Discrete Choice Modeling

William GreeneStern School of BusinessNew York University

Part 6Modeling Latent Parameter Heterogeneity

Parameter Heterogeneity Fixed and Random Effects Models

Latent common time invariant “effects” Heterogeneity in level parameter – constant term – in

the model General Parameter Heterogeneity in Models

Discrete: There is more than one time of individual in the population – parameters differ across types. Produces a Latent Class Model

Continuous; Parameters vary randomly across individuals: Produces a Random Parameters Model or a Mixed Model. (Synonyms)

Latent Class Models

There are Q types of people, q = 1,…,Q For each type,

Prob(Outcome|type=q) = f(y,x|βq) Individual i is and remains a member of class q An individual will be drawn at random from the

population. Prob(in class q) = πq From the modeler’s point of view:

Prob(Outcome) = Σq πq Prob(Outcome|type=q) = Σq πq f(y,x|βq)

Finite Mixture Model

Prob(Outcome|type=q) = f(y,x|βq) depends on parameter vector

Parameters are randomly, discretely distributed among population members, withProb(β = βq) = πq, q = 1,…,Q

Integrating out the variation across parameters, Prob(Outcome) = Σq πq f(y,x|βq)

Same model, slightly different interpretation

Estimation Problems Estimation of population features

Latent parameter vectors, βq, q = 1,…,Q Mixing probabilities, πq, q = 1,…,Q Probabilities, partial effects, predictions, etc. Model structure: The number of classes, Q

Classification: Prediction of class membership for individuals

An Extended Latent Class Model

it itx ,x βit it q

q

(1) There are Q classes, unobservable to the analyst(2) Class specific model: f(y | ,class q) g(y , )(3) Conditional class probabilities Common multinomial logit form for prior class

δ

Qqq=1

qq QJ

qj 1

q

probabilities to constrain all probabilities to (0,1) and ensure 1; multinomial logit form for class probabilities;

exp( ) P(class=q| ) , = 0exp( )

Note, = log( q Q/ ).

Log Likelihood for an LC Model

i

x x β

X ,β x βi

i

i,t i,t it i,t q

iT

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

i

Conditional density for each observation is P(y | ,class q) f(y | , )Joint conditional density for T observations isf(y ,y ,...,y | ) f(y | , )(T may be 1. This is not

iX x βi

i

TQi1 i2 i,T q it i,t qq 1 t 1

only a 'panel data' model.)Maximize this for each class if the classes are known. They aren't. Unconditional density for individual i isf(y ,y ,...,y | ) f(y | , )Log Likelihoo

1β β x βiTN Q

Q 1 Q q it i,t qi 1 q 1 t 1

dLogL( ,..., ,δ ,...,δ ) log f(y | , )

Example: Mixture of Normals

q q

2it q it q

itq j qq

Q normal populations each with a mean and standard deviation For each individual in each class at each period,

y y1 1 1f(y | class q) exp = .22Panel data,

T 2it qT

i1 iT t 1qq

T 2N Q it qT

q t 1i 1 q 1qq

T observations on each individual i,y1 1f(y ,...,y | class q) exp 22

Log Likelihood

y1 1logL log exp 22

Unmixing a Mixed SampleN[1,1] and N[5,1]

Sample ; 1 – 1000$Calc ; Ran(123457)$Create ; lc1=rnn(1,1) ; lc2=rnn(5,1)$ Create ; class=rnu(0,1)$Create ; if(class<.3)ylc=lc1 ; (else)ylc=lc2$ Kernel ; rhs=ylc $Regress ; lhs=ylc;rhs=one;lcm;pts=2;pds=1$

