Sample sensitivity for two-step and continuous
updating GMM estimators

Rikuto Onishi and Taisuke Otsu
November 27, 2020
KEO Discussion Paper No. 155

KEO Discussion Paper No. 155

SAMPLE SENSITIVITY FOR TWO-STEP AND CONTINUOUS UPDATING
GMM ESTIMATORSRIKUTO ONISHI AND TAISUKE OTSU
Abstract. This paper follows up the sensitivity analysis by Andrews, Gentzkow and Shapiro(2017) for biases in GMM estimators due to local violations of identifying assumptions, andproposes complementary bias measures that are sensitive to diﬀerent choices of GMM weightmatrices by considering a specific form of the local perturbation. Our method accommodates thetwo-step and continuous updating GMM estimators with or without centering. The proposedbias measures are illustrated by a consumption based asset pricing model using Japanese data.

1. Introduction
In a recent paper, Andrews, Gentzkow and Shapiro (2017) (hereafter, AGS) have introduced
a novel and practical measure of sensitivity of the GMM estimates for structural parametersagainst local perturbations of diﬀerent moments. More precisely, consider the GMM estimator✓ˆ = arg min✓ ĝ(✓)0 Ŵ ĝ(✓) for the parameters ✓0 , where ĝ(✓) is a vector of sample moments andŴ is a weight matrix. Then AGS’s sensitivity of ✓ˆ is defined as ⇤ =(G0 W G) 1 G0 W , where G

and W are probability limits of @ĝ(✓0 )/@✓0 and Ŵ , respectively.
AGS showed that under local violation of the identifying assumption in the sense ofpdnĝ(✓0 ) !

g̃ for a random vector g̃ with non-zero mean, the asymptotic bias of the GMM estimator isexpressed as ⇤E[g̃]. Based on this, they advocated to report the estimate of ⇤ to make thestructural parameter estimate ✓ˆ ‘transparent’, i.e., researchers can easily assess the potential
bias of the parameter estimators for various scenarios of misspecification represented by E[g̃].

Furthermore, Section 5 of AGS introduced the notion of the sample sensitivity, which is associatedˆˆ ) under the perturbed sample momentwith the derivative @ ✓(0)/@of the GMM estimates ✓(ĝ(✓, ) = ĝ(✓) + ⌘ with ⌘ = plimn!1 ĝ(✓0 ).
In this paper, we follow up the analysis in AGS by investigating the eﬀect of diﬀerent GMM
weight matrices Ŵ on the sample sensitivity. There are several motivations for this study. First,
the GMM estimation typically involves the first stage parameter estimates in the construction of
the weight matrix. Thus, researchers may want to allow the perturbation parameter to aﬀect on
the weight matrix. Indeed such analysis is conducted in the online appendix of AGS (Section 2.1),
where Ŵ is replaced with Ŵ ( ) and the corresponding sample sensitivity is derived. However,ˆ ✓ˆ1 ) = 1 Pn g(Xi , ✓ˆ1 )g(Xi , ✓ˆ1 )0 ,a typical form of the GMM weight matrix is the inverse of ⌦(ni=1

where g(X, ✓) is the moment functions for observables X and ✓ˆ1 is a first stage estimate, andthus it is not clear how the perturbation parameter in the sample moment ĝ(✓, ) will aﬀect onˆ ✓ˆ1 ) 1 and ✓ˆ1 . Second, for the GMM estimation, it is also common to employ (the inverse of)⌦(1˜ ✓ˆ1 ) = 1
the centered variance ⌦(
n

