1. ?????? (Discrete Choice Models)

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• 1. ?????? (Discrete Choice Models)
• 2. ?????? (Sample Selection Models)
• 3. ?????? (Duration Models)

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1 Discrete Choice Models  
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???????
??????(1?? 0???) ??(????) 1?? 0?? ???? ????? ??? ??? ?? ?? ???
0 0 35 ? 45?
0 0 25 ? 70?
1 0 84?12? 500? 41 ? 40?
1 0 84?12? 300? 46 ? 80?
1 1 84?12? 86?6? 350? 55 ? 150?
1 1 84?12? 87?2? 1000? 49 ? 270?
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Econometric model

• Single equation model
• System of equations model
• Simultaneous equation Model

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Single equation model
Can be written as
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System of equations model
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Simultaneous equations
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Microeconometrics

• Discrete choice models
• Sample selection models
• Duration models

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Discrete Choice Model

• Probit Model Logit Model
• Multinomial Choice Model
• Multinomial Logit Model
• Nested Logit Model
• Mixed Logit Model
• Multinomial Probit Model
• Bivariated Probit Model
• Multivariate Probit Model
• Sequential Choice Model
• Ordered Probit Model
• Count Data

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Sample Selection Model

• Censored model
• Sample selection model

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Duration Model

• Duration model
• Split population model

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Binary Choice Model
Individual i
Choose A
Dont Choose A
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Binary Choice Model
(Unobserved variable)
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Probit Model
N(0, 1)
Assume
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Binary Choice Model

• Boczar (1978, J. of Finance)

Personal loan debtor
Bank
Finance Company
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Binary choice model
Obtain a credit from a bank
Obtain a credit from a financial company
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The probability of choosing alternative 1 is
given by
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Probit Model
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The probability of choosing alternative 0 is
given by
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Probit Model
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Probit model
The loglikelihood for this model is given by
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Properties of Maximum Likilihood Estimator
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Probit, logit vs. OLS
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Modeling Decision

• This yes or no type decision leads to a dummy
  variable.
• The dependent variable of our model is a dummy
  variable.
• We will be modeling the probability function,
  P(Y1).

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Simplest ModelLinear Probability Model
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Picture of LPM
1
X
0
X0
X1
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Problems of LPM

• Predictions outside 0-1 range.
• Heteroscedasticity
• This can be solved and a estimated GLS estimator
  developed.
• Coefficient Determination has little meaning.
• Constant marginal effect.

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Interpreting the Probit Model
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The logit model
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The Log-Likelihood function

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LIMDEP Command   Read NVAR7Nobs200
filenames..   Regress LHSy1
RHSone,x1,x2,. Probit LHSy1
RHSone,x1,x2,. Logit LHSy1
RHSone,x1,x2,.
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PROBIT, LOGIT Goodness of Fit Measures?

• More often cited are R-square values based on
  likelihood ratios.
• Maddala  
• R2 1 - (LR / LUR) 2/n
• McFadden R-square
• R2 1 - (log(LUR ) / log(LR))

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Jacobson and Roszbach (2003, Journal of Banking
Finance) ----- Bivariate Probit Model
Providing a loan?
Loan defaults?
Yes
No
Yes
No
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Bivariate Probit Model
(if loan granted)
(if loan not granted)
(if loan does not default)
(if loan defaults)
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Bivariate Probit model
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Multivariate Probit Model
Cigarette
Alcohol
Marijuana
Cocaine
YES
No
No
YES
YES
YES
No
No
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Multivariate Normality
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Hausman and Wise (1978, Econometrica)
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Multivariate Probit Model

• J3, Clark (1961)
• J4, Hausman and Wise (1978, Econometrica)
• J gt 4
• McFadden (1989, Econometrica)
• ------ Simulation-Based Estimation
• ------ high dimensional integrals
• Stern (1997, Journal of Economic Literature)
• ----- Simulated Maximum Likelihood Estimator
• ----- Simulated Moment Estimator
• ----- GHK simulator

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LIMDEP CommandBivariated Probit Model   Read
NVAR7Nobs200 filenames..   Bivariate
Probit LHSY1, Y2
RHSone,x1,x2,.
RH2one,z1,z2,.
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Multivariate Probit Model
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Multinomial Choice Model Example Credit Card
Individual i
Alternatives
J

2
3
1
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Multinomial Choice Model
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Multinomial Choice Model
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Multinomial Logit Model
Let
be the probabilities associated these m categories
( j1,2,.m-1)
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If
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McFadden 1973
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Multinomial Logit Model
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If the ith individual falls in the jth category
otherwise
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Independence of Irrelevant Alternatives (IIA)

