A semi-nonparametric estimator is presented for a series of generalized models that nest the ordered probit model and thereby relax the distributional assumptions in that model. A new Stata command for the estimation of such models is presented. The approach is illustrated using examples.

Mark Stewart

English

Research Papers in Economics

Semi-nonparametric Estimation of
Extended Ordered Probit Models
Mark Stewart
Economics Department
University of Warwick
e-mail: Mark.Stewart@warwick.ac.uk
UK Stata Users’ Group Meeting, May 2003Outline
• A framework for ordered response models
• Semi-nonparametric estimation
• Computation issues
• The sneop command: syntax and sample output
• An illustration: job satisfactionFramework for ordered response models
Underlying latent variable model:
y∗
i = x0
iβ + εi
for i =1 ,...,n
x: vector of observations on set of exogenous variables
β: vector of unknown parameters
ε: random error term, independently distributed, distribution function F
yi =

  
  
1 if y∗
i <α 1
2 if α1 ≤ y∗
i <α 2
. . .
J if αJ−1 ≤ y∗
iProbabilities of observed outcomes
Pr[yi = j]=



F(α1 − x0
iβ) if j =1
F(αj − x0
iβ) − F(αj−1 − x0
iβ) if 2 ≤ j ≤ J − 1
1 − F(αJ−1 − x0
iβ) if j = J
More compactly:
Pr[yi = j]=F(αj − x0
iβ) − F(αj−1 − x0
iβ)
for all j,w h e r eα0 = −∞ and αJ =+ ∞
Deﬁnes class of cumulative probability models in which known transformation
of the cumulative probabilities is linear function of the x-variables
F−1{Pr[yi ≤ j]} = αj − x0
iβMaximum Likelihood Estimator
logL =
N X
i=1
J X
j=1
yij log[F(αj − x0
iβ) − F(αj−1 − x0
iβ)]
where
yij =
½
1 if yi = j
0 else
Maximized with respect to (β,α1,...,α J−1)
i.e. M + J − 1 parameters, where M = number of exogenous variables
[β (and hence M) does not include an intercept]Ordered Probit Model
Assumes εi ∼ N(0,σ2)
Adopting scale normalization σ =1and imposing zero intercept for
identiﬁcation, probabilities are given by
Pr[yi = j]=Φ(αj − x0
iβ) − Φ(αj−1 − x0
iβ)
where Φ is CDF of standard NormalSemi-nonparametric estimation
The “semi-nonparametric” series estimator of an unknown density, proposed by
Gallant and Nychka (1987), approximates the density using a Hermite form
fK(ε)=
1
θ
Ã
K X
k=0
γkεk
!2
φ(ε)
where φ(ε) is standard normal density and θ =
∞ R
−∞
³PK
k=0γkεk
´2
φ(ε)dε.
FK(u)=
u R
−∞
³PK
k=0γkεk
´2
φ(ε)dε
∞ R
−∞
³PK
k=0γkεk
´2
φ(ε)dε
Deﬁnes family of SNP distributions for increasing values of K.Semi-nonparametric estimator: properties
• Providing unknown density satisﬁes certain smoothness conditions, can be
approximated arbitrarily closely by this Hermite series by increasing K.
• Class of densities that can be approximated is very general. Any form of
skewness, kurtosis, etc. permitted, but upper bound on tails.
• Under these and other mild regularity conditions, providing K increases with
sample size, model parameters are estimated consistently by maximization of
the pseudo-likelihood function (Gallant and Nychka, 1987).Semi-parametric identiﬁcation
• Location normalization necessary. One used by Gabler et al. (1993) in binary
case to give error term zero mean very cumbersome.
• Restriction can be on either error distribution or systematic part of model.
Melenberg & van Soest (1996) set constant term equal to its probit estimate.
Equivalent used here in ordered response model context is to set ﬁrst
threshold, α1, equal to its ordered probit estimate.
• In K =1case, imposition of E(ε)=0or equivalent restriction used here
implies γ1 =0 , so model reduces to Ordered Probit.
• In general score for γ2 is zero at Ordered Probit estimates. Implies that
model for K =2is also equivalent to Ordered Probit. Model with K =3is
therefore ﬁrst model in series that is a generalization of the Ordered Probit.Computation
SNP density can be written as
fK(ε)=
1
θ
2K X
k=0
γ∗
kεkφ(ε)
where γ∗
k =
bk X
i=ak
γiγk−i,w i t hak =m a x ( 0 ,k− K) and bk =m i n ( k,K).
θ =
∞ Z
−∞
2K X
k=0
γ∗
kεkφ(ε)dε =
2K X
k=0
γ∗
kµk =1+
K X
k=1
γ∗
2kµ2k
where µk is k-th moment of standard normal.Computation: cumulative probabilities
Use higher order truncated moments of standard normal distribution
FK(u)=
1
θ
2K X
k=0
γ∗
k
u Z
−∞
εkφ(ε)dε =
1
θ
2K X
k=0
γ∗
kIk(u)
where
Ik(u)=
u Z
−∞
εkφ(ε)dε
These are given by the recursion
Ik(u)=( k − 1)Ik−2(u) − uk−1φ(u)
with I0(u)=Φ(u) and I1(u)=−φ(u).Computation: simpliﬁed recursion
The truncated moments can be written as
Ik(u)=µkΦ(u) − Ak(u)φ(u)
where the Ak(u) a r eg i v e nb yt h er e c u r s i o n
Ak(u)=( k − 1)Ak−2(u)+uk−1
with A0(u)=0and A1(u)=1 . Can write cumulative probabilities as
FK(u)=
1
θ
2K X
k=0
γ∗
k [µkΦ(u) − Ak(u)φ(u)] = Φ(u) −
1
θ
"
2K X
k=0
γ∗
kAk(u)
#
φ(u)
Term in square brackets is polynomial in u of order (2K − 1).The sneop command
