Markov Switching models have been successfully applied to many economic problems. The most popular version of these models implies that the change in the state is driven by a Markov Chain and that the state is an exogenous discrete unobserved variable. This hypothesis seems to be too restrictive. Earlier literature has often been concerned with endogenous switching, hypothesizing a correlation structure between the observed variable and the unobserved state variable. However, in this case the classical likelihood-based methods provide biased estimators. In this paper we propose a simple “estimation by simulation” procedure, based on indirect inference. Its great advantage is in the treatment of the endogenous switching, which is about the same as for the exogenous switching case, without involving any additional difficulty. A set of Monte Carlo experiments is presented to show the interesting performances of the procedure.Markov switching models; indirect inference; simulation estimation; Monte Carlo

Calzolari, Giorgio

Di Iorio, Francesca

Otranto, Edoardo

English

Research Papers in Economics

INDIRECT ESTIMATION OF MARKOV SWITCHING MODELS
WITH ENDOGENOUS SWITCHING
Edoardo Otranto1, Giorgio Calzolari2 and Francesca Di Iorio3
1 Dipartimento di Economia, Impresa e Regolamentazione
Universit` a di Sassari
Via Torre Tonda, 34-07100 Sassari
(e-mail: eotranto@uniss.it)
2 Dipartimento di Statistica “G. Parenti”
Universit` a di Firenze
Viale Morgagni, 59-50134 Firenze
(e-mail: calzolar@ds.unifi.it)
3 Dipartimento di Scienze Statistiche
Universit` a di Napoli Federico II
Via Rodin` o, 22-80138 Napoli
(e-mail: fdiiorio@unina.it)
ABSTRACT. Markov Switching models have been successfully applied to many economic problems.
The most popular version of these models implies that the change in the state is driven by a Markov
Chain and that the state is an exogenous discrete unobserved variable. This hypothesis seems to be
too restrictive. Earlier literature has often been concerned with endogenous switching, hypothesizing
a correlation structure between the observed variable and the unobserved state variable. However, in
this case the classical likelihood-based methods provide biased estimators. In this paper we propose a
simple “estimation by simulation” procedure, based on indirect inference. Its great advantage is in the
treatment of the endogenous switching, which is about the same as for the exogenous switching case,
without involving any additional difﬁculty. A set of Monte Carlo experiments is presented to show the
interesting performances of the procedure.
1 INTRODUCTION
Markov Switching (MS hereafter) models have been successful in the econometric literature,
since they are able to consider different states in the analyzed variable, and to provide an in-
ference on the regime. The most popular version of these models (Hamilton, 1990) supposes
that the change in the state is driven by a Markov chain with constant transition probability
matrix and with an exogenous latent state variable controlling the regime change. In some
applications the hypothesis of exogenous switching seems too strong; for example, in the
analysis of business cycle, the state variable would represent recession and growth periods,
and would be naturally correlated with the observable dependent variable.
The motivation for which the state variable has been considered exogenous with respect
to the dependent variable is that the estimation method used for MS models, based on the
nonlinear ﬁlter of Hamilton (1990), provides biased estimation in the case of endogenous
switching. The researchers have dealt with this problem only recently. In particular, Kim
et al. (2003) propose a bias correction in the likelihood; their approach is very interesting,because it is able to explicit the exact likelihood function, hypothesizing a particular Normal
model for the state variable; however when the model underlying the state variable is not
Normal, their approach provides only an approximation of the likelihood.
In this work, we propose to estimate the MS model with endogenous switching via indi-
rect estimation procedures (Gouri´ eroux et al. 1993, Gallant and Tauchen, 1996), that seem
robust under different model speciﬁcations for the state variable. Using “computational” care
