No CrossRef data available.
EFFICIENT METHOD OF MOMENTS IN MISSPECIFIED I.I.D. MODELS
Published online by Cambridge University Press: 08 June 2004
Abstract
The paper presents the asymptotic theory of the efficient method of moments when the model of interest is not correctly specified. The paper assumes a sequence of independent and identically distributed observations and a global misspecification. It is found that the limiting distribution of the estimator is still asymptotically normal, but it suffers a strong impact in the covariance matrix. A consistent estimator of this covariance matrix is provided. The large sample distribution on the estimated moment function is also obtained. These results are used to discuss the situation when the moment conditions hold but the model is misspecified. It also is shown that the overidentifying restrictions test has asymptotic power one whenever the limit moment function is different from zero. It is also proved that the bootstrap distributions converge almost surely to the previously mentioned distributions and hence they could be used as an alternative to draw inferences under misspecification. Interestingly, it is also shown that bootstrap can be reliably applied even if the number of bootstrap replications is very small.The authors express their gratitude to Professor A.R. Gallant for his support and for helpful discussions about this topic. They also thank Professor Donald Andrews for his revision and for the suggestion to read the work by Hall and Inoue (2003). They thank Hall and Inoue for sending them their technical report. Finally the authors thank two anonymous referees for their helpful comments, which led to a significant improvement of the paper. The views expressed in this paper are our sole responsibility. Víctor Aguirre-Torres acknowledges partial support from the Asociación Mexicana de la Cultura A.C. Manuel A. Domínguez acknowledges funding from CONACYT, research project J38076-D, and partial support from Asociación Mexicana de la Cultura A.C.
Information
- Type
- Research Article
- Information
- Copyright
- © 2004 Cambridge University Press
1. INTRODUCTION
An estimation method that has received much attention recently is the efficient method of moments (EMM) presented in Gallant and Tauchen (1996). This method uses the scores of an auxiliary model to generate moment constraints. Models with unobservable components are a particularly rich area of application of the method (see, e.g., Gallant and Long, 1997; Gallant, Hsieh, and Tauchen, 1997; Tauchen, 1997; Gallant and Tauchen, 1999a). It turns out that the theory presented in those papers assumes that the model of interest is correctly specified. In fact, a search of the literature on the topic of theoretical developments for EMM under misspecification showed only Tauchen (1997), where the author briefly discusses the problem from the point of view of finding the approximate slope of the overidentifying restrictions test as proposed in Geweke (1981).
On the other hand, Newey (1985) and Gallant (1987) study the properties of the closely related generalized method of moments (GMM) estimators under local misspecification, and Hall and Inoue (2003) derive the large sample distribution of the GMM estimator and associated test statistics under global misspecification.
With the inspiration yielded from the latter paper, the purpose of this work is to make a qualitative analysis of the consequences of model misspecification on EMM estimators. A characterization of the large sample limit of the estimator and the impact of misspecification on the variance-covariance matrix will be given. In addition, we provide conditions to get a misspecification consistent covariance matrix. The results also will cover the large sample distribution of the estimated moment function even if the moment constraints are correct but the model is misspecified. Furthermore, the work proves that bootstrap methods can fit the moment and parameter distribution under misspecification and hence can be used to draw inferences on them. Interestingly, we extend the procedures in Domínguez and Aguirre-Torres (2003) concerning the properties of the bootstrap when the number of the bootstrap replications is small so that the application of the bootstrap does not lead to a large increase in computation demands.
The paper deals with the independent and identically distributed (i.i.d.) case, whereas most of the applications of EMM are for time series data. Nonetheless, the results of the paper give an indication of the consequences of misspecification of the maintained model in time series models too. A referee also noticed that when the auxiliary model is a good approximation of the data then the scores are essentially serially uncorrelated, and then one may expect the results from the i.i.d. case to be relevant.
The paper is organized as follows. Section 2 presents the data generating process and the large sample properties of the quasi–maximum likelihood estimation (QMLE) for the auxiliary model; these properties are required to derive the asymptotic properties of the EMM procedure. Section 3 defines the EMM and discusses the almost sure limit of the estimator. Section 4 presents the results under global misspecification, including the distributional results for the estimator, the moment function, and the test for overidentifying constraints. Section 5 presents the results regarding bootstrap. Outlines of some proofs are given in the Appendix.
