1/53: topic 3.1 – models for ordered choices microeconometric modeling william greene stern school...

1/53: Topic 3.1 – Models for Ordered Choices

Microeconometric Modeling

William GreeneStern School of BusinessNew York UniversityNew York NY USA

3.1 Models for Ordered Choices


Concepts

• Ordered Choice• Subjective Well Being• Health Satisfaction• Random Utility• Fit Measures• Normalization• Threshold Values (Cutpoints0• Differential Item Functioning• Anchoring Vignette• Panel Data• Incidental Parameters Problem• Attrition Bias• Inverse Probability Weighting• Transition Matrix

Models

• Ordered Probit and Logit• Generalized Ordered Probit• Hierarchical Ordered Probit• Vignettes• Fixed and Random Effects OPM• Dynamic Ordered Probit• Sample Selection OPM


Ordered Discrete Outcomes

E.g.: Taste test, credit rating, course grade, preference scale Underlying random preferences:

Existence of an underlying continuous preference scale Mapping to observed choices

Strength of preferences is reflected in the discrete outcome Censoring and discrete measurement The nature of ordered data


Ordered Choices at IMDb


Health Satisfaction (HSAT)

Self administered survey: Health Care Satisfaction (0 – 10)

Continuous Preference Scale


Modeling Ordered Choices

Random Utility (allowing a panel data setting)

Uit = + ’xit + it

= ait + it

Observe outcome j if utility is in region j Probability of outcome = probability of cell

Pr[Yit=j] = F(j – ait) - F(j-1 – ait)


Ordered Probability Model

1

1 2

2 3

J -1 J

j-1

y* , we assume contains a constant term

y 0 if y* 0

y = 1 if 0 < y*

y = 2 if < y*

y = 3 if < y*

...

y = J if < y*

In general: y = j if < y*

βx x

j

-1 o J j-1 j,

, j = 0,1,...,J

, 0, , j = 1,...,J


Combined Outcomes for Health Satisfaction


Ordered Probabilities

j-1 j

j-1 j

j j 1

j j 1

j j 1

Prob[y=j]=Prob[ y* ]

= Prob[ ]

= Prob[ ] Prob[ ]

= Prob[ ] Prob[ ]

= F[ ] F[ ]

where F[ ] i

βx

βx βx

βx βx

βx βx

s the CDF of .


Coefficients

j 1 j kk

What are the coefficients in the ordered probit model?

There is no conditional mean function.

Prob[y=j| ] [f( ) f( )]

x

Magnitude depends on the scale factor and the coeff

xβ'x β'x

icient.

Sign depends on the densities at the two points!

What does it mean that a coefficient is "significant?"


Partial Effects in the Ordered Choice Model

Assume the βk is positive.

Assume that xk increases.

β’x increases. μj- β’x shifts to the left for all 5 cells.

Prob[y=0] decreases

Prob[y=1] decreases – the mass shifted out is larger than the mass shifted in.

Prob[y=3] increases – same reason in reverse.

Prob[y=4] must increase.

When βk > 0, increase in xk decreases Prob[y=0] and increases Prob[y=J]. Intermediate cells are ambiguous, but there is only one sign change in the marginal effects from 0 to 1 to … to J


Partial Effects of 8 Years of Education


An Ordered Probability Model for Health Satisfaction

+---------------------------------------------+| Ordered Probability Model || Dependent variable HSAT || Number of observations 27326 || Underlying probabilities based on Normal || Cell frequencies for outcomes || Y Count Freq Y Count Freq Y Count Freq || 0 447 .016 1 255 .009 2 642 .023 || 3 1173 .042 4 1390 .050 5 4233 .154 || 6 2530 .092 7 4231 .154 8 6172 .225 || 9 3061 .112 10 3192 .116 |+---------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant 2.61335825 .04658496 56.099 .0000 FEMALE -.05840486 .01259442 -4.637 .0000 .47877479 EDUC .03390552 .00284332 11.925 .0000 11.3206310 AGE -.01997327 .00059487 -33.576 .0000 43.5256898 HHNINC .25914964 .03631951 7.135 .0000 .35208362 HHKIDS .06314906 .01350176 4.677 .0000 .40273000 Threshold parameters for index Mu(1) .19352076 .01002714 19.300 .0000 Mu(2) .49955053 .01087525 45.935 .0000 Mu(3) .83593441 .00990420 84.402 .0000 Mu(4) 1.10524187 .00908506 121.655 .0000 Mu(5) 1.66256620 .00801113 207.532 .0000 Mu(6) 1.92729096 .00774122 248.965 .0000 Mu(7) 2.33879408 .00777041 300.987 .0000 Mu(8) 2.99432165 .00851090 351.822 .0000 Mu(9) 3.45366015 .01017554 339.408 .0000


