Duration Analysis. Joan Llull

Transcription

1 Duration Analysis Joan Llull Panel Data and Duration Models Barcelona GSE joan.llull [at] movebarcelona [dot] eu

2 Introduction Duration Analysis 2

3 Duration analysis Duration data: how long has an individual been in a state when exiting from it (e.g. weeks unemployed). Examples: unemployment duration, marriage duration, life expectancy, rms exit from the market,... We want to analyze: Why durations dier across individuals? How and why do exit probabilities vary over time? Long tradition in biometrics: surviving probabilities, hazard functions,... Duration Analysis 3

4 Duration data We denote these durations for the N observations as t 1, t 2,..., t N. These data are typically censored. E.g.: Individuals may not nd a job before the interview (we observe that t > t, but not t). Individuals may nd a job between selection and interview, and we cannot interview them (we observe t < t < t) One of the main motivations for duration analysis: dealing with censoring Duration Analysis 4

5 Figure I. Two examples of censored observations i. Example 1 ii. Example 2 Individual Jan90 Jul90 Jan91 Jul91 Jan92 Date Individual Jan90 Jul90 Jan91 Jul91 Jan92 Date Note: Black lines represent the time when the individual was unemployed. A dot indicates that the individual is still unemployed at that date, but we do not have further information about him/her. Vertical red dashed lines in Example 2 are interview dates. Duration Analysis 5

6 The Hazard Function Duration Analysis 6

7 The hazard function Hazard function: probability (or density) of exiting at t conditional on being alive. It can be time-varying (e.g. mortality) or constant. We analyze discrete and continuous hazard functions. Why do we care? Theoretically appealing (e.g. job arrival rate). Empirically convenient (binomial discrete choice+censoring). We start from unconditional hazards and then add regressors. Duration Analysis 7

8 Figure II. Mortality Hazard Rate Hazard function Age Note: The line depicts the hazard mortality rate, i.e. the probability of dying at age a conditional on survival until that age. Duration Analysis 8

9 Hazard function for a discrete variable Probability mass function for t: p(τ) = Pr(t = τ) for τ = 1, 2,... Cdf for t: F (t) = p(1) + p(2) p(t). Hazard function for t: Pr(t = τ) h(τ) = Pr(t = τ t τ) = Pr(t τ) = p(τ) 1 F (τ 1) We can recover p(t) and F (t) from h(t) recursively: F (τ) F (τ 1) =. 1 F (τ 1) 1 h(t) = 1 F (t) t 1 F (t 1) F (t) = 1 (1 h(s)) (Interpretation of these expressions) s=1 t 1 p(t) = (1 F (t 1))h(t) =h(t) (1 h(s)). s=1 Duration Analysis 9

10 Hazard function for a continuous variable Pdf function for t: f(τ) = lim dt 0 Pr(τ t<τ+dt) dt. Cdf for t: F (t) = t 0 f(u)du. Hazard function for t: / Pr(τ t < τ + dt t τ) Pr(τ t < τ + dt) h(τ) = lim = lim dt = f(τ) dt 0 dt dt 0 Pr(t τ) 1 F (τ). We can recover p(t) and F (t) with the integrated hazard H(t) = t 0 h(s)ds: H(t) = ln[1 F (t)] F (t) = 1 exp[ H(t)] f(t) = F (t) t =h(t) exp[ H(t)] =h(t)(1 F (t)). (Comparison with discrete through ln[1 F (t)]) Duration Analysis 10

11 Some frequently used hazard functions In discrete duration, we can estimate semi-parametric hazards. In continuous duration (and sometimes in discrete), we usually assume a given parametric function. Two frequent examples: Constant hazard. Weibull distribution. Duration Analysis 11

12 Constant hazard: the exponential distribution The simplest hazard function is a constant hazard: h(t) = λ. In the continuous case: H(t) = t 0 λdu = λt F (t) = 1 e λt f(t) = λe λt. In the discrete case: F (t) = 1 (1 λ) t p(t) = λ(1 λ) t 1. Having a constant hazard is the memoryless property of the exponential. Also E[T ] = 1/λ. Duration Analysis 12

13 Figure III. Examples of Hazard Functions (I): Constant i. h(t) ii. H(t) iii. f(t) iv. F (t) Hazard rate [h(t)] Duration [t] Integrated hazard [H(t)] Duration [t] Prob. density function [f(t)] Duration [t] Cum. distribution function [F(t)] Duration (t) Note: Black: λ = 0.12; gray: λ = These examples depict the hazard function (left), the integrated hazard (center-left), the pdf or probability mass function (center-right) and the cdf (right) of a constant hazard model with hazard equal to λ Duration Analysis 13

14 The Weibull distribution A generalization of the exponential distribution is: F (t) = 1 e (λt)α f(t) = αλ α t α 1 e (λt)α. This gives the following hazard function: h(t) = αλ α t α 1. It is monotonically increasing, decreasing, or constant depending on whether α ><= 1 Duration Analysis 14