2. QUASI–MAXIMUM LIKELIHOOD ESTIMATION
This section presents the data generating process and the corresponding large sample theory for the auxiliary model. The scores of this model are used to generate moment constraints for the estimation of the model of interest; hence the large sample properties of EMM depend partly on the results of this section.
The following assumption gives the data generating process.
Assumption 1. The data {Xt}t=1n are a sequence of i.i.d. random vectors with joint continuous density h(x). The observed data will be denoted by
.
The model of interest is
where
is the parameter space for ρ (notice that
is not the real line). Model p is correctly specified if there exists
such that
We are assuming that this model is not directly estimable by maximum likelihood and hence an alternative procedure must be used. To estimate ρ the EMM maximizes the pseudolikelihood of an auxiliary model that we denote by
and that uses the first-order condition of this problem to generate a set of moment restrictions. Formally, we consider the following notation.
Notation 1. Let
Recall from pseudomaximum likelihood theory that if the auxiliary model suits the data, θ0 identifies the data generating process. To obtain the large sample properties of the QMLE the following assumption is required.
Assumption 2. The parameter space Θ is closed and bounded. For the function log f (x|θ), its first and second partial derivatives with respect to θ are continuous in θ. The derivatives are also uniformly bounded in θ by an integrable function b1(x) such that ∫ b1(x)h(x) dx < ∞. The vector θ0 is unique, and it is an interior point of Θ. For every nonzero vector a,
Moreover, there exists an open neighborhood of θ0, say, NB0, such that
The uniformly bounded part of Assumption 2 insures that averages converge to integrals, whereas asking for the integral to be greater than zero allows the use of the Cramer-Wold device to show asymptotic normality. With the preceding assumptions the following theorem gives the large sample properties of the QMLE. The proofs of these theorems are a consequence of results given in Gallant (1987, Ch. 3) or similarly from White (1982). Hence these results are not proved.
THEOREM 1. Under Assumptions 1 and 2,
.
THEOREM 2. Under Assumptions 1 and 2,
It is important to notice that even under model misspecification Vθ could be consistently estimated by
which is a so-called sandwich formula. When the auxiliary model is correctly specified, then J0−1 = −I0 and Vθ becomes I0−1, the inverse of the information matrix.
In the case of the EMM, a common choice of the auxiliary model consists of a flexible semi-nonparametric (SNP) score generator that makes the efficiency of the estimator of ρ reach that of maximum likelihood when model p is correctly specified. The interested reader may consult Gallant and Nychka (1987) for a description of the SNP model suitable for cross-sectional data. By construction, the SNP score is expected to closely approximate any density function, and hence it approaches the efficiency of maximum likelihood as the number of moment functions entering the score increases and as the number of observations entering each moment function also increases (see also Gallant and Long, 1997). Although the score generator typically used for the EMM is the SNP, the discussion of the paper will be in general regarding the definition of the auxiliary model f (x|θ).
3. THE EFFICIENT METHOD OF MOMENTS
In this section we discuss the large sample properties of EMM under a global misspecified setup. As mentioned before, the model of interest is p(x|ρ), which we do not assume to be correctly specified; i.e., contrary to Gallant and Long (1997), Gallant et al. (1997), Tauchen (1997), and Gallant and Tauchen (1999a), we do not assume that the second equation after Assumption 1 holds.
Once the auxiliary model has been estimated, the moment function is taken as
A direct application of the method of moments to (1) would require finding ρ that solves the equation mn(ρ) = 0. Usually, the number of parameters in the auxiliary model exceeds that of the model of interest, and, instead of finding a root for (1), the estimator of ρ is defined as follows:
where
is given in Notation 1.
The advantage of using this approach in a situation where direct maximum likelihood is not possible is as follows: if we let
be a long simulated sequence from p(x|ρ) then (1) could be approximated by
It is assumed that N is large enough that the Monte Carlo integral approximates the analytic integral to within a negligible error of the same sort that is made in computing any mathematical expression on a digital computer.