Pn

ˆ

ĝ(✓ˆ1 )}{g(Xi , ✓ˆ1 ) ĝ(✓ˆ1 )}0 as the weight matrix.
˜ ✓ˆ1 ) 1 is more stable than ⌦(ˆ ✓ˆ1 ) 1
Indeed Hall (2000) argued that this centered weight matrix ⌦(i=1 {g(Xi , ✓1 )under global misspecification. Although AGS’s sensitivity focuses on local misspecification, it
would be useful to see the eﬀect of centering in the weight matrix for the sample sensitivity.
See also Hall and Inoue (2003) for fundamental roles of the GMM weight matrix under global
misspecification. Third, another popular choice of the GMM weight matrix is the one for thecontinuous updating GMM estimator (Hansen, Heaton and Yaron, 1996). In this case, the weightmatrix involves the unknown parameter ✓, say Ŵ (✓, ), and hence the framework of AGS needs
to be adjusted. Finally, in the applied econometrics literature, it is well known that the above
variants of the GMM estimators show rather diﬀerent finite sample properties (see, e.g., thespecial issue of the Journal of Business & Economic Statistics, Vol. 14, No. 3). Therefore, it
is of interest to provide bias measures which are also sensitive to diﬀerent choices of the weightmatrix and estimation method.
In this paper, we address the above issues by considering a more specific form of the local
perturbation, that is g(Xi , ✓, ) = g(Xi , ✓) + ⌘ for each i. Although this perturbation is lessgeneral than the perturbed moment ĝ(✓, ) studied by AGS, it allows us to derive explicit bias
formulae for the above variants of the GMM estimators. Also the derived bias estimators are
all consistent for AGS’s (population) bias ⇤⌘. Therefore, our bias estimates may be consideredas characterizations of the higher-order terms to estimate ⇤⌘ under the particular form of local
perturbation, and provide useful complements to AGS’s sensitivity analysis for researchers who
are also concerned with the choice of the GMM estimation method in practice.
This paper is organized as follows. Section 2 presents our main result on the bias measuresof the diﬀerent GMM estimates. Section 3 illustrates our methodology by a consumption based
asset pricing model using Japanese data.
2. Main results
We first fix the notation. Let gi (✓) = g(Xi , ✓) be a d-dimensional vector of moment functions
for a k-dimensional vector of parameters ✓ with d > k, and data {Xi }ni=1 . Define the samplePˆ
moment function ĝ(✓) = n 1 ni=1 gi (✓), uncentered sample second moments matrix ⌦(✓)
=
P
P
n
n
1
0
1
˜
n
gi (✓)gi (✓) , and centered sample second moments matrix ⌦(✓) = n{gi (✓)
i=1ĝ(✓)}{gi (✓)i=1ĝ(✓)}0 . We consider the GMM estimators solving
min ĝ(✓)0 W ĝ(✓),✓(1)

with five diﬀerent choices for W :(i): Ŵ which does not involve gi (✓) (first stage GMM, denoted by ✓ˆ1 ),ˆ ✓ˆ1 ) (two-step GMM, denoted by ✓ˆ2 ),(ii): ⌦(˜ ✓ˆ1 ) (two-step GMM with centering, denoted by ✓˜2 ),(iii): ⌦(ˆ(iv): ⌦(✓)
(continuous updating GMM, denoted by ✓ˆc ),˜(v): ⌦(✓)(continuous updating GMM with centering, denoted by ✓˜c ).
If the model is correctly specified (i.e., E[gi (✓0 )] = 0 for a unique true value ✓0 ), all theseestimators are consistent and asymptotically normal under certain regularity conditions, and the2

estimators (ii)-(v) are asymptotically eﬃcient. This paper is concerned with bias measures ofthese estimators against local misspecification from the model assumption.
In particular, we consider the following perturbations for the moment functions(2)

gi (✓, ) = gi (✓) + ⌘,for each i = 1, . . . , n, whereis a scalar perturbation tuning parameter and ⌘ = plimn!1 ĝ(✓0 ) isa d-dimensional constant vector. Researchers can choose ⌘ for each specific scenario of violationof the identifying assumptions.
To begin with, let us introduce the bias formula for the first stage GMM estimate ✓ˆ1 derived
P
in Section 5 of AGS. Let ĝ(✓, ) = n 1 n gi (✓, ) and ✓ˆ1 ( ) = arg min✓ ĝ(✓, )0 Ŵ ĝ(✓, ) be thei=1

first stage GMM estimator using the perturbed moment function gi (✓, ). Then we obtain (see,Section 2.1 in the online appendix of AGS)