Ordered Probability Partial Effects+----------------------------------------------------+| Marginal effects for ordered probability model || M.E.s for dummy variables are Pr[y|x=1]-Pr[y|x=0] || Names for dummy variables are marked by *. |+----------------------------------------------------++---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ These are the effects on Prob[Y=00] at means. *FEMALE .00200414 .00043473 4.610 .0000 .47877479 EDUC -.00115962 .986135D-04 -11.759 .0000 11.3206310 AGE .00068311 .224205D-04 30.468 .0000 43.5256898 HHNINC -.00886328 .00124869 -7.098 .0000 .35208362 *HHKIDS -.00213193 .00045119 -4.725 .0000 .40273000 These are the effects on Prob[Y=01] at means. *FEMALE .00101533 .00021973 4.621 .0000 .47877479 EDUC -.00058810 .496973D-04 -11.834 .0000 11.3206310 AGE .00034644 .108937D-04 31.802 .0000 43.5256898 HHNINC -.00449505 .00063180 -7.115 .0000 .35208362 *HHKIDS -.00108460 .00022994 -4.717 .0000 .40273000 ... repeated for all 11 outcomes These are the effects on Prob[Y=10] at means. *FEMALE -.01082419 .00233746 -4.631 .0000 .47877479 EDUC .00629289 .00053706 11.717 .0000 11.3206310 AGE -.00370705 .00012547 -29.545 .0000 43.5256898 HHNINC .04809836 .00678434 7.090 .0000 .35208362 *HHKIDS .01181070 .00255177 4.628 .0000 .40273000


Ordered Probit Marginal Effects


Analysis of Model Implications

Partial Effects Fit Measures Predicted Probabilities

Averaged: They match sample proportions. By observation Segments of the sample Related to particular variables


Predictions from the Model Related to Age


Fit Measures There is no single “dependent variable” to

explain. There is no sum of squares or other

measure of “variation” to explain. Predictions of the model relate to a set of

J+1 probabilities, not a single variable. How to explain fit?

Based on the underlying regression Based on the likelihood function Based on prediction of the outcome variable


Log Likelihood Based Fit Measures


A Somewhat Better Fit


Different Normalizations

NLOGIT Y = 0,1,…,J, U* = α + β’x + ε One overall constant term, α J-1 “cutpoints;” μ-1 = -∞, μ0 = 0, μ1,… μJ-1, μJ = + ∞

Stata Y = 1,…,J+1, U* = β’x + ε No overall constant, α=0 J “cutpoints;” μ0 = -∞, μ1,… μJ, μJ+1 = + ∞


α̂

ˆjμ


ˆ α

ˆˆ jμ α


Generalizing the Ordered Probit with Heterogeneous Thresholds

i

-1 0 j j-1 J

ij j

Index =

Threshold parameters

Standard model : μ = - , μ = 0, μ >μ > 0, μ = +

Preference scale and thresholds are homogeneous

A generalized model (Pudney and Shields, JAE, 2000)

μ = α +

β x

j i

ik i

ij i j ik ik

Note the identification problem. If z is also in x (same variable)

then μ - = α + γz -βz +... No longer clear if the variable

is in or (or both)

γ z

β x

x z


Hierarchical Ordered Probit

i

-1 0 j j-1 J

Index =




A generalized model (Harris and Zhao (2000), NLOGIT (2007))

β x

]

],

ij j j i

ij j i j j-1 j

μ = exp[α +

An internally consistent restricted modification

μ = exp[α + α α + exp(θ )

γ z

γ z


Ordered Choice Model


HOPit Model


Differential Item Functioning


A Vignette Random Effects Model


Vignettes


Panel Data Fixed Effects

The usual incidental parameters problem Practically feasible but methodologically

ambiguous Partitioning Prob(yit > j|xit) produces estimable

binomial logit models. (Find a way to combine multiple estimates of the same β.

