Eric Zivot

Saraswata Chaudhuri

English

Research Papers in Economics

A Comment on Weak Instrument Robust Tests in
GMM and the New Keynesian Phillips Curve
Eric Zivot∗and Saraswata Chaudhuri†
October 27, 2008
1 Introduction
The paper by Kleibergen and Mavroeidis (2008a), hereafter KM, is an excellent survey
of the current state of the art in the weak instrument robust econometrics for testing
subsets of parameters in GMM, and provides an important and relevant application
of the econometric theory to the analysis of the new Keynesian Phillips curve. We are
extremely grateful to have the opportunity to comment on this very nice paper. Our
comments will focus on the weak instrument robust tests for subsets of parameters,
and in particular on the projection-based test that KM referred to as the Robins
(2004) test.
We show that KM’s implementation of the Robins test is ineﬃcient, and provide an
eﬃcient implementation that performs nearly as well the MQLR test recommended by
∗Professor and Gary Waterman Distinguished Scholar, Department of Economics, University of
Washington. Box 353330, Seattle, WA, 98195-3330. email: ezivot@u.washington.edu. Support
from the Gary Waterman Distinguished Scholar Fund and the University of Washington Center for
Statistics in the Social Sciences Seed Grant is gratefully acknowledged.
†Assistant Professor, Department of Economics, University of North Carolina Chapel Hill. email:
saraswata_chaudhuri@unc.edu.
1KM. Our comment proceeds as follows. Section 2 reviews the tests used for inference
on subsets of parameters in GMM, and discusses in detail the implementation of
the Robins test which we call the new method of projection. Section 3 reports the
results of a small simulation study to demonstrate that the new method of projection
performs nearly as well as the tests recommended by KM. Section 4 contains our
concluding remarks.
2 Inference on Subsets of Parameters in GMM
In this section we describe inference on subsets of parameters in the GMM framework.
We follow the notation and assumptions of KM regarding the GMM framework.
Interest centers on a p−dimensional vector of parameters θ identiﬁed by a set of
k ≥ p moment conditions
E [ft(θ)] = 0.
Let θ =( α0,β0)0, where α is pα ×1 and β is pβ ×1. The parameters of interest are β,
and α are considered nuisance parameters. The weak-identiﬁcation robust methods of
inference on θ are based on the (eﬃcient) continuous updating (CU) GMM objective
function
Q(θ)=TfT(θ)
0ˆ Vff(θ)
−1fT(θ), (1)
where fT(θ)=T−1 PT
t=1 ft(θ), and ˆ Vff(θ) is a consistent estimator of the k × k di-
mensional covariance matrix Vff(θ) of the vector of sample moments. Let qt(θ)=
vec
³
∂ft(θ)
∂θ0
´
and deﬁne ¯ ft(θ)=ft(θ)−E[ft(θ)] and ¯ qt(θ)=qt(θ)−E[qt(θ)]. Assump-
2tion 1 of KM states that
1
√
T
T X
t=1
⎡
⎢
⎣
¯ ft(θ)
¯ qt(θ)
⎤
⎥
⎦
d →
⎡
⎢
⎣
ψf(θ)
ψθ(θ)
⎤
⎥
⎦ ∼ N(0,V(θ)),V (θ)=
⎡
⎢
⎣
Vff(θ) Vfθ(θ)
Vθf(θ) Vθθ(θ)
⎤
⎥
⎦.
The gradient of (1) with respect to θ is given by
OθQ(θ)=
∂Q(θ)
∂θ0 =2 fT(θ)
0ˆ Vff(θ)
−1 ˆ DT(θ),
where ˆ DT(θ)=
PT
t=1 Dt(θ) and Dt(θ)=d e v e c k
h
qt(θ) − ˆ Vθf(θ)ˆ Vff(θ)−1ft(θ)
i
.F o r
the deﬁnition of the devec operator see Chaudhuri (2007).
2.1 Tests for the Full Parameter Vector
Valid tests of the hypothesis H0 : θ = θ0 w e r ed e v e l o p e di nS t o c ka n dW r i g h t
(2000) and Kleibergen (2005). Stock and Wright’s S-statistic is a generalization
of the Anderson-Rubin statistic (see Anderson and Rubin (1949)) and is given by
S(θ)=Q(θ). Kleibergen’s K-statisic is a score-type statistic based on Q(θ) and may
be expressed as
K(θ)=
1
4
(OθQ(θ))
h
ˆ DT(θ)
0ˆ Vff(θ)
−1 ˆ DT(θ)
i−1
(OθQ(θ))
0 . (2)
Under the null H0 : θ = θ0,S (θ0)
A ∼ χ2
k and K(θ0)
A ∼ χ2
p.
2.2 Tests for Subsets of Parameters
For testing hypotheses on subsets of parameters of the form H0 : β = β0, subset
versions of the S and K-statistics were also considered by Stock and Wright (2000) and
Kleibergen (2005). These statistics are based on the plug-in principle and utilize the
3constrained CU-GMM estimate ˜ α(β0)=a r gm i n α Q(α,β0). Letting ˜ θ0 =(˜ α(β0)0,β0
0)0,
