The proof of part
X
11
ˆy1
ST sSTy1 qop
1
where ST sVqBT ␮ ␮) . We then use the form of STy1 to deduce the result
Ž .
)
stated in the theorem.
5
5
A
The proof is based on two matrix results, which for convenience are stated first. Let
s' Ž
.
Ž
.
Ž .
A denote the largest and smallest eigenvalues of A respectively.
min
tr AЈA , and ␭
A , ␭
max
11 I am greatly indebted to an anonymous referee for suggesting the strategy used to prove this
statement.

COVARIANCE MATRIX ESTIMATION
1525
It can be shown that for q=q symmetric matrices A and B:12
Ž
Ž
.
.
5
5
Ž
.
13
14
A Fq␭
A ,
max
<
Ž
.
Ž
.<
5
5
␭
A y␭
B
F AyB .
min
min
To begin the proof, note that for any T, we have13
BT
X
STy1 sVy1
y
Vy1␮ ␮)Vy1
Ž
.
15
)
X
1qBT ␮)Vy1
␮
)
y1 5
Ž .
5
5
Ž .
ˆ
5
and so ST sO 1 . Now, Lemma 2 implies that ST yST sop 1 . Since V is positive definite, it
Ž
. Ž .
follows that ␭
ST )0; see Magnus and Neudecker 1991, p. 208 . These two properties combined
min
ˆ
. x
ST )0 s1, and so
Ž
.
w
Ž
with 14 imply that limT ªϱ P ␭
min
ˆ
ˆy1
Ž
.
5
5
y1 5
ST Fq␭
Ž
.
ꢀ
.4y1
ˆ
Ž
min
Ž .
sOp 1 .
16
ST sq ␭
ST
max
Therefore,
y1 5
5
Ž
.
y1 5
5
y1 5 5
5 5 y1 5
ST yST ST
ˆy1
ˆy1
ˆ
ˆ
ˆ
ST yST s ST ST yST ST F ST
Ž . Ž . Ž .
sOp 1 op 1 O 1 sop
Ž .
1
ˆy1
and so we have established that ST sSTy1 qop 1 . To complete the proof of part
Ž .
Ž .
i note that as
p
X
X
y1
y1
Ž
.
Ž
.
␮ . The desired result then follows directly
)
Tªϱ, we have BTr 1qBT ␮)V
␮
ª1r ␮)V
)
Ž
.
from 15 .
Ž .
S) sVy1r2Ј Iq yx xЈx y1 xЈ Vy1r2
The proof of part ii follows immediately from
Ž
.
Ž
.
17
where Vy1r2 is the q=q matrix that satisfies Vy1 sVy1r2ЈVy1r2 and xsVy1r2␮) .
Ž
.
PROOF OF THEOREM 2:14 Let the minimand on the second step GMM estimation be QT ␪ . By
Ž
.
ˆ
Ž
Ž ..
Ž
.
Ž
. Ž .
sOp TrbT . By the
)
definition QT
␪
2
FQT
␪
, and so it is sufficient to prove that TQT
␪
T
)
Cauchy-Schwarz inequality and Lemma 2, we have
Ž
.
Ž
.
<
< < . <
Ž
<
5
5
Ž
. <
␪ .
)
18
TQT
␪
F TrBT BT QT
␪
F TrbT bTrBT BT QT
)
)
Ž
.
Ž
.
Ž .
Since TrbT sO TrbT we need to show that the remaining two terms in 18 are Op 1 . We first
Ž
.
Ž .
Ž
.
show that bTrBT sO 1 . Let cT sBTrbT . From Assumption 5 iii it follows that cT scqo 1
where 0-c-ϱ. Therefore bTrBT scyT 1 s cqo 1 scy1 qo 1 sO 1 because c is a finite
Ž
Ž ..y1
Ž .
sOp 1 . Since
Ž . Ž .
Ž
.
␪
)
positive constant. We now show that BT QT
sBT1r2 gT
␪
ЈST BT1r2 gT
ˆy1
Ž
.
Ž
.
Ž
.
␪ ,
)
BT QT
␪
)
)
we first consider BT1r2 gT
␪
Ž
.