15 Figure IV. Examples of Hazard Functions (II): Weibull i. h(t) ii. H(t) iii. f(t) iv. F (t) Hazard rate [h(t)] Duration [t] Integrated hazard [H(t)] Duration [t] Prob. density function [f(t)] Duration [t] Cum. distribution function [F(t)] Duration (t) Note: Black: λ = 0.12; gray: λ = Solid: α = 1; dotted: α = 0.5; dashed: α = 1.5; dot-dashed: α = 3. These examples depict the hazard function (left), the integrated hazard (center-left), the pdf or probability mass function (center-right) and the cdf (right) of a Weibull hazard model with parameters λ and α. Duration Analysis 15

16 Conditional Hazard Functions Duration Analysis 16

17 Including covariates to the hazard model We want to see how duration is aected by covariates x. We are interested in the conditional hazard: h(t, x) = f(t x) 1 F (t x). Duration Analysis 17

18 Proportional hazard model It was introduced by Cox (1972). It is a.k.a. Cox regression or Cox model. Main advantage: simplicity. It is a factorized model: h(t, x) = λ(t) exp(x β). λ(t) can be either constant λ(t) = 1 or Weibull λ(t) = αt α 1 (the scale cannot be separately identied from the constant) λ(t) is called the baseline hazard. Duration Analysis 18

19 Discrete durations Now conditional hazard rates are probabilities. One approach: treat spells as continuous and use the PH model. However, it is natural to use discrete choice models to specify the exit rate: h(τ, x) = Pr(t = τ t τ, x) = G(γ τ + x β τ ), where G(.) is a cdf (e.g. normal or logistic). Dierent ways to specify baseline hazard: Leave them free: γ t = T j=1 γj 1{t = j}. (why T?) Specify them as polynomials: e.g. γ t = γ 0 + γ 1 ln t + γ 2(ln t) Assuming γ t free delivers a semi-parametric estimation of λ(t) that allows us to test parametric assumptions. Duration Analysis 19

20 Likelihood functions Duration Analysis 20

21 Complete durations Assume that we observe {t 1, t 2,..., t N }. The log-likelihood function is: L N = where: N ln f(t i x i ), i=1 f(t x) = h(t, x) exp( H(t, x)) = λ(t) exp(x β) exp { Λ(t) exp(x β)}. Dierent expressions for λ(t) and Λ(t) depending on baseline hazards. Duration Analysis 21

22 Our two examples of censored durations Example 1: We observe t if t < t; otherwise we only observe that t > t. The log-likelihood function is: N { L N = wi ln f(t i x i) + (1 w i) ln(1 F (t } i x i)), w i = 1{t i < t i}. i=1 Example 2: We observe whether whether t < t 1, t 1 < t < t 2, or t > t 2. The log-likelihood function is: N { ( ) L N = wi 1 ln F (t 1 i x i) + wi 2 ln F (t 2 i x i) F (t 1 i x i) i=1 where w 1 i = 1{t i < t 1 i }, and w 2 i = 1{t 1 i < t i < t 2 i }. ( )} +(1 wi 1 wi 2 ) ln 1 F (t 2 i x i), Duration Analysis 22

23 An additional example Example 2 modied: Same as example 2, but with initial (known) durations d 1, d 2,..., d N. The log-likelihood function is: { N L N = wi 1 ln F (di + t1 i x i) F (d i x i) 1 F (d i x i) i=1 + wi 2 ln F (di + t2 i x i) F (d i + t 1 i x i) 1 F (d i x i) } +(1 wi 1 wi 2 ) ln 1 F (di + t2 i x i), 1 F (d i x i) where w 1 i = 1{d i < t i < d i + t 1 i }, and w 2 i = 1{d i + t 1 i < d i + t i < t 2 i }. Duration Analysis 23

24 Discrete durations The log-likelihood of a discrete duration model can be written as: L N = N T w iτ {y iτ ln G(γ τ + x iβ τ ) + (1 y iτ ) ln(1 G(γ τ + x iβ τ ))}, i=1 τ=1 where y iτ = 1{t i = τ}, and w iτ = 1{t i τ}. Two types of contributions: Spells that end at time τ: τ 1 ln Pr(t = τ x) = ln h(τ, x) + ln(1 h(s, x)). Spells that are incomplete at time T : s=1 ln Pr(t > T x) = ln(1 h(s, x)). T s=1 This problem is a sequence of binary choice models. If censoring occurs at dierent durations: T i = min{ t i, T }. Duration Analysis 24

25 Unobserved Heterogeneity Duration Analysis 25

26 State dependence vs unobserved heterogeneity Consider a regressor x = {0, 1}, and a constant proportional hazard model such that: h(t, x = 0) = h 0 h(t, x = 1) = h 1 h 1 > h 0. The aggregate hazard is: h(τ) = h 1 Pr(x = 1 t τ) + h 0 Pr(x = 0 t τ). This aggregate hazard is not constant anymore given that Pr(x = 1 t τ) decreases over time spurious state dependence. Duration Analysis 26

