Consistent Parameter Estimation for
Conditional Moment Restrictions
Shih-Hsun Hsu
Department of Economics
National Taiwan University
Chung-Ming Kuan
Institute of Economics
Academia Sinica
Preliminary; Please do not cite or quote.
† Author for correspondence: Chung-Ming Kuan, Institute of Economics, Academia Sinica, Taipei 115,
Taiwan; [email protected].
†† The research support from the National Science Council of the Republic of China (NSC95-2415-H-
001-034) is gratefully acknowledged.
Abstract
In estimating conditional moment restrictions, a well known difficulty is that the
estimator based on a set of implied unconditional moments may lose its consistency
when the parameters are not globally identified. In this paper, we consider a continuum
of unconditional moments that are equivalent to the postulated conditional moments and
project them along the exponential Fourier series. By constructing an objective function
using these Fourier coefficients, a consistent GMM estimator is readily obtained. Our
estimator compares favorably with that of Domınguez and Lobato (2004, Econometrica)
in that our method utilizes the full continuum of unconditional moments. We establish
the asymptotic properties of the proposed estimator. It is also shown that an efficient
estimator can be derived from the proposed consistent estimator via a Newton-Raphson
step. Our simulations demonstrate that the proposed estimator outperforms that of
Domınguez and Lobato (2004) in terms of bias, standard error and mean squared error.
JEL classification: C12, C22
Keywords: conditional moment restrictions, Fourier coefficients, generically compre-
hensive function, GMM, parameter identification
1 Introduction
Many economic and econometric models can be characterized in terms of conditional mo-
ment restrictions. Estimating the parameters in such restrictions thus plays a major role
in empirical studies. A typical estimation approach is to find a finite set of unconditional
moment restrictions implied by the conditional moment restrictions and apply a suitable
estimation method, such as the generalized method of moment (GMM) of Hansen (1982)
or the empirical likelihood method of Qin and Lawless (1994). The instrumental-variable
estimation method for regression models is a leading example. Yet there also exist es-
timation methods based directly on the conditional moments, e.g., Ai and Chen (2003)
and Kitamura, Tripathi, and Ahn (2004).
A crucial assumption in the former apporach is that the parameters in the conditional
moment restrictions can be gloabally identified by the implied, unconditional restrictions.
With this assumption, the consistency of an estimator can be easily established under
suitable regularity conditions. Much research interest therefore focuses on estimator
efficiency, e.g., Chamberlain (1987), Newey (1990, 1993), Carrasco and Florens (2000),
and Donald, Imbens, and Newey (2003). Domınguez and Lobato (2004), however, pointed
out that the assumption of global identifiability may fail to hold in nonlinear models.
It is well known that, except in some special cases, conditional moment restrictions
are equivalent to infinitely many unconditional restrictions in general (Bierens, 1982,
1990; Chamberlain, 1987; Stinchcombe and White, 1998). The identification problem
may arise when one arbitrarily selects a finite number of unconditional moments for
estimation. Domınguez and Lobato (2004) also demonstrated that this may happen
even when the unconditional moments are based on the so-called “optimal” instruments.
When the parameters of interest are not identified, the corresponding (GMM) estimator
is inconsistent.
To circumvent the potential non-identification and inconsistency problems, Domınguez
and Lobato (2004) proposed a different approach to estimating conditional moment re-
strictions. They considered a continuum of unconditional moment restrictions that are
equivalent to the conditional restrictions, where the unconditional moments depend on
the “instruments” generated from the indicator function. There are some disadvantages
of their approach. In view of Stinchcombe and White (1998), any “generically compre-
hensively revealing” function, such as the exponential and logistic functions, can also
induce equivalent, unconditional moment restrictions. Compared with these functions,
1
the indicator function, which takes only the values one and zero, may not well present
the information in the conditioning variables. Moreover, Domınguez and Lobato (2004)
did not utilize the full continuum of unconditional restrictions but included only a finite
number of them in estimation. This may result in further efficiency loss, as argued by
Carrasco and Florens (2000).
In this paper, we propose a new approach for consistent estimation of conditional
moment restrictions. We also consider a continuum of unconditional moments that are
equivalent to the postulated conditional moments. Instead of dealing with these mo-
ments directly, we project them along the exponential Fourier series. By constructing an
objective function using these Fourier coefficients, a consistent GMM estimator is readily
obtained. Because all unconditional moments are incorporated into each Fourier coef-
ficient, our estimation method in effect utilizes all possible information. Moreover, the
“remote” Fourier coefficients provide little information on the parameters of interest and
may be excluded from the objective function. The objective function is hence relatively
simple while still involving the full continuum of moment conditions. By choosing the ex-
ponential function to generate “instruments,” the resulting unconditional moments and
the Fourier coefficients have analytic forms. These properties greatly facilitate estima-
tion in practice. Compared with Carrasco and Florens (2000), our method is also simpler
because we do not require preliminary estimation of the eigenvalues and eigen-functions
of a covariance operator, nor do we need to determine a regularization parameter.
In this paper, we show that under quite general conditions, the proposed estimator is
consistent and asymptotically normally distributed. It is also shown that an efficient esti-
mator is readily obtained from the proposed consistent estimator via a Newton-Raphson
step, as in, e.g., Newey (1990) and Domınguez and Lobato (2004). Our simulations
demonstrate that, under various settings, the proposed estimator compares favorably
with that of Domınguez and Lobato (2004) in terms of bias, standard error and mean
squared error.
This paper is organized as follows. The proposed approach is introduced in Section
2. The asymptotic properties of the proposed estimator are established in Section 3.
We report Monte Carlo simulation results in Section 4 and discuss some extensions of
the proposed method. Section 6 concludes this paper. All proofs are deferred to the
Appendix.
2
2 The Proposed Estimation Approach
We are primarily interested in estimating θo, the q × 1 vector of unknown parameters,
in the following conditional moment restriction:
IE[h(Y ,θo)|X] = 0, w.p.1, (1)
where h is a p× 1 vector of functions of the variable Y (r× 1) and the parameter vector
θ (q × 1), and X is an m× 1 vector of conditioning variables. It is well known that (1)
is equivalent to the unconditional moment restriction:
IE[h(Y ,θo)f(X)] = 0, (2)
for all measurable function f . Here, each f(X) may be interpreted as an “instrument”
that helps to identify θo. Clearly, estimation based on all measurable functions of X
is practically infeasible. It is thus typical to select a finite number of unconditional
moments in (2) and estimate θo using, say, the GMM method. However, Domınguez
and Lobato (2004) showed that if these restrictions are not chosen properly, there is no
guarantee that θo can be identified globally. This identification problem in turn renders
the associated GMM estimator inconsistent.
In light of Bierens (1982, 1990) and Stinchcombe and White (1998), the parameters
of interest can be identified when the unconditional moments are based on a class of
instruments that can span an appropriate space of functions of X. Domınguez and
Lobato (2004) chose the indicator function to generate a continuum of instruments in
(2), i.e.,
f(X) = 1(X ≤ τ ) =m∏
j=1
1(Xj ≤ τj),
with 1(B) the indicator function of the event B. The indicator function is not the only
choice of functions that can generate the desired instruments; any “generically com-
prehensively revealing” function as defined in Stinchcombe and White (1998) will also
do. Letting A denote the affine transformation such that A(X, τ ) = τ0 +∑m
j=1 Xjτj .
Stinchcombe and White (1998) showed that, for a real analytic function G that is not a
polynomial, (1) is equivalent to
IE[h(Y ,θo)G
(A(X, τ )
)]= 0, for almost all τ ∈ T ⊂ Rm+1, (3)
3
where T has a nonempty interior and could be a small set in Rm+1. Observe that (3)
is a continuum of unconditional moment restrictions indexed by the parameter τ in T ,
cf. (2). In particular, the function G may be the exponential function (Bierens, 1982,
1990) or the logistic function (White, 1989).
