Let S be the number of components in a finite discrete mixing distribution. We prove that the number of waves of panel being greater than or equal to 2S is a sufficient condition for global identification of a dynamic binary choice model in which all the parameters are heterogeneous. This model results in a mixture of S binary first order Markov ChainsDiscrete choice, Markov processes, Global identification

Jesús M. Carro

Martin Browning

English

Research Papers in Economics

 
 
Working Paper 11-17  Departamento de Economía  
Economic Series  Universidad Carlos III de Madrid 
January, 2011                         Calle Madrid, 126 
  28903 Getafe (Spain) 
  Fax (34) 916249875 
 
 
 
The identification of a mixture of first order 
binary Markov Chains
* 
 
 
  Martin Browning  Jesus M. Carro 
  Department of Economics  Departmento de Economia, 
  University of Oxford  Universidad Carlos III de Madrid. 
Martin.Browning@economics.ox.ac.uk  jcarro@eco.uc3m.es 
 
 
 
Abstract 
Let S be the number of components in a finite discrete mixing distribution. We 
prove that the number of waves of panel being greater than or equal to 2S is a sufficient 
condition for global identification of a dynamic binary choice model in which all the 
parameters are heterogeneous. This model results in a mixture of S binary first order 
Markov Chains. 
 
 
JEL classification: C23, C24, J64 
 Keywords: discrete choice, Markov processes, global identification 
 
