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This chapter provides the mathematical foundations for stochastic intensity, on which most of the methodology is based. The intensity of an event such as default is its conditional mean arrival rate, measured in events per year, given all information currently available to the observer. The chapter also presents the multi-firm version of the doubly-stochastic hypothesis, under which the sole source of default correlation between two firms is the dependence of their default intensities on common or correlated observable risk factors. The doubly-stochastic property rules out contagion as well as correlation induced by unobservable risk factors. The chapter includes a mathematical device for testing a model of the default intensity processes of a large number of borrowers.

This chapter considers two versions of a rational model of speculative bubbles and stock market crashes. According to the first version, stock market prices are driven by the crash hazard that may increase sometimes due to the collective behavior of “noise traders.” The second version assumes the opposite: the crash hazard is driven by prices that may soar sometimes, again due to investors' speculative or imitative behavior. The chapter first provides an overview of what a model is before discussing the basic principles of model construction in finance. It then describes the basic ingredients of the two models of speculative bubbles and market crashes, along with the main properties of the risk-driven model. It also examines how imitation and herding drive the crash hazard rate and concludes with an analysis of the price-driven model, how imitation and herding drive the market price, and how the price return drives the crash hazard rate.

The chapter discusses several modeling approaches aimed at predicting prepayments and default rates given economic scenario, loan, borrower and collateral characteristics. Among those methods are predicting lifetime defaults and losses, roll rates, hazard rates, and dynamic transition rates. A dynamic transition model advocated by the authors combines attractive features of other models and enables initialization of the model with known loan statuses. Loans can generally be current, delinquent, severely delinquent, or terminated with transitions among those states modeled as S-curves of various drivers (the chapter provides examples and explanations). If a loan terminates, it may cause losses also modeled off economic drivers, both observed and unobserved.

This chapter, which examines internal migration in the United States in the nineteenth century, studies the characteristics of intercounty migrants and estimates the hazard rate of changing county of residence within a year. It investigates whether return migration was common and the characteristics of return migrants, and finally, examines the costs of migration, in terms of mortality. The study uses a large longitudinal data set of residential histories for Union Army veterans, allowing the investigation of not just the migration decision through a richer specification than previous researchers have been able to use, but also the return migration decision, which may be workers' optimal reaction to temporary economic shocks. Longitudinal microdata is also used to study the relationship between migration and life expectancy. Because migration, particularly to urban areas, may have decreased the life expectancy of workers, a “mortality wage premium” may partly account for wage differentials between cities and rural areas.

This chapter (co-written with Anke Hoeffler) seeks to identify factors that contribute to post-conflict peace stabilization based on a quantitative analysis using duration (survival) analysis and a qualitative analysis examining the peace consolidation process in six conflict-affected countries. Duration analysis, a statistical method, allows us to analyse the duration of peace. The hazard rate—the rate at which peace ends—can be modelled as a function of various co-variates, such as economic growth, aid, elections, military personnel and expenditure, regional autonomy, etc. The country case studies provide more detailed information on how some countries achieved lasting peace while others failed. The country cases that are included in this analysis are: Burundi, El Salvador, Liberia, Nepal, Sierra Leone, and Timor-Leste (East Timor).

This chapter investigates change-point (hazard rate) probability models for the random survival time in some population of interest. A parametric probability distribution is assumed with parameters to be estimated from a sample of observed survival times. If a change-point parameter, denoted by τ‎, is included to represent the time at which there is a discrete change in hazard rate, then the model is non-standard. The profile log-likelihood, with τ‎ as profiling parameter, has a discontinuous jump at every τ‎ equal to a sampled value, becoming unbounded as τ‎ tends to the largest observation. It is known that maximum likelihood estimation can still be used provided the range of τ‎ is restricted. It is shown that the alternative maximum product of spacings method is consistent without restriction on τ‎. Censored observations which commonly occur in survival-time data can be accounted for using Kaplan-Meier estimation. A real data numerical example is given.