@ ✓ˆ1 (0)= {Ĝ(✓ˆ1 )0 Ŵ Ĝ(✓ˆ1 ) + Â1 } 1 Ĝ(✓ˆ1 )0 Ŵ ⌘,
(3)
@
⇣
⌘0
⇣
⌘0
@ Ĝ(✓ˆ1 )@ Ĝ(✓ˆ1 )ˆ
where Ĝ(✓) = @ĝ(✓)
,
Â
=
Ŵ
ĝ(
✓
),
.
.
.
,
Ŵ ĝ(✓ˆ1 ) , and the matrix Ĝ(✓ˆ1 )0 Ŵ Ĝ(✓ˆ1 )+1
1@✓ 0@✓ (1)@✓ (k)
Â1 is assumed to be non-singular.

As clarified in AGS, the component{Ĝ(✓ˆ1 )0 Ŵ Ĝ(✓ˆ1 ) + Â1 } 1 Ĝ(✓ˆ1 )0 Ŵ in (3) is a consistentestimator for the sensitivity ⇤. However, as shown in AGS, if the weight also depends on , thenˆ

the derivative @ ✓@1 (0) may not take a linear form in ⌘ without an intercept. Thus, we hereafter
present analogous derivatives for the GMM estimators (ii)-(v).
PˆDefine ⌦(✓,
) = n 1 n gi (✓, )gi (✓, )0 , andi=1

ˆ ✓ˆ1 ( ), ) 1 ĝ(✓, ),✓ˆ2 ( ) = arg min ĝ(✓, )0 ⌦(✓ˆ✓ˆc ( ) = arg min ĝ(✓, )0 ⌦(✓,
) 1 ĝ(✓, ),˜ ✓ˆ1 ( )) 1 ĝ(✓, ),✓˜2 ( ) = arg min ĝ(✓, )0 ⌦(✓1˜✓˜c ( ) = arg min ĝ(✓, )0 ⌦(✓)
ĝ(✓, ).

✓✓

˜
Note that the weight matrices of ✓˜2 ( ) and ✓˜c ( ) are written by using ⌦(✓)
(because gi (✓, )ĝ(✓, ) = gi (✓)ĝ(✓)). The derivatives of these estimators with respect toare summarized in

the following proposition.

Proposition. Under the setup of this section, it holds(i): Bias of the first stage GMM:
@ ✓ˆ1 (0)=@{Ĝ(✓ˆ1 )0 Ŵ Ĝ(✓ˆ1 ) + Â1 } 1 Ĝ(✓ˆ1 )0 Ŵ ⌘.

(ii): Bias of the two step GMM:
@ ✓ˆ2 (0)=@ˆ ✓ˆ1 ) 1 Ĝ(✓ˆ2 ) + Â2 } 1 {Ĝ(✓ˆ2 )0 ⌦(ˆ ✓ˆ1 ) 1 ⌘ + B̂2 },{Ĝ(✓ˆ2 )0 ⌦(3where
Â2 =
B̂2 =
Ĉ2 =

"

@ Ĝ(✓ˆ2 )
@✓(1)

!0

ˆ ✓ˆ1 )⌦(

1ĝ(✓ˆ2 ), . . . ,@ Ĝ(✓ˆ2 )
@✓(k)

!0

ˆ ✓ˆ1 )⌦(

1#ĝ(✓ˆ2 ) ,ˆ ✓ˆ1 ) 1 Ĉ2 ⌦(ˆ ✓ˆ1 ) 1 ĝ(✓ˆ2 ),Ĝ(✓ˆ2 )0 ⌦(
"
!
!0 #
n
ˆ1 (0)
ˆ1 (0)
1X@✓@✓
Gi (✓ˆ1 )gi (✓ˆ1 )0 + gi (✓ˆ1 ) Gi (✓ˆ1 )
n
@
@
i=1

+ĝ(✓ˆ1 )⌘ 0 + ⌘ĝ(✓ˆ1 )0 .

