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Econometric Analysis of Panel Data

17. Nonlinear Effects Models and

Models for Binary Choice (Cont.)

Application – Doctor Visits

German Health Care Usage Data , 7,293 Individuals, Varying Numbers of Periods

Variables in the file are

Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,

individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary

choice. This is a large data set. There are altogether 27,326 observations. The number of observations ranges

from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Note, the variable

NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of

the data for the person. (Downlo0aded from the JAE Archive)

DOCTOR = 1(Number of doctor visits > 0)

HSAT = health satisfaction, coded 0 (low) - 10 (high)

DOCVIS = number of doctor visits in last three months

HOSPVIS = number of hospital visits in last calendar year

PUBLIC = insured in public health insurance = 1; otherwise = 0

ADDON = insured by add-on insurance = 1; otherswise = 0

HHNINC = household nominal monthly net income in German marks / 10000.

(4 observations with income=0 were dropped)

HHKIDS = children under age 16 in the household = 1; otherwise = 0

EDUC = years of schooling

AGE = age in years

MARRIED = marital status

EDUC = years of education

Application: Innovations

Bertschek and Lechner, J of Econometrics, 1998

Pooled Estimation

it

i t it^ it

Ignoring panel data nature; P(y 1 | ) F( ).

Estimation is based on the marginal distribution.

logL= (1 y ) log(1 F( )) y logF( )

Partial likelihood methods. (Include a 'robust' covaria

= =^ ′

− − ′^ + ′ ∑ ∑

it it

it it

x x β

x β x β

nce

matrix.)

What assumptions are needed to make this work?

Strict exogeneity?

Dynamic completeness? (No lagged effects)

Somewhat strong assumptions.

Latent common effects are usually not ignored.

FIML

i5 i5 i1 i

N

i 1 i1^ i

N (2y^ 1)^ (2y^ 1)

i 1^1

5 / 2 1 / 2 1

i1 i2 12 i1 i5 15

i1 i2 12 i2 i2 22

logL log Prob[y ,..., y ]

log ... g( | *)dv ...dv

g( | *) (2 ) | * | exp[ (1 / 2) ( *) ]

1 q q ... q q

q q 1 ... q q

... ... ... ..

=

=

= π −

ρ ρ

ρ ρ

∑

∑ (^) ∫ ∫

x β x β v Σ

v Σ Σ v' Σ v

Σ

i1 i5 1T i2 i5 2,T 1

it it

.

q q q q ... 1

q 2y 1

           (^) ρ ρ   

= −

See Greene, W., “Convenient Estimators for the Panel Probit Model: Further Results,”

Empirical Economics, 29, 1, Jan. 2004, pp. 21-48.

GMM

it i

i1 i

i2 i

i5 i

From the marginal distributions:

E[y ( ) | ] 0 (note: strict exogeneity)

Suggests orthogonality conditions

(y ( ))

(y ( ))

E

...

(y ( ))

− Φ ′ =

 − Φ ′  (^)  

    − Φ ′     =    

     − Φ ′    ^ 

it

i

i

i

x β X

x β x (^0)

x β x (^0)

...

x β x (^0)

5*K moments.



GMM Estimation-

i1 i

1270 i2 i

i 1

i5 i

Step 2. Minimize GMM criterion q = ( ) ( )

(y ( ))

1 (y^ (^ )) ( )= 1270 ...

(y ( ))

 − Φ ′ 

  − Φ ′  

 

   − Φ ′   

∑

i

i

i

g β 'W g β

x β x

x β x

g β

x β x

Unobserved Heterogeneity

it i it

it it i it

it it u

Assuming strict exogeneity

y *= u

Prob[y 1 | x ] Prob[u - ]

Using the same model format:

Prob[y 1 | x ] F / 1+ F( )

Ignoring heterogeneity, we estimate not.

Partial ef

= = ′^ σ = ′

it

it

it it

x β

x β

x β x δ

fects are f( ) not f( )

is underestimated, but f( ) is overestimated. Which

way does it go? Maybe ignoring u is ok? Not if we want

to compute probabilities or do statistical inference a

it it

it

δ x δ β x β

β x β

bout β.

