Econometric model

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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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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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EXAMPLE
Cigarette
Alcohol
Marijuana
Cocaine
Yes
No
Yes
Yes
Yes
No
No
No
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Hausman and Wise (1978, Econometrica)
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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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????
?
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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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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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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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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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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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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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B.3 Simultaneous Equations Models

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

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