Also, the method has a built-in specification test via the following test statistic:
which is based on a “norm” of the vector of moments evaluated at the optimum. In this regard, Gallant and Tauchen (1996) show that this test statistic has an asymptotic central chi-square distribution if the second equation after Assumption 1 holds, so if the optimized vector of moments is too far away from zero relative to sampling error, then that would be considered evidence against model p. This is the so-called test of overidentifying restrictions because the number of degrees of freedom of the chi-squared distribution is equal to the number of excess parameters in θ relative to those in ρ.
The discussion of the large sample properties of the EMM estimator, defined in (2), requires the following notation.
Notation 2. Let
Notice that N(ρ,θ) will be a square symmetric matrix under regularity conditions. A referee pointed out that it is important to observe that from the definition of ρ0, its value depends on the weighting matrix I0 and also to notice that in the case of EMM, once f (x|θ) is defined, then I0 is completely defined too and that
is a natural choice for its estimation. Besides Assumptions 1 and 2 we require Assumption 3.
Assumption 3. The parameter space
is closed and bounded. For the function [log f (x|θ)] p(x|ρ), its first and second partial derivatives with respect to ρ and θ are continuous in (x,θ,ρ) and uniformly bounded in (θ,ρ) by an integrable function b2(x) such that ∫h(x)b2(x) dx < ∞. The vector ρ0 is unique, and it is an interior point of
. The matrices M0 and N0 are full column rank. The Hessian of s0(ρ) evaluated at ρ0 is nonsingular.
THEOREM 3. Under Assumptions 1–3,
.
Notice that Theorem 3 does not require that moment conditions hold true; i.e., it may happen that
but, because ρ0 is a minimum of s0(ρ), then first-order conditions imply
and then, differentiation of the quadratic s0 results in
This last equation does not imply that m0(ρ0) = 0 either.
A random vector that plays an important role in the large sample results is the moment function m when only the θ argument is random. The next lemma gives its large sample distribution under misspecification.
LEMMA 1. Under Assumptions 1–3,
where Vm0 = N0Vθ N0.
The matrix Vm0 represents the variance-covariance of the moment vector when only the θ argument is random. A referee suggested pointing out that the matrix Vm0 is different from expression (17) of Gallant and Tauchen (1996); in fact, expression (18) of Gallant and Tauchen (1996) results from Lemma 1 under correct specification.
4. ESTIMATED METHOD OF MOMENTS UNDER GLOBAL MISSPECIFICATION
Theorem 3 establishes the almost sure convergence of
, the argument that minimizes the limiting quadratic s0. Besides, as a consequence of equation (3), we have that under misspecification mn(ρ0) is not necessarily centered on zero.
Notice however that when p is correctly specified, then m0(ρ0) = 0 is necessarily true because of the uniqueness assumption (for an argument, see, e.g., White, 1982). On the basis of these considerations on the almost sure limit of
, let us now turn to consider its asymptotic distribution.
4.1. The Large Sample Distribution of the Parameter Estimator
The starting point for dealing with the large sample approximation of
is the first-order condition for sn(ρ),
where
.
The next step is to apply a Taylor series approximation to
. Let
be the ith coordinate of
. From Serfling (1980, Theorem B, p. 44), there exists a ρin such that
and
If we let Mn be the matrix that is obtained by stacking (∂/∂ρT)min(ρin), then stacking the preceding scalar equations gives the vector approximation
Notice that Mn converges almost surely to M0. Now let
; then substituting (6) into (5) gives
This equation holds whether the model is correctly or incorrectly specified and relates
to mn(ρ0). Under correct specification (7) is all that is needed to obtain the large sample distribution of
because, from the remarks at the beginning of this section, m0(ρ0) = 0 and therefore from Lemma 1 mn(ρ0) is itself asymptotically normal centered on zero. When the model is incorrectly specified it may happen that m0(ρ0) ≠ 0, but in this case we can write
The first term of (8) converges in law to a normal distribution from Lemma 1. The presence of the second term will require dealing with the joint distribution of
, which in turn leads to considering the joint distribution of
. For that reason we require the next assumption.