Random Effects

 Uit = α + β’xit + (εit + σv vi )

 Joint probability for individual i | v i =

 Unobserved component vi must be eliminated

 Maximize wrt α, β and σv

 How to do the integration?

 Analytic integration – quadrature; most familiar software

 Simulation

( ' )

Ti

∏ (^) t = F^ α + β^ x it + σ v vi

1 log log ( ' )

N Ti i

i t it^ v^ i^ i

v v

v L F v f dv

  = ^ α + β + σ      σ σ  

∑ (^) ∫ ∏ x

Quadrature – Butler and Moffitt

( )

( )

N T i i 1 t 1 it^ v^ i^ i^ i

2 N i 1 i

N (^2) i 1 i

logL log F( ' x v ) v dv

1 -v = log g(v) exp dv 2 2

1 = log g( 2u) exp -u du

The integral can be computed using Hermite quadrature.

(See class notes

∞

= (^) −∞ =

∞

= (^) −∞

∞

= (^) −∞

= ^ α + β + σ φ  

    π (^)  

π

N H i 1 h 1 h^ h

5, Feb. 8)

1 log w g( 2z )

Most programs estimate this model using this method.

= =

≈ π

Random Effects is Equivalent to a Random

Constant Term

Uit = α + β’xit + (εit + σv vi )

= (α + σα vi ) + β’xit + εit

= αi + β’xit + εit

αi is random with mean α and variance

View the simulation as sampling over αi

2

α

σ

1 log ( ' )

N R Ti

i r t ir^ it

F R

  α + β  

∑ ∑ ∏ x

Why not make all the coefficients random?

No Effects – Pooled Model

| Binomial Probit Model |

| Number of observations 27322 |

| Log likelihood function -17408.37 |

| Restricted log likelihood -18016.64 |

| Chi squared 1216.548 |

| Degrees of freedom 7 |

| Prob[ChiSqd > value] = .0000000 |

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|

Index function for probability

Constant -.03345234 .06125319 -.546.

HHNINC -.08611064 .04766266 -1.807 .0708.

HHKIDS -.15550242 .01834218 -8.478 .0000.

EDUC -.01433684 .00357801 -4.007 .0001 11.

MARRIED .07154569 .02064966 3.465 .0005.

AGE .01113647 .00081150 13.723 .0000 43.

FEMALE .32490404 .01727948 18.803 .0000.

WORKING -.09339954 .01938730 -4.818 .0000.

Simulation

| Random Coefficients Probit Model | | Log likelihood function -16154.13 | | Number of parameters 9 | | Restricted log likelihood -17408.37 | | Chi squared 2508.469 | | Degrees of freedom 1 | | Prob[ChiSqd > value] = .0000000 | | Unbalanced panel has 7289 individuals. | | PROBIT (normal) probability model | | Simulation based on 20 Halton draws | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Nonrandom parameters HHNINC .03584846 .03558953 1.007 .3138. HHKIDS -.16946521 .01364814 -12.417 .0000. EDUC -.01676742 .00272734 -6.148 .0000 11. MARRIED .02965951 .01535582 1.931 .0534. AGE .01765054 .00061319 28.785 .0000 43. FEMALE .43479769 .01295456 33.563 .0000. WORKING -.09023285 .01416939 -6.368 .0000. Means for random parameters Constant -.22761867 .04667520 -4.877. Scale parameters for dists. of random parameters

Constant .87528074 .00790787 110.685 .0000 RHO = .40456 Docsity.com

Fixed Effects

 Dummy variable coefficients

Uit = αi + β’x it + εit

 Can be done by “brute force” for 10,000s of individuals

 F(.) = appropriate probability for the observed outcome

 Compute β and αi for i=1,…,N (may be large)

 Group mean deviations does not work here. This must be done the hard

way. (Infeasible? Generally viewed as such.)

 See “Estimating Econometric Models with Fixed Effects” at

N T i

i 1 t 1 it^ i^ it

logL logF(y , ' x )

= (^) ∑ ∑ α + β