In this paper, we propose an alternative to constructing a consistent estimator of θo
that improves on the approach of Domınguez and Lobato (2004). We first consider the
case where X is univariate, denoted as X. Without loss of generality, we transform X
to a bounded random variable using a generalized logistic mapping:
X =exp (X)
c + exp (X), c > 0,
so that 0 < X < 1/c. Because this mapping is one-to-one, the conditional moment
restriction (1) is equivalent to
IE[h(Y ,θo)|X] = 0, w.p.1; (4)
see e.g., Bierens (1994, Theorem 3.2.1). For a real function G that is analytic but not
a polynomial, Stinchcombe and White (1998) showed that the unonditional restriction
equivalent to (4) is
IE[h(Y ,θo)G
(A(X, τ )
)]= 0, for almost all τ ∈ T ⊂ R2. (5)
Then, θo can be globally identified via an L2-norm of (5):
θo = argminθ ∈ Θ
∥∥∥IE[h(Y ,θ)G(A(X, ·)
)]∥∥∥2
L2
= argminθ ∈ Θ
∫T
∣∣∣IE[h(Y ,θo)G(A(X, τ )
)]∣∣∣2 dP (τ ),(6)
where P (τ ) is a distribution function of τ in T .
For the choice of G, we set G(A(X, τ )
)= exp
(A(X, τ )
), as in Bierens (1982, 1990).
This choice has some advantages over the indicator function. First, while the indicator
function takes only the values one and zero, the exponential function is more flexible
and hence may better present the information in the conditioning variables. That is,
the exponential function may generate better instruments for identifying θo. Second,
the exponential function is “generically comprehensively revealing,” but the indicator
function is not. By the “genericity” property, we can focus on the instruments indexed
by τ in a small set T . By contrast, one must deal with the instruments generated
4
by the indicator function for all possible τ ∈ R. Moreover, in the class of generically
comprehensively revealing function, exp(A(X, τ )
)and exp(τX) only differ by a constant
and hence play the same role in function approximation (Stinchcombe and White, 1998).
Choosing the exponential function thus reduces the dimension of integration in (6) by
one, i.e., T ⊂ R. As far as implementation is concerned, the exponential function results
in an analytic form for the objective function, which greatly facilitates estimation in
practice.
Apart from the choice of the instrument-generating function G, our approach dif-
fers from that of Domınguez and Lobato (2004) in an important respect. Instead of
directly working on the unconditional moments, we project IE[h(Y ,θ) exp(τX)] along
the exponential Fourier series:
IE[h(Y ,θ) exp(τX)
]=
∞∑k=−∞
Ck(θ) exp (ikτ),
where Ck(θ) is a Fourier coefficient such that
Ck(θ) =12π
∫ π
−πIE[h(Y ,θ) exp(τX)
]exp (−ikτ) dτ
=12π
IE[h(Y ,θ)
∫ π
−πexp(τX) exp(−ikτ) dτ
], k = 0,±1,±2, . . .
Note that each Ck(θ) incorporates the continuum of the original instruments exp(τX)
into a new instrument with the following analytic form:
ϕk(X) =1√2π
∫ π
−πexp(τX) exp(−ikτ) dτ
=1√2π
(−1)k · [exp (πX)− exp(−πX)]
(X − ik)
=(−1)k · 2 sinh (πX)√
2π(X − ik),
where sinh(w) = (exp (w)− exp (−w))/2.
Setting dP (τ) = dτ and T = [−π, π] in (6), we have from Parseval’s Theorem that∥∥∥IE[h(Y ,θ)G(·X)]∥∥∥2
L2
= 2π
∞∑k=−∞
∣∣Ck(θ)∣∣2 =
∞∑k=−∞
∣∣∣IE[h(Y ,θ)ϕk(X)]∣∣∣2 .
It follows from (6) that θo can be globally identified as
θo = argminθ ∈ Θ
∞∑k=−∞
∣∣∣IE[h(Y ,θ)ϕk(X)]∣∣∣2 . (7)
5
That is, θo is identified using a countable collection of unconditional moments.
By Bessel’s inequality, Ck(θ) → 0 as |k| tends to infinity, so that the moment condition
IE[h(Y ,θ)ϕk(X)
]becomes less informative for identifying θo when |k| is large. This
suggests that there is no need to include all the Fourier coefficients in estimating θo. As
such, we may replace IE[h(Y ,θ)ϕk(X)
]with its sample counterpart and compute an
estimator of θo as
θ(KT) = argminθ ∈ Θ
KT∑k=−KT
∣∣∣∣∣ 1TT∑
t=1
h(yt,θ)ϕk(xt)
∣∣∣∣∣2
= argminθ ∈ Θ
KT∑k=−KT
∣∣∣∣∣ 1TT∑
t=1
h(yt,θ)(−1)k · 2 sinh (πxt)√
2π(xt − ik)
∣∣∣∣∣2
,
(8)
where KT depends on T but is less than T , and yt and xt are the realizations of Y and
X, respectively. This is in fact a GMM estimator based on (2KT + 1) unconditional
moments and the identity weighting matrix. The estimator θ(KT) now can be computed
quite straightforwardly using a numerical optimization algorithm.
By choosing the indicator function to generate the desired instruments, Domınguez
and Lobato (2004) showed that θo can be identified as
θo = argminθ ∈ Θ
∥∥IE[h(Y ,θ)1(X ≤ ·)]∥∥2
L2
= argminθ ∈ Θ
∫R
∣∣IE[h(Y ,θ)1(X ≤ τ)]∣∣2 dP (τ),
where P (τ) is a distribution function of τ ∈ R, cf. (6) and (7). Setting P (τ) as PX(τ),
the distribution function of X, it is easy to obtain the sample counterpart of the L2-norm
above. The estimator proposed by Dimınguez and Lobato (2004) is then
θDL(T ) = argminθ ∈ Θ
T∑k=1
(1T
T∑t=1
h(yt,θ)1(xt ≤ τk)
)2
, (9)
where τk = xk, k = 1, . . . , T , are just the sample observations. This is also a GMM
estimator based on T unconditional moments induced by the indicator function. Note
that when PX(τ) is used in the L2-norm, θDL(T ) utilizes only a finite number of the
unconditional moments. By contrast, the proposed estimator (8) depends on a few new
instruments ϕk, yet each ϕk utlizes the full continuum of the original instruments required
for identifying θo.
6
Carrasco and Florens (2000) also proposed an efficient GMM estimation method
based on a continuum of unconditional moments. Adapting to the current context, their
approach amounts to projecting the continuum of the induced unconditional moments
along the eigen-functions of a covariance operator. This approach thus requires prelimi-
nary estimation of a covariance operator and its eigenvalues and eigen-functions. Also, to
ensure the invertibility of the estimated covariance operator, a regularization parameter
must be set by researchers. As a result, their method is not easy to implement and may
be arbitrary in practice.
3 Asymptotic Properties
In this section, we establish the asymptotic properties of the proposed estimator θ(KT).
For notation simplicity, we define µk(θ) = IE[h(Y ,θ)ϕk(X)] and
mT,k(θ) =1T
T∑t=1
h(yt,θ)ϕk(xt).
When integration and differentiation can be interchanged, let
∇θµk(θ) = IE[∇θ′h(Y ,θ)ϕk(X)
],
∇θmT,k(θ) =1T
T∑t=1
∇θ′h(yt,θ)ϕk(xt).
Our maintained assumption is: θo is the unique solution to IE[h(Y ,θ)|X] = 0.
We impose the following conditions that are quite standard in the GMM literature
and establish a lemma concerning the limiting behavior of the sum of |mT,k(θ)−µk(θ)|.
[A1] The observed data (y′t, xt)′, t = 1, . . . , T, are independent realizations of (Y ′, X)′.
[A2] Θ ∈ Rq is compact such that for each θ ∈ Θ, h(·,θ) is measurable, and for each
y ∈ Rr, h(y, ·) is continuous on Θ.
