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PanelDataNotes-20A-HazardModels

# PanelDataNotes-20A-HazardModels - Econometric Analysis of...

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Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

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Econometric Analysis of Panel Data 20A. Hazard and Duration Models
Modeling Duration Time until business failure Time until exercise of a warranty Length of an unemployment spell Length of time between children Time between business cycles Time between wars or civil insurrections Time between policy changes Etc.

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Hazard Models for Duration Basic hazard rate model Parametric models Duration dependence Censoring Time varying covariates Sample selection
The Hazard Function For the random variable T = time until an event occurs, T   0. f(t) = density; F(t) = cdf =  Prob[T   t]; S(t) = 1-F(t) =  survival Probability of an event occurring at or before time t is F(t) A condition al probability: for small   >  0, h(t)=  Prob(event occurs in time t to t+  | has not already occurred) h(t)=  Prob(event occurs in time t to t+  | occurs after time t) F(t+ )-F(t)      =   1 F(t) Consider as  -    0, the function F(t+ )-F(t) f(t) (t) =     (1 F(t)) S(t) f(t) (t) the "hazard function" and  (t) Prob[t T t+ | T t] S(t) (t) is a characteristic of the distribution λ - λ = ∆λ λ

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Hazard Function t 0 t 0 t 0 Since  (t) =  f(t)/S(t) =  -dlogS(t)/dt, F(t) =  1 - exp - (s)ds , t 0. (Leibnitz' s Theorem) dF(t) / dt exp - (s)ds ( 1) (t)              (t)exp - (s)ds Thus,  F(t) is a function of the ha λ λ = - λ - λ = λ λ zard;          S(t) =  1 - F(t) is also,     and f(t) = S(t) (t) λ
A Simple Hazard Function The Hazard function Since f(t) =  dF(t)/dt and S(t) =  1-F(t),  f(t)         h(t)= = -dlogS(t)/dt S(t) Simplest Hazard Model - a function with no "memory" (t) =  a constant,  f(t) dlogS(t) / dt.  S(t) The second λ λ = λ = -  simplest differential equation; dlogS(t) / dt S(t) Kexp( t),  K =  constant of integration Particular solution requires S(0)= 1, so K=1 and S(t)=exp(- t) F(t) =  1-exp(- t) or f(t)= exp( t), t 0. Exponent = -λ ⇒ = λ λ λ ial model.

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Duration Dependence When d (t)/dt   0, there is 'duration dependence' λ
Parametric Models of Duration p-1 p-1 p

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• Fall '11
• WillamGreene
• Likelihood function, Survival analysis, Weibull, hazard function, Weibull distribution, Proportional hazards models

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