the subset S and K statistics are given by S(˜ θ0) and K(˜ θ0), respectively. Under the
null H0 : β = β0 and under the assumption that α is well identiﬁed, Stock and
Wright (2000) and Kleibergen (2005) showed that S(˜ θ0)
A ∼ χ2
k−pα and K(˜ θ0)
A ∼ χ2
pβ.
This result is based on the fact that when α is well identiﬁed, ˜ α(β0) is
√
n consistent
for α under H0 : β = β0.W h e n α is not well identiﬁed, ˜ α(β0) is no longer
√
n
consistent for α and hence the S and K-statistics are not asymptotically chi-square
distributed. However, Theorem 1 of KM showed that irrespective of the identiﬁcation
of α, the S and the K-statistics are always bounded from above by the χ2
k−pα and χ2
pβ
distributions, respectively.
2.3 Usual Method of Projection
Dufour (1997), Dufour and Jasiak (2001) and Dufour and Taamouti (2005, 2007)
showed that the usual projection approach could always be used to obtain valid infer-
ence for subsets of parameters provided there exists an asymptotically (boundedly)
pivotal statistic for testing the joint hypothesis H0 : θ = θ0. Let R(θ) denote such a
statistic and assume that R(θ)
A ∼ χ2
v. Suitable choices for R(θ) are S(θ), for which
v = k, and K(θ), for which v = p. The usual method of projection rejects H0 : β = β0
at level (at most) ζ if
inf
α∈Θα
R(α,β0) >χ
2
v(1 − ζ),
where Θα denotes the parameter space for α, and χ2
v(1−ζ) denotes the 1−ζ quantile
of the chi-square distribution with v degrees of freedom. The asymptotic size of the
projection test cannot exceed ζ irrespective of the identiﬁcation of α or β or both.
However, the power of the test can be very low if v is large compared to pβ.
42.4 New Method of Projection
Chaudhuri (2007), Chaudhuri et al. (2008) and Chaudhuri and Zivot (2008) pro-
posed a new method of projection for making inferences on subsets of parameters in
the presence of potentially unidentiﬁed nuisance parameters that are based on ideas
presented in Robins (2004). The new method of projection requires (i) a uniform
asymptotic (1−ξ)·100% conﬁdence set, Cα(1−ξ,β0),f o rα when the null hypothesis
H0 : β = β0 is true, and (ii) an asymptotically pivotal statistic R(θ). In most cases,
as described in Table 1, R(θ)
A ∼ χ2
v for some v depending upon the choice of Rβ(θ).
Then the new method of projection rejects H0 : β = β0 if
1. either Cα(1 − ξ,β0)=∅
2. or infα0∈Cα(1−ξ,β0) R(α0,β 0) >χ 2
v(1 − ζ).
Under the null hypothesis H0 : β = β0, Cα(1 − ξ,β0) asymptotically contains α
with probability at least 1−ξ, and hence it follows from Bonferroni’s inequality that
t h ea s y m p t o t i cs i z eo ft h en e wp r o j e c t i o nt y p et e s tc a n n o te x c e e dζ + ξ. The new
method of projection can be expected to be generally less conservative than the usual
method of projection because the inﬁmum for the new method is only computed over
Cα(1−ξ,β0) whereas the inﬁmum is computed over the whole space Θα for the usual
method. Similar projection methods have also been employed by Dufour (1990),
Berger and Boos (1994), and Silvapulle (1996).
To implement the new method of projection in the context of GMM, Cα(1−ξ,β0)
can be constructed by inverting the S or K tests as
C
S
α(1−ξ,β0)={α : S(α,β0) ≤ χ
2
k(1−ξ)} or C
K
α (1−ξ,β0)={α : K(α,β0) ≤ χ
2
p(1−ξ)}.
An advantage of using CK
α (1−ξ,β0) is that it will never be empty, and the asymptotic
5properties of the test will only depend on R(θ) when α is well identiﬁed. However, it
will also include saddlepoints α∗ where K(α∗,β 0)=0and these points are associated
with spurious declines in power of the K-statistic. In contrast, the set CS
α(1 − ξ,β0)
can be empty and this will occur for values β0 at which the overidentifying restrictions
are rejected (at level ξ). As we show in the next section, this can lead to improved
power properties of the new method of projection.
While the new method of projection can be implemented using any asymptotically
pivotal statistic R(θ), Robins (2004) showed that there are certain advantages of using
an eﬃcient score-type statistic for R(θ).T h e e ﬃcient score for β (given α),i nt h e
