The ZD-GARCH model: A new way to study heteroscedasticity
Introduction
Heteroscedasticity is often observed in economic and financial time series data. Modeling heteroscedasticity accurately is important for making valid inference. In the last three decades, a lot of conditionally heteroscedastic models have been proposed since the seminar work of Engle (1982) and Bollerslev (1986). However, few attempts have been made in the literature to capture conditional heteroscedasticity and heteroscedasticity together parametrically.
This paper reaches this goal by proposing a new first-order zero-drift generalized autoregressive conditional heteroscedasticity (ZD-GARCH(1, 1))model: with initial values and , where , , is a sequence of independent and identically distributed (i.i.d.) random variables, and is independent of . If , then the initial value would suffice. Model (1.1) nests the widely used exponentially weighted moving average (EWMA) model in “RiskMetrics” (Morgan, 1997), and the EWMA model has been widely applied to calculate the daily volatility of many assets. Clearly, when , model (1.1) can capture the conditional heteroscedasticity of , since designed as the conditional variance of changes over time. Moreover, let . If , we have in model (1.1) so that In this case, in model (1.1) is not only homoscedastic when , but also heteroscedastic with an exponentially decayed (or explosive) variance when (or ). Obviously, the heteroscedastic structure of in (1.2) is different from the parametric ones assumed in Breusch and Pagan (1979) and White (1980) or the nonparametric ones studied in Dahlhaus (1997), Dahlhaus and Subba Rao (2006), Engle and Rangel (2008), Giraitis et al. (2014), Giraitis et al. (2017), and many others. Thus, when , model (1.1) provides a new way to study conditional heteroscedasticity and heteroscedasticity together.
Needless to say, model (1.1) is motivated by the classical GARCH model: where all notations are inherited from model (1.1) except for . Model (1.3) introduced by Engle (1982), Bollerslev (1986) and Taylor (1986) has become the workhorse of financial applications. It can be used to describe the volatility dynamics of almost any financial return series; see comments in Engle (2004, p.408). Due to the importance of this model, numerous works were devoted to its probabilistic structure and statistical inference, see, e.g., Francq and Zakoïan (2010) for a comprehensive review. However, they all assume that is positive. The case with , i.e., model (1.1), is a meaningful one but is hardly touched, except for Hafner and Preminger (2015) who studied an ARCH(1) model without intercept (i.e., model (1.1) with ). We fill this gap from a theoretical viewpoint. Compared with Hafner and Preminger (2015), we develop a much more involved technique due to the existence of .
Bougerol and Picard (1992a) showed that model (1.3) is stationary if and only if the top Lyapunov exponent , where If and , then in model (1.3). When , we have and in model (1.3) is heteroscedastic with an exponentially explosive variance. However, when , no consistent estimator is available for as shown in Francq and Zakoïan (2012), and hence no prediction can be made in practice. Model (1.1) avoids this dilemma due to the absence of . Moreover, Section 2 demonstrates that except for a different scale, the sample path of model (1.1) has a similar shape to that of model (1.3) when . In view of this, model (1.1) could be more convenient than model (1.3) when studying heteroscedasticity.
This paper gives an omnifaceted investigation of model (1.1). First, we show that, after a suitable renormalization, the limit of the sample path of or converges weakly to a geometric Brownian motion regardless of the sign of . This result represents a sharp difference from those for model (1.3) in Li et al. (2014) and Li and Wu (2015). From this result, we find that diverges to infinity or converges to zero almost surely (a.s.) at an exponential rate when or , respectively, while oscillates randomly between zero and infinity over time when . Following the terminology in Hafner and Preminger (2015), we call model (1.1)stable if and unstable otherwise. Second, we study the generalized quasi-maximum likelihood estimator (GQMLE) of unknown parameter . It is shown that the GQMLE is strongly consistent and asymptotically normal in a unified framework. Third, we study an estimator of , and propose a -test to check the model stability (i.e., ). If model (1.1) is stable, a unit root test is further proposed to check the absence of . Fourth, we propose a portmanteau test for model checking. Simulation studies are carried out to assess the performance of all proposed estimators and tests in the finite sample. Finally, applications demonstrate that a stable ZD-GARCH(1, 1) model is more appropriate than a non-stationary GARCH(1, 1) model in fitting the KV-A stock returns in Francq and Zakoïan (2012).
The remainder of the paper is organized as follows. Section 2 investigates the limit of sample path of in model (1.1). Section 3 studies the GQMLE and its asymptotics. Section 4 presents the inference for and . Section 5 proposes a portmanteau test for model checking. Simulation results are reported in Section 6. Empirical examples are illustrated in Section 7. Concluding remarks and discussions are made in Section 8. All of proofs are relegated to Appendix.