Assumption 4. Let
Then, each element of c(x,θ), c(x,θ)cT(x,θ), and (∂/∂θT)c(x,θ) is continuous and dominated by the function b1(x) from Assumption 2.
The following lemma establishes the asymptotic joint normal distribution of
.
LEMMA 2. Under Assumptions 1–4,
which is partitioned accordingly.
To find the expression of the covariance matrix for
additional notation is required.
Notation 3. Let k be any natural number, Ik be the identity matrix1
Notice that because 0 is not a natural number, we can keep the notation I0 for the information matrix. We realize that the notation is confusing, but we prefer to keep the usual notation for both the identity and the information matrices.
Qi(ρ,θ) = (∂2/∂θT∂ρ)mi(ρ,θ), the second crossed partial derivative of mi(ρ,θ),
The previous notation makes the implicit assumption that the matrix W0 + G1 is nonsingular. It will be shown in the proof of the next theorem that in fact it is equal to ½(∂2/∂ρT∂ρ)s0(ρ0), which is the Hessian of s0(ρ) evaluated at the unique minimum. A referee pointed out that this Hessian may be singular. From Assumption 3 it is assumed to be nonsingular.
THEOREM 4. Under Assumptions 1–4,
What was the impact of model p being incorrectly specified? The expression m0(ρ0) ≠ 0, implies that G1 ≠ 0, G2 ≠ 0, and G3 ≠ 0. Here G1 and G2 depend on second-order derivatives of the moment function with respect to ρ and θ, and the covariance matrix of
depends on the joint covariance matrix of
.
We now discuss the implications of the construction of a misspecification consistent estimator of Vρ. First it is necessary to obtain a consistent estimator of
. This could be done by computing
To estimate G1, it is necessary to estimate H(ρ0,θ0), and this matrix involves blocks of the form (∂2/∂ρT∂ρ)mi(ρ0,θ0). The problem is similar in nature to that of estimating M0 = (∂/∂ρT)m(ρ0,θ0), which has been already addressed by Gallant and Tauchen (1999b) using numerical derivatives. Similar considerations apply to matrices G2 and Q(ρ0,θ0). Finally, estimation of N0 could be performed as follows. If
is a long simulated sequence from
, a natural estimator is
Once again, if the analytical expressions are not available, then numerical derivatives could be used.
Hall (2000) and Hall and Inoue (2003) assume that misspecification is equivalent to the fact that the limiting moment function is different from zero. This is so because, as remarked by a reviewer, in the GMM setup the model consists of the moment conditions, so that misspecification is the same as m0(ρ0) ≠ 0. But in our case this is not necessarily so, because the model may be misspecified and a finite set of moments could be zero. This results from the fact that a finite set of moments does not, in general, define the distribution. In such a situation the score is unable to detect misspecification. In the case of EMM the impact due to misspecification is summarized in the next corollary.
COROLLARY 1. Under Assumptions 1–4, and if m0(ρ0) = 0, then
Notice that under correct specification, N0 = J0, and then the matrix Vρ from Corollary 1 collapses into W0−1, which is the result mentioned in Gallant and Tauchen (1996). Hence Corollary 1 shows that when the moment conditions hold but the model is misspecified, then
is centered around zero, but the formula for the covariance matrix differs substantially from W0−1, the covariance under correct specification. Observe that Corollary 1 also suggests an improvement in the estimation of the variance covariance matrix that is consistent, even in the presence of model misspecification when there is no evidence that m0(ρ0) ≠ 0. If we let
and if
is a consistent estimator of N0, then
is a consistent estimator of Vρ of Corollary 1.
4.2. The Large Sample Distribution of the Moment Function
The estimated moment function
plays an important role in the diagnosis of the model of interest when there is evidence that m0(ρ0) ≠ 0 holds true; hence it may be important to obtain its large sample distribution under model misspecification. Its distribution can be derived from the asymptotic distribution of
. In fact, subtracting m0(ρ0) from both sides of (6) gives
Performing a Taylor series approximation similar to the one that led to (6), there exists
such that it converges with probability one to
. Substitution of this last equality into (10) produces
From formula (A.1) in the proof of Theorem 4 in the Appendix there exist matrices
such that
Then substituting (12) into (11) produces an expression of
as a matrix times the vector
, and this is enough to derive its large sample distribution. The following notation is useful to describe the corresponding results.