YLC

.045

.090

.135

.180

.224

.000-2 0 2 4 6 8 10-4

Kernel dens ity es timate for YLC

Dens

ity

Mixture of Normals+---------------------------------------------+| Latent Class / Panel LinearRg Model || Dependent variable YLC || Number of observations 1000 || Log likelihood function -1960.443 || Info. Criterion: AIC = 3.93089 || LINEAR regression model || Model fit with 2 latent classes. |+---------------------------------------------++--------+--------------+----------------+--------+--------+----------+|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|+--------+--------------+----------------+--------+--------+----------++--------+Model parameters for latent class 1 ||Constant| 4.97029*** .04511814 110.162 .0000 ||Sigma | 1.00214*** .03317650 30.206 .0000 |+--------+Model parameters for latent class 2 ||Constant| 1.05522*** .07347646 14.361 .0000 ||Sigma | .95746*** .05456724 17.546 .0000 |+--------+Estimated prior probabilities for class membership ||Class1Pr| .70003*** .01659777 42.176 .0000 ||Class2Pr| .29997*** .01659777 18.073 .0000 |+--------+------------------------------------------------------------+| Note: ***, **, * = Significance at 1%, 5%, 10% level. |+---------------------------------------------------------------------+

Estimating Which Class

Prior class probability Joint conditional density

X x βJoint density for data and class

i

i

q

iT

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

Prob[class=q]= for T observations is

P(y ,y ,...,y | ,class q) f(y | , ) membership is the product

i

X x βPosterior probability for class, given the data

y XX X

i

i

ii

Ti1 i2 i,T i q it i,t qt 1

ii1 i2 i,T i

i1 i2 i,T i

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

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

i i

i i

i i

y X zy X z

posterior (conditional) probabilityx β

y X y Xx β

i

iQ

iq 1

Tq it i,t qt 1

i i Qq it i,t qq 1

P( ,class q| , ) P( ,class q| , )

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

f(y | , )

Best guess = the class with the largest posterior probability.

iTt 1

iq w

Posterior for Normal Mixture

i

i

T it qq t 1

q qi

TQ it qqq 1 t 1

q q

y ˆ1ˆ ˆ ˆˆ ˆw(q| data) w(q| i)y ˆ1ˆ ˆ ˆ

Estimated Posterior Probabilities

How Many Classes?

β(1) Q is not a 'parameter' - can't 'estimate' Q with and (2) Can't 'test' down or 'up' to Q by comparing log likelihoods. Degrees of freedom for Q+1 vs. Q classes is not well define

1

2

3

d.(3) Use AKAIKE IC; AIC = -2 logL + 2#Parameters. For our mixture of normals problem, AIC 10827.88 AIC 9954.268 AIC 9958.756

More Difficult When the Populations are Close Together

The Technique Still Works----------------------------------------------------------------------Latent Class / Panel LinearRg ModelDependent variable YLCSample is 1 pds and 1000 individualsLINEAR regression modelModel fit with 2 latent classes.--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Model parameters for latent class 1Constant| 2.93611*** .15813 18.568 .0000 Sigma| 1.00326*** .07370 13.613 .0000 |Model parameters for latent class 2Constant| .90156*** .28767 3.134 .0017 Sigma| .86951*** .10808 8.045 .0000 |Estimated prior probabilities for class membershipClass1Pr| .73447*** .09076 8.092 .0000Class2Pr| .26553*** .09076 2.926 .0034--------+-------------------------------------------------------------

Heckman and Singer RE Model Random Effects Model Random Constants with Discrete Distribution

it itx ,x ,βit it q

q

(1) There are Q classes, unobservable to the analyst(2) Class specific model: f(y | ,class q) g(y , )(3) Conditional class probabilities Common multinomial logit form for prior clas

j

δ

Qqq=1

qq QQ

q 1

q

s probabilities to constrain all probabilities to (0,1) and ensure 1; multinomial logit form for class probabilities;

exp( ) P(class=q| ) , = 0exp( )

Note, = lo q Qg( / ).

