1/53: Topic 3.1 – Models for Ordered Choices
Microeconometric Modeling
William Greene
Stern School of Business
New York University
New York NY USA
3.1 Models for Ordered
Choices
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Concepts
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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
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Ordered Probit and Logit
Generalized Ordered Probit
Hierarchical Ordered Probit
Vignettes
Fixed and Random Effects OPM
Dynamic Ordered Probit
Sample Selection OPM
3/53: Topic 3.1 – Models for Ordered Choices
Ordered Discrete Outcomes
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E.g.: Taste test, credit rating, course grade, preference scale
Underlying random preferences:
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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
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Health Satisfaction (HSAT)
Self administered survey: Health Care Satisfaction (0 – 10)
Continuous Preference Scale
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Modeling Ordered Choices
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Random Utility (allowing a panel data setting)
Uit =  + ’xit + it
= ait +
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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)
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Ordered Probability Model
y*  βx  , we assume x contains a constant term
y  0 if y*
 0
y = 1 if 0
< y*  1
y = 2 if 1
< y*  2
y = 3 if 2
< y*  3
...
y = J if  J-1
< y*   J
In general : y = j if  j-1
< y*   j , j = 0,1,...,J
-1  , o  0,  J  ,  j-1   j, j = 1,...,J
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Combined Outcomes for Health Satisfaction
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Ordered Probabilities
Prob[y=j]=Prob[ j-1  y*   j ]
= Prob[ j-1  βx     j ]
= Prob[βx     j ]  Prob[βx     j1 ]
= Prob[   j  βx ]  Prob[   j1  βx ]
= F[ j  βx ]  F[ j1  βx]
where F[] is the CDF of .
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Coefficients
 What are the coefficients in the ordered probit model?
There is no conditional mean function.
Prob[y=j|x ]
 [f( j1  β'x)  f( j  β'x)] k
x k
Magnitude depends on the scale factor and the coefficient.
Sign depends on the densities at the two points!
 What does it mean that a coefficient is "significant?"
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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.
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
Prob[y=4] must increase.
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Partial Effects of 8 Years of Education
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Analysis of Model Implications
Partial Effects
 Fit Measures
 Predicted Probabilities
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Averaged: They match sample proportions.
By observation
Segments of the sample
Related to particular variables
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Panel Data
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Fixed Effects
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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
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Standard application
Extension to random parameters – see above
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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)
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A Study of Health Status in the Presence of Attrition
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Model for Self Assessed Health
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British Household Panel Survey (BHPS)
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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
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Balanced panel – analyze dynamics
Unbalanced panel – examine attrition
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Dynamic Ordered Probit Model
Latent Regression - Random Utility
h *it = xit +  H i ,t 1 + i + it
xit = relevant covariates and control variables
It would not be
appropriate to include
hi,t-1 itself in the model
as this is a label, not a
measure
H i ,t 1 = 0/1 indicators of reported health status in previous period
H i ,t 1 ( j ) = 1[Individual i reported h it  j in previous period], j=0,...,4
Ordered Choice Observation Mechanism
h it = j if  j 1 < h *it   j , j = 0,1,2,3,4
Ordered Probit Model - it ~ N[0,1]
Random Effects with Mundlak Correction and Initial Conditions
 i =  0  1H i ,1 + 2 xi + u i , u i ~ N[0, 2 ]
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Random Effects Dynamic Ordered Probit Model
Random Effects Dynamic Ordered Probit Model
hit *  xit    Jj1 jhi,t 1( j)  i  i,t
hi,t  j if  j-1 < hit * <  j
hi,t ( j)  1 if hi,t = j
Pit,j  P[hit  j]  ( j  xit    Jj1 jhi,t 1( j)  i )
 ( j1  xit    Jj1 jhi,t 1( j)  i )
Parameterize Random Effects
i   0   Jj11,jhi,1( j)  x i  ui
Simulation or Quadrature Based Estimation
lnL= i=1 ln
N
i
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Ti
t 1
Pit,j f(  j )d j
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Data
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Variable of Interest
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Dynamics
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Probability Weighting Estimators
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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.
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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.
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Estimated Partial Effects by Model