Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme of the text is the use of artificial regressions for estimation, reference, and specification testing of nonlinear models, including diagnostic tests for parameter constancy, serial correlation, heteroscedasticity, and other types of mis-specification. Explaining how estimates can be obtained and tests can be carried out, the authors go beyond a mere algebraic description to one that can be easily translated into the commands of a standard econometric software package. Covering an unprecedented range of problems with a consistent emphasis on those that arise in applied work, this accessible and coherent guide to the most vital topics in econometrics today is indispensable for advanced students of econometrics and students of statistics interested in regression and related topics. It will also suit practising econometricians who want to update their skills. Flexibly designed to accommodate a variety of course levels, it offers both complete coverage of the basic material and separate chapters on areas of specialized interest.

Estimation and Inference in Econometrics

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

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Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Statistical Analysis with Missing Data

Foreword Stephen P. Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng Introduction to MCMC, Charles J. Geyer A short history of Markov chain Monte Carlo: Subjective recollections from in-complete data, Christian Robert and George Casella Reversible jump Markov chain Monte Carlo, Yanan Fan and Scott A. Sisson Optimal proposal distributions and adaptive MCMC, Jeffrey S. Rosenthal MCMC using Hamiltonian dynamics, Radford M. Neal Inference and Monitoring Convergence, Andrew Gelman and Kenneth Shirley Implementing MCMC: Estimating with confidence, James M. Flegal and Galin L. Jones Perfection within reach: Exact MCMC sampling, Radu V. Craiu and Xiao-Li Meng Spatial point processes, Mark Huber The data augmentation algorithm: Theory and methodology, James P. Hobert Importance sampling, simulated tempering and umbrella sampling, Charles J.Geyer Likelihood-free Markov chain Monte Carlo, Scott A. Sisson and Yanan Fan MCMC in the analysis of genetic data on related individuals, Elizabeth Thompson A Markov chain Monte Carlo based analysis of a multilevel model for functional MRI data, Brian Caffo, DuBois Bowman, Lynn Eberly, and Susan Spear Bassett Partially collapsed Gibbs sampling & path-adaptive Metropolis-Hastings in high-energy astrophysics, David van Dyk and Taeyoung Park Posterior exploration for computationally intensive forward models, Dave Higdon, C. Shane Reese, J. David Moulton, Jasper A. Vrugt and Colin Fox Statistical ecology, Ruth King Gaussian random field models for spatial data, Murali Haran Modeling preference changes via a hidden Markov item response theory model, Jong Hee Park Parallel Bayesian MCMC imputation for multiple distributed lag models: A case study in environmental epidemiology, Brian Caffo, Roger Peng, Francesca Dominici, Thomas A. Louis, and Scott Zeger MCMC for state space models, Paul Fearnhead MCMC in educational research, Roy Levy, Robert J. Mislevy, and John T. Behrens Applications of MCMC in fisheries science, Russell B. Millar Model comparison and simulation for hierarchical models: analyzing rural-urban migration in Thailand, Filiz Garip and Bruce Western

Handbook of Markov Chain Monte Carlo

A detailed review of the education sector in Australia as in the data provided by the 2006 edition of the OECD's annual publication, 'Education at a Glance' is presented. While the data has shown that in almost all OECD countries educational attainment levels are on the rise, with countries showing impressive gains in university qualifications, it also reveals that a large of share of young people still do not complete secondary school, which remains a baseline for successful entry into the labour market.

/pdf/education-at-a-glance-m5vl7o68j5.pdf

Education at a Glance

A relevant problem in categorical data analysis is that the inference on the association between two variables may be affected by the presence of the so-called nuisance parameters, which correspond to the probabilities of the marginal distributions. The most well-known strategy to cope with this problem goes back to the Fisher's exact test and consists of conditioning the inference on the observed margins which are sufficient statistics for the nuisance parameters. The problem is particular relevant when we want to test the hypothesis that the variables of interest satisfy a certain type of positive association. We mention, in particular, two forms of association which are known as Positive Quadrant Dependence (PQD), introduced by Lehman, and Total positivity of order 2 (TP2), introduced by Karlin. These may be expressed by constraining certain association parameters to be non-negative. In particular, the parameters used to express PDQ are the so-called global log-odds ratios and those used to express TP2 are the so-called local log-odds ratios. 