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Ordered Probit Model
Example Blume, Lim, Mackinlay (1998, Jornal of
Finance) Corporate bond rating (????)
AAA
AA
A
BBB
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Ordered Probit Model
N(0, 1)
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????
?
????
?
????
?????? Never Fail
????? Eventually Fail
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Sequential Choice Model Example???
Auction
No
Yes
No
Yes
Yes
No
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Sequential Response Model
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Sequential Choice Model
First Auction
No 1-F(ß1x)
YesF(ß1x)
No1-F(ß2x)
YesF(ß2x)
YesF(ß3x)
No1-F(ß3x)
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Then the probabilities can be written as
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Model Selection Joint decision vs.
Sequential decision
EXAMPLE
Bivariate Probit Model ? Multinomial Choice
Model? Ordered Probit Model? Sequential
Choice Model?
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Model Selection
EXAMPLE
Ioannides and Rosenthal (1994, The Review of
Economics and Statistics) Estimating the
consumption and investment demand for housing and
their effect on housing tenure status
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Multinomial Choice Model?
(?????)
(??????)
(??????)
(???????)
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Ordered Probit Model?
Intensity of Utility
(???????)
(??????)
(??????)
(?????)
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Sequential Choice Model?
???
??
??
???
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Bivariate Probit Model ?
??
???
??
???
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Bivariate Probit Model
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Bivariate Probit model
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Multivariate Probit Model
Cigarette
Alcohol
Marijuana
Cocaine
YES
No
No
YES
YES
YES
No
No
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Multivariate Normality
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Multivariate Probit Model

• J3, Clark (1961)
• J4, Hausman and Wise (1978, Econometrica)
• J gt 4
• McFadden (1989, Econometrica)
• ------ Simulation-Based Estimation
• ------ high dimensional integrals
• Stern (1997, Journal of Economic Literature)
• ----- Simulated Maximum Likelihood Estimator
• ----- Simulated Moment Estimator
• ----- GHK simulator

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ApplicationSurvey bias models

• Censored model Deaton and Irish (1984)
• Probit model Bollinger and David (1997),
  Abrevaya and Hausman (1999a)
• Multinomial logit model Hsiao and Sun (1999)
• Ordered probit model Dustman and van Soest
  (2004)
• Duration model Torelli and Trivellato (1993),
  Abrevaya and Hausman (1999b)

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Bollinger and David (1997)

• Binary choice model

Did not use the substance
Used the substance
Lied
Did not lie
Lied
Did not lie
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Hsaio and Sun (1999)

• Binary choice model

Did not use the substance
Used the substance
lied
Did not lie
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Table 1 Replied
Substance Use qi qi yi(1 - wi
)
yi 1 yi 0
(used the substance) (did not use the substance)
wi 1 (lied) 0 Not Applicable
wi 1 (lied) (reply did not use) Not Applicable
wi 0 (did not lie) 1 0
wi 0 (did not lie) (reply used) (reply did not use)
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Model 1 Uniform one-sided survey response bias
model
The loglikelihood for this model is given by
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Model 2 Heterogeneous and independent one-sided
survey response bias model
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Model 3 Heterogeneous and dependent one-sided
survey response bias model
Assume
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Partial observability model Poirier (1980)
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Partial observability model Poirier (1980)

• Zero-inflated Poisson model
• Double hurdle model
• Split population model

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Leung and Yu(2003) Empirical Analysis

• 1998 U.S. National Household Survey on Drug Abuse
  (NHSDA)
• Tobacco
• Alcohol
• Marijuana
• Cocaine

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Table 4 Model 1 Uniform One-sided Survey
Response Bias Model
Variable Tobacco Alcohol Marijuana Cocaine
------- ---- ---- ----
------- ---- ---- ----
a 0.1386 0.1614 0.0003 0.4701
-0.0586 -0.0192 -0.3202 -0.7591
Loglikelihood -13852.738 -15181.128 -6073.143 -1519.109
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Count regression

• Appropriate when the dependent variable
• is a non-negative integer (0,1,2,3,)

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• Distributions and Models
• Poisson Model
• Negative Binomial Model
• Zero-inflated Poisson Model
• Zero-inflated Negative Binomial Model

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Poisson Regression
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Why not use linear regression?

• Typical count data in health care
• Large number of 0 values and small values
• Discrete nature of data
• Result
• Unusual distribution

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Normal distribution vsPoisson distribution
Bell shaped curve
Normal distribution
Poisson distribution
Not bell shapednext slide
Intensity of process
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Poisson with ? 0.5
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When Count Data Cannot be Treated Normally
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When they probably can.
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What happens when mean ? variance?