Notation 4. Let
THEOREM 5. Under Assumptions 1–4,
Correct specification necessarily implies m0(ρ0) = 0, but, as we pointed out before, it may happen that m0(ρ0) = 0 but there is model misspecification. In that case
is centered around zero, but misspecification impacts Vm, G5 = W0−1, G2 = 0, and G3 = 0, and the only block of Vm that contributes is Vθ, because in that case matrix A becomes
and then the large sample distribution of
is
This would be the large sample distribution of the estimated moments for a misspecified model and when the score generator is not able to detect misspecification. If in fact the model is correctly specified then N0 = J0, and the covariance matrix Σm0 simplifies to
and hence from (13) and Theorem 5
converges in law to N(0,I0 − M0W0−1M0T), which is the result obtained in Gallant and Tauchen (1996).
To obtain a covariance estimator for
that is consistent under model misspecification, the same computational requirements apply as in the case of
.
4.3. Behavior of the Test for Overidentifying Restrictions
An interesting aspect of this methodology is the availability of a global test for misspecification of model p. It is given in terms of
, the estimated moment vector, the logic being that if the size of this vector is too large then that would be evidence against model p. To formalize this idea a test statistic has been proposed, namely,
The following theorems characterize the large sample behavior of L0. They emphasize the fact that m0(ρ0) ≠ 0 is crucial in the ability of the test to detect misspecification.
THEOREM 6. Under Assumptions 1–4,
(1) If m0(ρ0) ≠ 0, then for every c > 0 we have that P(L0 > c) → 1 when n → ∞.
(2) If m0(ρ0) = 0, then
The first statement in Theorem 6 shows that L0 is consistent in detecting model misspecification if m0(ρ0) ≠ 0, at least for the i.i.d. situation.
Theorem 6 also shows that if model p is misspecified and the score generator is poor, in the sense that m0(ρ0) = 0, then L0 converges in law to a quadratic form of a normal random vector with mean zero. This quadratic form may or may not have a chi-squared distribution, depending on whether or not a condition holds true. To discuss this condition the following lemma is useful.
LEMMA 3. Let U0 be defined as in Notation 4. Then
(1) U0 is an idempotent matrix.
(2) U0TI0−1U0 = I0−1U0 = U0TI0−1.
The following theorem establishes a necessary and sufficient condition for the limiting distribution of Theorem 6(2) to be chi-squared.
THEOREM 7. Under Assumptions 1–4, if m0(ρ0) = 0, then L0 has a central chi-squared distribution if and only if
in which case the degrees of freedom is trace(I0−1Σm0).
It is not clear whether (16) is true in general, but the following corollary states a sufficient condition for the quadratic form to have a chi-squared distribution. Condition (17) is obtained by “canceling” the first five matrices on both sides of (16).
COROLLARY 2. Under Assumptions 1–4, if m0(ρ0) = 0,L0 has a central chi-squared distribution if
in which case the degrees of freedom is dim(θ) − dim(ρ).
Theorem 7 helps us understand the following situation. It may happen that a researcher fits a simple score generator, say, f1, and the corresponding test of overidentifying constraints, say, L01, is not significant. At first glance this may be interpreted as model p being correctly specified. Then the experimenter may fit a more complex score, say, f2, and the associated test of overidentifying constraints, say L02, is significant, thus giving the indication that model p is incorrectly specified. Which is the right conclusion? The explanation for this apparent paradox is that for the first score generator m01(ρ01) is indistinguishable from zero, even if model p is incorrectly specified, and the corresponding L01 behaves as if it were coming from a central chi-squared distribution. Then in the second case L02 finds evidence that m02(ρ02) ≠ 0. One may say that for this example misspecification resides in some higher order moments that are captured by the second score generator. This example and the theory shown here stress the importance of distinguishing between correct specification and m0(ρ0) = 0.