. By definition, we have
)
T
1r2
Ž
.
Ž
.
Ž
.
w
Ž
.
x
f ¨t , ␪ y␮ .
)
19
BT1r2 gT
␪
sBT
␮
) q BTrT 1r2Ty1r2
Ý
)
)
t
s1
T
Under our conditions Ty1r2 Ýts1 f ¨t, ␪ y␮ sOp 1 , and so it follows from Assumption
w Ž
sBT
.
x
)
Ž .
)
␮
1r2
5 ii ᎐ iii and 19 that BT1r2 gT
␪
) qop 1 . Therefore, it follows that
Ž . Ž
.
Ž
.
Ž
.
Ž .
)
X
ˆy1
Ž
Ž
.
.
Ž
.
Ž .
20
21
BT QT
␪
sBT ␮)ST
␮
) qop
1
)
sBT ␮X)STy1
␮
) qBT ␮ ST ySTy1
X
ˆy1
Ž
.
␮
Ž .
) qop 1 .
)
12
Ž
.
Ž
.
Ž .
Equation 13 follows from Magnus and Neudecker 1991, Theorem 14, p. 211 . Equation 14
Ž
.
follows from Golub and van Loan 1989, Corollary 8.1.3, p. 411 .
13
Ž
.
For example, see Morrison 1976, p. 69 .
14 I am greatly indebted to two anonymous referees whose comments greatly shortened this proof.

1526
ALASTAIR R. HALL
Ž
.
Using 15 it can be shown that
X
BT ␮)Vy1
␮
)
BT ␮X)STy1
␮
) s
sn1, T
,
say.
X
1qBT ␮)Vy1
␮
)
Ž
.
Assumption 5 iii and Lemma 2 together imply that BT ªϱ as Tªϱ and so limT ªϱ n1, T s1. Now
Ž
.
consider the second term in 21 , that is
X
)
X
y1
ˆy1
ˆy1
ˆ
.
BT ␮ ST ySTy1
␮
) sBT ␮)ST ST yST ST
␮
) sn2, T , say.
Ž
.
Ž
y1
Using 16 and Lemma 2, it follows that n2, T sOp 1 op 1 BT ST ␮) . Now BT STy1
␮
) s BTr 1q
Ž
.
␮
Ž . Ž .
w Ž
X
y1
BT ␮)Vy1
) sO 1 and so n2, T sop 1 . Therefore, we have
.x
V ␮
Ž .
Ž .
)
Ž
.
Ž . Ž . Ž .
BT QT
␪
sn1, T qn2, T qop 1 sn1, T qop 1 sOp
1
)
Ž
.
and so the desired result follows from 18 .
ˆ
ˆ
ˆ
Ž Ž ..
PROOF OF THEOREM 3: Part i : To simplify the presentation let ft sf ¨t, ␪ 1 , fsgT ␪ 1 ,
T
Ž .
Ž ..
Ž
Ž ..
T
and fi s Tyi y1 ÝTtsiq1 f ¨t, ␪ 1 . Using these definitions, it can be shown that
ˆ
Ž
.
Ž
T
X
⌫i s⌫i 0 yAi, T qCi, T qCi, T
˜
˜
Ž
.
22
where
Tyi
T
Tyi
T
w
xw
x
w
xw
x
fy␮ Ј.
)
Ai, T
s
fy␮
fy␮
Ј
and Ci, T
s
fi y␮
)
)
)
Ž
.
Using similar arguments to Andrews’ 1991 proof of this Theorem 1, it can be shown that
T
y1
p
˜
Ž .
␻ irbT ⌫
i
Ž
.
23
0 ª V.
Ý
syTq1
i
Therefore, the desired result will follow if
T
y1
p
Ž
Ž
.
.
Ž
.
24
25
As
Cs
␻ irbT Ai, T ª 0,
Ý
i
syTq1
y1
T
p
Ž
.
␻ irbT Ci, T ª0.
Ý
i
syTq1
Ž
.
Ž .
Equations 24 and 25 can be deduced from the following properties that are implied by the
conditions of the theorem:
T
y1
Tyi
T
Ž .
a
Ž
.