[A3] IE[supθ∈Θ |h(Y ,θ)|2] < ∞.
Lemma 3.1 Suppose that [A1]–[A3] hold. Then for any ε > 0,
IP
KT∑k=−KT
|mT,k(θ)− µk(θ)|2 ≥ ε
≤ ∆(2KT + 1)Tε
uniformly in θ ∈ Θ, where ∆ is some real number. 2
7
This result shows that, as long as KT = o(T ),
KT∑k=−KT
|mT,k(θ)− µk(θ)|2 IP−→ 0,
and hence KT →∞ implies, uniformly for all θ in Θ,
KT∑k=−KT
∣∣mT,k(θ)∣∣2 IP−→
∞∑k=−∞
|µk(θ)|2 . (10)
Given the maintained assumption, θo = argminθ∈Θ
∑∞k=−∞ |µk(θ)|2, and
θ(KT) = argminθ ∈ Θ
KT∑k=−KT
∣∣mT,k(θ)∣∣2 . (11)
The consistency result now follows from (10) and Theorem 2.1 of Newey and McFad-
den (1994).
Theorem 3.2 (Consistency) Given [A1]–[A3], suppose that KT →∞ as T →∞ such
that KT = o(T ). Then θ(KT) IP−→ θo. 2
In what follows, for a complex number (function) f , let f denote its complex conjugate
and Re(f) and Im(f) denote its real and imaginary parts, respectively. For the vector
of complex functions f , its complex conjugate, real part and imaginary part are defined
elementwise. For two p×1 vectors of functions f and g in L2[−π, π], their inner product
is defined as
〈f , g〉 =∫ π
−πf(τ)′g(τ) dτ =
∫ π
−π
p∑i=1
fi(τ)gi(τ) dτ.
Given (11), θ(KT) satisfies the first order condition:
0 =KT∑
k=−KT
∇θmT,k
(θ(KT)
)′mT,k
(θ(KT)
)+∇θmT,k
(θ(KT)
)′mT,k
(θ(KT)
)=
KT∑k=−KT
2 Re(∇θmT,k
(θ(KT)
)′mT,k
(θ(KT)
)).
A mean value expansion of mT,k
(θ(KT)
)about θo gives
mT,k
(θ(KT)
)= mT,k(θo) +∇θmT,k
(θ†)(
θ(KT)− θo
),
8
where θ† is on the line segment joining θ(KT) and θo which may be different for each
row of ∇θmT,k
(θ†). It follows that
KT∑k=−KT
2 Re(∇θmT,k
(θ(KT)
)′[mT,k(θo) +∇θmT,k
(θ†)(
θ(KT)− θo
)])= 0. (12)
Passing to the limit, we have, by the multiplication theorem (e.g., Stuart, 1961),
KT∑k=−KT
Re(∇θmT,k(θo)∇θmT,k(θo)
) IP−→∞∑
k=−∞∇θµk(θo)µk(θo),
KT∑k=−KT
Re(∇θmT,k(θo)
√TmT,k(θo)
)=
∞∑k=−∞
∇θµk(θo)√
TmT,k(θo) + oIP(1).
The normalized estimator can then be expressed as
√T(θ(KT)− θo
)= −
[ ∞∑k=−∞
∇θµk(θo)′∇θµk(θo)
]−1 [ ∞∑k=−∞
∇θµk(θo)′√T mT,k(θo)
]+ oIP(1).
(13)
Asymptotic normality then follows if the term in the first brackets is a finite valued, pos-
itive definite matrix, and the term in the second brackets obeys a central limit theorem.
We impose the following conditions.
[A4] θo is in the interior of Θ.
[A5] For each y, h(y, ·) is continuously differentiable in a neighborhood N of θo such
that IE[supθ∈N ‖∇θh(Y ,θ)‖2
]< ∞, where ‖ · ‖ is a matrix norm.
[A6] The q × q matrix Mq, with the (i, j)-th element⟨IE[∇θi
h(Y ,θo) exp(·X)], IE
[∇θj
h(Y ,θo) exp(·X)]⟩
,
is symmetric and positive definite.
[A7] As T goes to infinity,
1√T
T∑t=1
h(yt,θo) exp (·xt)D−→ Z,
9
where Z is a Gaussian random element in L2[−π, π] that has a zero mean and the
covariance operator K such that
Kf(τ ) = IE[⟨
h(Y ,θo) exp(·X), f⟩(
h(Y ,θo) exp(·X))]
,
for any f (p× 1) in L2[−π, π].
The covariance operator K in [A8] can be expressed as
Kf(τ ) =
(p∑
i=1
∫ π
−πκji(τ, s)fi(s) ds
)j=1,...,p
,
with the kernel
κji(τ, s) = IE[hj(Y ,θo) exp(τX)hi(Y ,θo) exp(sX)
], j, i = 1, . . . , p.
It can also be seen that 〈Z,f〉 has a normal distribution with mean zero and variance
〈Kf ,f〉. Note that the functional convergence result in [A8] is the same as the Assump-
tion 11 in Carrasco and Florens (2000). One may, of course, impose more primitive
conditions on h and the data that ensure such convergence result. For example, [A8]
would hold when h(yt,θo) exp(·xt) are i.i.d. with bounded second moment. More discus-
sions and results on functional convergence can be found in Chen and White (1998) and
Carrasco and Florens (2000).
Let Ωq denote a q × q matrix with the (i, j)-th element⟨IE[∇θi
h(Y ,θo) exp( ·X)], K IE
[∇θj
h(Y ,θo) exp( ·X)]⟩
.
We are ready to present the limiting distribution of the normalized estimator.
Theorem 3.3 (Asymptotic Normality) Given Assumptions [A1]–[A7], suppose that
KT →∞ as T →∞ such that KT = o(T 1/2). Then
√T(θ(KT)− θo
) D−→ N (0, V),
where V = M−1q ΩqM−1
q .
10
4 Simulations
In this section we examine the finite-sample performance of the proposed estimator via
Monte Carlo simulations. We compare the performance of the nonlinear least squares
(NLS) estimator:
θNLS = argminθ ∈ Θ
1T
T∑t=1
h(yt,θ)2.
the estimator of Domınguez and Lobato (2004): θDL in (9), and the proposed estimator:
θ(KT) with KT = 5 in (8). We consider three performance criteria: bias, standard error
(SE), and mean squared error (MSE). In all experiments, we compute the parameter es-
timate using the GAUSS optimization procedure, OPTMUM, with the BFGS algorithm;
for each optimization, three initial values are randomly drawn from the standard normal
distribution. The number of replications is 5000. It should be noted that neither θDL nor
θ(KT) is an efficient estimator here.
4.1 The Experiment in Domınguez and Lobato (2004)
Following Domınguez and Lobato (2004), we consider a simple nonlinear model:
Y = θ2oX + θoX
2 + ε, ε ∼ N (0, 1),
where θo = 5/4. The associated conditional moment restriction is IE[ε|X] = 0. In this
case, in addition to θNLS, θDL and θ(KT), we also consider the GMM estimator with the
“feasible” optimal instrument: 2θX + X2. The resulting GMM estimator is
θOPIV = argminθ ∈ Θ
(1T
T∑t=1
(yt − θ2xt − θx2t )(2θxt + x2
t )
)2
.
We consider two cases in which X follows N (0, 1) distribution and N (1, 1) distribu-
tion, respectively. Domınguez and Lobato (2004) showed that in the latter case, θo can
not be identified using the feasible optimal instrument because θ = −5/4 and θ = −3
also satisfy the implied, unconditional moment restriction. When X is N (0, 1), θo = 5/4
is the only real solution to the unconditional moment restriction implied by the feasible
optimal instrument.1 The simulation results are summarized in Tables 1. It can be seen
that in both cases, the NLS estimator is the best estimator in terms of all 3 criteria, and1The other two solutions are complex: −0.625± 1.0533i.