in dealing with the discontinuous response implied by the switching regime, the implemen-
tation of indirect inference is quite straightforward. In particular, we show, via Monte Carlo
experiments, that the auxiliary model we propose provides good performance of indirect es-
timation both for exogenous and endogenous switching and both for the Normal and non
Normal cases.
InthenextsectionwebrieﬂydescribetheMSmodelwedealwith;insection3theindirect
estimation method is introduced and in section 4 we show some simulation experiments to
evaluate this approach, comparing it with the procedures proposed by Kim et al. (2003) and
Hamilton (1990).
2 THE MODEL
For the sake of simplicity, let us consider the simple MS model:
yt = a0+a1st +et t = 1;:::;T (1)
et » N
¡
0;s2
st
¢
wheretheinterceptandthevarianceoftheerrortermsdependuponthestatevariablest,which
can assume two values (name them 0 and 1), representing the regimes. The state variable is
not observable and its distribution is unknown; we suppose that st follows an ergodic Markov
chain, in which the probability of a particular realization of st depends only on the realization
of st¡1. The transition probabilities are in the matrix P, where the generic element pij =
Pr(st = jjst¡1 =i); i; j = 0;1 represents the probability of being in state j at time t given that
the state at timet¡1 was i. Of course, model (1) can be generalized, including autoregressive
terms or exogenous variables, depending or not on the variable st, and considering a generic
number k of regimes or more complex probability structures (for instance as in Otranto,
2005).
Hamilton (1990) estimates this model developing a non linear Kalman ﬁlter to integrate
the effect of the lagged states, simplifying the likelihood function. Such a model considers
the state variable st exogenous with respect to yt.
To introduce the MS model with endogenous switching we need to specify a certain
degree of correlation between st and yt; as in Kim et al. (2003), we suppose a probit repre-
sentation of the state process:
st =
½
0 if s¤
t < 0
1 if s¤
t > 0
where:
s¤
t = J0+J1st¡1+ht t = 1;:::;T (2)
ht » N (0;1)From this representation, it follows that the transition probabilities are obtained as:
p00 = F(¡J0)
p11 = 1¡F(¡J0¡J1)
where F is the standard Normal cumulative distribution function. In addition, we suppose
that et in (1) and ht in (1) are correlated with correlation parameter r. Of course, to obtain
unbiased (consistent) estimators of the unknown parameters in (1), we have to take into ac-
count, in the likelihood speciﬁcation, the expected value and the variance of et conditional on
st and st¡1. Kim et al. (2003) propose a bias correction, which provides the exact likelihood
in the case of Normal ht, whereas it is only an approximation in the other cases. Its use in
other cases may lead to biased (inconsistent) estimators. In the next section we propose an
indirect inference methodology to obtain consistent estimators in more generale cases.
3 INDIRECT ESTIMATION
Among the many interesting simulation based estimation procedures proposed in the litera-
ture, Indirect Inference (Gouri´ eroux et al., 1993) and the Efﬁcient Method of Moments (Gal-
lant and Tauchen, 1996) deserve a special attention from the computational point of view for
their simplicity of implementation.
The underlying idea of these methods is that the estimation of a model of interest yt =
f(yt¡1;xt;et;a), where xt are exogenous variables, a 2 A is the vector of parameters of inter-
est, et are the errors with known parametric distribution, yt are endogenous variables and yt¡1
lagged endogenous variables, can be conducted using a misspeciﬁed model (called auxiliary)
yt = g(yt¡1;xt;nt;b), where b 2 B is a parameter vector, nt are errors.
The Indirect Inference of a proposed by Gouri´ eroux et al. (1993) is given by:
ˆ a = argmin[ˆ b¡ ˜ b(a)]0W¡1
1 [ˆ b¡ ˜ b(a)] (3)
where ˆ b are the parameter estimates of the auxiliary model using the observed data yt gener-
ated by the model of interest, ˜ b(a) are the parameter estimates of the auxiliary model using
data ˜ y(a), simulated from the model of interest conditional on xt, et and a, and W1 is a posi-
tive deﬁnite matrix.
Gallant and Tauchen (1996) proposed a different version of (3), also called Efﬁcient
Method of Moments (EMM hereafter), given by:
ˆ a = argmin
¶L
¶b0(˜ y(a); ˆ b)W¡1
2
¶L
¶b
(˜ y(a); ˆ b) (4)
where ¶L
¶b(˜ y(a); ˆ b) is the score function of the auxiliary model on the simulated data ˜ y(a)
evaluated in ˆ b, and W2 is a positive deﬁnite matrix.
Estimators (3) and (4) associated with the optimal choice of the weighting matrix W1 and
W2 have the same asymptotic efﬁciency and are both consistent and asymptotically Normal
under general conditions (Gouri´ eroux and Monfort, 1996).The simplicity of the indirect estimation “costs” in ﬁnite sample a larger variance of the
estimated parameters. The same variance can be reduced considering the average of h (=100
in our experiments) estimates of the auxiliary model obtained on h independent simulated
data sets from the model of interest: ˜ b(a) = 1
h å
h
j=1 ˜ bj(a), since the expression of the vari-
ance includes a multiplying factor (1+ 1
h). Gouri´ eroux et al. (1993) showed that, without ex-
ogenous variables, this variance reduction can be obtained with a single h¤T size simulated
series from the model of interest. The same result can be found for (4). Moreover, Gallant
and Tauchen (1999) proved that if the auxiliary model encompasses the true data generating
process, than the Quasi-Likelihood becomes a sufﬁcient statistic and the EMM becomes fully
efﬁcient, while if the auxiliary model is a close approximation to the data generating process,
the EMM efﬁciency can be expected to be close to the Maximum Likelihood estimator.
One of the remarkable features of indirect estimation methods is that they can work (and
sometimes they can work “well”) even with surprisingly simple auxiliary models, the only
requirement being the existence of a regular and well behaved binding function b(a) for
a 2 A (see Gouri´ eroux et al., 1993): in practice, some well behaved relationship must exist
between parameters of the model of interest (a) and parameters (or score) of the auxiliary
model ˜ b(a).
For our problem we suggest to use as auxiliary model a Normal bivariate model very
similar to (1)-(1), avoiding the unobservable variables. The model is:
yt = a0+a1zt +et
zt = J0+J1zt¡1+ht (5)
et » N (0;Hst)
where et=[et;ht]0 and
Hst =
·
s2
0(1¡zt)+s2
1zt rs0(1¡zt)+s1zt
rs0(1¡zt)+s1zt 1
¸
The variable zt is obtained by the logistic function:
zt =
exp[cyt]
1+exp[cyt]
andsubstitutesthelatentvariabless¤
t andst in(1)-(1).Theconstantcischosenexperimentally
(values around 5–10 have given good results in the experiments). Supposing st =1 the “high”
regime and st = 0 the “low” regime, the logistic transformation pushes large values towards
1 (likely related to st = 1), and pushes towards 0 all small values of yt (likely related to
st = 0). It has, however, the advantage of being a continuous transformation (discontinuities
cause great, sometimes overwhelming, computational difﬁculties in indirect estimation, often
making minimization difﬁcult or impossible; see Di Iorio and Calzolari, 2005).
What must be done to obtain an estimate of the auxiliary model parameters (ˆ b), is simply
to maximize the pseudo-likelihood of model (??):
T
Õ
t=1
j2pHstj
¡0:5exp
½
¡0:5
·
yt ¡a0¡a1zt
zt ¡J0¡J1zt¡1
¸0
H¡1
st
·
yt ¡a0¡a1zt
zt ¡J0¡J1zt¡1
¸¾Its score, which has a simple closed form expression, is then used in the EMM procedure,
as in equation (4).
Table 1. Simulation results: means and variances (in parentheses) of the estimates. Number of replica-
tions=1000; T=1000
Normal case
Par. a0 a1 J0 J1 s2
0 s2
1 r
True -2 4 -1.3 2.6 1 1 0
EMM -1.9951 3.9868 -1.2951 2.6039 0.9981 1.0109 -0.0018
(0.0049) (0.0127) (0.0086) (0.0216) (0.0096) (0.0097) (0.0097)
KPS -1.9966 4.0036 -1.3028 2.5811 1.0067 0.9848 -0.0040
(0.0027) (0.0056) (0.0068) (0.0142) (0.0069) (0.0056) (0.0073)
H -1.9950 3.9999 -1.3037 2.5828 1.0059 0.9844
(0.0023) (0.0043) (0.0067) (0.0140) (0.0068) (0.0056)
True -2 4 -1.3 2.6 1 1 0.9
EMM -1.9936 3.9872 -1.2932 2.5986 1.0050 1.0097 0.9061
(0.0028) (0.0057) (0.0058) (0.0150) (0.0049) (0.0059) (0.0014)
KPS -2.0008 3.9932 -1.2976 2.6007 0.9975 1.0113 0.9021
(0.0023) (0.0047) (0.0064) (0.0125) (0.0061) (0.0040) (0.0006)
H -2.3120 4.6173 -1.2980 2.5935 0.9023 0.9165
(0.0018) (0.0049) (0.0073) (0.0115) (0.0045) (0.0032)
True -2 4 -1.3 2.6 0.25 1 0.9
EMM -1.9974 3.9960 -1.3188 2.6570 0.2487 1.0000 0.9171
(0.0008) (0.0111) (0.0129) (0.0802) (0.0004) (0.0059) (0.0015)