LCM for Health Status Self Assessed Health Status = 0,1,…,10 Recoded: Healthy = HSAT > 6 Using only groups observed T=7 times; N=887 Prob = (Age,Educ,Income,Married,Kids) 2, 3 classes

Too Many Classes

Two Class Model----------------------------------------------------------------------Latent Class / Panel Probit ModelDependent variable HEALTHYUnbalanced panel has 887 individualsPROBIT (normal) probability modelModel fit with 2 latent classes.--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Model parameters for latent class 1Constant| .61652** .28620 2.154 .0312 AGE| -.02466*** .00401 -6.143 .0000 44.3352 EDUC| .11759*** .01852 6.351 .0000 10.9409 HHNINC| .10713 .20447 .524 .6003 .34930 MARRIED| .11705 .09574 1.223 .2215 .84539 HHKIDS| .04421 .07017 .630 .5287 .45482 |Model parameters for latent class 2Constant| .18988 .31890 .595 .5516 AGE| -.03120*** .00464 -6.719 .0000 44.3352 EDUC| .02122 .01934 1.097 .2726 10.9409 HHNINC| .61039*** .19688 3.100 .0019 .34930 MARRIED| .06201 .10035 .618 .5367 .84539 HHKIDS| .19465** .07936 2.453 .0142 .45482 |Estimated prior probabilities for class membershipClass1Pr| .56604*** .02487 22.763 .0000Class2Pr| .43396*** .02487 17.452 .0000

Partial Effects in LC Model----------------------------------------------------------------------Partial derivatives of expected val. withrespect to the vector of characteristics.They are computed at the means of the Xs.Conditional Mean at Sample Point .6116Scale Factor for Marginal Effects .3832B for latent class model is a wghted avrg.--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Elasticity--------+------------------------------------------------------------- |Two class latent class model AGE| -.01054*** .00134 -7.860 .0000 -.76377 EDUC| .02904*** .00589 4.932 .0000 .51939 HHNINC| .12475** .05598 2.228 .0259 .07124 MARRIED| .03570 .02991 1.194 .2326 .04934 HHKIDS| .04196** .02075 2.022 .0432 .03120--------+------------------------------------------------------------- |Pooled Probit Model AGE| -.00846*** .00081 -10.429 .0000 -.63399 EDUC| .03219*** .00336 9.594 .0000 .59568 HHNINC| .16699*** .04253 3.927 .0001 .09865 |Marginal effect for dummy variable is P|1 - P|0. MARRIED| .02414 .01877 1.286 .1986 .03451 |Marginal effect for dummy variable is P|1 - P|0. HHKIDS| .06754*** .01483 4.555 .0000 .05195--------+-------------------------------------------------------------

Conditional Means of ParametersJ

ijj=1ˆˆ.E[ | All information for individual i] = w

using posterior (conditional) estimated class probabilities.

jEst

Heckman and Singer Model – 3 Points

Heckman/Singer vs. REM-----------------------------------------------------------------------------Random Effects Binary Probit ModelSample is 7 pds and 887 individuals.--------+-------------------------------------------------------------------- | Standard Prob. 95% Confidence HEALTHY| Coefficient Error z |z|>Z* Interval--------+--------------------------------------------------------------------Constant| .33609 .29252 1.15 .2506 -.23723 .90941(Other coefficients omitted) Rho| .52565*** .02025 25.96 .0000 .48596 .56534--------+--------------------------------------------------------------------Rho = 2/(1+s2) so 2 = rho/(1-rho) = 1.10814.Mean = .33609, Variance = 1.10814

For Heckman and Singer model, 3 points a1,a2,a3 = 1.82601, .50135, -.756363 probabilities p1,p2,p3 = .31094, .45267, .23639Mean = .61593 variance = .90642

An Extended Latent Class Model

itxit

Class probabilities relate to observable variables (usuallydemographic factors such as age and sex).(1) There are Q classes, unobservable to the analyst(2) Class specific model: f(y | ,class q) g(

it

i

ii

i

,x βz

zδz δ δ 0zδ

it q

qiq QQ

qq 1

y , )(3) Conditional class probabilities given some information, ) Common multinomial logit form for prior class probabilities

exp( ) P(class=q| , ) , = exp( )