27 Figure V. An Example with Unobserved Heterogeneity Hazard function Note: Duration (t) Black dashed: h 1 (hazard rate when x = 1); gray dashed: h 0 (hazard rate when x = 0); gray dashed: observed (unconditional) hazard. Duration Analysis 27

28 The PH with unobserved heterogeneity Lancaster (1979) includes unobserved heterogeneity as a multiplicative random eect: h(t, x, ν) = λ(t) exp(x β)ν, where ν is independent of x with E[ν] = 1 and with pdf g(ν) (e.g. gamma). The cdf conditional on ν is: F (t x, ν) = 1 exp [ t 0 ] h(u, x, ν)du, and the unconditional cdf is: F (t x) = 0 F (t x, v)g(v)dv, Important remark: identication requires inclusion of exogenous regressors! Duration Analysis 28

29 Unobserved heterogeneity with discrete data Similarly to the continuous case: Pr(t = τ x) = Pr(t = τ x, v)g(v)dv = τ 1 h(τ, x, v) [1 h(s, x, v)]g(v)dv, s=1 where, for instance: h(t, x, ν) = G(γ t + x β t + ν). A frequently used specication for g(v) is a discrete-support mass point distribution: {ν 1,..., ν m} with probabilities {p 1,.., p m}: { } m τ 1 Pr(t = τ x) = h(τ, x, ν j) [1 h(s, x, ν j)]p j, j=1 s=1 and ν j and p j are treated as additional parameters. Duration Analysis 29

30 Multiple-exit Discrete Duration Models Duration Analysis 30

31 Transition intensities vs conditional hazard rates Let d 1 and d 2 be indicators for the two potential alternatives where to exit (e.g temporary or permanent job). We can dene the following intensities of transition to each state: φ j(τ) = Pr(t = τ, d j = 1 t τ), j = 1, 2. The unconditional hazard rates are: h(τ) = Pr(t = τ t τ) = φ 1(τ) + φ 2(τ). Also, we can dene conditional hazard rates as: h j(τ) = Pr(y jτ = 1 t τ, y kτ = 0), j k where y jτ = y τ d j = 1{t = τ, d j = 1}. The mapping between the two expressions is as follows: h j(τ) = Pr(yjτ = 1 t τ) Pr(y kτ = 0 t τ) = φj(t) 1 φ k (t). We can model either of the two. For instance, a MNL for φ's delivers a binary logit for h's with the same parameters. Duration Analysis 31

32 Competing risk models The models for conditional hazards h 1 (t) and h 2 (t) are often called competing risk models. This name derives from considering two latent variables t 1 and t 2, such that the observed duration is t = min{t 1, t 2}. If t 1 and t 2 are independent, h 1 (t) and h 2 (t) can be interpreted as exit rates of latent durations: h j (τ) = Pr(t j = τ t j τ). This implies that when we analyze exits to alternative 1, we take exits to 2 as censored observations. Duration Analysis 32

33 FIML estimation The log-likelihood is analogous to the discrete case: N T L N = w iτ {y iτ (d 1i ln φ 1(τ, x i) + d 2i ln φ 2(τ, x i)) i=1 τ=1 where y iτ = 1{t i = τ}, and w iτ = 1{t i τ}. Three types of contributions: Spells that end at time τ exiting to option 1: +(1 y iτ ) ln(1 φ 1(τ, x i) φ 2(τ, x i))}, τ 1 ln Pr(t = τ, d 1 = 1 x) = ln φ 1(τ, x) + ln(1 φ 1(s, x) φ 2(s, x)). s=1 Spells that end at time τ exiting to option 2: τ 1 ln Pr(t = τ, d 2 = 1 x) = ln φ 2(τ, x) + ln(1 φ 1(s, x) φ 2(s, x)). Spells that are incomplete at time T : s=1 ln Pr(t > T x) = ln(1 φ 1(s, x) φ 2(s, x)). T s=1 Duration Analysis 33

34 LIML estimation Alternatively, we can estimate the model separately for the two alternatives in a competing risk fashion. In this case, the conditional log-likelihood function for alternative j is: N T L Nj = w iτ {y ijτ ln h j(τ, x i) + (1 y ijτ ) ln(1 h j(τ, x i))}, or, equivalently: i=1 τ=1 N T L Nj = w iτ {y iτ d j ln h j(τ, x i) + (1 y iτ d j) ln(1 h j(τ, x i))}. i=1 τ=1 where y ijτ = 1{t i = τ, d ij = 1}, y iτ = 1{t i = τ}, and w iτ = 1{t i τ}, with t i = min{t 1i, t 2i}. Duration Analysis 34