Dynamic Models Part 1

Transcription

1 Dynamic Models Part 1 Christopher Taber University of Wisconsin December 5, 2016

2 Survival analysis This is especially useful for variables of interest measured in lengths of time: Length of life after a medical procedure Unemployment spell Life of a company How long someone has health insurance

3 Let T i be the dependent variable we are trying to explain (kind of annoying notation in this class-it is not a treatment it is typically an outcome) There are a whole bunch of different-and equivalent ways to characterize the distribution of T i. Distribution function Density function F(t) Pr(T i t) f (t) F(t) t

4 Survivor function Hazard Function Integrated Hazard S(t) Pr(T i > t) = 1 F(t) λ(t) Pr(T i t + δ T i t) lim δ 0 δ = f (t) S(t) = dlogs(t) dt Λ(t) ˆ t 0 λ(s)ds = log(s(t))

5 Hazard Function In some ways it would be natural to just look at the data like we typically do and analyze log(t i ) It is nicer to think about the model in terms of the hazard function Reasons: Right truncation Time varying covariates Just easier to think about

6 Specifying the model The easiest place to start is just a constant hazard rate λ which gives an exponential distribution with pdf f (t) = λe λt S(t) =e λt

7 We can easily add X s into this model and the most common specification is λ i = exp(x iβ) We can allow the X s to change over time but lets not worry about that yet

8 Cox Proportional Hazard An issue here is that we have still restricted the model so that the hazard rate has to be constant We relax this by letting the hazard rate take the proportional hazard form: λ i (t) =λ 0 (t) exp(x iβ) We would then typically specificy a parametric functional form for λ 0 (t)

9 Weibull The most common is the Weibull λ 0 (t) =pt p 1 Weibull hazard 2.5 alpha=1 alpha=0.5 alpha= Time Observed heterogeneity

10 This also has the nice property that we can write ( ) β log(t i ) =X i + ε i p p where ε i is extreme value This makes parameters easy to interpret

11 To see why note that the integrated hazard is so ˆ t Λ(t) = e X i β ps p 1 ds =e X i β t p 0 S (t) =e Λ(t) =e ex i β t p =e ex i β+p log(t)

12 The extreme value distribution is so if Pr (ε i < x) =1 e ex log(t i ) = X i β p + ε i p Then ( Pr (T i < t) =Pr X i β p + ε ) i p < log(t) =Pr ( ε i < p log(t) + X iβ ) =1 e ep log(t)+x i β

13 Estimation Estimate the model by maximum likelihood When we see the end of the spell likelihood is f (T i X i ; θ) = λ(t i X i ; θ)s(t i X i ; θ) If data is right sensored at τ, likelihood is Pr(T i > τ X i ; θ) = S(τ X i ; θ) Note that we have allowed observed heterogeneity across individuals, but no unobserved heterogeneity

14 Unobserved Heterogeneity We can write the mixed proportional hazard model as λ i (t) =λ 0 (t) exp(x iβ + v i ) =λ 0 (t) exp(x iβ)v i

15 For example hazard out of unemployment falls with t this can be due to: Duration dependence λ 0 (t) falls with t Unobserved heterogeneity

16 Identification Seems subtle, but as long as V i is independent of X i the model λ i (t) = λ 0 (t)φ(x)v i is identified To see why define the integrated baseline hazard as Λ 0 (t) ˆ t 0 λ 0 (t)dt. Pr(T i > t X i, V i ) = e Λ 0(t)φ(X i )V i.

17 Let F V to be the distribution of V i and define g( ) = log(φ( )) Then generalizing what we did before with the Weibull ˆ Pr(T i t X i = x) = 1 e Λ0(t)φ(x)V df v ˆ = 1 exp( exp(log(λ 0 (t)) g(x) + log(v))df v F v (log(λ 0 (t)) g(x)) where F v is defined implicitly by this relationship. Given the separability between log(λ 0 (t)) and g(x) one can show that with sufficient scale/location normalizations the model is identified

18 Likelihood The likelihood function is analogous we just need to integrate across the distribution of V: ˆ λ 0 (T i )φ(x i )Ve Λ 0(T i )φ(x i )V df v For data right truncated at τ ˆ e Λ 0(τ)φ(X i )V df v

19 Left truncation We have showed that right truncation (ongoing spells at the end of the sample) is not a problem However left truncation or ongoing spells at the beginning of the sample are a big deal The problem is that even if I know the length of the ongoing spells, I don t observe spells that began at the same time The distribution of V here is selected: we are going to oversample small values of V for any length of ongoing spell You either need to throw out ongoing spells or somehow model the initial state. (this is a case where Indirect Inference might be easier to compute then maximum likelihood)

20 Time Varying X s In principle its easy, in practice it could be a pain Let X i (t) denote X over time Let the integrated hazard (apart from V) be Λ i (t) ˆ t 0 λ 0 (s)φ(x i (s))ds Then the likelihood function is: ˆ λ 0 (T i )φ(x i (T i ))Ve Λ i(t i )V df v For data right truncated at τ ˆ e Λ i(τ)v df v

