Maximum likelihood estimation of random effects models

Iterated GLS has a remarkable property when applied to the random effects model in its usual parameterization. The values for the parameter that measures relative variance, obtained through successive iterations, form a monotonic sequence. This property provides convenient checks for multiple maxima of the likelihood function and for existence of a local maximum that satisfies the non-negativity condition.     

Linear instrumental variable estimation of limited dependent variable models with endogenous explanatory variables

The purpose of this paper is to present and analyze an instrumental variables estimator for limited dependent variable models that does not require functional form assumptions for the distribution of disturbances. This estimator is a weighted instrumental variables estimator, where the weight is the ratio of a multivariate normal density to the actual density of the instrumental variables. A semi-non-parametric estimator of the weights is presented and some conjectures concerning the asymptotic distribution of the estimator are discussed.     

Efficient estimation of limited dependent variable models with endogenous explanatory variables

This paper discusses asymptotically efficient estimation of the parameters of limited dependent variable models with endogenous explanatory variables. General results on asymptotic efficiency of two-stage and Amemiya GLS estimators are derived and used to obtain a simple, asymptotically efficient estimator of the structural coefficients. This estimator can be calculated by applying GLS to estimates of the reduced form coefficients that are obtained by using reduced form residuals as additional explanatory variables. It is also shown that it is possible to obtain asymptotically efficient estimators of the other coefficients by a modified minimum chi-square method.     

Testing for exogeneity in limited dependent variable models using a simplified likelihood ratio statistic

A simple likelihood-ratio statistic for the weak exogeneity of the continuously observed endogenous variables is presented for the limited information simultaneous equations models in which a single endogenous variable is censored. The statistic is a likelihood ratio test statistic for the exclusion of the reduced form residuals of the continuously observed endogenous variables and is asymptotically locally most powerful. The procedure is illustrated by an application to a model of female labour supply.     

Maximum likelihood estimation of partially observed diffusion models

This paper develops a maximum likelihood (ML) method to estimate partially observed diffusion models based on data sampled at discrete times. The method combines two techniques recently proposed in the literature in two separate steps. In the first step, the closed form approach of Aït-Sahalia (2008) is used to obtain a highly accurate approximation to the joint transition probability density of the latent and the observed states. In the second step, the efficient importance sampling technique of Richard and Zhang (2007) is used to integrate out the latent states, thereby yielding the likelihood function. Using both simulated and real data, we show that the proposed ML method works better than alternative methods. The new method does not require the underlying diffusion to have an affine structure and does not involve infill simulations. Therefore, the method has a wide range of applicability and its computational cost is moderate.     

Maximum likelihood estimation for dynamic factor models with missing data

This paper concerns estimating parameters in a high-dimensional dynamic factor model by the method of maximum likelihood. To accommodate missing data in the analysis, we propose a new model representation for the dynamic factor model. It allows the Kalman filter and related smoothing methods to evaluate the likelihood function and to produce optimal factor estimates in a computationally efficient way when missing data is present. The implementation details of our methods for signal extraction and maximum likelihood estimation are discussed. The computational gains of the new devices are presented based on simulated data sets with varying numbers of missing entries.     

Maximum likelihood and restricted maximum likelihood estimation for a class of Gaussian Markov random fields

This work describes a Gaussian Markov random field model that includes several previously proposed models, and studies properties of its maximum likelihood (ML) and restricted maximum likelihood (REML) estimators in a special case. Specifically, for models where a particular relation holds between the regression and precision matrices of the model, we provide sufficient conditions for existence and uniqueness of ML and REML estimators of the covariance parameters, and provide a straightforward way to compute them. It is found that the ML estimator always exists while the REML estimator may not exist with positive probability. A numerical comparison suggests that for this model ML estimators of covariance parameters have, overall, better frequentist properties than REML estimators.     

Identification in nonparametric limited dependent variable models with simultaneity and unobserved heterogeneity

We extend the identification results for nonparametric simultaneous equations models in Matzkin (2008) to situations where the observations on the vector of dependent variables might be limited, and where the number of exogenous unobservable variables is larger than the number of dependent variables.     

Estimation of some limited dependent variable models with application to housing demand

A model which extends the switching regression models and combines several different limited dependent variable models into a general framework is introduced. Methods to get consistent estimates and asymptotic efficient estimates are derived. Our estimation procedures are then used to study a housing expenditure model which takes into account the simultaneous determination of whether or not to own, and how much to spend.     

Maximum likelihood estimation of stochastic frontier models by the Fourier transform

The paper is concerned with several kinds of stochastic frontier models whose likelihood function is not available in closed form. First, with output-oriented stochastic frontier models whose one-sided errors have a distribution other than the standard ones (exponential or half-normal). The gamma and beta distributions are leading examples. Second, with input-oriented stochastic frontier models which are common in theoretical discussions but not in econometric applications. Third, with two-tiered stochastic frontier models when the one-sided error components follow gamma distributions. Fourth, with latent class models with gamma distributed one-sided error terms. Fifth, with models whose two-sided error component is distributed as stable Paretian and the one-sided error is gamma. The principal aim is to propose approximations to the density of the composed error based on the inversion of the characteristic function (which turns out to be manageable) using the Fourier transform. Procedures that are based on the asymptotic normal form of the log-likelihood function and have arbitrary degrees of asymptotic efficiency are also proposed, implemented and evaluated in connection with output-oriented stochastic frontiers. The new methods are illustrated using data for US commercial banks, electric utilities, and a sample from the National Youth Longitudinal Survey.     

