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Econometric model

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

Single equation model

Can be written as

System of equations model

Simultaneous equations

Microeconometrics

- Discrete choice models
- Sample selection models
- Duration models

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

Sample Selection Model

- Censored model
- Sample selection model

Duration Model

- Duration model
- Split population model

Binary Choice Model

Individual i

Choose A

Dont Choose A

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Binary Choice Model

(Unobserved variable)

Probit Model

N(0, 1)

Assume

Binary Choice Model

- Boczar (1978, J. of Finance)

Personal loan debtor

Bank

Finance Company

Binary choice model

Obtain a credit from a bank

Obtain a credit from a financial company

The probability of choosing alternative 1 is

given by

Probit Model

The probability of choosing alternative 0 is

given by

Probit Model

Probit model

The loglikelihood for this model is given by

Properties of Maximum Likilihood Estimator

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Probit, logit vs. OLS

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).

Simplest ModelLinear Probability Model

Picture of LPM

1

X

0

X0

X1

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.

Interpreting the Probit Model

The logit model

The Log-Likelihood function

LIMDEP Command Read NVAR7Nobs200

filenames.. Regress LHSy1

RHSone,x1,x2,. Probit LHSy1

RHSone,x1,x2,. Logit LHSy1

RHSone,x1,x2,.

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))

Jacobson and Roszbach (2003, Journal of Banking

Finance) ----- Bivariate Probit Model

Providing a loan?

Loan defaults?

Yes

No

Yes

No

Bivariate Probit Model

(if loan granted)

(if loan not granted)

(if loan does not default)

(if loan defaults)

LIMDEP CommandBivariated Probit Model Read

NVAR7Nobs200 filenames.. Bivariate

Probit LHSY1, Y2

RHSone,x1,x2,.

RH2one,z1,z2,.

Multivariate Probit Model

EXAMPLE

Cigarette

Alcohol

Marijuana

Cocaine

Yes

No

Yes

Yes

Yes

No

No

No

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Hausman and Wise (1978, Econometrica)

Multinomial Choice Model Example Credit Card

Individual i

Alternatives

J

2

3

1

Multinomial Choice Model

Multinomial Choice Model

Multinomial Logit Model

Let

be the probabilities associated these m categories

( j1,2,.m-1)

If

McFadden 1973

Multinomial Logit Model

If the ith individual falls in the jth category

otherwise

Independence of Irrelevant Alternatives (IIA)

Ordered Probit Model

Example Blume, Lim, Mackinlay (1998, Jornal of

Finance) Corporate bond rating (????)

AAA

AA

A

BBB

Ordered Probit Model

N(0, 1)

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????

?

????

?

????

?????? Never Fail

????? Eventually Fail

Sequential Choice Model Example???

Auction

No

Yes

No

Yes

Yes

No

Sequential Response Model

Sequential Choice Model

First Auction

No 1-F(ß1x)

YesF(ß1x)

No1-F(ß2x)

YesF(ß2x)

YesF(ß3x)

No1-F(ß3x)

Then the probabilities can be written as

Model Selection Joint decision vs.

Sequential decision

EXAMPLE

Bivariate Probit Model ? Multinomial Choice

Model? Ordered Probit Model? Sequential

Choice Model?

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

Multinomial Choice Model?

(?????)

(??????)

(??????)

(???????)

Ordered Probit Model?

Intensity of Utility

(???????)

(??????)

(??????)

(?????)

Sequential Choice Model?

???

??

??

???

Bivariate Probit Model ?

??

???

??

???

Count regression

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

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

Poisson Regression

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

Normal distribution vsPoisson distribution

Bell shaped curve

Normal distribution

Poisson distribution

Not bell shapednext slide

Intensity of process

Poisson with ? 0.5

When Count Data Cannot be Treated Normally

When they probably can.

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

Overdispersion

Poisson Regression models

Negative Binomial Regression models

u is Weibull distribution

Overdispersion and Zero Inflation

Zero-inflated Poisson

Example

- Bao article
- Predicting the use of outpatient mental health

services do modeling approaches make a

difference? Inquiry. 2002 Summer39(2)168-83.

Observed data

Poisson and Zero-Inflated Poisson

Negative Binomial Model

Zero-Inflated Negative Binomial Model

TOBIT Model

TOBIT MODEL

if

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

where

p.f.

TOBIT MODEL

let

TOBIT MODEL

NOTE

TOBIT MODEL

let

TOBIT MODEL

TOBIT MODEL

The log-likelihood function

Sample Selection Model

Self- Selection Model

Sample Selection Model

Heckmans Two-step Estimator (1979)

Duration Models

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

D

C

C

D

D

End of study

- Hazard Rate

- Survival Rate

Hazard Rate and Survival Rate

Duration Model

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

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,.

Bivariate Probit Model

Bivariate Probit model

Multivariate Probit Model

Cigarette

Alcohol

Marijuana

Cocaine

YES

No

No

YES

YES

YES

No

No

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Multivariate Normality

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

TOBIT MODEL

if

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

TOBIT MODEL

where

p.f.

TOBIT MODEL

let

TOBIT MODEL

NOTE

TOBIT MODEL

let

TOBIT MODEL

TOBIT MODEL

NOTE

by LHopital rule

Duration Model

D

C

C

D

D

End of study

Duration model

- Censored data
- Unobserved heterogeneity
- Time-varying covariates

2.2 Hazard Analysis

Survival rate and Hazard rate

2.2 Nonparametric Hazard Analysis

- Kaplan-Meier estimator
- Life table estimator

Figure 2 Kaplan-Meier Estimates of Survival

Function

Figure 3 Life Table Estimates of Survival

Function

The density and survival functions

f (Ti?wi) the probability density function

of the failure time S

(ti?wi) the probability of survival

The specifications for f (Ti?xi) and S

(ti?xi)

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

Figure 4 Life Table Estimates of Hazard

Functions

Eventually fail assumption

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?

?

????

????

Eventually Fail Assumption

????

?

????

?

????

?????? Never Fail

????? Eventually Fail

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

Schmidt and Witte (1989) --- Likelihood function

4.3 Multivariate Split Population Duration Model

Multivariate probit model

Multivariate duration model

Unobserved heterogeneity

The frailty ( m 1,2) is assumed to follow

a gamma distribution with mean 1 and variance

Whether Part

- individuals probability

of eventual failure for a type k event (k

1,2,3,4). - follows a Weibull distribution

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.

The cumulative hazard, the survival function, and

the density function are

The likelihood function is given by

B.3 Simultaneous Equations Models

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

- ????

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