Random Parameter Models: Latent Class Analysis and Estimation, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

12. Random Parameter Models

Agenda

 ‘True’ Random Parameter Variation

 Discrete – Latent Class

 Continuous

 Classical

 Bayesian

Discrete Parameter Variation

,j

The Latent Class Model

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

j = 1,...,J. J 'classes' differentiated by ( , )

(a) Analyst does not know class memberships. ('latent

βj σ ε

J 1 J j=1 J

i,t it i,t ,j

.')

(b) 'Mixing probabilities' (from the point of view of the

analyst) are ,..., , with 1

(2) Conditional density is

P(y | class j) f(y | x , , (^) ε )

π π Σ π =

= = βj σ

Estimating an LC Model

i i

i,t it i,t ,j

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

i

Conditional density for each observation is

P(y | class j) f(y | x , , )

Joint conditional density for T observations is

f(y , y ,..., y | , ) f(y | x , , )

(T may be 1. Th

ε

ε (^) = ε

σ = (^) ∏ σ

j

i j j

X , β β

i

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

is is not 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 | x , , ε )

= =

X (^) i = (^) ∑ π (^) ( βj σ )

( )

i

i

,1 ,J N J T i 1 j 1 j^ t 1 it^ i,t^ ,j

LogLikelihood

LogL(( , ),..., ( , ))

log f(y | x , , )

ε ε

= = = ε

∏

∑ ∑ (^) ∏

1 J

j

Mixture of Normals

2 it j it j it j j^ j^ j

T (^2)

T it j i1 iT t 1 j j

T N J (^) T it j i 1 j 1 j^ t 1 j j

1 1 y^1 y

f(y | class j) exp =

1 1 y

f(y ,..., y | class j) exp

1 1 y

logL log exp

=

= = =

= = ^ −    φ 

σ π ^ ^ σ^ ^  σ^ ^ σ 

 ^ ^  ^ 

  ^  − μ  

= = ^  ^ − Σ   

   ^ σ  

 ^ ^   

 ^  

Mixture of Normals

+---------------------------------------------+ | Latent Class / Panel LinearRg Model | | Dependent variable YLC | | Number of observations 2500 | | Iterations completed 15 | | Log likelihood function -4972.129 | | Number of parameters 5 | | Akaike IC= 9954.258 Bayes IC= 9983.378 | | Sample is 1 pds and 2500 individuals. | | LINEAR regression model | | Model fit with 2 latent classes. | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Model parameters for latent class 1 Constant 4.98725355 .02868463 173.865. Sigma 1.01530880 .02232943 45.470. Model parameters for latent class 2 Constant .96832859 .04770253 20.299. Sigma 1.00358273 .03436303 29.205. Estimated prior probabilities for class membership Class1Pr .69545563 .01040892 66.813. Class2Pr .30454437 .01040892 29.258.

Posterior for Normal Mixture

i

i

T (^) it j

j (^) t 1 j j

i J T it j

j 1 j t 1 j j

1 y^ ˆ

w(j | data )ˆ^ w(j | i)ˆ

1 y^ ˆ

=

= =

σ ^ σ 

σ ^ σ 

Estimated Posterior Probabilities

Estimating β i

J

j=1 i

(1) Use ˆ from the class with the largest estimated probability

(2) Probabilistic

ˆ (^) = Posterior Prob[class=j|data ]ˆ ∑

j

i j

β

β β

How Many Classes?

(1) J is not a 'parameter' - can't 'estimate' J with

and

(2) Can't 'test' down or 'up' to J by comparing

log likelihoods. Degrees of freedom for J+

vs. J classes is not well define

π β

1

2

1

d.

(3) Use AKAIKE IC; AIC = -2 logL + 2#Parameters.

AIC 10827.

AIC 9954.

AIC 9958.

×

=

= <===

=

Banking Data

Bank Cost Data, 500 Banks, 5 Years:

Variables in the file are

Cit = total cost of transformation of financial and physical resources into loans and

investments = the sum of the five cost items described below;

Y1it = installment loans to individuals for personal and household expenses;

Y2it = real estate loans;

Y3it = business loans;

Y4it = federal funds sold and securities purchased under agreements to resell;

Y5it = other assets;

An LCM for US Banks

+---------------------------------------------+ | Latent Class / Panel LinearRg Model | | Number of observations 2500 | | Log likelihood function -722.4603 | | Number of parameters 23 | | Akaike IC= 1490.921 Bayes IC= 1624.874 | | Sample is 5 pds and 500 individuals. | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Model parameters for latent class 1 Constant 2.12699463 .29651372 7.173. Q1 .12099446 .03964929 3.052. Q2 .36291987 .03752392 9.672. Q3 .10728655 .05245420 2.045. Q4 .12785217 .02482950 5.149. Q5 .39535779 .06081496 6.501. Sigma .71931764 .02537027 28.353. Model parameters for latent class 2 Constant 2.51877624 .06958519 36.197. Q1 .05918445 .00899501 6.580. Q2 .44083356 .00930001 47.401. Q3 .23897724 .01492919 16.007. Q4 .04896772 .00484760 10.101. Q5 .16105964 .01307985 12.314. Sigma .18434496 .00520057 35.447. Model parameters for latent class 3 Constant 3.83600468 .10233076 37.486. Q1 .08904293 .01502856 5.925. Q2 .33710302 .01266856 26.609. Q3 -.01256845 .01987228 -.632. Q4 .06333872 .00782013 8.099. Q5 .42847054 .02326421 18.418. Sigma .23914408 .00872954 27.395. Estimated prior probabilities for class membership Class1Pr .24778109 .02112395 11.730. Class2Pr .45386105 .03497825 12.976. Class3Pr .29835786 .03472726 8.591.

Implementing EM

0 0 0 0 j 1 2 J 0 0 0 0 j 1 1 1

0 j

Given initial guesses

(1) Compute w(j|i) = posterior class probabilitiesˆ

Reestimate each using a weighted log likelihood

For the linear regression:

1 N -1 N j i 1 i 1

N 1 2 i 1^ j ,j (^) N i 1 i

j ,j N j i=

ˆ [ w(j|i)ˆ ] [ w(j|i)ˆ ]

w(j|i)(ˆ ˆ )

T

(2) With the reestimate s and s, recompute w(j|i).ˆ

New ˆ=(1/N) w(j|i).ˆ Now, return to step

= =

= ε =

ε

= Σ ′^ Σ ′

i i i i

i i

β X X X y

y X β

Iterate until convergence.

Continuous Parameter Variation

(The Random Parameters Model)

it it

i i

i

i i

i i

i i i i

y , each observation

, T observations

E[ | ] =

Var[ | ] constant but nonzero

f( | )= g( , ), a density that does not involve

= ′ + ε

= +

= +

=

it i

i i i

i

x β

y X β ε

β β u

u X 0

u X Γ

u X u Γ X