terminology of van der Vaart (1998), is the part of the score (gradient of the objective
function with respect to) for β that is orthogonal to the score for α.T h e e ﬃcient
score statistic for β is a quadratic form in the eﬃcient score for β with respect to
an estimator of its asymptotic variance. In the context of GMM, Chaudhuri (2007)
and Chaudhuri and Zivot (2008) decomposed the K-statistic (2) into two orthogonal
statistics: a K-statistic for α (given β k n o w n )a n da ne ﬃcient (score) K-statistic for
β
K(θ)=Kα(θ)+Kβ.α(θ),
where
Kα(θ)=
1
4
(∇αQ(θ))
³
ˆ DTα(θ)
0ˆ Vff(θ)
−1
2 ˆ DTα(θ)
´−1
(∇αQ(θ))
0 ,
Kβ.α(θ)=
1
4
(∇β.αQ(θ))
³
ˆ DTβ(θ)
0ˆ Vff(θ)
−1
2Nˆ Vff(θ)
−1
20 ˆ DTα(θ)
ˆ Vff(θ)
−1
2 ˆ DTβ(θ)
´−1
(∇β.αQ(θ))
0 ,
and ∇β.αQ(θ) is the estimated eﬃcient score for β deﬁned as
∇β.αQ(θ)=fT(θ)
0ˆ Vff(θ)
−1
2Nˆ Vff(θ)
−1
20 ˆ DTα(θ)
ˆ Vff(θ)
−1
2 ˆ DTβ(θ).
6Cα(1 − ξ,β0) R(α,β) v
CK
α (1 − ξ,β0) S(α,β0) k
CK
α (1 − ξ,β0) K(α,β0) p
CKα
α (1 − ξ,β0) Kβ.α(α,β0) pβ
CS
α(1 − ξ,β0) S(α,β0) k
CS
α(1 − ξ,β0) K(α,β0) p
CS
α(1 − ξ,β0) Kβ.α(α,β0) pβ
Table 1: Conﬁdence sets, test statistics and degrees of freedom for new projection
type tests.
The above expressions use the partition ˆ DT(θ)=
h
ˆ DTα(θ), ˆ DTβ(θ)
i
and ˆ Vθf =
h
ˆ Vαf(θ)0, ˆ Vβf(θ)0
i0
.
It can be shown that under H0 : θ = θ0,K α(θ0)
A ∼ χ2
pα and Kβ.α(θ0)
A ∼ χ2
pβ.
Furthermore, if θ0 belongs to the
√
n-neighborhood of θ,t h e nKβ.α(θ0)=Kβ.α(θ)+
op(1).T h i sl a t t e rp r o p e r t yo fKβ.α(θ) makes it ideally suited for use in the new method
of projection. Indeed, Chaudhuri (2007) proved that if Cα(1 − ξ,β0) is non-empty
with probability approaching one and if α is well identiﬁed then the new method
of projection type test that rejects H0 : β = β0 when infα0∈Cα(1−ξ,β0) Kβ.α(θ0) >
χ2
pβ(1 − ζ) is asymptotically equivalent to the size (at most) ζ K-test for β against
local alternatives. This means that the new method of projection with R(θ)=Kβ.α(θ)
is size controlled when α is not identiﬁed and can be made asymptotically equivalent
to Kleibergen’s K-test when α is well identiﬁed.
Table 1 summarizes the possible ways of implementing the new method of pro-
jection type tests for testing H0 : β = β0. KM illustrated the use of the new method
of projection with Cα(1 − ξ,β0)=CK
α (1 − ξ,β0) and R(θ)=S(α,β0) and concluded
that the Robins test, proposed in Chaudhuri (2007) and Chaudhuri et al. (2008),
does not outperform the usual method of projection based on R(θ)=S(α,β0). How-
ever, this is not what Chaudhuri (2007) and Chaudhuri et al. (2008) refer to as the
Robins test. In the context of GMM, Chaudhuri (2007) and Chaudhuri and Zivot
7(2008) recommend using Cα(1−ξ,β)=CS
α(1−ξ,β) and R(θ)=Kβ.α(θ). The power
of this method is largely driven by the choice of the statistic R(θ). In addition, the
choice R(θ)=Kβ.α(θ) (i.e., the eﬃcient K-statistic) can make this test asymptotically
equivalent the K test when α is well identiﬁed. In the next section we show, using
the same simulation experiment as KM, that this latter implementation of the new
method of projection performs comparably to the tests recommended by KM.
3 Simulations
To illustrate the ﬁnite sample properties of the new method of projection based on
CS
α(1−ξ,β0) and Kβ.α(α,β0) we utilize the same simulation experiment described in
Section 4 of KM. We are grateful to Frank Kleibergen and Sophocles Mavroeidis for
sharing their Matlab code with us.
The data generating process is
πt = λxt + γfEt[πt+1]+ut,
xt = ρ1xt−1 + ρ2xt−2 + vt,
πt+1 =( α0ρ1 + α1)xt + α0ρ2xt−1 + ηt+1,
where ηt = ut + α0vt. There error terms ηt and vt are jointly normal with unit
variances and correlation ρηv =0 .2. The parameter of interest is γf and λ is the
nuisance parameter. Identiﬁcation of the structural parameters λ and γf is controlled
by the concentration parameter μ2 which is a complicated nonlinear function of the
model parameters.
KM’s Figure 3 illustrated the power curves for testing H0 : γf =1 /2 against H1 :
γf 6=1 /2 at the 5% level for the subset S, usual method of projection based on S, and
80 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
γ
f
R
e
j
e
c
t
i
o
n
 