A direct consequence of Corollary 2 is that under correct specification
because under correct specification, it was shown in (14) that Σm0 = I0 − M0W0−1M0T and hence the left-hand side of (17) becomes
but, from Notation 4, that is precisely U0T. Therefore from Corollary 2 the convergence stated in (18) is correct. This result is obtained in Gallant and Tauchen (1996).
5. BOOTSTRAPPING EFFICIENT METHOD OF MOMENTS
It was shown in the previous section that the asymptotic distribution of the EMM estimator is centered (multivariate) normal. Unfortunately, because its variance-covariance matrix is quite difficult to estimate, it also is difficult to generate a pivotal quantity that would allow the application of standard inferential procedures. In addition, there are considerations regarding the sensitivity of the results to the numerical approximation of the derivatives required. For these reasons, in this section we propose fitting the distribution of the EMM estimators using bootstrap.
To design a suitable resampling scheme that is able to do the job, two features of the problem in hand deserve our attention, namely, that the model is misspecified and that the sample is random. Hence, in the first place, if the researcher wants to prevent the effects of misspecification, he or she should use a resampling scheme that does not use the specified model. And in the second place, the resampling plan needs to generate a bootstrap sample that is random. Therefore, the natural resampling plan for obtaining the artificial sample is by sampling with replacement from the original sample because, first, the bootstrap sample will replicate the behavior of the data generating process without imposing any additional structure and, second, the observations of the artificial sample are conditionally independent. Formally, we consider that {Xt*}t=1n is a bootstrap sample drawn with replacement from
.
5.1. Bootstrapping Quasi–Maximum Likelihood Estimator
Bootstrap for extremum estimators is addressed by Arcones and Giné (1992). However, the conditions they require do not fit our assumptions. Davidson and MacKinnon (1999) and Andrews (2002) have considered the problem from a computational point of view, but there are no references in those papers to any other paper that provides the behavior of bootstrap extremum estimators. Thus, we tailor bootstrap results applicable to our problem. As usual in extremum estimation theory, we first study the convergence of the criterion function, and then the consistency result is obtained from the continuous mapping theorem. We first obtain the result for the auxiliary model.
Notation 5. Let
Here
denotes convergence in probability or in distribution of the bootstrap random variable for almost every realization of the original sample (for a formal definition of these modes of convergence, see, e.g., Bickel and Freedman, 1981, Theorem 2.1).
LEMMA 4. Under Assumptions 1 and 2,
As a consequence of the triangular inequality, we can state the following corollary.
COROLLARY 3. Under Assumptions 1 and 2,
Now, we extend the preceding pointwise convergence results to uniform convergence.
LEMMA 5. Under Assumptions 1 and 2,
Given the uniform convergence of the criterion functions, the next is a standard result in the extremum estimator theory.
COROLLARY 4. Then, under Assumptions 1 and 2,
We now state the consistency of the bootstrap distribution.
THEOREM 8. Under Assumptions 1 and 2, then
5.2. Bootstrapping Efficient Method of Moments
Once the asymptotic normality result of the bootstrap for the auxiliary model is available, we focus on the properties of the bootstrap for the parameters of interest. Again, we state first the convergence properties of the minimizing function. Henceforth, let ∥·∥ denote the euclidean norm in the corresponding finite-dimensional vector space.
LEMMA 6. Under Assumptions 1–3,
COROLLARY 5. Under Assumptions 1–3,
COROLLARY 6. Under Assumptions 1–3,
To show that the proposed bootstrap is able to fit the distribution of interest, we will mimic the steps given in Sections 3 and 4 using the same arguments but conditionally on the sample.
LEMMA 7. Under Assumptions 1–4,
Finally, we can state the distributional result for
.
THEOREM 9. Under Assumptions 1–4,
5.3. Applications of Bootstrap to Efficient Method of Moments
In this section we show two ways to use Theorem 9 to make inference on the misspecified model. Notably the second form will be much less demanding on the requirements of the bootstrap replications.
The first and most obvious form is using the bootstrap to obtain the percentiles of the distribution of
. This is set via the continuous mapping theorem; if w is a continuous function from the parameter space
into the real line, then using a Polya theorem type result, Serfling (1980, p. 18), we can write
Because inference for ρ0 is usually related to functions w, we can state the following corollary.
