Properties of the CUE estimator and a modification with moments
Introduction
2SLS is by far the most-used estimator for the simultaneous equation problem. However, it is now well-recognized that 2SLS can exhibit substantial finite sample (second-order) bias when the model is over-identified and the first stage partial r-squared is low. The initial recommendation to solve this problem was to do LIML, e.g. Bekker (1994) or Staiger and Stock (1997). However, Hahn, Hausman, and Kuersteiner (HHK, 2004) demonstrated that the “moment problem” of LIML led to undesirable estimates in this situation. Morimune (1983) analyzed both the bias in 2SLS and the lack of moments in LIML. While it was long known that LIML did not have finite sample moments, it was less known that this lack of moments led to the undesirable property of considerable dispersion in the estimates, e.g. the interquartile range was much larger than 2SLS. HHK developed a jackknife 2SLS (J2SLS) estimator that attenuated the 2SLS bias problem and had good dispersion properties. They found in their empirical results that the J2SLS estimator or the Fuller estimator, which modifies LIML to have moments, did well on both the bias and dispersion criteria. Since the Fuller estimator had smaller second-order MSE, HHK recommended using the Fuller estimator.
However, Bekker and van der Ploeg (2005) and Hausman, Newey and Woutersen (HNW, 2005) recognized that both Fuller and LIML are inconsistent with heteroscedasticity as the number of instruments becomes large in (Bekker, 1994; Chao and Swanson, 2005) many instrument sequences. Since econometricians recognize that heteroscedasticity is often present, this finding presents a problem. Hausman, Newey, Woutersen, Chao and Swanson (HNWCS, 2007) solved this problem by proposing jackknife LIML (HLIML) and jackknife Fuller (HFull) estimators that are consistent in the presence of heteroscedasticity. Newey and Windmeijer (2009a) showed that the continuous updating estimator (CUE), which is the GMM-like generalization of LIML from Hansen et al. (1996), is efficient relative to jackknife instrumental variables with many weak instruments. LIML does not have moments so HNWCS (2007) recommend using HFull, which does have moments.
A problem is that if serial correlation or clustering exists, neither HLIML, nor HFull, nor the CUE with heteroskedasticity consistent weighting matrix are consistent. The CUE will solve this problem, if the weighting matrix is made robust to serial correlation or clustering. The CUE also allows treatment of nonlinear specifications which the above estimators need not allow for. However, CUE suffers from the moment problem and exhibits wide dispersion.1 GMM does not suffer from the no-moments problem, but like 2SLS, GMM has finite sample bias that grows with the number of moments.
In this paper we modify the first-order conditions solved by the CUE by adding a term of order order to address the no-moments/large dispersion problem. To first order the variance of the estimator is the same as GMM or CUE, so no large sample efficiency is lost. The resulting estimator demonstrates considerably reduced bias relative to GMM and reduced dispersion relative to CUE. Thus, we expect the new estimator will be useful for empirical research. We next consider a similar approach but use a class of functions which permits us to specify an estimator with all integral moments existing. Lastly, we demonstrate how this approach can be extended to the entire family of Generalized Empirical Likelihood (GEL) estimators.
In the next section we specify a general nonlinear model of the type where GMM is often used. We then consider the finite sample bias of GMM. We discuss how CUE partly removes the bias but evidence demonstrates the presence of the no-moments/large dispersion problem. We next consider LIML and associated estimators under large instrument (moment) asymptotics and discuss why LIML, Fuller and HLIM and HFull will be inconsistent when time series or spatial correlation is present. We then give a specific modification to CUE and specify the “regularized” CUE estimator, RCUE, and we prove that a version of the estimator has moments in the case of linear moment restrictions. Lastly, we investigate our new estimators empirically. We find that they do not have either the bias arising from a large number of instruments (moments) or the no moment problem of CUE and LIML and perform well when heteroscedasticity and serial correlation are present. Thus, while further research is required to determine the optimal parameters of the modifications we present, the current estimators should allow improved empirical results in applied research.
Section snippets
Setup and motivation
Consider a sample of strictly stationary observations of random vectors . The -dimensional parameter of interest is defined by an -dimensional vector of unconditional moment restrictions for the moment functions . Denote Also, let and the th column of the Jacobian of the moment functions . Define the sequence as the smallest eigenvalue of
A modification of CUE with finite moments
It has been shown that the finite sample distribution of LIML does not have integral moments, Fuller (1977), and we would expect CUE to inherit these properties for most settings, especially if identification is weak, see Guggenberger (2005). To describe a modification of the CUE that would solve this moment problem, we use a transformation of the observations similarly to Kitamura and Stutzer (1997) to allow for autocorrelation or clustering. For a matrix let
Large sample properties
In this section we derive large sample properties of the RCUE estimator defined in Eq. (1), and show that the proposed estimator is asymptotically equivalent to the CUE estimator under many weak moment asymptotics. More specifically, we will consider the distribution of the estimator when the number of grows to infinity with sample size, and some eigenvalues of the matrix shrink to zero as specified in Assumption 1 below. Many weak moment asymptotics are motivated by theoretical and
Monte Carlo study
We performed a simulation study to investigate the properties of the proposed estimators, “Regularized” CUE () defined by the first-order conditions in Eq. (5), and (specified in Eq. (6)), and compare their performance with GMM, LIML, and CUE. In particular, we find that RCUE fixes the no-moments problem present in CUE while significantly reducing the over-identification bias of GMM in both the linear and nonlinear cases.
Extensions
In this section, we will propose a way of generalizing our approach to the class of Generalized Empirical Likelihood (GEL) estimators. As shown by Newey and Smith (2004), the CUE is also the solution to the dual problem where Taking derivatives, we obtain the first-order condition where denotes the generalized (Moore–Penrose Inverse) of a square matrix , and
Conclusion
In this paper we attempt to solve two problems: the bias in GMM which increases with the number of instruments (moment condition) and the “no moment” problem of CUE that leads to wide dispersion of the estimates. To date, much of the discussion of these problems has been under the assumption of homoscedasticity and absence of serial correlation, which limits their use in empirical research. HNWCS (2007) propose a solution to the homoscedasticity problem for the linear model. In this paper we
Acknowledgment
Newey thanks the NSF for research support.
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