Health Satisfaction Model----------------------------------------------------------------------Latent Class / Panel Probit Model Used mean AGE and FEMALEDependent variable HEALTHY in class probability modelLog likelihood function -3465.98697--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Model parameters for latent class 1Constant| .60050** .29187 2.057 .0396 AGE| -.02002*** .00447 -4.477 .0000 44.3352 EDUC| .10597*** .01776 5.968 .0000 10.9409 HHNINC| .06355 .20751 .306 .7594 .34930 MARRIED| .07532 .10316 .730 .4653 .84539 HHKIDS| .02632 .07082 .372 .7102 .45482 |Model parameters for latent class 2Constant| .10508 .32937 .319 .7497 AGE| -.02499*** .00514 -4.860 .0000 44.3352 EDUC| .00945 .01826 .518 .6046 10.9409 HHNINC| .59026*** .19137 3.084 .0020 .34930 MARRIED| -.00039 .09478 -.004 .9967 .84539 HHKIDS| .20652*** .07782 2.654 .0080 .45482 |Estimated prior probabilities for class membership ONE_1| 1.43661*** .53679 2.676 .0074 (.56519)AGEBAR_1| -.01897* .01140 -1.664 .0960FEMALE_1| -.78809*** .15995 -4.927 .0000 ONE_2| .000 ......(Fixed Parameter)...... (.43481)AGEBAR_2| .000 ......(Fixed Parameter)......FEMALE_2| .000 ......(Fixed Parameter)......--------+-------------------------------------------------------------

The EM Algorithm

i

i,q

i,qTN Q

c i,q i,t i,ti 1 q 1 t 1

Latent Class is a ' ' modeld 1 if individual i is a member of class qIf d were observed, the complete data log likelihood would be

logL log d f(y | data ,class q)

missing data

(Only one of the Q terms would be nonzero.)Expectation - Maximization algorithm has two steps(1) Expectation Step: Form the 'Expected log likelihood' given the data and a prior guess of the parameters.(2) Maximize the expected log likelihood to obtain a new guess for the model parameters.(E.g., http://crow.ee.washington.edu/people/bulyko/papers/em.pdf)

Implementing EM for LC Models

, β β β ββ

β

0 0 0 0 0 0 0 0q 1 2 Q q 1 2 Q

q

q

q

Given initial guesses , ,..., , ,...,E.g., use 1/Q for each and the MLE of from a one classmodel. (Must perturb each one slightly, as if all are equaland all are

0β δβ

β

0

q

q iq it

the same, the model will satisfy the FOC.)ˆ ˆˆ(1) Compute F(q|i) = posterior class probabilities, using ,

Reestimate each using a weighted log likelihood ˆ Maximize wrt F log f(y |

itx βδ

β

iN Tqi=1 t=1

qN

q i=1

, )(2) Reestimate by reestimating

ˆ =(1/N) F(q|i) using old and new ˆ ˆ Now, return to step 1.Iterate until convergence.

Random Parameters Models

it i

i i

i i

x ,β β β + u β = β |

it

Parameters Vary Randomly with a Continuous Distribution A General Model Structure f(y | )

= a set of random parameters = f( ) h( ) = a set of parameters in the

ΣΣ

i

i

it it i i iβ

β

x , x ,β β βit it

distribution of Typical application "repeated measures" = panel Typical application assumes normal distribution The "mixed" model

f(y | ) f(y | )h( , )d forms the basis of

Σ Σ

a likelihood function for the observed data.(NOTE: Random (heterogeneous) parameters is not to be confused with the Bayesian notion of "random parameters.")

A Mixed Probit Model

it i it i

i i2

i

x ,β x ββ β+uu 0 Σ Σ ΓΛ Γ' Λ =

Γ

it it

Random parameters probit modelf(y | ) [(2y 1) ]

~N[ , ], = diagonal matrix of standard deviations

= lower triangular matrix or

i

2it i iβ

β,Γ,Λ x β β ΓΛ Γ' βiTNiti=1 t 1

if uncorrelatedLogL( )= log [(2y 1) ] N[ , ]d

I

Application – HealthyGerman Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsVariables in the file areData 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.  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). 