COROLLARY 7. Under Assumptions 1–4,
where, in general, FZ denotes the distribution function of the random variable Z.
Hence, we can use the percentiles of
to make inference about ρ0. If, as usual, the percentiles of w are estimated using Monte Carlo experiments, a large number of bootstrap replications will be needed, and because in each bootstrap replication a further Monte Carlo experiment is needed to compute
, the computational demands of the method can become enormous.
To reduce the computation requirements, we also suggest a different application of the bootstrap method to make inference on the misspecified model. This method is based on the results in Domínguez and Aguirre-Torres (2003), where it is shown that if
are B bootstrap replications of
, then, if ρ(i) is the ith entry of vector ρ,
where tB is a t-student distribution with B degrees of freedom. Notice that B in (19) remains fixed, and it may even be equal to one. A thorough discussion on the choice of B is given in Domínguez and Aguirre-Torres (2003). Equation (19) could easily be used as a pivotal quantity to get confidence intervals or to make hypothesis tests. The procedure can be straightforwardly generalized for inference on the whole vector, i.e., if A is a full row rank r × p matrix, then, if B > r,
where T2 is Hotelling's central T2 distribution. Hence (20) could be used to get confidence regions or to test hypotheses on Aρ0.
6. CONCLUSION
The paper studies EMM when the model of interest suffers from severe misspecification. This is modeled using a global misspecification. This analysis shows the severe consequences due to the fact that the moment constraints do not hold. The covariance matrices of the parameters and sample moments become increasingly complex, making it very difficult to obtain estimators that are consistent under model misspecification. Complexity arises from the fact that covariance depends on second-order derivatives of the moment functions with respect to both arguments and also on the joint covariance matrix of the parameter of the auxiliary model and its information matrix. The analysis also stresses the importance of differentiating correct specification from moment constraints; they are not equivalent, and the consequences of this assumption are also discussed. Also, as an alternative it is shown how a modification of bootstrap could be used to draw inferences in the presence of misspecification. This modification of the traditional bootstrap is particularly useful for a procedure such as EMM that is based on simulation.
APPENDIX
This Appendix contains the outline of the proofs of the main results of the paper. A detailed discussion is available at http://cursos.itam.mx/vaguirre/ProofsEMMmiss.
Outline of proof of Lemma 2. Using a Taylor series argument, the vector
is made tail equivalent to
where R0 = ∫(∂/∂θ)a(x,θ0)h(x) dx. Then, from Theorem 2 in Chapter 3 of Gallant (1987),
has a joint singular normal distribution. From Assumption 2, I0 is positive definite; hence the matrix inverse transformation is continuous and differentiable on a neighborhood around I0. Then,
is a differentiable transformation of a vector that is asymptotically normal. █
Outline of proof of Theorem 4. Consider (8). Performing Taylor series approximations similar to those that led to (6), there exist matrices
such that they converge with probability one to N0, G1, and G2 given in Notation 3, and
from Lemma 2 the right-hand side is asymptotically normal. It is now necessary to consider the behavior of
. This sequence of matrices converges with probability one to W0 + G1, which, by virtue of Corollary 30 in Chapter 4 of Dhrymes (1984), is equal to ½(∂2/∂ρT∂ρ)s0(ρ0). This is the Hessian of s0, which, from Assumption 3, is positive definite at ρ0 because it is a minimum. Then from Slutsky's theorem
converges in law to N(0,Vρ), where Vρ is given in Notation 3.