11
θOPIV has the worst performance with severe biases and larger standard errors. As to
the comparison between θDL and θ(KT), when X ∼ N (1, 1), the performance of θDL is
dominated by that of θ(KT) under all 3 criteria. When X ∼ N (0, 1), θ(KT) outperforms
θDL under all 3 criteria only for the samples T = 50 and 100.
4.2 Model with An Endogenous Explanatory Variable
We extend the previous experiment to the cases where the explanatory variable is en-
dogenous. The model specification is:Y = θ2
oZ + θoZ2 + ε
Z = X + ν,
[ε
ν
]∼ N
(0,
[1 ρ
ρ 1
] ),
where θo = 5/4, and X ∼ N (0, 1) and is independent of ε and ν. In this spec-
ification, X can be treated as an instrumental variable of Z and satisfies the con-
ditional moment restriction IE[ε|X] = 0. In Table 2, we summarize the results for
ρ = 0.01, 0.1, 0.3, 0.5, 0.7, 0.9 and the sample sizes 50, 100 and 200. It can be seen
that the NLS estimator has more severe bias (and smaller SE) when ρ becomes larger.
It is also quite remarkable that the proposed estimator θ(KT) outperforms θDL under all
3 criteria for any ρ and any sample size. The bias of θ(KT) is also less than that of
θNLS in almost all cases, except for the case that ρ = 0.01 and T = 100. Moreover, the
proposed estimator may outperform the NLS estimator in terms of MSE when ρ is not
too small. For example, when ρ ≥ 0.3, θ(KT) has a smaller MSE than the NLS estimator
for T = 100 and 200.
4.3 Noisy disturbance
In this experiment, we examine the effect of the disturbance variance on the performance
of various estimators. The model is again
Y = θ2oX + θoX
2 + ε, ε ∼ N (0, σ2),
where θo = 5/4, X is a uniform random variable on (−1, 1), and σ2 =0.1, 1, 4, 9, and 16.
The results are summarized in Table 3. We first observe that the bias, SE and MSE of
these estimators all increase as σ2 increases. In all cases, the proposed estimator is the
one with the smallest bias, while the estimtor of Domınguez and Lobato (2004) has the
largest bias. In terms of the MSE, the proposed estimator, in general, dominates θDL and
may also outperform the NLS estimator when σ2 is not too large. For example, for the
12
sample T = 100, the proposed estimator has smaller MSE than does the NLS estimator
(θDL) when σ2 ≤ 1 (σ2 ≤ 9). For T = 200, the proposed estimator has smaller MSE than
does the NLS estimator when σ2 ≤ 4 but dominates θDL in all cases.
4.4 The Proposed Estimator with Various KT
In this experiment, we examine how the choice of KT may affect the performance of
θ(KT). The model specification is the same as that in Section 4.2 where the explanatory
variable is endogenous. We consider the cases that ρ equals 0.01, 0.1, 0.5 and 0.9, and the
sample T = 50, 100, 200 and 400. We simulate θ(KT) with KT = 1, 2, . . . , 10, 15, 20, 30,
The results are in Table 4–1–1 to Table 4–4–4. For the three performance criteria (bias,
SE and MSE), we also calculate their percentage changes when KT increase. In Table
4–1–1, for instance, when KT increases from 1 to 2, the bias increases 14.795%, SE
decreases 1.15011% and MSE decreases 2.87911%. The performance of θNLS and θDL
are also given for comparison. It is easily seen that, when KT increases, SE decreases
in all cases while the bias also decreases in most cases (except in Tables 4-1-1, 4-2-2
and 4-3-2). The percentage changes are typically small. In most case, the percentage
change of the bias is less 0.1% when KT is greater than 5. These results suggest that
for the propposed estimator, the first few Fourier coefficients indeed contain the most
information for identifying θo.
5 Extensions
In this section we discuss two extensions of the previous results. We first discuss how to
obtain an efficient estimator from the proposed consistent estimator via a single Newton-
Raphson iterative step. We also extend the model with a univariate conditioning variable
to the model with multiple conditioning variables.
5.1 Efficient Estimator
We follow Newey (1990, 1993) and Dominguez and Lobato (2004) to construct a two-
step efficient estimator. Let QT (θ,KT) be the objective function for the efficient GMM
estimator and ∇θQT (θ,KT) and ∇θθ′QT (θ,KT) be its gradient vector and the Hessian
matrix, respectively. Since θ(KT) is consistent for θo, it will be in a neighborhood of
θo for sufficiently large T . Therefore, a Newton-Raphson step from θ(KT) will lead to a
13
consistent and efficient estimator:
θe(KT) = θ(KT)−
[∇θθ′QT (θ(KT),KT)
]−1∇θQT (θ(KT),KT).
Note that this approach is valid provided that θo can be identified by the limit of
QT (θ,KT).
5.2 Multiple Conditioning Variables
Consider the conditional moment restriction IE[h(Y ,θo)|X] = 0 with multiple condi-
tioning variables X. This is equivalent to
IE[h(Y ,θo) exp(τ ′X)] = 0, almost all τ ∈ [−π, π]m ⊂ Rm.
Let S := k = [k1 k2 . . . km]′ ∈ Rm with ki = 0,±1,±2, · · · ,±∞. The Fourier repre-
sentation of the unconditional moment is
IE[h(Y ,θ) exp(τ ′X)] =∑k∈S
Ck(θ) exp (ik′τ ),
where for k ∈ S,
Ck(θ) =1
(2π)m
∫[−π,π]m
IE[h(Y ,θ) exp(τ ′X)
]exp (−ik′τ ) dτ
=1
(2π)m/2IE[h(Y ,θ)ϕk1
(X1)ϕk2(X2) · · ·ϕkm
(Xm)],
with
ϕkj(Xj) =
1√2π
∫ π
−πexp(τjXj) exp (−ikjτj) dτj =
(−1)kj · 2 sinh (πXj)√2π(Xj − ikj)
,
as in the model with a univariate conditioning variable.
Let ϕk(X) = ϕk1(X1)ϕk2
(X2) · · ·ϕkm(Xm). Again by Parseval’s Theorem, we have∥∥∥IE[h(Y ,θ) exp(·X)]
∥∥∥2
L2
= (2π)m∑k∈S
∣∣Ck(θ)∣∣2 =
∑k∈S
∣∣∣IE [h(Y ,θ)ϕk(X)]∣∣∣2 .
It follows that θo can be globally identified as
θo = argminθ ∈ Θ
∑k∈S
∣∣∣IE [h(Y ,θ)ϕk(X)]∣∣∣2 , (14)
Then, a consistent estimator of θo can be computed as
θ(KT) = argminθ ∈ Θ
∑k∈S(KT)
∣∣∣∣∣ 1TT∑
t=1
h(yt,θ)ϕk(xt)
∣∣∣∣∣2
. (15)
14
Appendix
Proof of Lemma 3.1: Let ∆ be a generic constant which varies in different cases and
ηtk = h(yt,θ)ϕk(xt)− IE[h(Y ,θ)ϕk(X)
],
for t = 1, . . . , T and k = 0, . . . ,KT. First note that as X is bounded, so is sinh(πX).
Thus, ϕ0(X) is bounded, and with [A3],
IE[|ηt0|2
]≤ IE
[|h(Y ,θ)|2 |ϕ0(X)|2
]≤ ∆.
For |k| ≥ 1, it is easy to see |ϕk(X)| ≤ ∆/|k|. It follows that
IE[|ηtk|2
]≤ IE
[|h(Y ,θ)|2 |ϕk(X)|2
]≤ ∆/k2.