(iii): Bias of the two-step GMM with centering for the weight@ ✓˜2 (0)
=
@
where
Ã2 =
B̃2 =
C̃2 =

"

˜ ✓ˆ1 ) 1 Ĝ(✓˜2 ) + Ã2 } 1 {Ĝ(✓˜2 )0 ⌦(˜ ✓ˆ1 ) 1 ⌘ + B̃2 },{Ĝ(✓˜2 )0 ⌦(@ Ĝ(✓˜2 )
@✓(1)

!0

˜ ✓ˆ1 ) 1 ĝ(✓˜2 ), . . . ,⌦(@ Ĝ(✓˜2 )
@✓(k)

!0

#

˜ ✓ˆ1 ) 1 ĝ(✓˜2 ) ,⌦(˜ ✓ˆ1 ) 1 C̃2 ⌦(˜ ✓ˆ1 ) 1 ĝ(✓˜2 ),Ĝ(✓˜2 )0 ⌦(
"
!
!0 #
n
ˆ1 (0)
ˆ1 (0)
1X@✓@✓
0
Gi (✓ˆ1 )gi (✓ˆ1 ) + gi (✓ˆ1 ) Gi (✓ˆ1 )
n
@
@
i=1
!
!0
ˆ1 (0)
ˆ1 (0)@✓@✓
0Ĝ(✓ˆ1 )ĝ(✓ˆ1 ) + ĝ(✓ˆ1 ) Ĝ(✓ˆ1 ).
@@(iv): Bias for the continuous updating GMM:
@ ✓ˆc (0)
=
@
where

h

{Ĝ(✓ˆc )0

Âc =

"

ˆ ✓ˆc )0 }⌦(ˆ ✓ˆc ) 1 Ĝ(✓ˆc ) + Âc⇠(@G(✓ˆc )0@✓(1)ˆ ✓ˆc )0
@ ⇠(
@✓(1)

{Ĝ(✓ˆc )0@G(✓ˆc )0@✓(k)ˆ ✓ˆc )0
@ ⇠(
@✓(k)

{Ĝ(✓ˆc )0

i 1[{Ĝ(✓ˆc )0ˆ ✓ˆc )0 }⌦(ˆ ✓ˆc ) 1 (⌘⇠(ˆ ˆˆ ✓ˆc )0 }⌦(ˆ ✓ˆc ) 1 @ ⌦(✓c ) ⌦(ˆ ✓ˆc ) 1 ĝ(✓ˆc ), . . .
⇠(
@✓(1)
!
#
ˆ ✓ˆc )@⌦(ˆ ✓ˆc )0 }⌦(ˆ ✓ˆc ) 1ˆ ✓ˆc ) 1 ĝ(✓ˆc )
⇠(
⌦(
@✓(1)

ˆ 0)0
@ ⇠(✓,ˆ ✓ˆc ) 1 ĝ(✓ˆc ),ˆ ✓ˆc ) 1 ĝ(✓ˆc ),⌦(D̂c = {ĝ(✓ˆc )⌘ 0 + ⌘ĝ(✓ˆc )0 }⌦(@2ˆ ✓ˆc ) 3ˆ ✓ˆc ) 1 @ ⌦(ĝ(✓ˆc )0 ⌦(nn@✓ (1) 7ˆ ✓ˆc )6
@ ⌦(1 X ˆ (j) ˆ 0 1 X (j) ˆ
..
6
7
= 4
,
=
g
(
✓
)G
(
✓
)
+
Gi (✓c )gi (✓ˆc )0 ,
i
c
c
.
i
5
(j)
n
n
@✓
ˆ ✓ˆc )i=1
i=1
ˆ ✓ˆc ) 1 @ ⌦(ĝ(✓ˆc )0 ⌦(@✓ (k)⇣⌘ˆ ˆˆ ✓ˆc ) 1 {ĝ(✓ˆc )⌘ 0 + ⌘ĝ(✓ˆc )0 } ⌦(ˆ ✓ˆc ) 1 @ ⌦(✓c )=⌘ 0 ĝ(✓ˆc )0 ⌦(
@✓(j)
!
n
n
X
X
1
1
ˆ ✓ˆc ) 1+ĝ(✓ˆc )0 ⌦(⌘ Ĝ(j) (✓ˆc )0 +Ĝ(j) (✓ˆc )⌘ 0 ,
n
n

B̂c =

ˆ ✓ˆc )0⇠(@ ⇠ˆ(j) (✓, 0)0@D̂c ) + B̂c ],

i=1

i=1

where the superscript j refers to the j-th row of an object.