In this paper we illustrate an exact conditional approach which has been recently developed by Bartolucci and Scaccia to test certain forms of positive association, such as PQD and TP2, between two ordinal variables. The approach is based on the use of a test statistic measuring the goodness-of-fit of the model formulated according to the type of association of interest. By conditioning the inference on the observed margins, we remove the uncertainty on the conclusion of the test. Since the multivariate generalized hypergeometric distribution, which results from this type of conditioning, is normally intractable, Markov chain Monte Carlo methods are used to obtain maximum likelihood estimates of the parameters of the constrained model. The Pearson's chi-squared statistics is used as a test statistic; a p-value for this statistic is computed by simulation, when the data are sparse, or by exploiting the asymptotic theory based on the chi-bar squared distribution. We also outline the extension of the present approach to deal with bivariate contingency tables, stratified according to one or more explanatory discrete variables. 

The approach is illustrated through an application based on data on social mobility.

Exact Conditional Testing of Certain Forms of Positive Association for Bivariate Ordinal Data

The evaluation of nursing homes is usually based on the administration of questionnaires made of a large number of polytomous items. In such a context, the Latent Class (LC) model represents a useful tool for clustering subjects in homogenous groups corresponding to different degrees of impairment of the health conditions. It is known that the performance of model-based clustering and the accuracy of the choice of the number of latent classes may be affected by the presence of irrelevant or noise variables. In this paper, we show the application of an item selection algorithm to real data collected within a project, named ULISSE, on the quality-of-life of elderly patients hosted in italian nursing homes. This algorithm, which is closely related to that proposed by Dean and Raftery in 2010, is aimed at finding the subset of items which provides the best clustering according to the Bayesian Information Criterion. At the same time, it allows us to select the optimal number of latent classes. Given the complexity of the ULISSE study, we perform a validation of the results by means of a sensitivity analysis to different specifications of the initial subset of items and of a resampling procedure.

Item selection by Latent Class-based methods

We develop a recursion for hidden Markov model of any order h, which allows us to obtain the posterior distribution of the latent state at every occasion, given the previous h states and the observed data. With respect to the well-known Baum-Welch recursions, the proposed recursion has the advantage of being more direct to use and, in particular, of not requiring dummy renormalizations to avoid numerical problems. We also show how this recursion may be expressed in matrix notation, so as to allow for an efficient implementation, and how it may be used to obtain the
manifest distribution of the observed data and for parameter estimation within the Expectation-Maximization algorithm. The approach is illustrated by an application
to nancial data which is focused on the study of the dynamics of the volatility level of log-returns.

An alternative to the Baum-Welch recursions for hidden Markov models

We propose a hidden Markov model for multivariate continuous longitudinal responses with covariates that accounts for three different types of missing pattern: (I) partially missing outcomes at a given time occasion, (II) completely missing outcomes at a given time occasion (intermittent pattern), and (III) dropout before the end of the period of observation (monotone pattern). The missing‐at‐random (MAR) assumption is formulated to deal with the first two types of missingness, while to account for the informative dropout, we rely on an extra absorbing state. Estimation of the model parameters is based on the maximum likelihood method that is implemented by an expectation‐maximization (EM) algorithm relying on suitable recursions. The proposal is illustrated by a Monte Carlo simulation study and an application based on historical data on primary biliary cholangitis.

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/bimj.202200016

A hidden Markov model for continuous longitudinal data with missing responses and dropout

We propose a general recursive algorithm for the computation of the conditional probability function of the quadratic exponential model for binary panel data given the total of the responses, which is a sufficient statistic for the individual intercept parameter. This recursion permits to implement conditional and pseudo-conditional maximum likelihood estimators of the parameters of this model, and related models such as the dynamic logit model, even when one or more statistical units are observed at many occasions. In this way we solve a typical problem in dealing with distributions with a complex normalizing constant. The advantage in terms of computational load with respect to standard techniques is assessed by simulation and illustrated by an application based on a popular dataset about brand loyalty.

Francesco Bartolucci

Papers

Exact Conditional Testing of Certain Forms of Positive Association for Bivariate Ordinal Data

Item selection by Latent Class-based methods

An alternative to the Baum-Welch recursions for hidden Markov models

A hidden Markov model for continuous longitudinal data with missing responses and dropout

Recursive Computation of the Conditional Probability Function of the Quadratic Exponential Model for Binary Panel Data