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

Estimates of a Mixed Probit Model

Partial Effects are Also Simulated

Simulating Conditional Means for Individual Parameters

, ,1 1

,1 1

1 ˆ ˆˆ ˆ( ) [ ( ) ]ˆ ( | , )

1 ˆ ˆ[ ( ) ]

i

i

TRi r it i r itr t

i i i TRit i r itr t

F qRE

F qR

Lw Lw xy X

Lw x

Posterior estimates of E[parameters(i) | Data(i)]

Summarizing Simulated Estimates

Correlated Parameters----------------------------------------------------------------------Random Coefficients Probit ModelDependent variable HEALTHYPROBIT (normal) probability modelSimulation based on 25 random draws--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Means for random parametersConstant| .22395 .18073 1.239 .2153 AGE| -.03919*** .00257 -15.256 .0000 44.3352 EDUC| .15526*** .01173 13.236 .0000 10.9409 HHNINC| .28023** .12572 2.229 .0258 .34930 MARRIED| .03971 .05918 .671 .5023 .84539 HHKIDS| .06313 .04713 1.340 .1804 .45482----------------------------------------------------------------------Partial derivatives of expected val. withrespect to the vector of characteristics.They are computed at the means of the Xs.Conditional Mean at Sample Point .6351Scale Factor for Marginal Effects .3758 AGE| -.01473*** .00102 -14.420 .0000 -1.02820 EDUC| .05835*** .00444 13.149 .0000 1.00526 HHNINC| .10532** .04722 2.231 .0257 .05793 MARRIED| .01492 .02228 .670 .5029 .01987 HHKIDS| .02373 .01754 1.353 .1761 .01699--------+-------------------------------------------------------------

Cholesky Matrix--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Means for random parametersConstant| .22395 .18073 1.239 .2153 AGE| -.03919*** .00257 -15.256 .0000 44.3352 EDUC| .15526*** .01173 13.236 .0000 10.9409 HHNINC| .28023** .12572 2.229 .0258 .34930 MARRIED| .03971 .05918 .671 .5023 .84539 HHKIDS| .06313 .04713 1.340 .1804 .45482 |Diagonal elements of Cholesky matrixConstant| .66612*** .21850 3.049 .0023 AGE| .01041*** .00183 5.687 .0000 EDUC| .07307*** .00592 12.346 .0000 HHNINC| .18897* .10133 1.865 .0622 MARRIED| .47889*** .03140 15.252 .0000 HHKIDS| .44804*** .03126 14.334 .0000 |Below diagonal elements of Cholesky matrixlAGE_ONE| -.00211 .00298 -.706 .4799lEDU_ONE| .07359*** .01403 5.246 .0000lEDU_AGE| -.01881** .00778 -2.417 .0156lHHN_ONE| -.32031** .15453 -2.073 .0382lHHN_AGE| .05302 .12989 .408 .6831lHHN_EDU| .44021*** .13082 3.365 .0008lMAR_ONE| -.19247** .07503 -2.565 .0103lMAR_AGE| -.24710*** .06002 -4.117 .0000lMAR_EDU| .01475 .05933 .249 .8037lMAR_HHN| .07949* .04724 1.683 .0924lHHK_ONE| -.07220 .05686 -1.270 .2041lHHK_AGE| .21508*** .04456 4.827 .0000lHHK_EDU| .31374*** .04369 7.181 .0000lHHK_HHN| -.11592*** .04023 -2.881 .0040lHHK_MAR| -.35853*** .04154 -8.631 .0000--------+-------------------------------------------------------------

Estimated Parameter Correlation Matrix

Modeling Parameter Heterogeneity

i,t i i,t i

i i i i i i

x ,β θ ,x ,β θ

β =β Δz u (Hierarchical M l0 odeu X z

i,t i,t

Conditional Model, linear or nonlinear density : f(y | , ) g(y , )Individual heterogeneity in the means of the parameters

+ , E[ | , ]

i i k

i i i

i

)

z zδu z Φ

z

i,k ik k

ik

Heterogeneity in the variances of the parameters Var[u | ] exp( ) Var[ | ] = = diag( )(Different variables in may appear in means and variances.)Free correlation: i i i iu z Σ ΓΦ Γ'

Γ Var[ | ] = = ,

= a lower triangular matrix with 1s on the diagonal.

