On relative efficiency of Quasi-MLE and GMM estimators of covariance structure models

Optimal GMM is known to dominate Gaussian QMLE in terms of asymptotic efficiency (Chamberlain, 1984). I derive a new condition under which QMLE is as efficient as GMM for a general class of covariance structure models. The condition trivially holds for normal data but also identifies non-normal cases for which Gaussian QMLE is efficient.

Second Order Bias of Quasi-MLE for Covariance Structure Models

Several recent papers (e.g., Newey et al., 2005; Newey and Smith, 2004; Anatolyev, 2005) derive general expressions for the second-order bias of the GMM estimator and its first-order equivalents such as the EL estimator. Except for some simulation evidence, it is unknown how these compare to the second-order bias of QMLE of covariance structure models. The paper derives the QMLE bias formulas for this general class of models. The bias -- identical to the EL second-order bias under normality -- depends on the fourth moments of data and remains the same as for EL even for non-normal data so long as the condition for equal asymptotic efficiency of QMLE and GMM derived in Prokhorov (2009) is satisfied.(Q)MLE, GMM, EL, Covariance structures

GMM Redundancy Results for General Missing Data Problems

We consider questions of efficiency and redundancy in the GMM estimation problem in which we have two sets of moment conditions, where two sets of parameters enter into one set of moment conditions, while only one set of parameters enters into the other. We then apply these results to a selectivity problem in which the first set of moment conditions is for the model of interest, and the second set of moment conditions is for the selection process. We use these results to explain the counterintuitive result in the literature that, under an ignorability assumption that justifies GMM with weighted moment conditions, weighting using estimated probabilities of selection is better than weighting using the true probabilities. We also consider estimation under an exogeneity of selection assumption such that both the unweighted and the weighted moment conditions are valid, and we show that when weighting is not needed for consistency, it is also not useful for efficiency.c13

Efficient estimation of parameters in marginals in semiparametric multivariate models

Recent literature on semiparametric copula models focused on the situation when the marginals are specified nonparametrically and the copula function is given a parametric form. For example, this setup is used in Chen, Fan and Tsyrennikov (2006) [Efficient Estimation of Semiparametric Multivariate Copula Models, JASA] who focus on efficient estimation of copula parameters. We consider a reverse situation when the marginals are specified parametrically and the copula function is modelled nonparametrically. This setting is no less relevant in applications. We use the method of sieve for efficient estimation of parameters in marginals, derive its asymptotic distribution and show that the estimator is semiparametrically efficient. Simulations suggest that the sieve MLE can be up to 40% more efficient relative to QMLE depending on the strength of dependence between the marginals. An application using insurance company loss and expense data demonstrates empirical relevance of this setting.

Bartlett-type Correction of Distance Metric Test

We derive a corrected distance metric (DM) test of general restrictions. The correction factor depends on the value of the uncorrected statistic and the new statistic is Bartlett-type. In the setting of covariance structure models, we show using simulations that the quality of the new approximation is good and often remarkably good. Especially at around the 95th percentile, the distribution of the corrected test statistic is strikingly close to the relevant asymptotic distribution. This is true for various sample sizes, distributions, and degrees of freedom of the model. As a by-product we provide an intuition for the well-known observation in labor economic applications that using longer panels results in a reversal of the original inference.Distance Metric, GMM, Asymptotic expansion, Bartlett-type correction

Efficient estimation of parameters in marginal in semiparametric multivariate models

We consider a general multivariate model where univariate marginal distributions are known up to a common parameter vector and we are interested in estimating that vector without assuming anything about the joint distribution, except for the marginals. If we assume independence between the marginals and maximize the resulting quasi-likelihood, we obtain a consistent but inefficient estimate. If we assume a parametric copula (other than independence) we obtain a full MLE, which is efficient but only under correct copula specification and badly biased if the copula is misspecified. Instead we propose a sieve MLE estimator which improves over OMLE but does not suffer the drawbacks of the full MLE. We model the unknown part of the joint distribution using the Bernstein-Kantorovich polynomial copula and assess the resulting improvement over QMLE and over misspecified FMLE in terms of relative efficiency and robustness. We derive the asymptotic distribution of the new estimator and show that it reaches the semiparametric efficiency bound. Simulations suggest that the sieve MLE can be almost as efficient as FMLE relative to QMLE provided there is enough dependence between the marginals. An application using insurance company loss and expense data demonstrates empirical relevance of the estimator

Consistent Estimation of Linear Regression Models Using Matched Data

Economists often use matched samples, especially when dealing with earnings data where a number of missing observations need to be imputed. In this paper, we demonstrate that the ordinary least squares estimator of the linear regression model using matched samples is inconsistent and has a nonstandard convergence rate to its probability limit. If only a few variables are used to impute the missing data, then it is possible to correct for the bias. We propose two semiparametric bias-corrected estimators and explore their asymptotic properties. The estimators have an indirect-inference interpretation and they attain the parametric convergence rate if the number of matching variables is no greater than three. Monte Carlo simulations confirm that the bias correction works very well in such cases

Consistent Estimation of Linear Regression Models Using Matched Data

Economists often use matched samples, especially when dealing with earnings data where a number of missing observations need to be imputed. In this paper, we demonstrate that the ordinary least squares estimator of the linear regression model using matched samples is inconsistent and has a non-standard convergence rate to its probability limit. If only a few variables are used to impute the missing data then it is possible to correct for the bias. We propose two semi-parametric bias-corrected estimators and explore their asymptotic properties. The estimators have an indirectinference interpretation and their convergence rates depend on the number of variables used in matching. We can attain the parametric convergence rate if that number is no greater than three. Monte Carlo simulations confirm that the bias correction works very well in such cases