Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument for flexible regression analysis. In many practical situations, however, it is desirable to restrict the flexibility of nonparametric estimation in order to accommodate a presumed monotonic relationship between a covariate and the response variable. For example, consumers usually will buy less of a brand if its price increases, and therefore one expects a brand's unit sales to be a decreasing function in own price. We follow a Bayesian approach using penalized B-splines and incorporate the assumption of monotonicity in a natural way by an appropriate specification of the respective prior distributions. We illustrate the methodology in an empirical application modeling demand for a brand of orange juice and show that imposing monotonicity constraints for own- and cross-item price effects improves the predictive validity of the estimated sales response function considerably

Switching regressions and activity analysis.

We study the use of switching regression models to characterize the coefficients in linear production technologies with a finite number of activities. Maximum likelihood-based methods are proposed and different switching specifications are discussed. The viability of these newly proposed technniques is established. The methods developed combine the advantages of the two major approaches to frontier estimation: the functional flexibility of the linear programing-nonparametric and nonstatistical-approach and the statistical nature of the econometric-both parametric and statistical-approach. This combination comes at the expense of some analytical complexity.Switching regression models; Activity analysis; Linear production models;

Classical and Bayesian Inference for Threshold Regression Models

Η παρούσα διπλωματική εργασία πραγματεύεται την Κλασσική και Μπεϋζιανή θεωρία για το Μοντέλο Παλινδρόμησης με μια ή δυο μεταβλητές τύπου κατωφλίου (threshold). Τα Threshold Μοντέλα Παλινδρόμησης χρησιμοποιούνται σε μια πληθώρα εφαρμογών, κυρίως στον κλάδο της Οικονομετρίας, και ανήκουν στην ευρύτερη οικογένεια μοντέλων παλινδρόμησης με δομικές αλλαγές που εισήγαγε ο Quandt (1960). Στη βιβλιογραφία το ενδιαφέρον εστιάζεται συνήθως στα "μη συνεχή" Threshold Μοντέλα Παλινδρόμησης, λόγω της μη κανονικής ασυμπτωτικής κατανομής των στατιστικών συναρτήσεων που εμπεριέχουν τις παραμέτρους. Εδώ θα ασχοληθούμε, μεταξύ άλλων, και με την περίπτωση αυτή τόσο σε θεωρητικό αλλά και σε πρακτικό επίπεδο. Στα Μοντέλα Παλινδρόμησης το ενδιαφέρον επικεντρώνεται στην εκτίμηση των παραμέτρων, καθώς και στην ασυμπτωτική κατανομή στατιστικών συναρτήσεων που τις εμπεριέχουν. Απώτερος σκοπός είναι η κατασκευή διαστημάτων εμπιστοσύνης και ο έλεγχος υποθέσεων σχετικά με τη στατιστική σημαντικότητα κάθε παραμέτρου. Από την πλευρά της Μπεϋζιανής θεωρίας, κυρίαρχο ρόλο παίζει ο ορισμός των εκ των προτέρων (prior) κατανομών, δεδομένης της δοθείσας πληροφορίας, με σκοπό την εύρεση των εκ των υστέρων (posterior) κατανομών. Η εύρεση των posterior κατανομών είναι ένα σύνθετο πρόβλημα καθώς αυξάνει το πλήθος των παραμέτρων του μοντέλου και απαιτεί τον υπολογισμό σύνθετων ολοκληρωμάτων ή τη χρήση αλγορίθμων προσομοίωσης όταν αυτό δεν είναι εφικτό. Τέτοιοι αλγόριθμοι είναι οι Markov Chain Monte Carlo (MCMC) αλγόριθμοι και μια ειδική περίπτωση αυτών, ο δειγματολήπτης Gibbs (Gibbs sampler). ΄Ολες αυτές οι μέθοδοι παρουσιάζονται αναλυτικά στην παρούσα διπλωματική εργασία και καλύπτουν ένα ευρύ φάσμα μοντέλων παλινδρόμησης. Παρ’ ότι η εκτίμηση ενός μοντέλου είναι ο βασικός στόχος ενός στατιστικού, αυτό που προηγείται είναι η εύρεση του κατάλληλου μοντέλου για ένα δοθέν δείγμα. Η σύγκριση μοντέλων είναι λοιπόν το πρώτο βήμα που πρέπει να ακολουθήσει κανείς προκειμένου η συμπερασματολογία να είναι ολοκληρωμένη και ακριβής. Πρόκειται ουσαστικά για έναν έλεγχο υποθέσεων ο οποίος υποδεικνύει τη σχέση που περιγράφει καλύτερα τη σύνδεση της εξαρτημένης μεταβλητής με τις επεξηγηματικές, δηλαδή το είδος του μοντέλου. Τέτοιοι έλεγχοι πραγματοποιούνται από την πλευρά της Κλασσικής θεωρίας με τη χρήση κατάλληλων στατιστικών συναρτήσεων, όπως είναι το LR στατιστικό μέτρο, και από την πλευρά της Μπεϋζιανής θεωρίας με τον υπολογισμό της εκ των υστέρων πιθανότητας για κάθε μοντέλο. ΄Εχοντας καταλήξει λοιπόν, με τον ένα ή τον άλλο τρόπο, στο πιο κατάλληλο μοντέλο, τότε συνεχίζει κανείς με την ανάλυση και τη συμπερασματολογία για τις παραμέτρους του.This dissertation is concerned with Classical and Bayesian theory for the threshold regression model with one or two threshold variables. Threshold regression models have a wide variety of applications, mainly in the field of econometrics, and belong to the family of regression models with structural breaks that were introduced by Quandt (1960). In the literature most of the interest is focused on the discontinuous Threshold Regression Models, because of the non-standard asymptotic distribution of the statistical functions of the threshold parameters. Among others, we will examine this particular issue not only in theoretical but also in a more applied level. Estimating the model parameters and obtaining asymptotic distributions of the respective estimators concentrates most of the interest in the area of Regression Analysis. The main goal is the construction of confidence intervals and hypotheses testing regarding the significance of each parameter. From the scope of Bayesian analysis, it is of great importance to take advantage of all the available information in order to define the prior distribution and finally get the posterior. The computation of the posterior distribution is a procedure that becomes more complex as the number of parameters increases, since it demands the calculation of composite integrals and the utilization of simulation techniques when the former is not applicable. Such methods are the Markov Chain Monte Carlo (MCMC) algorithms and a special case of them, the Gibb's sampler. All these methods are presented extensively in this dissertation and cover a wide variety of regression models. Although the estimation of a model's parameters is the primary objective for a statistician, the selection of the most appropriate model for a given dataset comes first. Therefore, model comparison is the first step that one needs to do for precise and complete inference results. In essence, this is a hypotheses test that concerns the kind of relationship between the dependent variable and the explanatories, namely the type of model. Such tests are accomplished from the scope of Classical theory by using appropriate statistical tools, such as the LR statistic, and from the scope of Bayesian theory with the computation of each model's posterior probability. Having selected, either way, the most appropriate model, shall one proceed to statistical inference regarding its parameters