Random Effects Standard application Extension to random parameters – see above


Incidental Parameters Problem

Table 9.1 Monte Carlo Analysis of the Bias of the MLE in Fixed Effects Discrete Choice Models (Means of empirical sampling distributions, N = 1,000 individuals, R = 200 replications)


A Study of Health Status in the Presence of Attrition


Model for Self Assessed Health

British Household Panel Survey (BHPS) Waves 1-8, 1991-1998 Self assessed health on 0,1,2,3,4 scale Sociological and demographic covariates Dynamics – inertia in reporting of top scale

Dynamic ordered probit model Balanced panel – analyze dynamics Unbalanced panel – examine attrition


Dynamic Ordered Probit Model

*, 1

, 1

, 1

Latent Regression - Random Utility

h = + + +

= relevant covariates and control variables

= 0/1 indicators of reported health status in previous period

H ( ) =

it it i t i it

it

i t

i t j

x H

x

H

*1

1[Individual i reported h in previous period], j=0,...,4

Ordered Choice Observation Mechanism

h = j if < h , j = 0,1,2,3,4

Ordered Probit Model - ~ N[0,1]

Random Effects with

it

it j it j

it

j

20 1 ,1 2

Mundlak Correction and Initial Conditions

= + + u , u ~ N[0, ]i i i i i H x

It would not be appropriate to include hi,t-1 itself in the model as this is a label, not a measure


Random Effects Dynamic Ordered Probit Model

Jit it j 1 j i,t 1 i i,t

i,t j-1 it j

i,t i,t

Jit,j it j it j 1 j i,t 1 i

Random Effects Dynamic Ordered Probit Model

h * h ( j)

h j if < h * <

h ( j) 1 if h = j

P P[h j] ( h ( j) )

x

x

i

i

Jj 1 it j 1 j i,t 1 i

Ji 0 j 1 1,j i,1 i i

TN

it,j j ji=1 t 1

( h ( j) )

Parameterize Random Effects

h ( j) u

Simulation or Quadrature Based Estimation

lnL= ln P f( )d

x

x


Data


Variable of Interest


Dynamics


Attrition


Testing for Attrition Bias

Three dummy variables added to full model with unbalanced panel suggest presence of attrition effects.


Probability Weighting Estimators

A Patch for Attrition (1) Fit a participation probit equation for each wave. (2) Compute p(i,t) = predictions of participation for each

individual in each period. Special assumptions needed to make this work

Ignore common effects and fit a weighted pooled log likelihood: Σi Σt [dit/p(i,t)]logLPit.


Attrition Model with IP Weights

Assumes (1) Prob(attrition|all data) = Prob(attrition|selected variables) (ignorability) (2) Attrition is an ‘absorbing state.’ No reentry. Obviously not true for the GSOEP data above.Can deal with point (2) by isolating a subsample of those present at wave 1 and the monotonically shrinking subsample as the waves progress.


Estimated Partial Effects by Model


Partial Effect for a Category

These are 4 dummy variables for state in the previous period. Using first differences, the 0.234 estimated for SAHEX means transition from EXCELLENT in the previous period to GOOD in the previous period, where GOOD is the omitted category. Likewise for the other 3 previous state variables. The margin from ‘POOR’ to ‘GOOD’ was not interesting in the paper. The better margin would have been from EXCELLENT to POOR, which would have (EX,POOR) change from (1,0) to (0,1).


Ordered Choice Model Extensions


Model Extensions

Multivariate Bivariate Multivariate

Inflation and Two Part Zero inflation Sample Selection Endogenous Latent Class


Generalizing the Ordered Probit with Heterogeneous Thresholds

i

-1 0 j j-1 J

ij j

Index =




A generalized model (Pudney and Shields, JAE, 2000)

μ = α +

β x

j i

ik i

ij i j ik ik

Note the identification problem. If z is also in x (same variable)

then μ - = α + γz -βz +... No longer clear if the variable

is in or (or both)

γ z

β x

x z


Generalized Ordered Probit-1

it it it

it it i

Pudney, S. and M. Shields, "Gender, Race Pay and Promotion in

the British Nursing Profession," J . Applied Ec'trix, 15/4, July 2000.

Ordered Probit Kernel

y * x

y 0 if y * 0; Prob[y

t it

it it 1 it 1 it it

it 1 it 2 it 2 it 1 it

it it J 1 it J 1 it

0] [-x ]

y 1 if 0 < y * ; Prob[y 1] = [ -x ]- [-x ]

y 2 if < y * Prob[y 2] = [ -x ]- [ -x ]

...

y J if y * Prob[y J] 1- [ -x ]

ij i j

it i j it j i j 1 it j 1

Heterogeneous Thresholds and Latent Regression

z

Prob[y j] = [z -x ( )]- [z -x ( )]

Problems:

(1) Coefficients on variables in both Z and X are unid

j j 1

entified

(2) How do you make sure that is positive?