Define
. Then, the difference
may be expressed as
Outline of the proof of Lemma 4. Denote Zt = log f (Xt|θ) and Zt* = log f (Xt*|θ). Let ε,δ > 0. Then
, where Mt,n equals the number of times that Zt is selected in the bootstrap sample. Let k be such that E(|Z1|1(|Z1| > k)) < εδ/2. Define Zt,1 = Zt1(|Zt| > k) and Zt,2 = Zt1(|Zt| ≤ k). Then
On the one hand
using Tchebychev inequality for all sequences {Zt}. On the other hand, using Markov inequality
for n large enough because
█
Outline of the proof of Lemma 5. Use the dominated convergence theorem to show that ∀ε > 0,θ ∈ Θ ?bE? kε,θ > 0 such that
By compactness, there exists {B(θj,kε,θj) : θj ∈ Θ, j = 1,2,…,m}, a collection of open balls, that covers Θ. If Bj = B(θj,kε,θj) ∩ Θ, we have
Lemma 4 and the definition of Bj help to show that these three terms vanish. █
Outline of the proof of Theorem 8. Assume for simplicity that Θ is a subset of the real line. Applying the mean value theorem to the bootstrap first-order condition,
where
is between
. As in the preceding proofs, it follows that
In addition,
. Thus the assumptions of the Lindeberg–Feller central limit theorem hold for the triangular array
for almost all samples. █
Outline of the proof of Lemma 6. Because m(ρ,θ) is uniformly continuous in θ, select δ such that
. Hence
The first addend is zero, and the second vanishes a.s. from Corollary 4. █
Outline of the proof of Corollary 5. Notice that
, then result (2) follows. The first result follows analogously using the latter lemma, and
See (A.2). █
References
REFERENCES
Andrews, D.
(2002)
Higher order improvements of a computationally attractive
k-step bootstrap for extremum estimators.
Econometrica
70,
119–162.Google Scholar
Arcones, M. &
E. Giné
(1992)
On the bootstrap of M-estimators and other statistical
functionals. In
R. Lepage &
L. Billard (eds.),
Exploring the Limits of Bootstrap, pp.
13–49.
Wiley.
Bickel, P. &
D. Freedman
(1981)
Some asymptotic theory for the bootstrap.
Annals of Statistics
9,
1196–1217.Google Scholar
Davidson, R. &
J. MacKinnon
(1999)
Bootstrap testing in nonlinear models.
International Econometric Review
40,
487–508.Google Scholar
Dhrymes, P.J
(1984)
Mathematics for Econometrics,
2nd ed.
Springer-Verlag.
Domínguez, M.A. &
V. Aguirre-Torres
(2003)
Broadening the Scope of the Bootstrap in Complex Problems.
Working paper DE-C03.8,
Department of Statistics, ITAM.
Gallant, A.R.
(1987)
Nonlinear Statistical Models.
Wiley.
Gallant, A.R.,
D. Hsieh, &
G. Tauchen
(1997)
Estimation of volatility models with diagnostics.
Journal of Econometrics
81,
159–192.Google Scholar
Gallant, A.R. &
J.R. Long
(1997)
Estimating stochastic differential equations efficiently by minimum
chi-squared.
Biometrika
84,
125–141.Google Scholar
Gallant, A.R. &
D.W. Nychka
(1987)
Seminonparametric maximum likelihood estimation.
Econometrica
55,
363–390.Google Scholar
Gallant, A.R. &
G. Tauchen
(1996)
Which moments to match?
Econometric Theory
12,
657–681.Google Scholar
Gallant, A.R. &
G. Tauchen
(1999a)
The relative efficiency of method of moments estimators.
Journal of Econometrics
92,
149–172.Google Scholar
Gallant, A.R. &
G. Tauchen
(1999b)
EMM: A program for efficient method of moments estimation. Version
1.4,
anonymous ftp at ftp.econ.duke.edu in directory
pub/get/emm.
Geweke, J.
(1981)
The approximate slopes of econometric tests.
Econometrica
49,
1427–1442.Google Scholar
Hall, A.R.
(2000)
Covariance matrix estimation and the power of the overidentifying
restrictions test.
Econometrica
68,
1517–1527.Google Scholar
Hall, A.R., &
Inoue, A.
(2003)
The large sample behaviour of the generalized method of moments
estimator in misspecified models.
Journal of Econometrics
114,
361–394.Google Scholar
Newey, W.K.
(1985)
Generalized method of moments specification testing.
Journal of Econometrics
29,
229–256.Google Scholar
Serfling, R.J.
(1980)
Approximation Theorems of Mathematical Statistics.
Wiley.
Tauchen, G.
(1997)
New minimum chi-square methods in empirical finance. In
D. Kreps &
K. Wallis (eds.),
Advances in Econometrics, pp.
279–317.
Cambridge University Press.
You have
Access
- 2
- Cited by