Strong Consistency of the Maximum Likelihood Estimator in Generalized Linear and Nonlinear Mixed-Effects Models

Generalized linear and nonlinear mixed-effects models are used extensively in biomedical, social, and agricultural sciences. The statistical analysis of these models is based on the asymptotic properties of the maximum likelihood estimator. However, it is usually assumed that the maximum likelihood estimator is consistent, without providing a proof. A rigorous proof of the consistency by verifying conditions from existing results can be very difficult due to the integrated likelihood. In this paper, we present some easily verifiable conditions for the strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Based on this result, we prove that the maximum likelihood estimator is consistent for some frequently used models such as mixed-effects logistic regression models and growth curve models.     

Estimation of a dynamic stochastic frontier model using likelihood-based approaches

This paper considers a panel stochastic production frontier model that allows the dynamic adjustment of technical inefficiency. In particular, we assume that inefficiency follows an AR(1) process. That is, the current year's inefficiency for a firm depends on its past inefficiency plus a transient inefficiency incurred in the current year. Interfirm variations in the transient inefficiency are explained by some firm-specific covariates. We consider four likelihood-based approaches to estimate the model: the full maximum likelihood, pairwise composite likelihood, marginal composite likelihood, and quasi-maximum likelihood approaches. Moreover, we provide Monte Carlo simulation results to examine and compare the finite-sample performances of the four above-mentioned likelihood-based estimators of the parameters. Finally, we provide an empirical application of a panel of 73 Finnish electricity distribution companies observed during 2008–2014 to illustrate the working of our proposed models.     

Analytic derivatives and the computation of GARCH estimates

In the context of univariate GARCH models we show how analytic first and second derivatives of the log-likelihood can be successfully employed for estimation purposes. Maximum likelihood GARCH estimation usually relies on the numerical approximation to the log-likelihood derivatives, on the grounds that an exact analytic differentiation is much too burdensome. We argue that this is not the case and that the computational benefit of using the analytic derivatives (first and second) may be substantial. Furthermore, we make a comparison of various gradient algorithms that are used for the maximization of the GARCH Gaussian likelihood. We suggest the implementation of a globally efficient computation algorithm that is obtained by suitably combining the use of the estimated information matrix with that of the exact Hessian during the maximization process. As this would appear a straightforward extension, we then study the finite sample performance of the exact Hessian and its approximations (that is, the estimated information, outer products and misspecification robust matrices) in inference.     

On price exogeneity in complete demand systems

Complete demand systems have always been estimated under the assumption that either prices or quantities were exogenous. In this paper, we introduce some explicit price adjustment process and test the assumption of exogeneity of prices. Comparing the model with endogenous prices to the model with exogenous prices will reveal that the assumption of the exogeneity of prices is not a very dramatic one as both estimated models are very similar in many experiments with American and Canadian data. Applying a system-wide Wu-Hausman test and a likelihood ratio test, will always lead us to reject the exogeneity of prices in three-commodity models, while at the more disaggregated level, prices may often be considered as exogenous.     

Bayesian parametric and semiparametric factor models for large realized covariance matrices

This paper introduces a new factor structure suitable for modeling large realized covariance matrices with full likelihood‐based estimation. Parametric and nonparametric versions are introduced. Because of the computational advantages of our approach, we can model the factor nonparametrically as a Dirichlet process mixture or as an infinite hidden Markov mixture, which leads to an infinite mixture of inverse‐Wishart distributions. Applications to 10 assets and 60 assets show that the models perform well. By exploiting parallel computing the models can be estimated in a matter of a few minutes.     

Inside the black box of high-growth firms in a crisis-hit economy: corporate strategy,employee human capital and R&D capabilities

In an attempt to open the black box of high-growth firms within turbulent economic environments, this paper explores the role of corporate strategy, employee human capital and R&D capabilities in achieving exceptional growth performance in a crisis-hit economy. Relative and absolute growth measures based on both employment and sales are computed utilizing survey data on 1500 firms in the midst of the Greek crisis. Our findings indicate that adopting a geographical diversification strategy significantly increases the likelihood of becoming a fast-growing firm, irrespective of the growth metric used. Entering in diverse product markets and taking advantage of R&D capabilities appear to additionally contribute to relative employment change in HGFs of smaller size. Based on the absolute employment growth indicator, we provide some evidence that HGFs of larger size are able to grow fast through product diversification, acquiring other firms or by investing on training low-skilled employees. Nevertheless, hiring already highly educated persons seems to matter only for sales HGFs, while research collaborations are found to negatively affect the probability of growing fast in terms of sales.