Y=Grade (rank)

Z=Sex, Race

X=Experience, Education, Training, History, Marital Status, Age


Generalized Ordered Probit-2

it it it

it it it it

it it 1 it 1 it it

it 1 it 2 it 2 it 1 it

it it

y * x

y 0 if y * 0; Prob[y 0] [-x ]

y 1 if 0 < y * ; Prob[y 1] = [ -x ]- [-x ]

y 2 if < y * Prob[y 2] = [ -x ]- [ -x ]

...

y J if y *

J 1 it J 1 it

ij j i

Prob[y J] 1- [ -x ]

exp( z )


A G.O.P Model

+---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Index function for probability Constant 1.73737318 .13231824 13.130 .0000 AGE -.01458121 .00141601 -10.297 .0000 46.7491906 LOGINC .17724352 .03275857 5.411 .0000 -1.23143358 EDUC .03897560 .00780436 4.994 .0000 10.9669624 MARRIED .09391821 .03761091 2.497 .0125 .75458666 Estimates of t(j) in mu(j)=exp[t(j)+d*z] Theta(1) -1.28275309 .06080268 -21.097 .0000 Theta(2) -.26918032 .03193086 -8.430 .0000 Theta(3) .36377472 .02109406 17.245 .0000 Theta(4) .85818206 .01656304 51.813 .0000 Threshold covariates mu(j)=exp[t(j)+d*z] FEMALE .00987976 .01802816 .548 .5837

How do we interpret the result for FEMALE?


Hierarchical Ordered Probit

i

-1 0 j j-1 J

Index =




A generalized model (Harris and Zhao (2000), NLOGIT (2007))

β x

]

],

ij j j i

ij j i j j-1 j

μ = exp[α +

An internally consistent restricted modification

μ = exp[α + α α + exp(θ )

γ z

γ z


Ordered Choice Model


HOPit Model


A Sample Selection Model


Zero Inflated Ordered Probit

it it it it it

it it

it it

p * =z u , p = 1[p * 0] (PROBIT Model)

Nonparticipants (p 0) always report y 0.

Participants (p 1) report y 0,1,2,...J (Ordered)

Behavioral Regime (Latent Class) = "Participation"

Consum

it it it

it it it it

it it 1 it 1 it it

it 1 it 2 it 2 it

y * x

y 0 if y * 0; Prob[y 0] [-x ]

y 1 if 0 < y * ; Prob[y 1] = [ -x ]- [-x ]

y 2 if < y * Prob[y 2] = [ -x ]-

er Behavior (Ordered Outcome)

1 it

it it J 1 it J 1 it

it it it it it

it it it

[ -x ]

...

y J if y * Prob[y J] 1- [ -x ]

Prob[y =0] =Prob[p =0]+Prob[p =1]Prob[y =0|p =1]

Prob[y =j>0]= Prob[p =1]Prob[y =j |p

Implied Probabilities

it=1]


Teenage Smoking

Harris, M. and Zhao, Z., "Modelling Tobacco Consumption with a Zero

Inflated Ordered Probit Model," (Monash University - under review,

Journal of Econometrics, 2005)

"How often do you currently smoke cigarettes, pipes or other tobacco

products in the last 12 months?"

0 = Not at all (76%)

1 = Less frequently than weekly (4%)

2 = Daily, less than 20/day (13.8%)

3 = Daily, more than 20/day (6.2%)

Splitting Equation: Young & Female, Log(Age), Male, married, Working,

Unemployed, English speaking, ...

Smoking Equation: Prices of alcohol, marijuana, tobacco, Age, Sex,

Married, English speaking, ...


A Bivariate Latent Class Correlated Generalised Ordered

Probit Model with an Application to Modelling Observed Obesity Levels

William GreeneStern School of Business, New York University

With Mark Harris, Bruce Hollingsworth, Pushkar Maitra Monash University

Stern Economics Working Paper 08-18.

http://w4.stern.nyu.edu/emplibrary/ObesityLCGOPpaperReSTAT.pdf

Forthcoming, Economics Letters, 2014


Obesity

The International Obesity Taskforce (http://www.iotf.org) calls obesity one of the most important medical and public health problems of our time.

