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We derive marginal conditions of optimality (i.e., Euler equations) for a general class of Dynamic Discrete Choice (DDC) structural models. These conditions can be used to estimate structural parameters in these models without having to solve for approximate value functions. This result extends to discrete choice models the GMM-Euler equation approach proposed by Hansen and Singleton (1982) for the estimation of dynamic continuous decision models. We first show that DDC models can be represented as models of continuous choice where the decision variable is a vector of choice probabilities. We then prove that the marginal conditions of optimality and the envelope conditions required to construct Euler equations are also satisfied in DDC models. The GMM estimation of these Euler equations avoids the curse of dimensionality associated to the computation of value functions and the explicit integration over the space of state variables. We present an empirical application and compare estimates using the GMM-Euler equations method with those from maximum likelihood and two-step methods.

A major stumbling block in multivariate discrete data analysis is the problem of evaluating the outcome probabilities that enter the likelihood function. Calculation of these probabilities involves high-dimensional integration, making simulation methods indispensable in both Bayesian and frequentist estimation and model choice. We review several existing probability estimators and then show that a broader perspective on the simulation problem can be afforded by interpreting the outcome probabilities through Bayes’ theorem, leading to the recognition that estimation can alternatively be handled by methods for marginal likelihood computation based on the output of Markov chain Monte Carlo (MCMC) algorithms. These techniques offer stand-alone approaches to simulated likelihood estimation but can also be integrated with traditional estimators. Building on both branches in the literature, we develop new methods for estimating response probabilities and propose an adaptive sampler for producing high-quality draws from multivariate truncated normal distributions. A simulation study illustrates the practical benefits and costs associated with each approach. The methods are employed to estimate the likelihood function of a correlated random effects panel data model of women's labor force participation.

Research on strategic choices available to the firm are often modeled as a limited number of possible decision outcomes and leads to a discrete limited dependent variable. A limited dependent variable can also arise when values of a continuous dependent variable are partially or wholly unobserved. This chapter discusses the methodological issues associated with such phenomena and the appropriate statistical methods developed to allow for consistent and efficient estimation of models that involve a limited dependent variable. The chapter also provides a road map for selecting the appropriate statistical technique and it offers guidelines for consistent interpretation and reporting of the statistical results.

This paper aims to introduce a new target tracking method based on risk theory in a 2-D discrete environment. After that, the related sensor scheduling method is proposed. This can make up the blank of target tracking and sensor management in the 2-D discrete environment.

The definition of risk is proposed based on risk decision theory firstly. Then the target tracking model in a two-dimensional discrete environment is built. The motion state updating and estimation method of target’s motion state based on Bayes theory is given. Thirdly, the method of computing sensor emission interception risk is provided. Afterwards, the optimization rule of obtaining the minimum risk is followed to model the sensor scheduling objective function. The lion algorithm is adjusted and improved combined with Chaos theory to generate the optimal sensor management projects.

The risk-based sensor target tracking method and sensor management method are both effective in a 2-D discrete environment.

To the best of the authors’ knowledge, this paper is the first to study the target tracking method and sensor scheduling method in a 2-D environment. Furthermore, the lion algorithm is improved combined with Chaos theory to show a better optimization performance.

Recent studies have examined tax policy issues using labour supply models characterised by a discretised budget set. Microsimulation modelling using a discrete hours approach is probabilistic. This makes analysis of the distribution of income difficult as even for a small sample with a modest range of labour supply points the range of possible labour supply combinations over the sample is extremely large. This paper proposes a method of approximating measures of income distribution and compares the performance of this method to alternative approaches in a microsimulation context. In this approach a pseudo income distribution is constructed, which uses the probability of a particular labour supply value occurring (standardised by the population size) to refer to a particular position in the pseudo income distribution. This approach is compared to using an expected income level for each individual and to a simulated approach, in which labour supply values are drawn from each individual’s hours distribution and summary statistics of the distribution of income are calculated by taking the average over each set of draws. The paper shows that the outcomes of various distributional measures using the pseudo method converge quickly to their true values as the sample size increases. The expected income approach results in a less accurate approximation. To illustrate the method, we simulate the distributional implications of a tax reform using the Melbourne Institute Tax and Transfer Simulator.

There is a growing literature in nonparametric econometrics in the recent two decades. Given the space limitation, it is impossible to survey all the important recent developments in nonparametric econometrics. Therefore, we choose to limit our focus on the following areas. In Section 2, we review the recent developments of nonparametric estimation and testing of regression functions with mixed discrete and continuous covariates. We discuss nonparametric estimation and testing of econometric models for nonstationary data in Section 3. Section 4 is devoted to surveying the literature of nonparametric instrumental variable (IV) models. We review nonparametric estimation of quantile regression models in Section 5. In Sections 2–5, we also point out some open research problems, which might be useful for graduate students to review the important research papers in this field and to search for their own research interests, particularly dissertation topics for doctoral students. Finally, in Section 6 we highlight some important research areas that are not covered in this paper due to space limitation. We plan to write a separate survey paper to discuss some of the omitted topics.

Patients and health professionals often make decisions which involve a choice between discrete alternatives. This chapter reviews the econometric methods which have been developed for modelling discrete choices and their application in the health economics literature. We start by reviewing the multinomial and mixed logit models and then consider issues such as scale heterogeneity, estimation in willingness to pay space and attribute non-attendance.

Choice under risk has a large stochastic (unpredictable) component. This chapter examines five stochastic models for binary discrete choice under risk and how they combine with “structural” theories of choice under risk. Stochastic models are substantive theoretical hypotheses that are frequently testable in and of themselves, and also identifying restrictions for hypothesis tests, estimation and prediction. Econometric comparisons suggest that for the purpose of prediction (as opposed to explanation), choices of stochastic models may be far more consequential than choices of structures such as expected utility or rank-dependent utility.

The search for flexible models has led the simple multinomial logit model to evolve into the powerful but computationally very demanding mixed multinomial logit (MMNL) model. That flexibility search lead to discrete choice hybrid choice models (HCMs) formulations that explicitly incorporate psychological factors affecting decision making in order to enhance the behavioral representation of the choice process. It expands on standard choice models by including attitudes, opinions, and perceptions as psychometric latent variables.

In this paper we describe the classical estimation technique for a simulated maximum likelihood (SML) solution of the HCM. To show its feasibility, we apply it to data of stated personal vehicle choices made by Canadian consumers when faced with technological innovations.

We then go beyond classical methods, and estimate the HCM using a hierarchical Bayesian approach that exploits HCM Gibbs sampling considering both a probit and a MMNL discrete choice kernel. We then carry out a Monte Carlo experiment to test how the HCM Gibbs sampler works in practice. To our knowledge, this is the first practical application of HCM Bayesian estimation.

We show that although HCM joint estimation requires the evaluation of complex multi-dimensional integrals, SML can be successfully implemented. The HCM framework not only proves to be capable of introducing latent variables, but also makes it possible to tackle the problem of measurement errors in variables in a very natural way. We also show that working with Bayesian methods has the potential to break down the complexity of classical estimation.

I review the burgeoning literature on applications of Markov regime switching models in empirical finance. In particular, distinct attention is devoted to the ability of Markov Switching models to fit the data, filter unknown regimes and states on the basis of the data, to allow a powerful tool to test hypotheses formulated in light of financial theories, and to their forecasting performance with reference to both point and density predictions. The review covers papers concerning a multiplicity of sub-fields in financial economics, ranging from empirical analyses of stock returns, the term structure of default-free interest rates, the dynamics of exchange rates, as well as the joint process of stock and bond returns.