Penalized likelihood based tests for regime switching in autoregressive models [Elektronische Ressource] / Florian Ketterer. Betreuer: Hajo Holzmann

Florian Ketterer

Sujets

• Mathematics

Informations

• Publié par: philipps-universitat_marburg
• Publié le: 01 janvier 2011
• Nombre de lectures: 33
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Penalized likelihood based tests
for regime switching
in autoregressive models
Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakultäten
der Philipps-Universität Marburg
vorgelegt von
Florian Ketterer
Dipl. Math. oec.
aus Achern
Erstgutachter: Prof. Dr. Hajo Holzmann
Zweitgutachter: Prof. Dr. Norbert Henze
Eingereicht: 08.04.2011
Tag der mündlichen Prüfung: 16.06.2011Contents
1 Markov-switching autoregressive and related models 5
1.1 Finite mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Markov-switching autoregressive models . . . . . . . . . . . . . . . . . . . 12
1.4 Related models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Standing assumptions and methodology . . . . . . . . . . . . . . . . . . . . 17
2 Feasible Tests for regime switching in autoregressive models 21
2.1 Testing for the number of components in a Markov-switching autoregressive
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Examples and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Markov-switching autoregressive models . . . . . . . . . . . . . . . 22
2.2.2 Penalized maximum likelihood estimation . . . . . . . . . . . . . . 24
2.3 Feasible quasi-likelihood based tests for regime switching . . . . . . . . . . 26
2.3.1 The modified quasi-likelihood ratio test . . . . . . . . . . . . . . . . 26
2.3.2 The EM-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Simulated sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Power comparison of several tests . . . . . . . . . . . . . . . . . . . 37
2.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Testing in a linear switching autoregressive model with normal innovations 59
3.1 Example 2.1.1 (reconsidered) . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 Penalized maximum likelihood . . . . . . . . . . . . . . . . . . . . . 60
3.2 The EM-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 Simulated sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.2 Power comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6.1 Orthogonality of Y , Z , U , V and W . . . . . . . . . . . . . . . . . 91t t t t tiv Contents
4 Testing in a Markov-switching intercept-variance model 93
4.1 Testing in a linear switching autoregressive model with possibly switching
intercept and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 The EM-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.1 Simulated sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.2 Power comparison of several tests . . . . . . . . . . . . . . . . . . . 99
4.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 Outlook 119
Bibliography 121Introduction
A large variety of time series models, such as linear autoregressive or autoregressive con-
ditional heteroscedastic (ARCH) models, are used to analyze the dynamic behavior of
economic or financial variables. Since time series often undergo changes in their behavior
over time, associated with events such as financial crises, such constant parameter time
series models might be inadequate for describing the data.
The Markov-switching model of Hamilton (1989) is one of the most popular regime switch-
ing models in the literature. This model involves multiple structures that characterize
the time series’ behavior in different regimes. While the original Markov-switching model
mainlyfocusesonthemeanbehaviorofthetimeseries,incorporatingtheswitchingmechan-
sim into linear autoregressive models, Cai (1994) and Hamilton and Susmel (1994) studied
various ARCH models with Markov switching, incorporating the switching mechanism into
conditional variance models. An important feature of the Markov-switching model is that
the switching mechanism is controlled by an unobservable state variable that follows a
first-order Markov chain. The determination of the number of states in the hidden Markov
chain is a task of major importance. In this thesis we are mainly concerned with the basic
methodological issue to test for regime switching, i.e. we are testing for the existence of
at least two states, in various Markov-switching autoregressive models. Since, under the
hypothesis, parameters of the full model are not identifiable the asymptotic distribution of
the corresponding likelihood ratio test is highly nonstandard. This problem already arises
in the closely related problem of testing for homogeneity in two-component mixtures. To
overcome this non-identifiability problem Chen, Chen and Kalbfleisch (2001) developed a
penalized likelihood ratio test which admits a simple asymptotic distribution. Additional
difficulties arise if the Markov dependence structure is incorporated into the test statistic.
Therefore, Cho and White (2007) propose a quasi likelihood ratio test (QLRT) for regime
switching in general autoregressive models which neglects the dependence structure of the
hidden Markov chain under the alternative. We extend their approach using penalized
likelihood based tests in order to obtain tractable asymptotic distributions of several test
statistics.
In Chapter 1 we introduce Markov-switching autoregressive and closely related models and
discuss the methodology we use.
The modified likelihood ratio test introduced by Chen, Chen and Kalbfleisch (2001) is well
established for testing for homogeneity in finite mixture models. In Chapter 2 we extend
this test to Markov-switching autoregressive models with a univariate switching parameter
which fulfill some regularity conditions. These regularity conditions are satisfied by2
(i) linear switching autoregressive models with switching variance and t- or normal in-
novations, linear switching autoregressive models with a univariate switching au-
toregressive parameter and t- or normal innovations, linear switching autoregressive
models with switching intercept and t-innovations and
(ii) switchingARCHmodelswithswitchinginterceptintheARCHpartwitht-ornormal
innovations.
Weshowthattheasymptoticdistributionofthemodified(quasi)likelihoodratiotestunder
2the hypothesis is given by a mixture of a point mass at zero and a distribution with1
equal weights. Finally, we introduce a closely related test, called EM-test, which admits
the same asymptotic distribution as the modified (quasi) likelihood ratio test.
For applications, the linear switching autoregressive model with switching intercept and
normal innovations is very important, cf. Hamilton (2008). It is desirable to develop
feasible methods for testing for homogeneity in this model. Studying asymptotic properties
of test statistics which are based on the (penalized) likelihood becomes very challenging
2@ f(x; ; ) @f(x; ; )since = holds for the normal distribution. Here, f(x; ; ) denotes the2@ @
density of a normal distribution with mean and standard deviation> 0. This problem
already arises when testing for homogeneity in homoscedastic normal mixture models, for
which Chen and Li (2009) investigated a method for testing. In Chapter 3 we extend their
approach to linear switching autoregressive models where the intercept switches according
to the underlying regime. We show that the asymptotic distribution of the corresponding
1 12 2 2test statistic under the hypothesis is a simple function of a shifted and a + 1 0 12 2
distribution. We also propose a test based on fixed proportions under the alternative.
Under the hypothesis, the asymptotic distribution of the corresponding test statistic is a
2 1 2 1 2function of a and a + distribution. We apply the methods developed in Chapter1 0 12 2
2 and 3 to the series of seasonally adjusted quarterly U.S. GNP data from 1947(1)–2002(3)
and find a regime switch in the volatility of the growth rate. Dividing the series in two
subseries 1947(1)–1984(1) and 1984(2)–2002(3), we cannot find clear evidence of a regime
switch in the intercept of a linear autoregressive model in these subseries.
In Chapter 4 we are concerned with testing for homogeneity in a linear switching autore-
gressivemodelwheretheinterceptaswellasthescaleparameterofthenormallydistributed
innovations are allowed to switch. To this end, we extend the EM-test introduced by Chen
and Li (2009) for testing for homogeneity in a normal mixture model with possibly distinct
means and variances unde