Section snippets
Sample path properties
This section studies the sample path properties of renormalized and in model (1.1). From model (1.1), we have , and hence The theorem below then follows directly from Donsker’s Theorem in Billingsley (1999, Theorem 8.2 on p. 90) .
Theorem 2.1 Suppose that
is a sequence of i.i.d. random variables with
and
; and
is a
The GQMLE and its asymptotics
Let be the unknown parameter of model (1.1) with the true parameter , where is the parametric space. Assume the data sample 1 is from model
Inference for and
Generally, plays a key role in determining the stationarity of nonlinear time series models. In model (1.1), plays an equally important role in determining its stability. Thus, it is necessary to do statistical inference for .
From the definition of in (1.4), a natural plug-in estimator of is defined as where is defined in Remark 3.2. Furthermore, we have the following theorem:
Theorem 4.1 If the conditions in Theorem 3.2are satisfied,
Model diagnostic checking
This section proposes a portmanteau test to check the adequacy of model (1.1). We first define the lag- autocorrelation function (ACF) of the th power of the absolute residuals as where , , is a positive integer, and .
Next, we introduce the following notations:
Simulation studies
In this section, we first assess the finite-sample performance of and . We generate 1000 replications from the following ZD-GARCH(1, 1) model: where is taken as , the standardized Student’s () or the standardized Student’s () such that . Here, we fix , and choose as in Table 1, where the values of correspond to the cases of , , and , respectively. For the indicator , we choose it to be 2, 1, 0.5, and 0.
Applications
This section restudies the daily stock data of Monarch Community Bancorp (NasdaqCM: MCBF), KV Pharmaceutical (NYSE: KV-A), Community Bankers Trust (AMEX: BTC), and China MediaExpress (NasdaqGS: CCME) in Francq and Zakoïan (2012). The log-return () of each stock data is non-stationary according to their strict stationarity test, and it was fitted by a non-stationary model (1.3) with the Gaussian QMLE. Let be the residuals of each fitted model (1.3) in Francq and Zakoïan (2012).
Concluding remarks and discussions
This paper introduced a ZD-GARCH(1, 1) model to study the conditional heteroscedasticity and heteroscedasticity together. Unlike the classical GARCH model, the ZD-GARCH(1, 1) model is always nonstationary. It is stable with its sample path oscillating randomly between zero and infinity over time when . Moreover, this paper studied the GQMLE of the ZD-GARCH(1, 1) model, and established its strong consistency and asymptotic normality, regardless of the sign of . Based on the GQMLE, an
Acknowledgments
The authors greatly appreciate the very constructive suggestions and comments of three anonymous referees, the Associate Editor, and the Co-Editor Y. Ait-Sahalia that greatly improve this article. The authors also thank Dr. Alice Cheung for her professional proof-editing help. Li’s research is supported by the Start-up Fund of Tsinghua University (No. 53330230117) and NSFC (No. 11401337 and 11571348). Zhang’s research is supported by NSFC (No. 11401123). Zhu’s research is supported by NSFC (No.
References (45)
Generalized autoregressive conditional heteroscedasticity
J. Econometrics
(1986)- et al.
Stationarity of GARCH processes and of some nonnegative time series
J. Econometrics
(1992) - et al.
Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations
J. Econometrics
(2015) - et al.
Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero
Stochastic Process. Appl.
(2007) - et al.
Inference on stochastic time-varying coefficient models
J. Econometrics
(2014) - et al.
An ARCH model without intercept
Econom. Lett.
(2015) - et al.
On dynamics of volatilities in nonstationary GARCH models
Statist. Probab. Lett.
(2014) Testing for (in)finite moments
J. Econometrics
(2016)- et al.
A bootstrapped spectral test for adequacy in weak ARMA models
J. Econometrics
(2015) - et al.
The efficiency of the estimators of the parameters in GARCH processes
Ann. Statist.
(2004)
Convergence of Probability Measures
Strict stationarity of generalized autoregressive processes
Ann. Probab.
The stochastic equation with stationary coefficients
Adv. Appl. Probab.
A simple test for heteroscedasticity and random coefficient variation
Econometrica
Fitting time series models to nonstationary processes
Ann. Statist.
Statistical inference for time-varing ARCH processes
Ann. Statist.
Distribution of the estimators for autoregressive time series with a unit root
J. Amer. Statist. Assoc.
Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation
Econometrica
Risk and volatility: Econometric models and financial practice
Am. Econ. Rev.
The spline GARCH model for unconditional volatility and its global macroeconomic causes
Rev. Financ. Stud.
Model checks using residual marked emprirical processes
Statist. Sinica
Joint and marginal specification tests for conditional mean and varnance models
J. Econometrics
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