By [A1], the data are i.i.d., and
KT∑k=−KT
IE
∣∣∣∣∣ 1TT∑
t=1
ηtk
∣∣∣∣∣2 =
1T 2
T∑t=1
IE[∣∣ηt0
∣∣2]+2T 2
KT∑k=1
T∑t=1
IE[∣∣ηtk
∣∣2]≤ ∆
T
(1 + 2
KT∑k=1
1k2
)≤ ∆
T.
By the implication rule and the generalized Chebyshev inequality, we have
IP
KT∑k=−KT
∣∣∣∣∣ 1TT∑
t=1
ηtk
∣∣∣∣∣2
≥ ε
≤ KT∑k=−KT
IP
∣∣∣∣∣ 1TT∑
t=1
ηtk
∣∣∣∣∣2
≥ ε
2KT + 1
≤
KT∑k=−KT
IE
∣∣∣∣∣ 1TT∑
t=1
ηtk
∣∣∣∣∣2 · 2KT + 1
ε
≤ ∆(2KT + 1)Tε
.
This result holds uniformly in θ because the bound ∆ does not depend on θ. 2
Proof of Theorem 3.2: We have seen from the text that Lemma 3.1 implies
KT∑k=−KT
∣∣mT,k(θ)∣∣2 IP−→
∞∑k=−∞
|mk(θ)|2 ,
uniformly for all θ in Θ. It follows from Theorem 2.1 of Newey and West (1994) that the
solution to the left-hand side, θ(KT), must converge in probability to the unique solution
to the right-hand side, θo. 2
15
References
Ai, C. and X. Chen (2003). Efficient estimation of models with conditional moment
restrictions containing unknown functions, Econometrica, 71, 1975–1843.
Bierens, H. J. (1982). Consistent model specification test Journal of Econometrics, 26,
323–353.
Bierens, H. J. (1990). A consistent conditional moment test of functional form, Econo-
metrica, 58, 1443–1458.
Bierens, H. J. (1994). Topics in Advanced Econometrics, New York: Cambridge Univer-
sity Press.
Chen, X. and H. White (1998). Central limit and functional central limit theorems
for Hilbert-valued dependent heterogeneous arrays with applications, Econometric
Theory, 14, 260–284.
Carrasco, M. and J.- P. Florens (2000). Generalization of GMM to a continuum of
moment conditions, Econometric Theory, 16, 797–834.
Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment
restrictions, Journal of Econometrics, 34, 305–334.
Domınguez, M. A. and I. N. Lobato (2004). Consistent estimation of models defined by
conditional moment restrictions, Econometrica, 72, 1601–1615.
Donald, S. G., G. W. Imbens, and W. K. Newey (2003). Empirical likelihood estimation
and consistent tests with conditional moment restrictions, Journal of Econometrics,
117, 55-93.
Hansen, L. P. (1982). Large sample properties of generalized method of moments esti-
mators, Econometrica, 50, 1029–1054.
Kitamura, Y., G. Tripathi, and H. Ahn (2004). Empirical likelihood-based inference in
conditional moment restriction models, Econometrica, 72, 1167–1714.
Newey, W. K. (1990). Efficient instrumental variables estimation of nonlinear models,
Econometrica, 58, 809–837.
Newey, W. K. (1993). Efficient estimation of models with conditional moment restric-
tions, in G. Maddala, C. Rao, and H. Vinod (eds.), Handbook of Statistics, 11,
16
pp. 2111–2245. Elsevier Science.
Qin, J. and J. Lawless (1994). Empirical likelihood and generalized estimating equations,
Annals of Statistics, 22, 300–325.
Stinchcombe, M. B. and H. White (1998). Consistent specification testing with nuisance
parameters present only under the alternative, Econometric Theory, 14, 295–325.
Stuart, R. D. (1961). An Introduction to Fourier Analysis, New York: Halsted Press.
White, H. (1989). An additional hidden unit test for neglected nonlinearity, Proceedings
of the International Joint Conference on Neural Networks, Vol. 2, pp. 451–455.
New York: IEEE Press.
17
Table 1: Example in Domınguez and Lobato (2004).
Sample X ∼ N (0, 1) X ∼ N (1, 1)
T Estimator Bias SE MSE Bias SE MSE
50 θNLS −0.0006 0.0501 0.0025 −0.0083 0.1881 0.0354
θDL −0.0390 0.2282 0.0536 −0.0336 0.3667 0.1355
θ(KT) −0.0061 0.1600 0.0256 −0.0249 0.3308 0.1100
θOPIV −0.2222 0.6288 0.4447 −1.6922 1.2783 4.4972
100 θNLS −0.0004 0.0342 0.0012 −0.0071 0.1713 0.0294
θDL −0.0152 0.1541 0.0240 −0.0316 0.3595 0.1302
θ(KT) −0.0059 0.1511 0.0228 −0.0217 0.3094 0.0962
θOPIV −0.1480 0.5096 0.2815 −1.7217 1.2619 4.5564
200 θNLS −0.0004 0.0239 0.0006 −0.0025 0.1035 0.0107
θDL −0.0017 0.0864 0.0075 −0.0191 0.2796 0.0785
θ(KT) −0.0045 0.1390 0.0193 −0.0116 0.2278 0.0520
θOPIV −0.0931 0.3994 0.1681 −1.6649 1.2859 4.4250
18
Table 2: Models with an endogenous explanatory variable.
T = 50 T = 100 T = 200
ρ Est. Bias SE MSE Bias SE MSE Bias SE MSE
0.01 θNLS 0.0009 0.0317 0.0010 0.0005 0.0212 0.0004 0.0011 0.0146 0.0002
θDL −0.0103 0.1165 0.0137 −0.0062 0.0809 0.0066 −0.0027 0.0561 0.0032
θ(KT) −0.0003 0.0561 0.0031 −0.0009 0.0365 0.0013 0.0001 0.0245 0.0006
0.1 θNLS 0.0097 0.0313 0.0011 0.0102 0.0210 0.0005 0.0103 0.0146 0.0003
θDL −0.0116 0.1153 0.0134 −0.0069 0.0816 0.0067 −0.0036 0.0570 0.0033
θ(KT) −0.0021 0.0550 0.0030 −0.0010 0.0358 0.0013 −0.0006 0.0242 0.0006
0.3 θNLS 0.0315 0.0310 0.0020 0.0311 0.0209 0.0014 0.0315 0.0144 0.0012
θDL −0.0125 0.1214 0.0149 −0.0061 0.0819 0.0067 −0.0032 0.0585 0.0034
θ(KT) − 0.0039 0.0565 0.0032 −0.0016 0.0358 0.0013 −0.0002 0.0244 0.0006
0.5 θNLS 0.0539 0.0311 0.0039 0.0527 0.0207 0.0032 0.0520 0.0143 0.0029
θDL −0.0125 0.1231 0.0153 −0.0045 0.0817 0.0067 −0.0017 0.0570 0.0033
θ(KT) −0.0056 0.0596 0.0036 −0.0021 0.0366 0.0013 −0.0010 0.0247 0.0006
0.7 θNLS 0.0746 0.0298 0.0064 0.0739 0.0196 0.0058 0.0731 0.0140 0.0055
θDL −0.0153 0.1242 0.0156 −0.0083 0.0840 0.0071 −0.0053 0.0588 0.0035
θ(KT) −0.0097 0.0574 0.0034 −0.0038 0.0366 0.0014 −0.0020 0.0247 0.0006
0.9 θNLS 0.0972 0.0285 0.0103 0.0953 0.0190 0.0094 0.0942 0.0134 0.0091
θDL −0.0166 0.1288 0.0169 −0.0086 0.0845 0.0072 −0.0042 0.0598 0.0036
θ(KT) −0.0117 0.0947 0.0091 −0.0053 0.0370 0.0014 −0.0019 0.0250 0.0006
19
Table 3: Models with different disturbance variances.