R
a
t
e
 
(
i
n
 
%
)
Weak identification (10,000 trials)
 
 
KLM (5%)
S (5%)
MQLR (5%)
JKLM (5%)
KJ (0.5% + 4.5%)
R (0.5% + 4.5%)
R (0.5% + 5%)
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
80
90
100
γ
f
R
e
j
e
c
t
i
o
n
 
R
a
t
e
 
(
i
n
 
%
)
Strong identification (10,000 trials)
Figure 1: Power curves of 5% level tests for H0 : γf =0 .5 against H1 : γf 6=0 .5. The
sample size is 1000 and the number of Monte Carlo simulations is 10000.
the new method of projection based on CK
λ (1−ξ,γf =1 /2) and S(λ,γf =1 /2) with
ξ =0 .02 and ζ =0 .03. The ﬁgure shows that the power curves of the usual method
of projection and an ineﬃcient application of the new method are indistinguishable,
and are dominated by the subset S statistic.
Figure 1 in this note shows the power curves of the new method of projection based
on CS
λ(1 − ξ,γf =1 /2) and Kλ.γf(λ,γf =1 /2) with ξ =0 .005 and ζ =0 .045,0.05,
along with the recommended tests of KM. The graphs show that new method of
projection actually performs as well as the MQLR and KJ tests recommended by
KM. For the strong identiﬁcation case, use of CS
λ(1−ξ,γf =1 /2) avoids the spurious
decline in power observed for the KLM statistic.
94C o n c l u s i o n
KM showed that the subset versions of the S, K and MQLR statistics are valid
tests even when the nuisance parameters are unidentiﬁed. This is an important
theoretical and practical result. Their simulation results calibrated to a stylized new
Keynesian Phillips curve showed that projection-type tests are too conservative and
are dominated by the subset S, K and MQLR statistics. We show that a version of
the Robins test, which we call the new method of projection, based on an eﬃcient
score type statistic performs nearly as well as the MQLR statistic and provides an
alternative approach to weak instrument robust inference for subsets of parameters
in models estimated by GMM.
A real practical drawback of the weak instrument robust tests is that they are
based on the CU-GMM objective function. The CU-objective function can be ill-
behaved, even for linear models, and ﬁnding the global minimum can be diﬃcult.
Moreover, most commercial software implementations of GMM do not support CU-
GMM. Until commonly used software implementations of GMM catch up with the
important theoretical developments surveyed by KM, it is not likely that weak in-
strument robust methods will be widely used in practice.
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12

A Comment on Weak Instrument Robust Tests in GMM and the New Keynesian Phillips Curve

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A Comment on Weak Instrument Robust Tests in GMM and the New Keynesian Phillips Curve

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