Essays in multivariate duration models

Effraimidis, Georgios

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URN: urn:nbn:de:bsz:180-madoc-325882
Document Type: Doctoral dissertation
Year of publication: 2012
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Berg, Gerard J. van den
Date of oral examination: 19 July 2012
Publication language: English
Institution: School of Law and Economics > Alexander v. Humboldt Professor in Econometrics and Empirical Economics (van den Berg 2009-2016)
Subject: 330 Economics
Subject headings (SWD): Ereignisdatenanalyse , Multivariate Analyse
Keywords (English): Multivariate Duration Analysis
Abstract: Duration analysis, which is also known as survival analysis, is a core subject of applied statistics and econometrics. Application of duration analysis techniques can be found in actuarial science, demography, economics, finance, marketing, and many other scientific fields. In the univariate case, the tools of duration analysis are used for the study of the distribution of a certain duration variable which is possibly associated with a set of explanatory covariates. This variable measures the time to the occurrence of an event of interest such as transition from unemployment to employment, retirement time, onset of a disease, purchase of a product. The main difference between duration analysis and standard regression analysis is that sometimes the duration variable is right-censored, namely, the only available information we have is that its realization exceeds a certain value. Multivariate duration analysis is the natural extension of the univariate analysis. In this set up, multiple duration variables, which specify the time to the occurrence of multiple events, are considered and their joint distribution is analyzed for describing the association among them. These variables can be either parallel or sequential. Parallel duration variables refer to cases in which the multiple duration variables are measured by using the same reference point of time. On the other hand, sequential duration variables refer to cases in which the measurement of each duration variable starts after the realization of some other duration variable. Death times of twins or the corresponding time of onset of several diseases are multivariate examples with parallel duration variables. On the other hand, unemployment duration and the subsequent employment duration is an example with sequential durations. The current PhD dissertation deals with multivariate duration models. In particular, it consists of three independent essays on multivariate duration models. In the next three paragraphs, a synopsis of each essay is given. The first essay, written jointly with Gerard J. van den Berg, considers bivariate frailty models in which the frailty terms enter multiplicatively on the corresponding hazard rates. The frailty terms capture unobserved or nonmeasurable characteristics that affect the duration outcomes. We assume that the joint distribution of the frailty terms is characterized by gamma marginals. In particular, the gamma distribution is widely used in empirical analysis for modelling the distribution for the unobserved heterogeneity terms. Both analytical and graphical arguments have been developed in the past which rationalize this specific choice. First, the focus of the paper is on the concepts of negative quadrant dependence and positive quadrant dependence between the duration variables. Second, two measures of association between the duration variables are considered; the Pearson's correlation coefficient and the Kendall's tau. In particular, (sharp) bounds for these measures are derived and the necessary conditions are discussed discussion is provided about the conditions which should be satisfied so that the bounds are approached very well. The second essay, written jointly with Carlos Hernandez Mireles and Gerard Tellis, is concerned with a new trivariate hazard rate model which can be applied to study the relationship among the timing of the corresponding events. The suggested model allows for three types of dependence among the timing of the underlying events: due to unobserved heterogeneity, lagged dependence, and due to causality. As shown in the paper, this model can be nonparametrically identified and as consequence the three different types of dependence are disentangled. The new model is adopted to study the endogenous relationship between the timing of three important events in the sales and prices of new products. Specifically, we investigate the causal relationship between the sales crash, price crash, and sales recovery. A sales crash is a significant and permanent cut in the sales of a new product. On the other hand, the sales recovery is a sales peak which is realized after the crash. Finally, the price crash is a deep and permanent reduction in the price of a new product. The last essay deals with competing risks models which are very popular in the scientific field of duration analysis. Such models deal with cases in which we observe only the minimum duration among several multiple durations for each individual unit under study. The goal of this paper is the development of statistical properties of the cumulative incidence function. This function, which is common in the empirical practice, specifies the probability that a particular duration variable will be realized by a certain point of time and before the other duration variables. The proposed estimator is nonparametric, that is, no parametric assumptions are made regarding the data generating process. In addition, the estimator allows for Missing At Random observations. More precisely, for some observations we have information about the value of the minimum duration variable, but not information about which duration variable is the one smallest value of realization.

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