                                                 
* The second author gratefully acknowledges that this research was supported by grants ECO2009-11165 
from the Spanish Minister of Science, MCINN (Consolider- Ingenio2010) and Consejería de Educación 
de la Comunidad de Madrid (Excelecon project). 1 Introduction
When considering observed persistence in time varying choices, Heckman (1981)
emphasized the importance of distinguishing between unobserved heterogeneity and
true state dependence. A preeminent example in empirical work is the modelling
of dynamic discrete choice models, for example, labour force participation. In al-
lowing for heterogeneity it is important to capture unobserved heterogeneity in the
state dependence parameter as well as in the ￿ intercept￿ ; see Browning and Carro
(2007). One convenient way to do this is to employ ￿nite mixture models. Here we
examine the identi￿cation of mixture model of S binary ￿rst order Markov Chains.
This mixture model corresponds to a dynamic binary choice model in which all the
parameters are heterogeneous.
Discrete ￿nite mixtures as a ￿ exible (￿ nonparametric￿ ) way to control for unob-
served heterogeneity have been widely used. It was popularized in economics by the
work of Heckman and Singer (1982) in duration models, but it is used in many other
non-linear models, including discrete choice models. The question of identi￿cation
of ￿nite mixtures has been studied for many decades in statistics and econometrics.
Teicher (1961), Blischke (1962), Blischke (1964) and Teicher (1963) are among the
￿rst examples. These studies considered the identi￿cation of mixtures of normal,
gamma, or binomial distributions, but they did not consider mixtures of ￿rst order
Markov Chains. A recent study in econometrics is Kasahara and Shimotsu (2009),
but for the model of our interest they do not give identi￿cation conditions for an
arbitrary number of periods. Moreover, their conditions are very di¢ cult to check in
actual data. In this paper we derive a explicit (and very simple) su¢ cient condition
for global identi￿cation in terms of the number of waves of a panel that is needed
to identify a mixture of S binary ￿rst order Markov Chains.
12 Su¢ cient conditions for identi￿cation
Let Yi = (yi0;yi1;::::yiT) be a realization of a binary variable yit that follows a time-
homogeneous ￿rst order Markov process. The transition probabilities that de￿ne
this process are:
Gs = Pr(yt = 1 j yt￿1 = 0;s) (1)
Hs = Pr(yt = 1 j yt￿1 = 1;s) (2)
where s indexes the S distributions we are mixing, and we have T + 1 realizations
of this process. We make all our analysis conditional on the initial observation. The
distribution of Yi conditional on yi0 is the following mixture
Pr(Yijyi0) =
S X
s=1
￿sjyi0G
n01
s (1 ￿ Gs)
n00 H
n11
s (1 ￿ Hs)
n10 (3)
where n01 is the number of 0 ! 1 transitions in path Yi, and similarly for the
other three transitions. ￿sjyi0 gives the mixing probabilities of each value of (Gs;Hs)
conditional on the initial observation. That is, we have one mixing distribution for
those Yi that start with yi0 = 0, and another one, possibly di⁄erent, for those with
yi0 = 1. The unconditional mixing proportions can be easily recovered using the
observed proportion of yi0 = 1. Note that
PS
s=1 ￿sjyi0 = 1, and therefore ￿Sjyi0 =
1 ￿
PS￿1
s=1 ￿sjyi0. 0 < ￿sjyi0;Gs;Hs < 1 for s = 1;::;S. Also, Gs and Hs take distinct
values for di⁄erent s.
The unknown parameters we want to identify are
n￿
￿sjyi0=0;￿sjyi0=1
￿S￿1
s=1 ;[Gs;Hs]
S
s=1
o
;
in all there are (4S ￿ 2) parameters. We provide su¢ cient conditions for global
identi￿cation of the mixture in (3). We say the mixture is identi￿ed if from the
realized proportions of the mixed distribution we can recover only one distinct value
of
n￿
￿sjyi0=0;￿sjyi0=1
￿S￿1
s=1 ;[Gs;Hs]
S
s=1
o
that yields that mixed distribution. Also, any
2sets of values of the unknowns that contain the same values but in di⁄erent order
(e.g. (G1;H1) in one set is (G3;H3) on another set) are the same solution. If that
solution is unique we say the model is identi￿ed regardless of the number of ways it
could be ordered.
The possible realizations of Yi are all the possible combination of zeros and ones
in the periods we have. To identify
￿
￿sjyi0=1;Hs
￿S
s=1, we take those realizations with
yi0 = 1, and construct moment conditions using the survivor function. That is,
we take the probability that in the u periods following the initial observation we
observe only ones: SH(u) =
PS
s=1 ￿sjyi0=1Hu
s . For instance, for a given value Hs,
the probability, conditional on yi0 = 1; of observing yi1 = yi2 = 1 (that is, u = 2) is,
from equation (2), equal to H2
s. Each value of u will give a moment condition, so
we have the following system of equations:
fH;u =
S X
s=1
￿sjyi0=1H
u
s ; (u = 0;:::;T) (4)
with SH(0) and fH;0 being trivially equal to one. The value fH;u is the population
proportion of realizations Yi whose ￿rst u+1 elements are equal to one. In order to
have at least as many (informative) equations as unknowns in system (4) we need
T ￿ 2S ￿ 1:
To show that T ￿ 2S ￿ 1 is a su¢ cient condition for global identi￿cation of the
mixture in (3), ￿rstly note that equations (4) are the same equations as equations
(6) on page 513 of Blischke (1964), except for the di⁄erent notation used. Therefore,
from the same arguments used in Blischke (1964), if T ￿ 2S ￿ 1 there is a unique
solution to this system and
n￿
￿sjyi0=1
￿S￿1
s=1 ;[Hs]
S
s=1
o
is identi￿ed from (4).
To identify
n￿
￿sjyi0=0
￿S￿1
s=1 ;[Gs]
S
s=1
o
, we do the same analysis taking those real-
izations with yi0 = 0, and use the survivor function with the number of consecutive
3zeros following yi0. This gives the following equations
fG;u =
S X
s=1
￿sjyi0=0 (1 ￿ Gs)
u ; (u = 0;:::;T) (5)
fG;u is the population proportion of realizations Yi whose ￿rst u + 1 elements are
equal to zero. Again, this is the same system of equations as equation (6) in Blischke
(1964) with pk
i in Blischke(1964) being (1 ￿ Gs)
u here. Therefore, if T ￿ 2S ￿ 1,
there is a unique solution to this system and
n￿
￿sjyi0=0
￿S￿1
s=1 ;[Gs]
S
s=1
o
is identi￿ed
from (5). This complete the identi￿cation of all the unknowns.
3 Concluding Remarks
We have shown that T ￿ 2S ￿ 1 is a su¢ cient condition for global identi￿cation
of a mixture to S binary ￿rst order Markov Chains. Since in our notation the ￿rst
observation of the process is 0 and T is the last observation, in terms of the number
of periods observed (= T + 1) this condition is
number of periods ￿ 2S (6)
Two ￿nal remarks are important:
1. Though we have made use of some results in Blischke (1964) the condition (6)
for identi￿cation of (3) is di⁄erent than the condition for identi￿cation of the
binomial mixtures studied in Blischke (1964). Our mixture requires one more
period to satisfy the su¢ cient condition for identi￿cation. This comparison is
relevant since the binomial mixture is a special case of our model: the case in
which Gs = Hs = ps for all s. This is the static version of our dynamic binary
choice model.
2. The condition derived in this paper is su¢ cient but not necessary. At least it
4is not necessary for all the values of the parameters. The moments we have
constructed are ignoring conditions in which both Gs and Hs are combined.
Browning and Carro (2011) exploit all the possible conditions to derive, among
other results, a necessary and su¢ cient condition for (generic local) identi￿-
cation. This condition requires a smaller number of periods than condition
(6) for identi￿cation. In particular, it requires T ￿ ￿1
2 +
q
￿7
4 + 4S. For
example, if S = 4 the latter requires at least 5 waves as opposed to the 8
needed for the su¢ cient condition derived here. However, the advantage of
the condition derived here (equation (6)) is that it is for global identi￿cation
and holds everywhere. The condition in Browning and Carro (2011) holds
almost everywhere and, as such, it has some exceptions. Moreover, though
it is conjectured that it holds also for global identi￿cation, the condition in
Browning and Carro (2011) is proved in general only for local identi￿cation.
References
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Binomial Distributions￿ . The Annals of Mathematical Statistics, 33 (2), pp. 444-
454
[2] Blischke (1964):￿Estimating the Parameters of Mixtures of Binomial Distribu-
tions￿ . Journal of the American Statistical Association, Vol. 59, No. 306, pp.
510-528.
[3] Browning, M. and J. M. Carro (2007). "Heterogeneity and Microeconometrics
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W. K. Newey, and T. Persson. Cambridge University Press.
5[4] Browning, M. and J. M. Carro. (2011): "Dynamic binary outcome models with
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[5] Heckman, J. J. (1981): ￿Heterogeneity and State Dependence￿ in Studies in
Labor Markets, edited by Sherwin Rosen, University of Chicago Press.
[6] Heckman, J. J. and Singer, B. (1984) "A Method for Minimizing the Impact of
Distributional Assumptions in Econometric-Models for Duration Data." Econo-
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[7] Kasahara, H. and K. Shimotsu (2009): ￿Nonparametric Identi￿cation of Finite
Mixture Models of Dynamic Discrete Choices￿ Econometrica, 77(1), 135-175
[8] Teicher (1961): ￿Identi￿ability of Mixtures￿ . The Annals of Mathematical Sta-
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[9] Teicher (1963): ￿Identi￿ability of Finite Mixtures￿ . The Annals of Mathematical
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6

The identification of a mixture of first order binary Markov Chains

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The identification of a mixture of first order binary Markov Chains

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