• Overdispersion when variance gt mean
• Sometimes called unobserved heterogeneity
• Zero-Inflated More zeros than expected by
  Poisson distribution
• Ex. If ?1 (mean1), then we expect 37 0s

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Overdispersion
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Poisson Regression models
Negative Binomial Regression models
u is Weibull distribution
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Overdispersion and Zero Inflation
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Zero-inflated Poisson
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Example

• Bao article
• Predicting the use of outpatient mental health
  services do modeling approaches make a
  difference? Inquiry. 2002 Summer39(2)168-83.

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Observed data
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Poisson and Zero-Inflated Poisson
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Negative Binomial Model
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Zero-Inflated Negative Binomial Model
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TOBIT Model
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TOBIT MODEL


if
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
where
p.f.
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TOBIT MODEL
let
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TOBIT MODEL
NOTE
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TOBIT MODEL
let
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TOBIT MODEL
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TOBIT MODEL
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The log-likelihood function

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Sample Selection Model
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Self- Selection Model
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Sample Selection Model
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Heckmans Two-step Estimator (1979)
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Application Heckmans Two-step Estimator vs.
Two-part model
Duan, Manning, Morris, Newhouse(1984,
JBES) Maddala (1985a, 1985b) Hay, Leu,
Fohrer(1987,JBES) Manning, Duan, Rogers (1987,
JE) Leung and Yu (1996, JE) Dow and Norton (2003,
HSORM) Dow and Norton (2005)
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Sequential Choice Model?
???
??
??
???
0
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Joint Choice Model
???
??
???
??
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Duration Models

• ? Censored Data
• ? Unobserved Heterogeneity
• ? Time-Varying Covariates

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D
C

C
D
D
End of study
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• Hazard Rate

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• Survival Rate

Hazard Rate and Survival Rate

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Duration Model
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• Distributions
• Parametric
• Expoential
• Weibull
• Log-normal
• Log-logistic
• Gamma
• Semi-parametric
• Coxs partial likelihood estimator

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LIMDEP Command---Duration Model   Read
NVAR7Nobs200 filenames..   Survival
LHSln(time), status (exit1) RHSone,x1,x2,.
modelExponential Survival LHSln(time),
status (exit1) RHSone,x1,x2,.
modelWeibull Coxs Semiparametric
Estimator Survival LHSln(time), status
(exit1) RHSone,x1,x2,.
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TOBIT MODEL


if
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
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TOBIT MODEL
where
p.f.
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TOBIT MODEL
let
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TOBIT MODEL
NOTE
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TOBIT MODEL
let
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TOBIT MODEL
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TOBIT MODEL
NOTE
by LHopital rule
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Duration Model
D
C

C
D
D
End of study
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Duration model

• Censored data
• Unobserved heterogeneity
• Time-varying covariates

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2.2 Hazard Analysis
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Survival rate and Hazard rate
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2.2 Nonparametric Hazard Analysis

• Kaplan-Meier estimator
• Life table estimator

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Figure 2 Kaplan-Meier Estimates of Survival
Function
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Figure 3 Life Table Estimates of Survival
Function
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The density and survival functions
f (Ti?wi) the probability density function
of the failure time S
(ti?wi) the probability of survival
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The specifications for f (Ti?xi) and S
(ti?xi)

• Exponential
• Weibull
• Log-logistic
• Log-normal

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Figure 4 Life Table Estimates of Hazard
Functions
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Eventually fail assumption
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?
????
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Eventually Fail Assumption
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????
?
????
?
????
?????? Never Fail
????? Eventually Fail
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Schmidt and Witte (1989) --- Split
population duration model
G (?xi) the probability of eventual failure f
(Ti?wi) the probability density function
of the failure time S (Ti?wi) the
probability of survival
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Schmidt and Witte (1989) --- Likelihood function
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4.3 Multivariate Split Population Duration Model
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Multivariate probit model
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Multivariate duration model
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Unobserved heterogeneity
The frailty ( m 1,2) is assumed to follow
a gamma distribution with mean 1 and variance

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Whether Part

• individuals probability
  of eventual failure for a type k event (k
  1,2,3,4).
• follows a Weibull distribution

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Duration Part

• Assume the survival function is log-logistic. The
  second frailty
• enters the hazard function as

, and is the failure time or the
where
censored time, whichever is earlier.
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The cumulative hazard, the survival function, and
the density function are
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The likelihood function is given by

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Partial likelihood Estimation
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Person Event Time Li
1 1 3
2 2 8
3 10
4 15
5 3 21
6 4 30
7 5 32
8 52
9 52
10 52
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B.3 Simultaneous Equations Models

• M. J. Lee (1995, Journal of Applied Econometrics)

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• ????