T = 50 T = 100 T = 200
σ2 Est. Bias SE MSE Bias SE MSE Bias SE MSE
σ2 = 0.1 θNLS −0.2827 0.7836 0.6939 −0.2511 0.7392 0.6094 −0.2523 0.7433 0.6160
θDL −0.4645 0.8678 0.9687 −0.3938 0.8108 0.8124 −0.3701 0.7900 0.7610
θ(KT) −0.1129 0.5687 0.3361 −0.0491 0.4016 0.1637 −0.0317 0.3467 0.1212
σ2 = 1 θNLS −0.5845 1.0572 1.4591 −0.4451 0.9481 1.0968 −0.3089 0.8158 0.7608
θDL −0.9491 1.0820 2.0711 −0.7899 1.0236 1.6715 −0.6692 0.9776 1.4033
θ(KT) −0.3478 1.1121 1.3574 −0.1724 0.8191 0.7005 −0.0880 0.5790 0.3429
σ2 = 4 θNLS −0.8508 1.2109 2.1899 −0.7320 1.1470 1.8513 −0.5897 1.0582 1.4673
θDL −1.2441 1.1875 2.9576 −1.1126 1.1187 2.4891 −0.9874 1.0752 2.1307
θ(KT) −0.5081 1.5507 2.6623 −0.3767 1.3357 1.9257 −0.2599 1.0442 1.1577
σ2 = 9 θNLS −0.9452 1.2880 2.5519 −0.8877 1.2281 2.2960 −0.7253 1.1491 1.8463
θDL −1.3698 1.2738 3.4985 −1.2821 1.1920 3.0644 −1.1359 1.1210 2.5467
θ(KT) −0.5013 1.8507 3.6759 −0.4600 1.6048 2.7864 −0.3355 1.3225 1.8612
σ2 = 16 θNLS −1.0299 1.3672 2.9295 −0.9329 1.2814 2.5121 −0.8481 1.2134 2.1915
θDL −1.4794 1.3772 4.0848 −1.3361 1.2439 3.3321 −1.2542 1.1681 2.9371
θ(KT) −0.5085 2.1182 4.7443 −0.3882 1.7852 3.3372 −0.3912 1.5377 2.5172
20
Table 4-1-1 : The performance of θ(KT) with various KT (ρ = 0.01, T = 50).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00009 . 0.05827 . 0.00340 .
2 −0.00011 14.79500 0.05743 −1.45011 0.00330 −2.87911
3 −0.00011 4.22366 0.05709 −0.58447 0.00326 −1.16549
4 −0.00011 1.96850 0.05692 −0.31031 0.00324 −0.61964
5 −0.00011 1.13476 0.05681 −0.19113 0.00323 −0.38189
6 −0.00012 0.73752 0.05673 −0.12917 0.00322 −0.25816
7 −0.00012 0.51757 0.05668 −0.09299 0.00321 −0.18588
8 −0.00012 0.38323 0.05664 −0.07008 0.00321 −0.14011
9 −0.00012 0.29519 0.05661 −0.05468 0.00320 −0.10933
10 −0.00012 0.23425 0.05658 −0.04384 0.00320 −0.08766
15 −0.00012 0.69575 0.05651 −0.13185 0.00319 −0.26353
20 −0.00012 0.34221 0.05647 −0.06610 0.00319 −0.13215
30 −0.00012 0.33899 0.05644 −0.06615 0.00318 −0.13224
θNLSE 0.00162 . 0.03138 . 0.00099 .
θDL −0.01044 . 0.11574 . 0.01350 .
21
Table 4-1-2 : The performance of θ(KT) with various KT (ρ = 0.01, T = 100).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 0.00016 . 0.03674 . 0.00135 .
2 −0.00117 −852.65119 0.10000 172.15227 0.01000 640.75682
3 −0.00116 −0.75644 0.09898 −1.01554 0.00980 −2.02070
4 −0.00116 −0.36554 0.09848 −0.51161 0.00970 −1.02056
5 −0.00115 −0.21339 0.09818 −0.30618 0.00964 −0.61139
6 −0.00115 −0.13930 0.09798 −0.20320 0.00960 −0.40597
7 −0.00115 −0.09789 0.09783 −0.14448 0.00957 −0.28874
8 −0.00115 −0.07247 0.09773 −0.10791 0.00955 −0.21570
9 −0.00115 −0.05578 0.09765 −0.08363 0.00953 −0.16718
10 −0.00115 −0.04424 0.09758 −0.06669 0.00952 −0.13333
15 −0.00115 −0.13049 0.09739 −0.19863 0.00948 −0.39684
20 −0.00115 −0.06399 0.09729 −0.09854 0.00947 −0.19697
30 −0.00115 −0.06309 0.09720 −0.09792 0.00945 −0.19573
θNLSE 0.00101 . 0.02107 . 0.00045 .
θDL −0.00425 . 0.07889 . 0.00624 .
22
Table 4-1-3 : The performance of θ(KT) with various KT (ρ = 0.01, T = 200).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 0.00019 . 0.02538 . 0.00064 .
2 0.00019 −1.03685 0.02505 −1.31430 0.00063 −2.61129
3 0.00019 −0.42775 0.02492 −0.53650 0.00062 −1.07012
4 0.00019 −0.22997 0.02484 −0.28627 0.00062 −0.57172
5 0.00019 −0.14266 0.02480 −0.17678 0.00061 −0.35323
6 0.00019 −0.09682 0.02477 −0.11964 0.00061 −0.23913
7 0.00019 −0.06990 0.02475 −0.08621 0.00061 −0.17234
8 0.00019 −0.05278 0.02473 −0.06501 0.00061 −0.12998
9 0.00019 −0.04125 0.02472 −0.05075 0.00061 −0.10148
10 0.00019 −0.03311 0.02471 −0.04070 0.00061 −0.08139
15 0.00019 −0.09977 0.02468 −0.12249 0.00061 −0.24483
20 0.00019 −0.05010 0.02467 −0.06144 0.00061 −0.12284
30 0.00019 −0.05018 0.02465 −0.06150 0.00061 −0.12297
θNLSE 0.00120 . 0.01487 . 0.00022 .
θDL −0.00201 . 0.05628 . 0.00317 .
23
Table 4-1-4 : The performance of θ(KT) with various KT (ρ = 0.01, T = 400).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00010 . 0.01732 . 0.00030 .
2 −0.00010 −4.18850 0.01709 −1.31251 0.00029 −2.60798
3 −0.00010 −1.83557 0.01700 −0.53430 0.00029 −1.06583
4 −0.00009 −1.00887 0.01695 −0.28474 0.00029 −0.56871
5 −0.00009 −0.63318 0.01692 −0.17570 0.00029 −0.35113
6 −0.00009 −0.43295 0.01690 −0.11886 0.00029 −0.23760
7 −0.00009 −0.31422 0.01689 −0.08562 0.00029 −0.17119
8 −0.00009 −0.23820 0.01688 −0.06456 0.00028 −0.12909
9 −0.00009 −0.18669 0.01687 −0.05039 0.00028 −0.10076
10 −0.00009 −0.15020 0.01686 −0.04041 0.00028 −0.08080
15 −0.00009 −0.45398 0.01684 −0.12157 0.00028 −0.24302
20 −0.00009 −0.22931 0.01683 −0.06096 0.00028 −0.12190
30 −0.00009 −0.23050 0.01682 −0.06102 0.00028 −0.12201
θNLSE 0.00110 . 0.01039 . 0.00011 .
θDL −0.00155 . 0.03981 . 0.00159 .
24
Table 4-2-1 : The performance of θ(KT) with various KT (ρ = 0.1, T = 50).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00140 . 0.05940 . 0.00353 .