4

(v): Bias of the continuous updating GMM with centering for the weight:h@ ✓˜c (0)
=
@
where
Ãc =

{Ĝ(✓˜c )0"˜ ✓˜c )0 }⌦(˜ ✓˜c ) 1 Ĝ(✓˜c ) + Ãc⇠(@G(✓˜c )0@✓(1)

˜ ✓˜c )0
@ ⇠(
@✓(1)

@G(✓˜c )0@✓(k)˜ ✓˜c )0
@ ⇠(
@✓(k)

{Ĝ(✓˜c ){Ĝ(✓˜c )0˜ ✓˜c , 0)0@ ⇠(˜ ✓˜c ) 1 ĝ(✓˜c ),⌦(@2
˜ ✓˜c ) 3˜ ✓˜c ) 1 @ ⌦(ĝ(✓˜c )0 ⌦(@✓ (1) 7
6
..
6
7,
= 4
.
5
˜ ✓˜c )
@ ⌦(
0
1
˜
˜
˜
ĝ(✓c ) ⌦(✓c ) @✓(k)

0

i 1h

i
˜ ✓˜c )0 }⌦(˜ ✓˜c ) 1 ⌘ + B̃c ,⇠({Ĝ(✓˜c )0˜ ✓˜c ) }⌦(ˆ ✓˜c )⇠(0˜ ˜

1 @ ⌦(✓c )
@✓(1)

˜ ˜
˜ ✓˜c )0 }⌦(ˆ ✓˜c ) 1 @ ⌦(✓c )
⇠(
@✓(k)

!

!

˜ ✓˜c ) 1 ĝ(✓˜c ), . . .
⌦(#˜ ✓˜c ) 1 ĝ(✓˜c ) ,
⌦(

B̃c =

˜ ✓˜c )0⇠(@ ⇠˜(j) (✓˜c , 0)0@˜ ✓˜c ) 1
= ⌘ 0 ⌦(˜ ✓˜c )ˆ ✓˜c )
@ ⌦(
@ ⌦(
=
@✓(j)
@✓(j)

Ĝ(j) (✓˜c )ĝ(✓˜c )0ĝ(✓˜c )Ĝ(j) (✓˜c )0 ,˜ ✓˜c )
@ ⌦(
@✓(j)

n

˜ ✓˜c ) 1+ĝ(✓˜c )0 ⌦(n

1 X (j) ˜ 0 1 X (j) ˜ 0
Gi (✓c )⌘ +⌘Gi (✓c )nn
i=1Ĝ(j) (✓˜c )⌘ 0i=1

⌘ Ĝ(j) (✓˜c )0!

The derivations are tedious but analogous to the one for the online appendix Proposition 2 of
AGS by using the implicit function theorem. We note that under local perturbations in the sensepˆ !
of the online appendix Proposition 1 of AGS, it holds ĝ(✓)
0 for ✓ˆ = ✓ˆ2 , ✓˜2 , ✓ˆc , ✓˜c , and thus allthese bias terms converge to ⇤⌘ under mild regularity conditions that guarantee the convergenceˆ˜of the sample moments, such as ⌦(·),⌦(·),
Ĝ(·), and their derivatives.

The derivatives for the two-step and continuous updating GMM estimators take somewhat
analogous forms. For example, the terms Â2 , Ã2 , Âc , Ãc emerge by the same reason as Â1 in thefirst stage GMM, i.e., the second term in the derivative of the first-order condition@{Ĝ(✓ˆ1 ( ), )0 Ŵ ĝ(✓ˆ1 ( ), )}
@
⇢
⇢
@ ˆ
@
0
ˆ
= Ĝ(✓1 ( ), ) Ŵĝ(✓1 ( ), ) +Ĝ(✓ˆ1 ( ), )@@0Ŵ ĝ(✓ˆ1 ( ), ).

