Slides_Microectrcs_Chapter5.pdf - Chapter 5 Duration Models Joan Llull Microeconometrics IDEA Fall 2017 joan.llull[at movebarcelona[dot eu Introduction

Slides_Microectrcs_Chapter5.pdf - Chapter 5 Duration Models...

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Chapter 5: Duration Models Joan Llull Microeconometrics. IDEA. Fall 2017 joan.llull [at] movebarcelona [dot] eu
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Introduction Chapter 5. Fall 2017 2
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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 ex- pectancy, rms exit from the market,. .. We want to analyze: Why durations di er across individuals? How and why do exit probabilities vary over time ? Long tradition in biometrics : surviving probabilities, hazard functions,. .. Chapter 5. Fall 2017 3
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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 . Chapter 5. Fall 2017 4
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Figure I . Two examples of censored observations i. Example 1 0 1 2 3 4 Individual Jan90 Jul90 Jan91 Jul91 Jan92 Date ii. Example 2 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. Chapter 5. Fall 2017 5
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The Hazard Function Chapter 5. Fall 2017 6
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The hazard function Hazard function : probability (or density) of exiting at t condi- tional 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. Chapter 5. Fall 2017 7
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Figure II . Mortality Hazard Rate 0.0 0.2 0.4 0.6 0.8 1.0 Hazard function 0 20 40 60 80 100 Age Note: The line depicts the hazard mortality rate, i.e. the probability of dying at age a conditional on survival until that age. Chapter 5. Fall 2017 8
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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 : h ( τ ) = Pr( t = τ | t τ ) = Pr( t = τ ) Pr( t τ ) = p ( τ ) 1 - F ( τ - 1) = F ( τ ) - F ( τ - 1) 1 - F ( τ - 1) . We can recover p ( t ) and F ( t ) from h ( t ) recursively: 1 - h ( t ) = 1 - F ( t ) 1 - F ( t - 1) F ( t ) = 1 - t Y s =1 (1 - h ( s )) , p ( t ) = (1 - F ( t - 1)) h ( t ) = h ( t ) t - 1 Y s =1 (1 - h ( s )) . (Interpretation of these expressions) Chapter 5. Fall 2017 9
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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 ) = R t 0 f ( u ) du . Hazard function for t : h ( τ ) = lim dt 0 Pr( τ t < τ + dt | t τ ) dt = lim dt 0 Pr( τ t < τ + dt ) Pr( t τ ) ± dt = f ( τ ) 1 - F ( τ ) .
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