Ž
.
␻ irbT
sO bT ;
Ý
syTq1
i
y1r2
Ž
.
Ž
.
;
b
fy␮) sOp
T
y1r2
Ž .
c
Ž
.
,
fi y␮) sOp
T
for all is1, 2 . . . bT .
1r2
Ž . Ž .
Ž
.
Ž .
Ž .
It is easily shown that a ᎐ c plus bT so T
and bT ªϱ imply 24 and 25 .
Ž .
Ž
.
Part ii : This follows directly from Wooldridge’s 1994 Theorem 7.1.
Part iii : From parts i and ii it follows that
Ž
.
Ž .
Ž .
X
Ty1 JT s␮
V ␮
y1
˜
Ž .
)) qop 1 .
))
X
y1
Assumptions 2 and 6 together imply that ⑀s␮
V ␮)) )0.
))

COVARIANCE MATRIX ESTIMATION
REFERENCES
1527
Ž
.
ANDREWS, D. W. K. 1991 : ‘‘Heteroscedasticity and Autocorrelation Consistent Covariance Matrix
Estimation,’’ Econometrica, 59, 817᎐858.
Ž
.
ᎏᎏᎏ 1999 : ‘‘Consistent Moment Selection Procedures for Generalized Method of Moments
Estimation,’’ Econometrica, 67, 543᎐564.
Ž
.
GALLANT, A. R. 1987 : Nonlinear Statistical Models. New York, NY: Wiley.
Ž
.
GALLANT, A. R., AND H. WHITE 1988 : A Unified Theory of Estimation and Inference in Nonlinear
Models. Oxford, UK: Basil Blackwell.
GOLUB, G. H., AND C. F. VAN LOAN Ž1989 : Matrix Computations, Second Edn. Baltimore, MD: Johns
.
Hopkins Press.
Ž
.
HALL, A. R. 1993 : ‘‘Some Aspects of Generalized Method of Moments Estimation,’’ in Handbook
of Statistics, Vol. 11, ed. by G. S. Maddala, C. R. Rao, and H. D. Vinod. Amsterdam, The
Netherlands: Elsevier Science Publishers, pp. 393᎐417.
Ž
.
HANSEN, L. P. 1982 : ‘‘Large Sample Properties of Generalized Method of Moments Estimators,’’
Econometrica, 50, 1029᎐1054.
MAGNUS, J. R., AND H. NEUDECKER 1991 : Matrix Differential Calculus with Applications in Statistics
and Econometrics. New York, NY: Wiley.
Ž
.
Ž
.
MORRISON, D. F. 1976 : Multi¨ariate Statistical Methods, Second Edn. Tokyo, Japan: McGraw-Hill.
NEWEY, W. K., AND D. L. MCFADDEN Ž1994 : ‘‘Large Sample Estimation and Hypothesis Testing,’’ in
.
Handbook of Econometrics, Vol. 4, ed. by R. Engle and D. L. McFadden. Amsterdam, The
Netherlands: Elsevier Science Publishers, pp. 2113᎐2247.
Ž
.
NEWEY, W. K., AND K. D. WEST 1987 : ‘‘A Simple Positive Semi-definite Heteroscedasticity and
Autocorrelation Consistent Covariance Matrix,’’ Econometrica, 55, 703᎐708.
Ž
.
ᎏᎏᎏ 1994 : ‘‘Automatic Lag Selection in Covariance Matrix Estimation,’’ Re¨iew of Economic
Studies, 61, 631᎐653.
Ž
.
OGAKI, M. 1993 : ‘‘Generalized Method of Moments: Econometric Applications,’’ in Handbook of
Statistics, Vol. 11, ed. by G. S. Maddala, C. R. Rao, and H. D. Vinod. Amsterdam, The
Netherlands: Elsevier Science Publishers, pp. 455᎐488.
WOOLDRIDGE, J. M. 1994 : ‘‘Estimation and Inference for Dependent Processes,’’ in Handbook of
Econometrics, Vol. 4, ed. by R. Engle and D. L. McFadden. Amsterdam, The Netherlands:
Elsevier Science Publishers, pp. 2641᎐2739.
Ž
.