Hierarchical Probit Model, ,1 ,2 i ,AverageAge Femalei k k k i k k i kw

----------------------------------------------------------------------Random Coefficients Probit Model--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Means for random parametersConstant| 2.80514*** .84261 3.329 .0009 AGE| -.06321*** .01397 -4.523 .0000 44.3352 EDUC| -.15340*** .05506 -2.786 .0053 10.9409 HHNINC| 2.56154*** .67822 3.777 .0002 .34930 MARRIED| .61453** .26650 2.306 .0211 .84539 HHKIDS| -.19855 .24303 -.817 .4140 .45482 |Scale parameters for dists. of random parametersConstant| .12981*** .02448 5.303 .0000 AGE| .01424*** .00050 28.712 .0000 EDUC| .00368** .00172 2.142 .0322 HHNINC| .52685*** .05165 10.201 .0000 MARRIED| .16399*** .02111 7.768 .0000 HHKIDS| .13928*** .02845 4.896 .0000 |Heterogeneity in the means of random parameterscONE_AGE| -.02875 .02082 -1.381 .1673cONE_FEM| -.98200*** .35328 -2.780 .0054cAGE_AGE| .00022 .00029 .740 .4592cAGE_FEM| .01552*** .00510 3.043 .0023cEDU_AGE| .00575*** .00130 4.438 .0000cEDU_FEM| -.00877 .02172 -.404 .6864cHHN_AGE| -.04540*** .01485 -3.057 .0022cHHN_FEM| -.03645 .25041 -.146 .8843cMAR_AGE| -.01556** .00610 -2.550 .0108cMAR_FEM| .20538* .11232 1.828 .0675cHHK_AGE| .01053* .00552 1.906 .0566cHHK_FEM| -.25666*** .08923 -2.876 .0040--------+-------------------------------------------------------------

Mixed Model Estimation Programs differ on the models fitted, the algorithms, the paradigm, and

the extensions provided to the simplest RPM, i = +ui. MLWin: http://www.cmm.bristol.ac.uk/MLwiN/index.shtml

Multilevel models Regression and some loglinear models

WinBUGS: Mainly for Bayesian Applications MCMC User specifies the model – constructs the Gibbs Sampler/Metropolis Hastings

SAS: Proc Mixed. Classical Uses primarily a kind of GLS/GMM (method of moments algorithm for loglinear

models) LIMDEP/NLOGIT

Classical Mixing done by Monte Carlo integration – maximum simulated likelihood Numerous linear, nonlinear, loglinear models, multinomial choice models

Stata: Classical - GLAMM Mixing done by quadrature. (Very, very slow for 2 or more dimensions) Several loglinear models Arne Hole has developed a basic RP multinomial logit estimator

Ken Train’s Free Gauss Code Monte Carlo integration Used by many researchers Mixed Multinomial Logit model only (but free!)

Biogeme – Michel Bierlaire free multinomial logit package R: nlme package – multilevel linear regression

Hierarchical Model

i,t i i,t i

i i i i i i

x ,β θ ,x ,β θ

β =β Δz u (Hierarchical M l0 odeu X z

i,t i,t

Conditional Model, linear or nonlinear density : f(y | , ) g(y , )Individual heterogeneity in the means of the parameters

+ , E[ | , ]

i i k

i i i

i

)

z zδu z Φ

z

i,k ik k

ik

Heterogeneity in the variances of the parameters Var[u | ] exp( ) Var[ | ] = = diag( )(Different variables in may appear in means and variances.)Free correlation: i i i iu z Σ ΓΦ Γ'

Γ Var[ | ] = = ,

= a lower triangular matrix with 1s on the diagonal.