On the other hand, the terms B̂2 , B̃2 , B̂c , B̃c are additional terms due to dependence of the weightmatrices on . Observe that the terms Â2 , Ã2 , Âc , Ãc and B̂2 , B̃2 , B̂c , B̃c involve the componentspˆ for the corresponding estimators. Therefore, under local perturbations with ĝ(✓)
ˆ !
ĝ(✓)
0, theseterms are asymptotically negligible. However, as is illustrated in the next section, these terms areuseful to distinguish bias properties of the diﬀerent GMM estimation methods in finite samples.
3. Illustration
In this section, we apply our proposed sensitivity measures to a consumption based asset
pricing model. Our illustrating example follows Hansen and Singleton (1982). Consider a representative investor who maximizes his/her expected discounted utility
5

,

max

{Ct+i }1
i=1

E

" 1
X

i Ct+i

1

#

It ,

i=0

where Ct denotes consumption at time t, and It denotes the information set of the investor attime t. There is an asset available to the investor, and we denote the net rate of return on theasset at time t as Rt . Then the first order condition impliesE"(1 + Rt )

✓

Ct+1
Ct

◆

1

1 It

#

= 0.

Let Zt ✓ It be a subset of the information set It . Our parameters of interest ✓ = ( , )0 are thesubjective discount factor

and the coeﬃcient in relative risk aversion . These parameters canbe estimated by using the moment condition E[gt (✓0 )] = 0, wheregt (✓) = Zt(

(1 + Rt )

✓

Ct+1
Ct

◆

1

)

1 .

(4)

For our illustration, we use data on the consumption growth rate and the average rate of
return on stocks in Japan. Using the Nikkei 225 stock price index and the seasonally adjusted
nondurable consumption data from Yahoo Finance and Statistics Bureau of Japan, we construct
the data on the consumption growth rate and the average rate of return on stocks. These timeseries data are adjusted for inflation by the consumer price index. The time period is fromJanuary 1981 to June 2016.
In the moment function (4), there are many possible choices of instruments Zt . Hansen andSingleton (1982) used lags of consumption growth and lags of the rate of return on assets, andestimated the parameters with a number of diﬀerent choices of instruments. Here we focus on one
set of instruments to simplify the presentation, that is Zt = (1, Ct /Ct 1 , Ct 1 /Ct 2 , Rt 1 , Rt 2 )0 .
The parameter estimates are summarized in Table 1. The obtained GMM estimates are reasonable in the empirical literature of this class of consumption based asset pricing model. The
estimate offor the first stage GMM with W = I takes a lower value compared to the other

GMM estimates.
Table 1. Parameter estimatesŴIˆ ✓ˆ1 )⌦(˜ ✓ˆ1 )
⌦(
ˆ
⌦(✓)
˜
⌦(✓)

Coeﬃcient in relative Discount factor
risk aversion
0.61870.99701.01100.99661.0116
0.99661.00300.99661.00300.9966

The present model assumes time-separable utility, and thus the current consumption decision
is independent from the past consumption path. There are some situations where this assumption
is not appropriate; in models with habit formation, the current consumption choice may dependon the agent’s own past consumption through the habit stock, for example. Based on this6

background, we consider the perturbed model gi (✓, ) = gi (✓) + ⌘. If the model is correctly
specified, ⌘ should equal to zero vector, which corresponds to the original model assumption.
Alternatively, a researcher may have doubts about the time-separability assumption and believethat lags of consumption growth cannot be valid instruments. Under this alternative view, thesecond and third elements of ⌘ should be non-zero (with the other elements being zeros).
ˆ
Under such local violations of identifying assumptions, Table 2 reports the bias estimates ⇤⌘based on AGS’s proposed measure of sensitivity
ˆ 0 Ŵ Ĝ(✓)}ˆ 1 Ĝ(✓)ˆ 0 Ŵ ,ˆ = {Ĝ(✓)⇤

with diﬀerent GMM weight matrices for Ŵ . Except for the case of Ŵ = I, the bias estimatesˆ are quite similar across diﬀerent choices of weight matrices. The local violation with ⌘ =⇤⌘(0, 0.01, 0.01, 0, 0)0 implies that the estimates ofbias forare biased upward by roughly 17%, and thatis negligibly small. For Ŵ = I, although the estimated values are diﬀerent, theproportions of the biases are comparable to the other cases. In this example, the bias estimatesˆ by AGS are not sensitive to the choice of the weights for the two-stage or continuous updating
⇤⌘
GMM.