2 −0.00139 −0.90246 0.05856 −1.40777 0.00343 −2.79517
3 −0.00138 −0.40118 0.05823 −0.56941 0.00339 −1.13538
4 −0.00138 −0.22127 0.05805 −0.30271 0.00337 −0.60442
5 −0.00138 −0.13922 0.05794 −0.18658 0.00336 −0.37276
6 −0.00137 −0.09541 0.05787 −0.12614 0.00335 −0.25209
7 −0.00137 −0.06937 0.05782 −0.09083 0.00334 −0.18156
8 −0.00137 −0.05267 0.05778 −0.06847 0.00334 −0.13688
9 −0.00137 −0.04134 0.05774 −0.05343 0.00334 −0.10683
10 −0.00137 −0.03329 0.05772 −0.04285 0.00333 −0.08566
15 −0.00137 −0.10103 0.05765 −0.12889 0.00332 −0.25757
20 −0.00137 −0.05115 0.05761 −0.06462 0.00332 −0.12919
30 −0.00137 −0.05154 0.05757 −0.06468 0.00332 −0.12930
θNLSE 0.01001 . 0.03115 . 0.00107 .
θDL −0.01199 . 0.12097 . 0.01477 .
25
Table 4-2-2 : The performance of θ(KT) with various KT (ρ = 0.1, T = 100).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00010 . 0.03738 . 0.00140 .
2 −0.00014 34.68240 0.03688 −1.34168 0.00136 −2.66474
3 −0.00015 10.51217 0.03668 −0.54726 0.00134 −1.09121
4 −0.00016 5.09125 0.03657 −0.29195 0.00134 −0.58287
5 −0.00016 2.99958 0.03650 −0.18027 0.00133 −0.36010
6 −0.00017 1.97508 0.03646 −0.12201 0.00133 −0.24378
7 −0.00017 1.39794 0.03643 −0.08792 0.00133 −0.17570
8 −0.00017 1.04109 0.03640 −0.06631 0.00132 −0.13252
9 −0.00017 0.80518 0.03638 −0.05176 0.00132 −0.10346
10 −0.00017 0.64118 0.03637 −0.04152 0.00132 −0.08298
15 −0.00018 1.92137 0.03632 −0.12494 0.00132 −0.24964
20 −0.00018 0.94721 0.03630 −0.06268 0.00132 −0.12526
30 −0.00018 0.94066 0.03628 −0.06275 0.00132 −0.12540
θNLSE 0.01065 . 0.02156 . 0.00058 .
θDL −0.00313 . 0.08059 . 0.00650 .
26
Table 4-2-3 : The performance of θ(KT) with various KT (ρ = 0.1, T = 200).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00057 . 0.02505 . 0.00063 .
2 −0.00057 −0.67359 0.02470 −1.38955 0.00061 −2.75905
3 −0.00056 −0.31231 0.02456 −0.56744 0.00060 −1.13139
4 −0.00056 −0.17450 0.02449 −0.30282 0.00060 −0.60458
5 −0.00056 −0.11025 0.02444 −0.18700 0.00060 −0.37357
6 −0.00056 −0.07563 0.02441 −0.12656 0.00060 −0.25291
7 −0.00056 −0.05497 0.02439 −0.09120 0.00060 −0.18228
8 −0.00056 −0.04171 0.02437 −0.06878 0.00059 −0.13748
9 −0.00056 −0.03271 0.02436 −0.05369 0.00059 −0.10733
10 −0.00056 −0.02632 0.02435 −0.04306 0.00059 −0.08609
15 −0.00056 −0.07969 0.02432 −0.12959 0.00059 −0.25896
20 −0.00056 −0.04021 0.02430 −0.06500 0.00059 −0.12994
30 −0.00056 −0.04041 0.02429 −0.06507 0.00059 −0.13007
θNLSE 0.01033 . 0.01436 . 0.00031 .
θDL −0.00318 . 0.05735 . 0.00330 .
27
Table 4-2-4 : The performance of θ(KT) with various KT (ρ = 0.1, T = 400).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00031 . 0.01742 . 0.00030 .
2 −0.00030 −3.50418 0.01720 −1.26751 0.00030 −2.52035
3 −0.00030 −1.56196 0.01711 −0.51695 0.00029 −1.03186
4 −0.00029 −0.86527 0.01707 −0.27570 0.00029 −0.55100
5 −0.00029 −0.54513 0.01704 −0.17019 0.00029 −0.34031
6 −0.00029 −0.37349 0.01702 −0.11515 0.00029 −0.23033
7 −0.00029 −0.27136 0.01700 −0.08296 0.00029 −0.16597
8 −0.00029 −0.20585 0.01699 −0.06256 0.00029 −0.12516
9 −0.00029 −0.16139 0.01698 −0.04883 0.00029 −0.09770
10 −0.00029 −0.12988 0.01698 −0.03916 0.00029 −0.07835
15 −0.00029 −0.39272 0.01696 −0.11782 0.00029 −0.23566
20 −0.00029 −0.19833 0.01695 −0.05908 0.00029 −0.11821
30 −0.00029 −0.19932 0.01694 −0.05914 0.00029 −0.11832
θNLSE 0.01020 . 0.01042 . 0.00021 .
θDL −0.00205 . 0.03974 . 0.00158 .
28
Table 4-3-1 : The performance of θ(KT) with various KT (ρ = 0.5, T = 50).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00569 . 0.06032 . 0.00367 .
2 −0.00569 −0.07632 0.05959 −1.21612 0.00358 −2.39747
3 −0.00569 −0.01686 0.05929 −0.49144 0.00355 −0.97192
4 −0.00569 −0.00447 0.05914 −0.26110 0.00353 −0.51684
5 −0.00569 −0.00100 0.05904 −0.16086 0.00352 −0.31855
6 −0.00569 0.00013 0.05898 −0.10873 0.00351 −0.21533
7 −0.00569 0.00051 0.05893 −0.07828 0.00350 −0.15504
8 −0.00569 0.00063 0.05890 −0.05900 0.00350 −0.11686
9 −0.00569 0.00063 0.05887 −0.04604 0.00350 −0.09119
10 −0.00569 0.00060 0.05885 −0.03691 0.00349 −0.07311
15 −0.00569 0.00232 0.05878 −0.11102 0.00349 −0.21982
20 −0.00569 0.00146 0.05875 −0.05565 0.00348 −0.11022
30 −0.00569 0.00166 0.05872 −0.05569 0.00348 −0.11029
θNLSE 0.05397 . 0.03125 . 0.00389 .
θDL −0.01208 . 0.12521 . 0.01582 .
29
Table 4-3-2 : The performance of θ(KT) with various KT (ρ = 0.5, T = 100).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00191 . 0.03708 . 0.00138 .
2 −0.00191 0.11425 0.03658 −1.34396 0.00134 −2.66222
3 −0.00191 0.05820 0.03638 −0.54672 0.00133 −1.08718
4 −0.00191 0.03550 0.03628 −0.29128 0.00132 −0.57993
5 −0.00191 0.02381 0.03621 −0.17973 0.00131 −0.35801
6 −0.00191 0.01703 0.03617 −0.12158 0.00131 −0.24224
7 −0.00191 0.01276 0.03614 −0.08758 0.00131 −0.17453
8 −0.00191 0.00991 0.03611 −0.06603 0.00131 −0.13160
9 −0.00191 0.00791 0.03609 −0.05154 0.00131 −0.10272
10 −0.00191 0.00646 0.03608 −0.04133 0.00131 −0.08238
15 −0.00191 0.02011 0.03603 −0.12436 0.00130 −0.24776
20 −0.00191 0.01047 0.03601 −0.06237 0.00130 −0.12428
30 −0.00191 0.01075 0.03599 −0.06242 0.00130 −0.12440
θNLSE 0.05283 . 0.02020 . 0.00320 .
θDL −0.00552 . 0.08383 . 0.00706 .
30
Table 4-3-3 : The performance of θ(KT) with various KT (ρ = 0.5, T = 200).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00161 . 0.02545 . 0.00065 .
