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Unformatted text preview: Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business 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. 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) = 1F(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 Hazard Function t t t 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) = 1F(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) = 1exp( t) or f(t)= exp( t),t 0. Exponent = λ ⇒ =λ λ λ λλ ≥ ial model. Duration Dependence When d (t)/dt 0, there is 'duration dependence' λ ≠ Parametric Models of Duration p1 p1 p There is a large menu of parametric models for survival analysis: Exponential: (t)= Weibull: (t)= p( t) ; p=1 implies exponential Loglogistic: (t)= p( t) /[1 ( t) ] Lognormal: (t)= [plog( t) λ λ λ λ λ λ λ λ + λ λ φ λ ]/ [plog( t)] Gompertz: (t)= p exp( t)...
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This note was uploaded on 01/05/2012 for the course B 55.9912 taught by Professor Willamgreene during the Fall '11 term at NYU.
 Fall '11
 WillamGreene

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