KPS -2.0004 3.9928 -1.2976 2.6007 0.2494 1.0112 0.9021
(0.0006) (0.0030) (0.0064) (0.0125) (0.0004) (0.0040) (0.0006)
H -2.1561 4.4619 -1.2991 2.5955 0.2252 0.9146
(0.0005) (0.0030) (0.0072) (0.0113) (0.0003) (0.0031)
Mixed case
True -2 4 -1.3 2.6 1 1 0.9
EMM -1.9980 3.9997 -1.3092 2.6274 0.9705 0.9554 0.9048
(0.0024) (0.0053) (0.0164) (0.0652) (0.0192) (0.0116) (0.0090)
KPS -2.0675 4.1361 -1.5420 3.0768 0.9656 0.9752 0.7820
(0.0018) (0.0042) (0.0069) (0.0147) (0.0164) (0.0148) (0.0033)
H -2.2544 4.5102 -1.4772 2.9604 0.9277 0.9313
(0.0019) (0.0039) (0.0069) (0.0168) (0.0138) (0.0121)
True -2 4 -1.3 2.6 0.25 1 0.9
EMM -1.9967 4.0098 -1.3427 2.7119 0.2483 0.9316 0.9254
(0.0005) (0.0049) (0.0171) (0.1027) (0.0021) (0.0185) (0.0036)
KPS -2.0321 4.0915 -1.5384 3.0728 0.2347 0.9994 0.7925
(0.0004) (0.0022) (0.0069) (0.0151) (0.0006) (0.0220) (0.0021)
H -2.1264 4.3802 -1.4776 2.9616 0.2254 0.9507
(0.0005) (0.0023) (0.0069) (0.0169) (0.0005) (0.0191)4 MONTE CARLO EXPERIMENT
In this work we follow the Gallant and Tauchen (1996) EMM approach that presents, in this
framework, some computational advantages. We propose some simulation experiments to
evaluate the performance of the indirect method in the estimation of a model (1)-(1). The
length of each simulated time series is T = 1000 and the number of Monte Carlo replications
is 1000.
We have performed several simulation experiments, considering the case of high corre-
lation and the case of no correlation, as well as the case of switching variance and ﬁxed
variance. In Table 1 we compare the results obtained by the EMM procedure with the method
of Kim et al. (KPS) and the classical estimation via Hamilton (1990) non linear ﬁlter (H),
which is biased when r 6= 0. In addition, we consider two cases: the case named “Normal” in
which both et and ht are Normal, and the case named “Mixed”, in which ht is generated as a
Student’s-t random variable with 4 degrees of freedom (successively standardized), and et as
a weighted sum of ht and a standard Normal variable, with weights calibrated to guarantee a
correlation equal to r and a variance of et equal to s2
st. The values of J0 and J1 correspond
to p00=p11=0.9032, representing a situation of strong persistence in the same regime (typical
situation in many real cases, e.g. Hamilton, 1990).
From Table 1 we can note the performance of EMM. It shows a loss in efﬁciency with
respect to KPS in the Normal case, where KPS considers the true likelihood function; the
loss is particularly evident in the estimation of r. However, the EMM method is the only
robust with respect to the Mixed case, in which KPS and H perform poorly, in particular in
the estimation of r, J0 and J1.
REFERENCES
DI IORIO, F., CALZOLARI, G. (2005 forthcoming): Discontinuities in Indirect Estimation: an Appli-
cation to EAR Models, Computational Statistics and Data Analysis
GALLANT, A.R., TAUCHEN, G. (1996): Which Moments to Match?, Econometric Theory, 12, 657-
681.
GALLANT,A.R.,TAUCHEN,G.(1999):TheRelativeEfﬁciencyoftheMethodofMomentsEstimator,
Journal of Econometrics , 92, 149-172.
GOURI´ EROUX, C., MONFORT, A., (1996): Simulation-Based Econometric Methods, Oxford Univer-
sity Press.
GOURI´ EROUX, C., MONFORT, A., RENAULT, E. (1993): Indirect Inference, Journal of Applied
Econometrics, 8, S65-S118.
HAMILTON, J.D. (1990): Analysis of Time Series Subject to Changes in Regime, Journal of Econo-
metrics, 45, 39-70.
KIM, C.-J., PIGER, J., STARTZ, R. (2003): Estimation of Markov Regime-Switching Regression Mod-
els with Endogenous Switching, Federal Reserve Bank of St. Louis, Working Paper, 2003-015.
OTRANTO, E. (2005 forthcoming): The Multi-Chain Markov Switching Model, Journal of Forecast-
ing.

Analysis of Time Series Subject to Changes in Regime,

Estimation of Markov Regime-Switching Regression Models with Endogenous Switching, Federal Reserve Bank of St. Louis, Working Paper,

forthcoming): Discontinuities in Indirect Estimation: an Application to EAR Models, Computational Statistics and Data Analysis

forthcoming): The Multi-Chain Markov Switching Model,

Indirect Inference,

Simulation-Based Econometric Methods,

Indirect estimation of Markov switching models with endogenous switching

Indirect estimation of Markov switching models with endogenous switching

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