2 −0.00159 −1.35402 0.02509 −1.39866 0.00063 −2.77740
3 −0.00158 −0.57188 0.02495 −0.57062 0.00062 −1.13800
4 −0.00158 −0.30861 0.02487 −0.30440 0.00062 −0.60790
5 −0.00157 −0.19151 0.02483 −0.18794 0.00062 −0.37555
6 −0.00157 −0.12995 0.02480 −0.12719 0.00062 −0.25423
7 −0.00157 −0.09378 0.02477 −0.09164 0.00062 −0.18322
8 −0.00157 −0.07078 0.02476 −0.06911 0.00062 −0.13819
9 −0.00157 −0.05529 0.02474 −0.05395 0.00061 −0.10788
10 −0.00157 −0.04436 0.02473 −0.04327 0.00061 −0.08653
15 −0.00157 −0.13356 0.02470 −0.13021 0.00061 −0.26027
20 −0.00156 −0.06702 0.02468 −0.06531 0.00061 −0.13059
30 −0.00156 −0.06710 0.02467 −0.06538 0.00061 −0.13073
θNLSE 0.05199 . 0.01416 . 0.00290 .
θDL −0.00514 . 0.05945 . 0.00356 .
31
Table 4-3-4 : The performance of θ(KT) with various KT (ρ = 0.5, T = 400).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00067 . 0.01778 . 0.00032 .
2 −0.00066 −1.65936 0.01754 −1.37195 0.00031 −2.72588
3 −0.00065 −0.67579 0.01744 −0.56017 0.00030 −1.11753
4 −0.00065 −0.35958 0.01739 −0.29892 0.00030 −0.59712
5 −0.00065 −0.22159 0.01735 −0.18459 0.00030 −0.36894
6 −0.00065 −0.14974 0.01733 −0.12493 0.00030 −0.24977
7 −0.00065 −0.10778 0.01732 −0.09002 0.00030 −0.18002
8 −0.00065 −0.08122 0.01731 −0.06789 0.00030 −0.13578
9 −0.00065 −0.06336 0.01730 −0.05300 0.00030 −0.10600
10 −0.00065 −0.05079 0.01729 −0.04251 0.00030 −0.08502
15 −0.00064 −0.15268 0.01727 −0.12792 0.00030 −0.25574
20 −0.00064 −0.07650 0.01726 −0.06416 0.00030 −0.12832
30 −0.00064 −0.07652 0.01725 −0.06423 0.00030 −0.12845
θNLSE 0.05172 . 0.01002 . 0.00278 .
θDL −0.00224 . 0.04186 . 0.00176 .
32
Table 4-4-1 : The performance of θ(KT) with various KT (ρ = 0.9, T = 50).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.01144 . 0.06326 . 0.00413 .
2 −0.01140 −0.39167 0.06256 −1.10364 0.00404 −2.15032
3 −0.01138 −0.14974 0.06228 −0.44675 0.00401 −0.87247
4 −0.01137 −0.07674 0.06213 −0.23750 0.00399 −0.46406
5 −0.01137 −0.04617 0.06204 −0.14637 0.00398 −0.28604
6 −0.01136 −0.03069 0.06198 −0.09895 0.00397 −0.19337
7 −0.01136 −0.02183 0.06194 −0.07125 0.00396 −0.13923
8 −0.01136 −0.01630 0.06190 −0.05371 0.00396 −0.10495
9 −0.01136 −0.01262 0.06188 −0.04191 0.00396 −0.08189
10 −0.01136 −0.01006 0.06186 −0.03360 0.00395 −0.06566
15 −0.01135 −0.02995 0.06179 −0.10109 0.00395 −0.19744
20 −0.01135 −0.01482 0.06176 −0.05068 0.00394 −0.09900
30 −0.01135 −0.01470 0.06173 −0.05072 0.00394 −0.09907
θNLSE 0.09695 . 0.02847 . 0.01021 .
θDL −0.01613 . 0.13133 . 0.01750 .
33
Table 4-4-2 : The performance of θ(KT) with various KT (ρ = 0.9, T = 100).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00627 . 0.08508 . 0.00728 .
2 −0.00621 −0.95742 0.08356 −1.78127 0.00702 −3.52203
3 −0.00618 −0.37986 0.08299 −0.68917 0.00692 −1.37021
4 −0.00617 −0.19960 0.08269 −0.35995 0.00687 −0.71685
5 −0.00616 −0.12208 0.08251 −0.21985 0.00684 −0.43814
6 −0.00616 −0.08209 0.08238 −0.14783 0.00682 −0.29472
7 −0.00616 −0.05888 0.08230 −0.10608 0.00681 −0.21152
8 −0.00615 −0.04425 0.08223 −0.07977 0.00680 −0.15907
9 −0.00615 −0.03445 0.08218 −0.06213 0.00679 −0.12392
10 −0.00615 −0.02757 0.08214 −0.04975 0.00678 −0.09923
15 −0.00614 −0.08263 0.08202 −0.14930 0.00676 −0.29764
20 −0.00614 −0.04125 0.08196 −0.07468 0.00675 −0.14893
30 −0.00614 −0.04116 0.08189 −0.07463 0.00674 −0.14883
θNLSE 0.09544 . 0.01901 . 0.00947 .
θDL −0.01039 . 0.08930 . 0.00808 .
34
Table 4-4-3 : The performance of θ(KT) with various KT (ρ = 0.9, T = 200).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 −0.00257 . 0.02498 . 0.00063 .
2 −0.00255 −0.48005 0.02464 −1.36135 0.00061 −2.68593
3 −0.00255 −0.18776 0.02451 −0.55611 0.00061 −1.10132
4 −0.00255 −0.09742 0.02443 −0.29676 0.00060 −0.58838
5 −0.00255 −0.05904 0.02439 −0.18324 0.00060 −0.36348
6 −0.00254 −0.03942 0.02436 −0.12401 0.00060 −0.24604
7 −0.00254 −0.02812 0.02434 −0.08935 0.00060 −0.17730
8 −0.00254 −0.02105 0.02432 −0.06738 0.00060 −0.13371
9 −0.00254 −0.01633 0.02431 −0.05259 0.00060 −0.10438
10 −0.00254 −0.01303 0.02430 −0.04218 0.00060 −0.08371
15 −0.00254 −0.03887 0.02427 −0.12692 0.00060 −0.25177
20 −0.00254 −0.01928 0.02425 −0.06365 0.00059 −0.12630
30 −0.00254 −0.01916 0.02424 −0.06371 0.00059 −0.12641
θNLSE 0.09420 . 0.01310 . 0.00904 .
θDL −0.00480 . 0.05960 . 0.00357 .
35
Table 4-4-4 : The performance of θ(KT) with various KT (ρ = 0.9, T = 400).
KT Bias Bias(+%) SE SE(+%) MSE MSE(+%)
1 -0.00111 . 0.01749 . 0.00031 .
2 -0.00109 -1.30692 0.01725 -1.38686 0.00030 -2.75386
3 -0.00109 -0.51684 0.01715 -0.56761 0.00030 -1.13158
4 -0.00109 -0.27137 0.01710 -0.30320 0.00029 -0.60523
5 -0.00108 -0.16602 0.01707 -0.18734 0.00029 -0.37415
6 -0.00108 -0.11169 0.01705 -0.12683 0.00029 -0.25338
7 -0.00108 -0.08015 0.01703 -0.09142 0.00029 -0.18266
8 -0.00108 -0.06027 0.01702 -0.06895 0.00029 -0.13779
9 -0.00108 -0.04694 0.01701 -0.05383 0.00029 -0.10758
10 -0.00108 -0.03758 0.01700 -0.04318 0.00029 -0.08630
15 -0.00108 -0.11276 0.01698 -0.12997 0.00029 -0.25963
20 -0.00108 -0.05637 0.01697 -0.06520 0.00029 -0.13029
30 -0.00108 -0.05631 0.01696 -0.06528 0.00029 -0.13044
θNLSE 0.09374 . 0.00916 . 0.00887 .
θDL −0.00252 . 0.04133 . 0.00171 .
36