Nonlinearity in the Fed's Monetary Policy Rule

This paper investigates the nature of nonlinearities in the monetary policy rule of the US Fed using the flexible approach of Hamilton (2001a). We find that while there is significant evidence of nonlinearity for the period to 1979, there is little such evidence for the subsequent period. Possible asymmetries in the Fed's reactions to inflation deviations from target and the output gap in the 1960s and 70s may tell part of the story, but do not capture the entire nature of the nonlinearity. The inclusion of the interaction between inflation deviations and the output gap, as recently proposed, appears to characterize the nonlinear policy rule more adequately.nonlinearities, monetary policy rule, Phillips curve, interaction

Switching regressions and activity analysis

We study the use of switching regression models to characterize the coefficients in linear production technologies with a finite number of activities. Maximum likelihood-based methods are proposed and different switching specifications are discussed. The viability of these newly proposed technniques is established. The methods developed combine the advantages of the two major approaches to frontier estimation: the functional flexibility of the linear programing-nonparametric and nonstatistical-approach and the statistical nature of the econometric-both parametric and statistical-approach. This combination comes at the expense of some analytical complexity

Relative Risk Aversion and Social Reproduction in Intergenerational Educational Attainment: Application of a Dynamic Discrete Choice Mode

The theory of Relative Risk Aversion (RRA) claims that educational decision-making is ultimately motivated by the individual’s desire to avoid downward social class mobility, and that this desire is stronger than the desire to pursue upward mobility. This paper implements a dynamic programming model which tests the central behavioral assumption in the RRA theory stating that (1) individuals are forward-looking when choosing education and (2) that the RRA mechanism comprises an important component in the educational decision-making process. Using data from the Danish Youth Longitudinal Study, we find strong evidence of RRA in educational decision-making over and above the effect of traditional social background variables.

Gaussian Copula Regression in the Presence of Thresholds

Park and Gupta’s (2012) introduction of the Gaussian Copula (GC) approach to deal with endogeneity has made a significant impact on empirical marketing research with many papers using this approach. Recent studies have however started to explore and examine the approach and its underlying assumptions more closely, resulting in a more critical picture of it. A particular challenge is the non-testable assumption that the dependency structure between the endogenous regressor and the error term should be described by a Gaussian copula. In general, there exists a limited understanding of what this assumption implies, what causes its violation, and potential remedies. Our study addresses this explicitly. We provide a detailed discussion of the dependency structure assumption and how thresholds in the data can lead to its violation and biased results. We use real and simulated data to show how threshold detection before applying the GC approach can overcome this problem and thereby provide researchers with a useful tool to increase the likelihood of the GC approach’s assumptions being met