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Econometric Analysis of Panel Data

14. Nonlinear Models

And Nonlinear Optimization

Agenda

 Nonlinear Models

 Estimation Theory for Nonlinear Models

 Estimators

 Properties

 M Estimation

 Nonlinear Least Squares

 Maximum Likelihood Estimation

 GMM Estimation

 Minimum Distance Estimation

 Minimum Chi-square Estimation

 Computation – Nonlinear Optimization

 Nonlinear Least Squares

 Newton-like Algorithms; Gradient Methods

 (Background: JW, Chapters 12-14, Greene, Chapters 16-18)

What is a Nonlinear Model?

 Model: E[g(y)| x ] = m( x , θ )

 Objective:

 Learn about θ from y , X

 Usually “estimate” θ

 Linear Model: Closed form; = h( y , X )

 Nonlinear Model

 Not wrt m( x , θ ). E.g., y=exp( θ’x + ε )

 Wrt estimator: Implicitly defined. h( y , X, )=0,

E.g., E[y|x]= exp( θ’x )

ˆ

θ

θ

ˆ

What is an Estimator?

 Point and Interval

 Classical and Bayesian

ˆ

f(data | mod el)

ˆ ˆ

I( ) sampling variability

θ =

θ = θ ±

ˆ

E[ | data,prior f( )] expectation from posterior

ˆ

I( ) narrowest interval from posterior density

containing the specified probability (mass)

θ = θ θ =

θ =

The Conditional Mean Function

y,x

m(x, ) E[y | x] for some in.

A property of the conditional mean:

E (y m(x, )) is minimized by E[y | x]

(Proof, pp. 343-344, JW)

θ = θ Θ

− θ

M Estimation

Classical estimation method

n

i

i=

n

2

i i i

i=

1

ˆ

arg min q( , )

n

Example : Nonlinear Least squares

1

ˆ

arg min [y -E(y | , )]

n

θ = θ

θ = θ

∑

∑

data

x

Estimation

n

P

i

i 1

0

P

P

0

q= q(data , ) q*=E[q(data, )]

n

Estimator minimizes q

True parameter minimizes q*

q q*

Does this imply?

Yes, if ...

=

Identification

4

1 0 1 0

1 2 3

Uniqueness :

If , then m(x, ) m(x, )

Examples

(1) (Multicollinearity)

(2) (Need for normalization) E[y|x] = m( x/ )

(3) (Indeterminacy) m(x, )= x x

β

θ ≠ θ θ ≠ θ

′

β σ

θ β + β + β

Consistency

n

P

i

i 1

0

P

P

0

1

q= q(data , ) q*=E[q(data, )]

n

ˆ

Estimator minimizes q

True parameter minimizes q*

q q*

ˆ

Does this imply?

Yes. Consistency follows from identification

and continuity with the other a

=

θ → θ

θ

θ

→

θ → θ

∑

ssumptions

Asymptotic Normality of M

Estimators

N

i=1 i

N i

i=

N

i=1 i

First order conditions:

(1/n) q(data , )

1 q(data , )

n

(data , ) (data, )

n

For any , this is the mean of a random

sample. We apply Lindberg-Feller CLT to assert

the limit

g g

ing normal distribution of n g (data, θ).

Asymptotic Normality

1

0 0

1

0

ˆ

n ( ) [ ( )] n (data, )

[ ( )] converges to its expectation (a matrix)

n (data, ) converges to a normally distributed

vector (Lindberg-Feller)

ˆ

Implies limiting normal distribution of n (

−

−

θ − θ = θ θ

θ

θ

θ − θ

H g

H

g





0

).

Limiting mean is 0.

Limiting variance to be obtained.

Asymptotic distribution obtained by the usual means.

Asymptotic Variance

a 1

0 0

0

1 1

0 0 0

0

i 0 i 0

ˆ

[ ( )] (data, )

Asymptotically normal

Mean

ˆ

Asy.Var[ ] [ ( )] Var[ (data, )] [ ( )]

(A sandwich estimator, as usual)

What is Var[ (data, )]?

1

E[ (data , ) (data , ) ']

n

Not known

−

− −

θ → θ + θ θ

= θ

θ = θ θ θ

θ

θ θ

H g

H g H

g

g g



n

i 1 i i

what it is, but it is easy to estimate.

1 1

ˆ ˆ

(data , ) (data , ) '

n n

=

× Σ g θ g θ

Nonlinear Least Squares

i

i

0 i

i i

(k+1) (k) 1

Gauss-Marquardt Algorithm

q the conditional mean function

= m(x , )

m(x , )

x 'pseudo regressors '

Algorithm - iteration

ˆ ˆ

[ ]

−

=

θ

∂ θ

= = = −

∂θ

θ = θ +

0 0 0 0

g

X 'X X 'e

Application - Income

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 7,293 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)

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