     

A new approach to maximum likelihood estimation of sum-constrained linear models in case of undersized samples

Maximum likelihood procedures for estimating sum-constrained models like demand systems, brand choice models and so on, break down or produce very unstable estimates when the number of categories ( n ) is large as compared with the number of observations ( T ). In applied research, this problem is usually resolved by postulating the contemporaneous covariance matrix of the dependent variables to be known apart from a constant of proportionality. In this paper we develop a maximum likelihood procedure for sum-constrained models with large numbers of categories, which does not require too many observations, but nevertheless allows for n covariance parameters to be estimated freely.     

Statistical Implementations of Agent-Based Demographic Models

A variety of demographic statistical models exist for studying population dynamics when individuals can be tracked over time. In cases where data are missing due to imperfect detection of individuals, the associated measurement error can be accommodated under certain study designs (e.g. those that involve multiple surveys or replication). However, the interaction of the measurement error and the underlying dynamic process can complicate the implementation of statistical agent-based models (ABMs) for population demography. In a Bayesian setting, traditional computational algorithms for fitting hierarchical demographic models can be prohibitively cumbersome to construct. Thus, we discuss a variety of approaches for fitting statistical ABMs to data and demonstrate how to use multi-stage recursive Bayesian computing and statistical emulators to fit models in such a way that alleviates the need to have analytical knowledge of the ABM likelihood. Using two examples, a demographic model for survival and a compartment model for COVID-19, we illustrate statistical procedures for implementing ABMs. The approaches we describe are intuitive and accessible for practitioners and can be parallelised easily for additional computational efficiency.     

Diversification preferences in the theory of choice

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von Neumann–Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper analyzes axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion.     

On existence and uniqueness of maximum likelihood estimates in quantal and ordinal response models

Summary For quantal and ordinal response models, conditions on existence and uniqueness of maximum likelhood estimates are presented. Results are derived from general results on direction sets and spaces associated with a proper concave function. If each summand of the log likelihood is in any direction either strictly concave or affine, necessary and sufficient conditions are obtained. If all cell counts are strictly positive, then it is shown that estimates always exist, and that they are unique if all parameters are identifiable. If estimates exist without being unique, results on uniquely estimable linear functions are given, paralleling corresponding results in linear regression. An extension of the maximum likelihood principle is outlined yielding similar results even if the likelihood does not attain its supremum. The logit model, the linear probability model, cumulative and sequential models and binomial response models are considered in detail.     

Deriving target selection rules from endogenously selected samples

The selection of the most profitable customers in a customer database for targeted activities is often done based on observed behaviour in the past. Consequently, databases arising from the responses to, for example, direct mailings in the past are not random samples. When not all heterogeneity across customers is observed, target selection will be based on unobserved heterogeneity and hence it is endogenous. We develop a method to adjust the likelihood function of latent class models to correct for this endogenous sampling process. We apply this technique to the selection of mail targets for a Dutch charity. Based on a joint model for the response rate and the amount donated, we create a target selection rule that maximizes expected revenues. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimation of dynamic models with nonparametric simulated maximum likelihood

We propose an easy-to-implement simulated maximum likelihood estimator for dynamic models where no closed-form representation of the likelihood function is available. Our method can handle any simulable model without latent dynamics. Using simulated observations, we nonparametrically estimate the unknown density by kernel methods, and then construct a likelihood function that can be maximized. We prove that this nonparametric simulated maximum likelihood (NPSML) estimator is consistent and asymptotically efficient. The higher-order impact of simulations and kernel smoothing on the resulting estimator is also analyzed; in particular, it is shown that the NPSML does not suffer from the usual curse of dimensionality associated with kernel estimators. A simulation study shows good performance of the method when employed in the estimation of jump-diffusion models.     

Determinants and Dynamics of Current Account Reversals: An Empirical Analysis

We use panel probit models with unobserved heterogeneity, state dependence and serially correlated errors in order to analyse the determinants and the dynamics of current account reversals for a panel of developing and emerging countries. The likelihood‐based inference of these models requires high‐dimensional integration for which we use efficient importance sampling. Our results suggest that current account balance, terms of trades, foreign reserves and concessional debt are important determinants of current account reversal. Furthermore, we find strong evidence for serial dependence in the occurrence of reversals. While the likelihood criterion suggest that state dependence and serially correlated errors are essentially observationally equivalent, measures of predictive performance provide support for the hypothesis that the serial dependence is mainly due to serially correlated country‐specific shocks related to local political or macroeconomic events.     

