consistent parameter estimation for conditional moment...

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Preliminary; Please do not cite or quote.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


∞∑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


∥∥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


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


∑k∈S

∣∣∣IE [h(Y ,θ)ϕk(X)]∣∣∣2 , (14)

Then, a consistent estimator of θo can be computed as


∑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

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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



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



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



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



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



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



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



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



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



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



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



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



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



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



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



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

consistent parameter estimation for conditional moment...

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