21 Competing Risk Model Sometimes can leave a spell for more than two reasons (other than data ending): Can die of cancer or something else Can leave unemployment to OLF rather than employment Can die before you become married

22 This is easy to deal with, just define two different hazards: λ 1 i (t) λ 2 i (t) We get to see the minimum of T 1 i, T2 i A lot like a straight Roy model Lets think about this with non time varying X s but allow for unobserved heterogeneity λ 1 i (t) =λ 01 (t)φ 1 (X i1 )V i1 λ 2 i (t) =λ 02 (t)φ 2 (X i1 )V i2 with Λ 01 (t) and Λ 02 (t) the corresponding integrated hazard

23 So we can write the likelihood as if leave because of risk 1 ˆ λ 01 (T i1 )φ 1 (X i1 )V 1 e Λ 01(T i1 )φ 1 (X i1 )V 1 e Λ 02(T i1 )φ(x i2 )V 2 df v if leave because of risk 2 ˆ λ 02 (T i2 )φ 2 (X i2 )V 2 e Λ 01(T i2 )φ 1 (X i1 )V 1 e Λ 02(T i2 )φ(x i2 )V 2 df v if right truncated at τ ˆ e Λ 01(τ)φ 1 (X i1 )V 1 e Λ 02(τ)φ(X i2 )V 2 df v

24 Example 1: Gibbons and Katz Gibbons and Katz, Layoffs and Lemons", Journal of Labor Economics, Oct They write down a model of Asymmetric Information in the labor market They compare displaced workers who lost their job either due to layoffs or plant closings Idea is that layoff is a bad signal relative to plant closing

25 Table 6 Effects of Selected Variables on the Duration of the First Spell of Joblessness following Displacement from January 1986 CPS Displaced Workers Survey, Males with Only One Spell of Joblessness since Displacement Dependent Variable = Log (Weeks of Joblessness) Weibull Duration Model Specification Variable (1) (2) (3) (4) Layoff Layoff x white collar (.086) (.108) (.106) (.126) (.168) Layoff X high union Layoff x fraction union (.147) Fraction union (.345) Previous tenure in years.037 (.266) (.294) (.033).033 (.007) (.007) (.007) (.007) Log of previous real weekly earnings (.100) (.099) (.099) (.099) Weibull scale parameter (a) (.033) (.032) (.032) (.032) Log likelihood -1, , , ,821.7 NOTE.-The reported models were estimated by maximum likelihood with left censoring explicitly

26 Example 2: Meyer Meyer, Unemployment Insurance and Unemployment Spells," Econometrica, July, 1990 Meyer looks at unemployment spells right before benefits run out After a period of time benefits run out He estimates baseline hazard flexibly

27 BRUCE D. MEYER TABLE V HAZARDMODEL ESTIMATES" Soecification Number of dependents 1 = married, spouse present Years of schooling Log UI benefit level Log pre-ui after tax wage Age Age Age Age State unemployment rate Exhaustion spline:b UI 1 Benefits previously expected to lapsec State fixed effects Nonparametric baseline Heterogeneity variance Sample size Log-likelihood value Standard errors are shown in parentheses. b ~ hexhaustion e spline variables are defined in the text.

28 Discrete Time Duration Models An alternative way to model duration data is to treat time as discrete For some of us, this is a more intuitive way to understand these models Now T i is an integer We can do something analogous to the constant hazard with X s by modeling the discrete hazard with a logit (or a probit or something else) λ it = Pr(T i = t T i > t 1, X i ) 1 = 1 + e X i β

29 To make it like the proportional hazard model we could allow the intercept to vary with t. More generally we can easily see how to allow both X i and β to vary with t λ it = Pr(T i = t T i > t 1, X it ) 1 = 1 + e X it βt And then adding unobserved heterogeneity is also straight forward 1 Pr(T i = t T i > t 1, X it ) = 1 + e X it βt+v i

30 We can write the likelihood without censoring as ˆ [T i 1 t=1 and with censoring as e X it βt+v 1 + e X it βt+v ] e X it i β Ti +v df v(v) ˆ τ t=1 e X it βt+v 1 + e X it βt+v df v(v)

31 Identification Cameron and Heckman, Lifecycle Schooling and Dynamic Selection Bias: Models and Evidence for Five Cohorts of American Males, JPE April 1998 This is easiest to think about with two periods. Suppose T i takes on three values 0, 1, 2 Conditional on T i > 0, T i > 0 1(g 1 (X i1 ) + v i1 > 0) T i = 2 1(g 2 (X i2 ) + v i2 > 0) This is like the standard selection model

32 Can identify the model in the following way (assuming sufficient normalizations): 1 From first period identify g 1 2 Use identification at infinite to get g 2 3 Given those construct joint distribution of (v i1, v i2 )

33 Example: Cameron and Heckman Same Cameron and Heckman Paper They estimate a dynamic duration model for schooling accounting for selection in a flexible way

34