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Econometric Analysis of Panel Data

22. Limited Dependent Variables

And Models for Count Data

Censoring and Corner

Solution Models

 Censoring model: T(y) = 0 if y < 0 and y*

otherwise.

 Corner solution: y = 0 if some exogenous

condition is met; y = g(x)+e if the condition is

not met. We then model P(y=0) and

E[y|x,y>0]. (See text, pp. 518-519)

Application: Fair, R., “A Theory of Extramarital

Affairs,” JPE, 1978.

 Same structural form

Conditional Mean Functions

2

y* , ~N[0, ]

y Max(0, y*)

E[y* | ] x

E[y| ]=Prob[y=0| ]×0+Prob[y>0| ]E[y|y>0, ]

=Prob[y>0| ] E[y|y*>0, ]

= + ε ε σ

= β

 φ σ   

Φ × σ

σ Φ σ

Φ σ

x β

x

x x x x

x x

x β x β/

x β +

x β/

x β/ x β ( )

" Inverse Mills ratio"

E[y| ,y>0]=

σφ σ

φ σ

Φ σ

φ σ

σ

Φ σ

+ x β/

x β/

x β/

x β/

x x β +

x β/

Conditional Means

XB

-1.

-.

.

1.

2.

-2.

-2.00 -1.20 -.40 .40 1.20 2.

EYSTAR EY

Variable

OLS is Inconsistent - Attenuation

E[y | ] = ( ) ( )

Nonlinear function of x. What is estimated by OLS

regression of y on x?

Slopes of the linear projection are approximately

equal to the derivatives of the conditional mea

x Φ x β/ σ x β + σφ x β/ σ

n

evaluated at the means of the data.

E[y | ]

Note the attenuation.

= Φ σ ×

x

x β/ β

x

Estimating the Tobit Model

n

i

i i

i=

i i i

Log likelihood for the tobit model for estimation of and :

1 y

logL= log (1-d ) log d log

d 1 if y 0, 0 if y = 0. Derivatives are very complicated,

Hessian

σ

  ′  ′  − −    

Φ + φ

       

σ σ σ

     

 

= >

i i

β

x β x β

n

i i i

i=

i i i

is nightmarish. Consider the Olsen transformation*:

=1/ , =- /. (One to one; =1/ ,

logL= log (1-d ) log d log y

log (1-d ) log d (log (1 / 2) log 2 (1 / 2) y

θ σ σ σ θ θ

  ′ ′ Φ + θφ θ +

 

Φ ′ + θ + π − θ + ′

∑ i i

i i

β β = -

x x

x x

γ γ/ .)

γ γ

γ γ

n 2

i=

n

i i i

i 1

n

i i i

i 1

)

logL

(1-d ) d e

logL 1

d e y

*Note on the Uniqueness of the MLE in the Tobit Model," Econometrica, 1978.

=

=

 

 

 ′  φ

∂

= −

 

′ ∂ Φ

 

 

∂  

= −

 

∂θ θ

 

i

i

i

x

x

x

γ

γ γ

Simplified Hessian

2

2

n

i i

i 1

2

n

i

i 1

2

n

i 2

i 1

logL

(1-d ) ( d

logL

d y

logL 1

d

=

=

=

φ φ ∂

∂ ∂θ

∂θ∂θ θ

i i

i i i

i i

i i

x x

x x x

x x

x

γ γ

γ)

γ γ γ γ

γ

i

2

n

i i i i

2 2

i 1

i i i

i i i i i i i i

d y y

((1 d ) d ) d y

logL

d y d (1 / y )

a (a ) / (a ), (a )

=

− δ − −   ∂

  − − θ +

∂ ∂ θ

θ

= λ = φ Φ δ = −λ + λ

i i

i i i i

i i

i

x x x

x

x

γ

γ

γ,

Recovering Structural Parameters

2

2

2

1

( , )

( , ) 1

ˆ ˆ

( , ) ˆ

Use the delta method to estimate Asy.Var

ˆ

( , ) ˆ ˆ

( , )

1 1

( , ) 1

( , )

1 1

ˆ ˆ

( , ˆ

Est.Asy.Var

 

−

 

θ  

θ

= 

 

σ θ

   

 

θ  

 

θ

 

 

σ θ

 

 θ 

−  

∂

 

 

θ σ θ −  

θ θ  

=  = = θ

 

′ ∂ θ − θ

   

 

 θ 

β β β I I G

0'

0'

β

γ γ γ γ γ γ γ

−γ γ

γ

γ

γ

ˆ

)

ˆ ˆ

( , ) Est.Asy.Var ( , ) ˆ ˆ

ˆ

ˆ

( , ) ˆ ˆ

 

 

θ

= θ × × θ  

 

 

θ

σ θ  

 

G G '

γ

γ γ

γ

A Decompositioin

McDonald and Moffitt's decomposition of

the partial effect

E[y| ] E[y| ,y>0]

=Prob[y>0| ]

x

Prob[y>0| ]

+E[y| ,y>0]

x x

x

x

x

x

x

Application: Fair’s Data

F22.2 Fair’s (1977) Extramarital Affairs Data, 601 Cross Section observations.

Source: Fair (1977) and http://fairmodel.econ.yale.edu/rayfair/pdf/1978ADAT.ZIP.

Several variables not used are denoted X1, ..., X5.)

y = Number of affairs in the past year, (0,1,2,3,4-10=7, more=12, mean = 1.

(Frequencies 451, 34, 17, 19, 42, 38)

z1 = Sex, 0=female; mean=.

z2 = Age, mean=32.

z3 = Number of years married, mean=8.

z4 = Children, 0=no; mean=.

z5 = Religiousness, 1=anti, …,5=very. Mean=3.

z6 = Education, years, 9, 12, 16, 17, 18, 20; mean=16.

z7 = Occupation, Hollingshead scale, 1,…,7; mean=4.

z8 = Self rating of marriage. 1=very unhappy; 5=very happy

Estimated Tobit Model

Neglected Heterogeneity

2 2

c

2 2

c

2 2

c 2 2

c

y* (c ) assuming c

Pr ob[y* 0 | x]

MLE estimates /( ) Attenuated for

Marginal effects are /( )

Consistently estimated by ML even though is not

ε

ε

ε

ε

x'β x

x'β

x'β

(The standard result.)

Regression with the Truncated

Distribution

i i

i

n

i

i 1

2

( )

E[y | ,y >0]= +

( )

= +

OLS will be inconsistent:

1

Plim b = plim plim

n n

A left out variable problem.

Approximately: plim b plim(1 - a )

=

′ φ σ

′ σ

′ Φ σ

′ σλ

 

× λ

 

 

≈ λ − λ

i

i i

i

i

i

x β/

x x β

x β/

x β

X'X

β + x

β

n

i 1

1

a= , (a)

n

General result: Attenuation

=

′ σ λ = λ

∑ i

x β/

Estimated Tobit Model