Extracting a common stochastic trend: Theory with some applications

This paper investigates the statistical properties of estimators of the parameters and unobserved series for state space models with integrated time series. In particular, we derive the full asymptotic results for maximum likelihood estimation using the Kalman filter for a prototypical class of such models—those with a single latent common stochastic trend. Indeed, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator and show that the conventional method of inference is valid for this class of models. The models we explicitly consider comprise a special–yet useful–class of models that may be employed to extract the common stochastic trend from multiple integrated time series. Such models can be very useful to obtain indices that represent fluctuations of various markets or common latent factors that affect a set of economic and financial variables simultaneously. Moreover, our derivation of the asymptotics of this class makes it clear that the asymptotic Gaussianity and the validity of the conventional inference for the maximum likelihood procedure extends to a larger class of more general state space models involving integrated time series. Finally, we demonstrate the utility of this class of models extracting a common stochastic trend from three sets of time series involving short- and long-term interest rates, stock return volatility and trading volume, and Dow Jones stock prices.     

Autocorrelation and dynamic methodology with an application to wage determination models

A demonstration is provided of rigorous, statistical methodology whereby both the type and order of an error process can be identified in dynamic, single equation econometric models. The paper relies heavily upon maximum likelihood estimation, nested likelihood ratio tests and the overfitting or exponentially weighted procedure for model selection. An application of the methodology to a class of quarterly wage determination models is included.     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

Using the Fraction of Missing Information to Identify Auxiliary Variables for Imputation Procedures via Proxy Pattern‐mixture Models

In many surveys, imputation procedures are used to account for non‐response bias induced by either unit non‐response or item non‐response. Such procedures are optimised (in terms of reducing non‐response bias) when the models include covariates that are highly predictive of both response and outcome variables. To achieve this, we propose a method for selecting sets of covariates used in regression imputation models or to determine imputation cells for one or more outcome variables, using the fraction of missing information (FMI) as obtained via a proxy pattern‐mixture (PMM) model as the key metric. In our variable selection approach, we use the PPM model to obtain a maximum likelihood estimate of the FMI for separate sets of candidate imputation models and look for the point at which changes in the FMI level off and further auxiliary variables do not improve the imputation model. We illustrate our proposed approach using empirical data from the Ohio Medicaid Assessment Survey and from the Service Annual Survey.     

An alternative quasi likelihood approach,Bayesian analysis and data-based inference for model specification

This paper studies an alternative quasi likelihood approach under possible model misspecification. We derive a filtered likelihood from a given quasi likelihood (QL), called a limited information quasi likelihood (LI-QL), that contains relevant but limited information on the data generation process. Our LI-QL approach, in one hand, extends robustness of the QL approach to inference problems for which the existing approach does not apply. Our study in this paper, on the other hand, builds a bridge between the classical and Bayesian approaches for statistical inference under possible model misspecification. We can establish a large sample correspondence between the classical QL approach and our LI-QL based Bayesian approach. An interesting finding is that the asymptotic distribution of an LI-QL based posterior and that of the corresponding quasi maximum likelihood estimator share the same “sandwich”-type second moment. Based on the LI-QL we can develop inference methods that are useful for practical applications under possible model misspecification. In particular, we can develop the Bayesian counterparts of classical QL methods that carry all the nice features of the latter studied in  White (1982). In addition, we can develop a Bayesian method for analyzing model specification based on an LI-QL.     

Optimal prepayment and default rules for mortgage-backed securities

We study the optimal stopping problems embedded in a typical mortgage. Despite a possible non-rational behaviour of the typical borrower of a mortgage, such problems are worth to be solved for the lender to hedge against the prepayment risk, and because many mortgage-backed securities pricing models incorporate this suboptimality via a so-called prepayment function which can depend, at time t, on whether the prepayment is optimal or not. We state the prepayment problem in the context of the optimal stopping theory and present an algorithm to solve the problem via weak convergence of computationally simple trees. Numerical results in the case of the Vasicek model and of the CIR model are also presented. The procedure is extended to the case when both the prepayment as well as the default are possible: in this case, we present a new method of building two-dimensional computationally simple trees, and we apply it to the optimal stopping problem.