Defined as a condition of excess body fat; associated with a large number of debilitating and life-threatening disorders

Health experts argue that given an individual’s height, their weight should lie within a certain range

Most common measure = Body Mass Index (BMI): Weight (Kg)/height(Meters)2

WHO guidelines: BMI < 18.5 are underweight 18.5 < BMI < 25 are normal 25 < BMI < 30 are overweight BMI > 30 are obese

Around 300 million people worldwide are obese, a figure likely to rise


Models for BMI

Simple Regression Approach Based on Actual BMI:BMI* = ′x + , ~ N[0,2]

No accommodation of heterogeneity Rigid measurement by the guidelinesInterval Censored Regression Approach

WT = 0 if BMI* < 25 Normal 1 if 25 < BMI* < 30

Overweight 2 if BMI* > 30 Obese

Inadequate accommodation of heterogeneityInflexible reliance on WHO classification


An Ordered Probit Approach

A Latent Regression Model for “True BMI”BMI* = ′x + , ~ N[0,σ2], σ2 = 1

“True BMI” = a proxy for weight is unobserved

Observation Mechanism for Weight TypeWT = 0 if BMI* < 0 Normal

1 if 0 < BMI* < Overweight

2 if BMI* > Obese


A Basic Ordered Probit Model

Prob( 0 | ) Prob( * 0)

Prob( 0)

( )

Prob( 1| ) Prob(0 * )

Prob( ) Prob( 0)

i i

i i

i

i i

i i i i

WT BMI

WT BMI

x

x

x

x

x x

( ) ( )

Prob( 2 | ) Prob( * )

Prob( )

1 Prob( )

1 ( )

i i

i i

i i

i i

i

WT BMI

x x

x

x

x

x


Latent Class Modeling

Irrespective of observed weight category, individuals can be thought of being in one of several ‘types’ or ‘classes. e.g. an obese individual may be so due to genetic reasons or due to lifestyle factors

These distinct sets of individuals likely to have differing reactions to various policy tools and/or characteristics

The observer does not know from the data which class an individual is in.

Suggests use of a latent class approach Growing use in explaining health outcomes (Deb and

Trivedi, 2002, and Bago d’Uva, 2005)


A Latent Class Model

For modeling purposes, class membership is distributed with a discrete distribution,

Prob(individual i is a member of class = c) = ic = c

Prob(WTi = j | xi) = Σc Prob(WTi = j | xi,class = c)Prob(class = c).


Probabilities in the Latent Class Model

, 1,Prob( = | )

There are two classes labeled c = 0 and c = 1.

i i c j c c i j c c icWT j

x x x

1, ,

Prob( = | ) i i

c j c c i j c c i cci

WT j

xx x

x


Class Assignment

1

1

Probit Model for Class Membership

Prob(Class = 1 | )

( )

Prob(Class = 0 | ) 1-

1 ( )

i i i

i

i i i

i

w

w

w

w

( )

Prob(Class = | ) [(2 1) ]i

i i ic c

w

w w

Class membership may relate to demographics such as age and sex.


Generalized Ordered Probit – Latent Classes and Variable Thresholds

,

,

,

Basic Ordered Choice Model

Prob( 0 | , ) ( )

Prob( 1| , ) ( ) ( )

Prob( 2 | , ) 1 ( )

Heterogeneity in Threshold Parameter

exp(

i c i

i i c c i c i

i i c c i

i c c

WT class c

WT class c

WT class c

x x

x x x

x x

)c iz


Data

US National Health Interview Survey (2005); conducted by the National Centre for Health Statistics

Information on self-reported height and weight levels, BMI levels

Demographic information Remove those underweight Split sample (30,000+) by gender


Model Components

x: determines observed weight levels within classes For observed weight levels we use lifestyle factors such as

marital status and exercise levels z: determines latent classes For latent class determination we use genetic proxies such

as age, gender and ethnicity: the things we can’t change

w: determines position of boundary parameters within classes

For the boundary parameters we have: weight-training intensity and age (BMI inappropriate for the aged?) pregnancy (small numbers and length of term unknown)


Correlation Between Classes and Regression

Outcome Model (BMI*|class = c) = c′x + c, c ~ N[0,1] WT|class=c = 0 if BMI*|class = c < 0

1 if 0 < BMI*|class = c < c

2 if BMI*|class = c > c.

Threshold|class = c: c = exp(c + γc′r) Class Assignment

c* = ′w + u, u ~ N[0,1]. c = 0 if c* < 0 1 if c* > 0. Endogenous Class Assignment (c,u) ~ N2[(0,0),(1,c,1)]

1/53: topic 3.1 – models for ordered choices microeconometric modeling william greene stern school...

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