Goodness-of-fit tests for parametric specifications of conditionally heteroscedastic models
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Goodness-of-fit tests for parametric specifications of conditionally heteroscedastic models M. Dolores Jiménez-Gamero1 · Sangyeol Lee2 · Simos G. Meintanis3,4 Received: 7 November 2018 / Accepted: 2 August 2019 © Sociedad de Estadística e Investigación Operativa 2019
Abstract We consider a goodness-of-fit test for certain parametrizations of conditionally heteroscedastic time series with unobserved components. Our test is quite general in that it can be employed to validate any given specification of arbitrary order and may even be invoked for testing not just GARCH models but also some related models such as autoregressive conditional duration models. The test statistic utilizes the characterization of Bierens (J Econom 20:105–134, 1982) and may be written down in a convenient closed-form expression. Consistency of the test is proved, and the asymptotic distribution of the test statistic under the null hypothesis is studied. Since this distribution depends on unknown quantities, two bootstrap resampling schemes are investigated and compared in order to approximate critical points and actually carry out the test. Finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets. Keywords GARCH model · Goodness-of-fit test · Bierens characterization · Conditional duration model · Weighted bootstrap
Simos G. Meintanis: on sabbatical leave from the University of Athens. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11749019-00676-0) contains supplementary material, which is available to authorized users.
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M. Dolores Jiménez-Gamero [email protected] Sangyeol Lee [email protected] Simos G. Meintanis [email protected]
1
Department of Statistics and Operations Research, University of Sevilla, Sevilla, Spain
2
Department of Statistics, Seoul National University, Seoul, South Korea
3
Department of Economics, National and Kapodistrian University of Athens, Athens, Greece
4
Unit for Business Mathematics and Informatics, North-West University, Potchefstroom, South Africa
123
M. D. Jiménez-Gamero et al.
Mathematics Subject Classification 62G10 · 62G20
1 Introduction Consider a multiplicative strictly stationary process xt = σt εt ,
(1)
where εt and σt are independent, {εt }t∈Z is a sequence of independent and identically distributed (IID) random variables with mean zero and unit variance, and suppose σt2 = Var(xt |It−1 ), where It denotes the information available at time t. A GARCH model is a specification of the conditional variance which takes the general form σt2 = h(ϒ t, p,q ),
(2)
2 ), . . . , log(σ 2 )) and h : Rr → R, with where ϒ t, p,q = (xt−1 , . . . , xt− p , log(σt−1 t−q r = p + q, being p ≥ 1 and q ≥ 0. The classical GARCH corresponds to
h(ϒ t, p,q ) = ω +
p j=1
2 γ j xt− j +
q
2 β j σt− j,
t ∈Z
(3)
j=1
Although (3) is indeed the most popular specification of the function h(·) in (2) for the multiplicative time series in (1), there exist numerous alternative instanc
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