Download Hierarchical Bayesian Estimation and Random Coefficients Model in Econometrics and more Slides Econometrics and Mathematical Economics in PDF only on Docsity!

Econometric Analysis of Panel Data

20. Individual Heterogeneity

and Random Parameter Variation

Heterogeneity

 Observational: Observable differences across

individuals (e.g., choice makers)

 C hoice strategy: How consumers make

decisions – the underlying behavior

 S tructural: Differences in model frameworks

 P references: Differences in model ‘parameters’

Distinguish Bayes and Classical

 Both depart from the heterogeneous ‘model,’

f(y (^) it| x it)=g(y (^) it, x it, β i)

 What do we mean by ‘randomness’

 With respect to the information of the analyst (Bayesian)  With respect to some stochastic process governing ‘nature’ (Classical)

 Bayesian: No difference between ‘fixed’ and ‘random’

 Classical: Full specification of joint distributions for

observed random variables; piecemeal definitions of ‘random’ parameters. Usually a form of ‘random effects’

Hierarchical Bayesian Estimation

i,t i,t

0

Sample data generation: f(y | ) g(y , ) Individual heterogeneity: , ~ N[ ] What information exists about 'the model?'

p( ) = N[ , ]

′ (^) =

i,t i,t i i i i

0

x , x , β Ω β = β u u 0, Γ

Prior densities for structural parameters : β β Σ , e.g., and (large) v^0 p( ) = Inverse Wishart[ , ] p( ) = whatever works for other parameters in model

p( )= N[ , ] End Result: Joint prior distribution for all param

γ

i

0 I Γ A Ω Priors for parameters of interest : β β Γ

0

eters p( β Γ Ω β , , , (^) i |prior 'beliefs' in β^0 , Σ , ,γ A, assumed densities)

Priors

i

i i i β j j j

β

β

Prior Densities

~N[ , ],

Implies = + , ~N[0, ] λ ~Inverse Gamma[v,s ] (looks like chi-squared), v=3, s =

Priors over structural model parameters

~N[ ,a ], =

β β V β

β β w w V

β β V β

V ~Wishart[v , 0 V 0 (^) ],v =8, 0 V 0 =8 I

Bayesian Posterior Analysis

 Estimation of posterior distributions for upper level parameters: and Vβ

 Estimation of posterior distributions for low (individual) level parameters, βi|data (^) i. Detailed examination of ‘individual’ parameters

 (Comparison of results to counterparts using classical methods)

β

Fixed Management and Technical

Efficiency in a Random Coefficients

Model

Antonio Alvarez, University of Oviedo Carlos Arias, University of Leon William Greene, Stern School of Business, New York University

The Production Function Model

2

2

1

2 1 2

ln ln (ln )

ln

it x (^) it xx it

m (^) i mm (^) i xm it i it

y x x

m m x m v

α β β

β β β

Definition: Maximal output, given the inputs Inputs: Variable factors, Quasi-fixed (land) Form: Log-quadratic - translog Latent Management as an unobservable input

Translog Production Model

(^12) 1 1 1

• 1 *2 1 * (^2 2 )

i

ln = ln -

ln ln ln

• ln

m * is an unobserved, time invariant

= = =

=

= α + β + β

β + β + β

it it it K K K k k^ itk^ k l kl^ itk^ itl K m i mm i (^) k km itk i it it

y y u

x x x

m m x m

v u effect.

( ) (^ )^ (^ )

1 *^1 * 2^2

= ln - ln

ln 0.

it it it K m (^) k km kit i i mm i i

u y y

= β + ∑ = β x m − m + γ m − m ≥

Random Coefficients Model

( ) ( )

ln ln

ln ln

ln ln ln

= α + β + β + β + β

• β + −

= α + β + β + ε

∑

∑ ∑

∑ ∑ ∑

K

it m i mm i (^) k k km i itk K K k l kl^ itk^ itl^ it^ it K K K i (^) k ki itk (^) k l kl itk itl it

y m m m x

x x v u

x x x

1

ln

K i k k i k

m τ x w

= (^) ∑ +

[Chamberlain/Mundlak:]

(1) Same random effect appears in each random parameter

(2) Only the first order terms are random

Discrete Parameter Variation

The Latent Class Model

(1) Population is a (finite) mixture of Q types of individuals.

q = 1,...,Q. Q 'classes' differentiated by ( ) (a) Analyst does not know class memberships. ('latent.')