Syntax
sneop depvar [varlist][ if exp][ in range][ weight][ ,order(#) robust
from(matname)]
sneop shares the features of all estimation commands; see help est.
Options
order(#) speciﬁes the order of the Hermite polynomial to be used. The
default is 3. Orders 1 and 2 give models equivalent to the Ordered Probit.
robust speciﬁes that the Huber/White/sandwich estimator of variance is to be
used in place of the traditional calculation.
from(matname) speciﬁes a matrix containing starting values. (This option can
be used to investigate whether a global maximum has been found.) The default
uses the ordered probit estimates as starting values.Sample Output
. use d:\micromet\ORM\fullauto\fullauto.dta
(Automobile Models)
. sneop rep77 foreign length mpg, order(5)
Order of SNP polynomial = 5
Number of categories = 5, rep77 assumed coded 1,...,5
1st threshold set to Ordered Probit estimate = 10.158904
initial: log likelihood = -78.020025
(output deleted)
Iteration 11: log likelihood = -77.099231
SNP Estimation of Extended Ordered Probit Model Number of obs = 66
Wald chi2(3) = 333.84
Log likelihood = -77.099231 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
rep77 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
foreign | 1.311066 .3460682 3.79 0.000 .6327847 1.989347
length | .0485854 .0030438 15.96 0.000 .0426197 .0545511
mpg | .1550694 .0200241 7.74 0.000 .1158229 .1943159
-------------+----------------------------------------------------------------
Tresholds: 1 | 10.158904
2 | 11.14811 .3948177 28.24 0.000 10.37428 11.92193
3 | 12.47549 .8385637 14.88 0.000 10.83193 14.11904
4 | 14.04136 1.023637 13.72 0.000 12.03507 16.04765
-------------+----------------------------------------------------------------
SNP coefs: 1 | -.388371 .9299086 -0.42 0.676 -2.210958 1.434216
2 | -.1782818 .1767031 -1.01 0.313 -.5246134 .1680498
3 | -.169297 .2396613 -0.71 0.480 -.6390245 .3004305
4 | .0252662 .0580207 0.44 0.663 -.0884523 .1389847
5 | .0189331 .0166684 1.14 0.256 -.0137365 .0516026
------------------------------------------------------------------------------
Likelihood ratio test of OP model against SNP extended model:
Chi2(3) statistic = 1.8415875 (p-value = .60592711)
------------------------------------------------------------------------------
Estimated moments of error distribution:
Variance = 1.3902935 Std Dev = 1.1791071
3rd. moment = 2.1202618 Coef of skewness = 1.2933907
4th. moment = 13.943124 Coef of kurtosis = 7.2135173
------------------------------------------------------------------------------ Job satisfaction illustration
[Clark & Oswald, “Satisfaction and comparison income”, J. Public Economics,
1996.] Data from 1991 wave of BHPS. Working respondents provided job
satisfaction scores: 1 (“not satisﬁed at all”) to 7 (“completely satisﬁed”).
Table 1: Likelihood-Ratio Tests for Diﬀerent Values of K
log- LR-test degr. of LR-test
K likelihood of OP freedom p-value of K-1 p-value
OP -6174.25
3 -6169.87 8.76 1 0.003 8.76 0.003
4 -6167.95 12.60 2 0.002 3.83 0.050
5 -6165.48 17.54 3 0.001 4.95 0.026
6 -6165.31 17.87 4 0.001 0.33 0.568
7 -6164.56 19.38 5 0.002 1.51 0.219
8 -6164.56 19.39 6 0.004 0.01 0.939
Note: 1 degree of freedom for LR test of K-1 in each row.Table 2: Job Satisfaction Models for Diﬀerent Values of K
OP SNP(3) SNP(5)
coef (s.e.) coef (s.e.) coef (s.e.)
log(earnings) 0.134 (.054) 0.096 (.050) 0.087 (.056)
log(comp. earn.) -0.283 (.064) -0.254 (.060) -0.323 (.068)
Male -0.156 (.044) -0.147 (.044) -0.130 (.046)
Age/10 -0.210 (.100) -0.154 (.092) -0.141 (.104)
Age2/1000 0.380 (.126) 0.310 (.115) 0.305 (.131)
Log-likelihood -6174.25 -6169.87 -6165.48
Standard deviation 1 0.979 1.369
Skewness coeﬃcient 0 0.034 0.064
Kurtosis coeﬃcient 3 4.600 4.665
Test sum=0 [χ2(1)] 7.40 9.26 15.83
p-value [0.007] [0.002] [0.000]
Notes: (1) Sample size = 3895. (2) Models also contain 34 other variables.Table 3: Derived statistics
OP SNP(3) SNP(5)
coef (s.e.) coef (s.e.) coef (s.e.)
log(earnings) 0.134 (.054) 0.096 (.050) 0.087 (.056)
log(comp. earn.) -0.283 (.064) -0.254 (.060) -0.323 (.068)
Trade oﬀ 2.12 (0.70) 2.66 (1.15) 3.70 (2.04)
Age/10 -0.210 (.100) -0.154 (.092) -0.141 (.104)
Age2/1000 0.380 (.126) 0.310 (.115) 0.305 (.131)
Age at minimum 27.6 (4.4) 24.8 (6.1) 23.1 (7.5)0
.
2
.
4
D
e
n
s
i
t
y
-4 -2 0 2 4
Estimated density Normal density
 
 
 
Figure 1: Estimated Density for K = 5 

Semi-nonparametric estimation of extended ordered probit models

http://fmwww.bc.edu/repec/usug2003/snp_uksug.pdf

Semi-nonparametric estimation of extended ordered probit models

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Research Papers in Economics