ˆ
Table 2. Bias in parameter estimates measured with ⇤⌘Ŵ and ✓ˆI and ✓ˆ1
ˆ ✓ˆ1 ) and ✓ˆ2⌦(
˜ ✓ˆ1 ) and ✓˜2
⌦(
ˆ
⌦(✓)
and ✓ˆc˜⌦(✓) and ✓˜cBias in
Bias incoeﬃcient in relative discount factor
risk aversion
0.009113
1.2425 ⇥ 10 6
0.016944
5.0308 ⇥ 10 6
0.016962
5.0082 ⇥ 10 6
0.016466
5.1106 ⇥ 10 6
0.016465
5.11 ⇥ 10 6

Notes: This table reports bias in parameter estimates measured with ⇤⌘. We consider a local violation of identifying
assumptions with ⌘ = (0, 0.01, 0.01, 0, 0)0 .

In Table 3, we now report the estimates of the proposed bias measures. For Ŵ = I, theˆ with Ŵ = I. As expected, our bias measures arebias estimates are very similar to AGS’s ⇤⌘sensitive to the choice of the GMM weight. Although the signs of the biases are same as AGS’s
ˆ their values are diﬀerent. For example, if we focus on , the bias of the continuous updating⇤⌘,
ˆ and takes the smallest value among the
GMM with centering (i.e., @ ✓˜c (0)/@ ) is similar to ⇤⌘,two step and continuous updating GMM methods. On the other hand, the bias of the two step
GMM (i.e., @ ✓ˆ2 (0)/@ ) implies a larger upward bias, around 58% of the estimate of . Also,
the centering substantially reduces the bias of @ ✓ˆ2 (0)/@ . In this example, our bias estimatesindicate that the two step GMM methods are more sensitive due to the sensitivity in the firststep estimation, and that the continuous updating GMM tends to be less sensitive.
7ˆ
Table 3. Bias in parameter estimates measured with @ ✓(0)/@ˆ@ ✓(0)@

Bias in
Bias incoeﬃcient in relative discount factor

✓ˆ1
✓ˆ2
✓˜2
✓ˆc
✓˜c

risk aversion
0.0092349
0.582070.130850.024685
0.018237

1.1902 ⇥ 10 6
4.8183 ⇥ 10 3
6.0069 ⇥ 10 42.7922 ⇥ 10 7
4.0453 ⇥ 10 6ˆNotes: This table reports bias in parameter estimates measured with @ ✓(0)/@. We consider a local violation ofidentifying assumptions with ⌘ = (0, 0.01, 0.01, 0, 0)0 .

References[1] Andrews, I., Gentzkow, M. and J. M. Shapiro (2017) Measuring the sensitivity of parameter estimates toestimation moments, Quarterly Journal of Economics, 132, 1553-1592.
[2] Hall, A. R. (2000) Covariance matrix estimation and the power of the overidentifying restrictions test, Econometrica, 68, 1517-1527.
[3] Hall, A. R. and A. Inoue (2003) The large sample behaviour of the generalized method of moments estimatorin misspecified models, Journal of Econometrics, 114, 361-394.
[4] Hansen, L. P., Heaton, J. and A. Yaron (1996) Finite-sample properties of some alternative GMM estimators,
Journal of Business & Economic Statistics, 14, 262-280.
[5] Hansen, L. P. and K. J. Singleton (1982) Generalized instrumental variables estimation of nonlinear rational
expectations models, Econometrica, 50, 1269-1286.
Faculty of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan.
Email address: rikutoo0802@keio.jp
Department of Economics, London School of Economics, Houghton Street, London, WC2A
2AE, UK, and Keio Economic Observatory (KEO), 2-15-45 Mita, Minato-ku, Tokyo 108-8345,
Japan.
Email address: t.otsu@lse.ac.uk

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