Estimation of Hazard Models with Dependence Across Observations: Correlation, Frailty and Contagion

This paper proposes nonparametric and semi-nonparametric estimation of hazard models with various types of dependence between observations. The methods are designed to examine the impact that di¤erent macroeconomic and …nancial conditions have on hazard rates. First, dependence between time-varying covariate processes across observations is examined. Among other possibilities, covariate processes that are common to all observations are permitted. This is motivated by situations where macroeconomic variables such as the interest or unemployment rate a¤ect all hazard rates. Second, I examine a global latent risk factor which increases clustering of defaults. This unobserved risk factor is referred to as frailty. Finally, defaults of certain observations are allowed to directly a¤ect the hazard rate of related observations. This phenomenon is given the general name contagion. The martingale nature of default is preserved in the presence of these types of dependence. Martingale CLT and FCLT results are derived. Two types of estimation are presented, both of which are based on the derived martingale results. First, a kernel approach is taken. Asymptotic results are derived while accounting for dependence between processes using mixing conditions. Second, a point process likelihood approach is taken. Sieve estimation is possible in the presence of dependent process, contagion and frailty. Again, mixing conditions are assumed. The path of the unobserved frailty component and the parametric impact of covariates are consistently estimated. The estimate of the frailty path is then used to estimate the underlying stochastic process the unobserved risk factor follows.