β q

J 1 Q q=1 q

i,t it

(b) 'Mixing probabilities' (from the point of view of the analyst) are ,..., , with 1

(2) Conditional density is

P(y | class q) f(y | , )

π π Σ π =

= = x (^) i,t βq

Latent Classes and

Random Parameters

′ ′

π

∑

Q i (^) q=1 i i,choice class i j=choice i,j class

i,q i,q

Pr(Choice ) = Pr(choice | class = q)Pr(class = q)

exp(x β )

Heterogeneity with resp

Pr(choice | class = q) = Σ exp(x β ) e Pr(class =

ect

q | i

to 'latent' cons

) = , e.g.,

umer classes

F =

( ) ,

′ ′

′ ′ ′ = = π ′

∑

i q q=classes i q

i,choice i i i j=choice i,j i i q i q i,q q=classes i q Q i (^) q=

xp(z ) Σ exp(z )

exp(x β

Simple discrete ran

) Pr(choice | β ) = Σ exp(x β ) exp(z ) Pr β β = q = 1,..., Q Σ exp(

dom param

z )

Pr(Choice

eter v

) =

ariatio

(

n

Pr c

δ δ

δ δ hoice | βi =β )Pr(β )q q

Estimating an LC Model

i i

i,t i,t it i,t i T i1 i2 i,T (^) t 1 it i,t i

Conditional density for each observation is

P(y | , class q) f(y | , )

Joint conditional density for T observations is f(y , y ,..., y | ) f(y | , ) (T may be 1. This is not

=

= =

= (^) ∏

q

i q q

x x β

X , β x β

( )

i i

Q T i1 i2 i,T (^) q 1 iq (^) t 1 it i,t

only a 'panel data' model.)

Maximize this for each class if the classes are known.

They aren't. Unconditional density for individual i is f(y , y ,..., y | , ) f(y | , )

LogLikeli

X (^) i zi = (^) ∑ (^) = π ∏ = x βq

N Q T i 1 Q (^) i 1 q 1 iq (^) t 1 it i,t

hood

LogL( β (^) 1 ,..., βQ , δ ,..., δ ) = (^) ∑ (^) = log (^) ∑ (^) = π ∏ = f(y | x , βq )

Estimating Which Class

i i

iq i T i1 i2 i,T i (^) t 1 it i,t

Prob[class=q| ]= for T observations is P(y , y ,..., y | , class q) f(y | , ) membership is the pro

=

π

= = (^) ∏

i

q

Prior class probability z Joint conditional density X x β Joint density for data and class i i

i i

i1 i2 i,T i q T it i,t t 1

i1 i2 i,T i i i1 i2 i,T i

duct P(y , y ,..., y , class q | , ) f(y | , )

P(class q | y , y ,..., y , , ) P(^ , class^ q |^ ,^ ) P(y , y ,..., y | , )

= = π =

i ∏ q

i i^ i i

X z x β Posterior probability for class, given the data X z y^ X^ z X z Q^ i q 1 i T i i iq^ t 1 it^ i,t

P( , class q | , ) P( , class q | , ) Use Bayes Theorem to compute the w(q | , , ) P(class j | , , ) f(y^ |^ ,^ )

=

=

= = = π

∑

i i i i

i i i i q

y X z y X z posterior (conditional) probability y X z y X z^ i x^ β Qq 1 iq Tt 1i it i,t iq

f(y | , ) w

= π =

∏ ∑ (^) ∏ x^ βq

Best guess = the class with the largest posterior probability.