Maximum likelihood estimation and uniform inference with sporadic identification failure

This paper analyzes the properties of a class of estimators, tests, and confidence sets (CSs) when the parameters are not identified in parts of the parameter space. Specifically, we consider estimator criterion functions that are sample averages and are smooth functions of a parameter θ$\theta$. This includes log likelihood, quasi-log likelihood, and least squares criterion functions.     

Maximum likelihood estimation of a GARCH-stable model

Maximum likelihood is used to estimate a generalized autoregressive conditional heteroskedastic (GARCH) process where the residuals have a conditional stable distribution (GARCH-stable). The scale parameter is modelled such that a GARCH process with normally distributed residuals is a special case. The usual methods of estimating the parameters of the stable distribution assume constant scale and will underestimate the characteristic exponent when the scale parameter follows a GARCH process. The parameters of the GARCH-stable model are estimated with daily foreign currency returns. Estimates of characteristic exponents are higher with the GARCH-stable than when independence is assumed. Monte Carlo hypothesis testing procedures, however, reject our GARCH-stable model at the 1% significance level in four out of five cases.     

Maximum likelihood estimation of monotone and concave production frontiers

In this paper we bring together the previously separate parametric and nonparametric approaches to production frontier estimation by developing composed error models for maximum likelihood estimation from nonparametrically specified classes of frontiers. This approach avoids the untestable restrictions of parametric functional forms and also provides a statistical foundation for nonparametric frontier estimation. We first examine the single output setting and then extend our formulation to the multiple output setting. The key step in developing the estimation problems is to identify operational constraint sets to ensure estimation from the desired class of frontiers. We also suggest algorithms for solving the resulting constrained likelihood function optimization problems.The refereeing process of this paper was handled through R. Robert Russell. Helpful comments from Bob Russell and two anonymous referees are gratefully acknowedged. We are, of course, solely responsible for any remaining errors or omissions.     

Maximum likelihood estimation based on the Laplace approximation for p2 network regression models

The class of p₂ models is suitable for modeling binary relation data in social network analysis. A p₂ model is essentially a regression model for bivariate binary responses, featuring within‐dyad dependence and correlated crossed random effects to represent heterogeneity of actors. Despite some desirable properties, these models are used less frequently in empirical applications than other models for network data. A possible reason for this is due to the limited possibilities for this model for accounting for (and explicitly modeling) structural dependence beyond the dyad as can be done in exponential random graph models. Another motive, however, may lie in the computational difficulties existing to estimate such models by means of the methods proposed in the literature, such as joint maximization methods and Bayesian methods. The aim of this article is to investigate maximum likelihood estimation based on the Laplace approximation approach, that can be refined by importance sampling. Practical implementation of such methods can be performed in an efficient manner, and the article provides details on a software implementation using R . Numerical examples and simulation studies illustrate the methodology.     

Estimation of limited dependent variable models by ordinary least squares and the method of moments

Under certain conditions, a broad class of qualitative and limited dependent variable models can be consistently estimated by the method of moments using a non-iterative correction to the ordinary least squares estimator, with only a small loss of efficiency compared to maximum likelihood estimation. The class of models is that obtained from a classical multinormal regression by any type of censoring or truncation and includes the tobit, probit, two-limit probit, truncated regression, and some variants of the sample selection models. The paper derives the estimators and their asymptotic covariance matrices.     

Bayesian inference in limited dependent variable models. An application to measuring strike duration

This paper presents a straightforward set of Bayesian techniques for analyzing models involving limited dependent variables; the techniques are demonstrated in an analysis of Kennan's (1985) data on contract strikes in US manufacturing. The data are analyzed by deriving posterior distributions—including probability distributions—of hazard functions for strike duration using numerical Monte Carlo methods. The distributions are employed to derive coverage intervals for hazard functions, to assess the relative plausibility of nonnested hypotheses concerning the shape of the functions, and to assess the impact of industrial production on duration.     

Maximum likelihood estimation of seemingly unrelated stochastic frontier regressions

This paper explores the changes in value added (VA) of a sample of schools for cohorts of students finishing secondary education between 2005 and 2008. VA estimates are based on distance measures obtained from DEA models. These measures are computed for each pupil in each school, and evaluate the distance between the school frontier in a given year and a pooled frontier comprising all schools analysed. The school VA is then computed by aggregating the VA scores for the cohort of pupils attending that school in a given year. The ratio between VA estimates for two consecutive cohorts, that attended the school in different years, is taken as the index of VA change. However, the evolution of school performance over time should consider not only the movements of the school frontier, but should also take into account other effects, such as the proximity of the students to the best-practices, represented by the school frontier, observed over time. For that purpose we developed an enhanced Malmquist index to evaluate the evolution of school performance over time. One of the components of the Malmquist index proposed measures VA change, and the other measures the ability of all school students to move closer to their own school best practices over time. The approach developed is applied to a sample of Portuguese secondary schools.     

Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion

Consider N independent stochastic processes \((X_i(t), t\in [0,T])\), \(i=1,\ldots , N\), defined by a stochastic differential equation with random effects where the drift term depends linearly on a random vector \(\Phi _i\) and the diffusion coefficient depends on another linear random effect \(\Psi _i\). For these effects, we consider a joint parametric distribution. We propose and study two approximate likelihoods for estimating the parameters of this joint distribution based on discrete observations of the processes on a fixed time interval. Consistent and \(\sqrt{N}\)-asymptotically Gaussian estimators are obtained when both the number of individuals and the number of observations per individual tend to infinity. The estimation methods are investigated on simulated data and show good performances.