Maximum Simulated Likelihood

TNiti=1 t 1

K 1 K

logL( )=

log f(y | , )h( | , )d

,..., , ,..., ,

i

it i i i iβ

1

θ,Ω

x ,β θ β z Ω β

Ω=β,Δ, δ δ Γ

Monte Carlo Integration

range of v

(1) Integral is of the form K = g(v|data, ) f(v| ) dvwhere f(v) is the density of random variable vpossibly conditioned on a set of parameters and g(v|data, ) is a function of data and

β Ω

Ωβ

r

parameters.(2) By construction, K( ) = E[g(v|data, )](3) Strategy: a. Sample R values from the population of v using a random number generator. b. Compute average K = (1/R) g(v |dat

Ω β

Rr=1 a, )

By the law of large numbers, plim K = K. β

Monte Carlo Integration

ii

RP

ir i i i u iur=1

1 f(u ) f(u )g(u )du =E [f(u )]R(Certain smoothness conditions must be met.)

ir

ir ir ir

-1 2ir ir

Drawing u by 'random sampling'u = t(v ), v ~ U[0,1]

E.g., u = σΦ (v )+μ for N[μ,σ ]Requires many draws, typically hundreds or thousands

Example: Monte Carlo Integral

2

1 2 3

1 2 3

exp( v / 2)(x .9v) (x .9v) (x .9v) dv2

where is the standard normal CDF and x = .5, x = -.2, x = .3.The weighting function for v is the standard normal.Strategy: Draw R (say 1000) standard

r

1 r 2 r 3 r

normal random draws, v . Compute the 1000 functions

(x .9v ) (x .9v ) (x .9v ) and average them.(Based on 100, 1000, 10000, I get .28746, .28437, .27242)

Simulated Log Likelihood for a Mixed Probit Model

i

i

it it

TNiti=1 t 1

TN RSiti=1 r 1 t 1

Random parameters probit modelf(y | ) [(2y 1) ]

~N[ , ]LogL( )= log [(2y 1) ] N[ , ]d

1LogL log [(2y 1) ( )]RWe now max

i

it i it i

i i2

i

2it i iβ

it ir

x ,β x ββ β+uu 0 ΓΛ Γ'

β,Γ x β β ΓΛ Γ' β

x β+ΓΛv

imize this function with respect to ( ).β,Γ,Λ

Generating a Random DrawMost common approach is the "inverse probability transform"Let u = a random draw from the standard uniform (0,1).Let x = the desired population to draw fromAssume the CDF of x is F(x).The random -1 draw is then x = F (u).Example: exponential, . f(x)= exp(- x), F(x)=1-exp(- x) Equate u to F(x), x = -(1/ )log(1-u).Example: Normal( , ). Inverse function does not exist in

closed form. There are good polynomial approxi-

mations to produce a draw from N[0,1] from a U(0,1). Then x = + v.This leaves the question of how to draw the U(0,1).

Drawing Uniform Random NumbersComputer generated random numbers are not random; theyare Markov chains that look random.The Original IBM SSP Random Number Generator for 32 bit computers.SEED originates at some large odd number d3 = 2147483647.0 d2 = 2147483655.0 d1=16807.0SEED=Mod(d1*SEED,d3) ! MOD(a,p) = a - INT(a/p) * pX=SEED/d2 is a random value between 0 and 1.Problems: (1) Short period. Based on 32 31 bits, so recycles after 2 1 values(2) Evidently not very close to random. (Recent tests have discredited this RNG)

Quasi-Monte Carlo Integration Based on Halton Sequences

0

Coverage of the unit interval is the objective,not randomness of the set of draws.Halton sequences --- Markov chainp = a prime number, r= the sequence of integers, decomposed as H(r|p)

I iiib p

b 10

, ,...1 r = r (e.g., 10,11,12,...) I i

iip

For example, using base p=5, the integer r=37 has b0 = 2, b1 = 2, and b3 = 1; (37=1x52 + 2x51 + 2x50). Then H(37|5) = 25-1 + 25-2 + 15-3 = 0.448.

Halton Sequences vs. Random Draws

Requires far fewer draws – for one dimension, about 1/10. Accelerates estimation by a factor of 5 to 10.