Theoretical Results on Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedastic Processes
Here we present a theoretical study on the main properties of
Fractionally Integrated Exponential Generalized Autoregressive Conditional
Heteroskedastic (FIEGARCH) processes. We analyze the conditions for
the existence, the invertibility, the stationarity and the ergodicity of
these processes. We prove that, if is a
FIEGARCH process then, under mild conditions,
is an ARFIMA, that is, an
autoregressive fractionally integrated moving average process. The
convergence order for the polynomial coefficients that describes the
volatility is presented and results related to the spectral representation
and to the covariance structure of both processes
are also discussed. Expressions for the kurtosis and the asymmetry measures
for any stationary FIEGARCH process are also derived. The -step
ahead forecast for the processes ,
are given with their respective mean square error forecast. The work also
presents a Monte Carlo simulation study showing how to generate, estimate
and forecast based on six different FIEGARCH models. The forecasting
performance of six models belonging to the class of autoregressive
conditional heteroskedastic models (namely, ARCH-type models) and radial basis models
is compared through an empirical application to Brazilian stock market exchange index.
Keywords: Long-Range Dependence, Volatility, Stationarity, Ergodicity, FIEGARCH Processes.
MSC (2000): 60G10, 62G05, 62G35, 62M10, 62M15, 62M20
Financial time series present an important characteristic known as volatility which can be defined/measured in different ways but it is not directly observable. A common approach, but not unique, is to define the volatility as the conditional standard deviation (or the conditional variance) of the process and use heteroskedastic models to describe it.
ARCH-type models, proposed by , constitute one of the main classes of econometric models used for representing the dynamic evolution of volatilities. Another popular one is the class of Stochastic Volatility (SV) models (see,  and references therein). In both cases, ARCH-type and SV models, the stochastic process can be written as
where is a sequence of independent identically distributed (i.i.d.) random variables, with zero mean and variance equal to one, and , where denotes the sigma field generated by the past informations until time . An important difference between these two classes is that, for ARCH-type models, or , while for SV models , where is a sequence of latent variables, independent of . Therefore, the volatility of a SV process is specified as a latent variable which is not directly observable and this can make the estimation challenging, which is a known drawback of this class of models.
By ARCH-type models we mean not only the ARCH model, proposed by , where
(which characterizes the volatility as a function of powers of past observed values, consequently, the volatility can be observed one-step ahead), but also the several generalizations that were lately proposed to properly model the dynamics of the volatility. Among the generalizations of the ARCH model are the Generalized ARCH (GARCH) processes, proposed by , and the Exponential GARCH (EGARCH) processes, proposed by . These models are given, respectively, by (1) and (2) below by setting . The usual definition of for a GARCH() model, namely,
is obtained from (1) by letting and , where and .
ARCH, GARCH and EGARCH are all short memory models. Among the generalizations that capture the effects of long-memory characteristic in the conditional variance are the Fractionally Integrated GARCH (FIGARCH), proposed by , and the Fractionally Integrated EGARCH (FIEGARCH), introduced by . For a FIGARCH, is given by
while for a FIEGARCH, is defined through the relation,
where is the backward shift operator defined by , for all , and is the operator defined by its Maclaurin series expansion as,
with the gamma function.
FIEGARCH models have not only the capability of modeling clusters of volatility (as in the ARCH and GARCH models) and capturing its asymmetry222By asymmetry we mean that the volatility reacts in an asymmetrical form to the returns, that is, volatility tends to rise in response to “bad” news and to fall in response to “good” news. (as in the EGARCH models) but they also take into account the characteristic of long memory in the volatility (as in the FIGARCH models, with the advantage of been weakly stationary if ). Besides non-stationarity (in the weak sense), another drawback of the FIGARCH models is that we must have and the polynomial coefficients in its definition must satisfy some restrictions so the conditional variance will be positive. FIEGARCH models do not have this problem since the variance is defined in terms of the logarithm function.
Some authors argue that the long memory behavior observed in the sample autocorrelation and periodogram functions of financial time series could actually be caused by the non-stationarity property. According to , long range behavior could be just an artifact due to structural changes. On the other hand,  also argue that, when modeling return series with large sample size, considering a single GARCH model is unfeasible and that the best alternative would be to update the parameter values along the time. As an alternative to the traditional heteroskedastic models,  presents a regime switching model that, combined with heavy tailed distributions, presents the long memory characteristic.
It is our belief that FIEGARCH models are a competitive alternative for modeling large sample sized data, especially because they avoid parameter updating. Also, as we prove in this work, FIEGARCH processes are weakly stationary if and only if and hence, non-stationarity can be easily identified. Moreover,  analyze the daily returns of the Tunisian stock market and rule out the random walk hypothesis. According to the authors, the rejection of this hypothesis seems to be due to substantial non-linear dependence and not to non-stationarity in the return series and, after comparing several ARCH-type models they concluded that a stationary FIEGARCH model provides the best fit for the data. Furthermore,  presents a sub period investigation of long memory and structural changes in volatility. The authors consider FIEGARCH models to examine the long run persistence of stock return volatility for 23 developing markets for the period of January 2000 to October 2007. No clear evidence that long memory characteristic could be attributed to structural changes in volatility was found.
Although, in practice, often a simple FIEGARCH model with suffices to fully describe financial time series (for instance,  and , consider FIEGARCH models and  considers FIEGARCH models), there are evidences that for some financial time series higher values of and are in fact necessary (,,). In this work we present a theoretical study on the main properties of FIEGARCH processes, for any .
One of the contributions of the paper is to extend, for any and , the results already known in the literature for or . In particular, we provide the expressions for the asymmetry and kurtosis measures of FIEGARCH process, for all . These results extends the one in  where only the case and was considered and only the kurtosis measure was derived.
Another contribution of this work is the ARFIMA representation of , when is a FIEGARCH process, which is derived in the paper. This results is very useful in model identification and parameter estimation since the literature of ARFIMA models is well developed (see  and references therein) and, to the best of our knowledge, this result is absent in the literature.
To derive the properties of , we first investigate the conditions for the existence of power series representation for and the behavior of the coefficients in this representation. This study is fundamental not only for simulation purposes but also to draw conclusions on the autocorrelation and spectral density functions decay of the non-observable process and the observable one . We also provide a recurrence formula to calculate the coefficients of the series expansion of , for any . This recurrence formula allows to easily simulate FIEGARCH processes.
The fact that is an ARFIMA process and the result that any FIEGARCH process is a martingale difference with respect to the natural filtration , where , are applied to obtain the -step ahead forecast for the processes and . We also present the -step ahead forecast for both and processes, with their respective mean square error forecast. To the best of our knowledge, formal proofs for these expressions are not given in the literature of FIEGARCH processes.
We also present a simulation study including generation, estimation and forecasting features of FIEGARCH models. Despite the fact that the quasi-likelihood is one of the most applied methods in non-linear process estimation, asymptotic results for FIEGARCH processes are still an open question (see )333The asymptotic properties for the quasi-likelihood method are well established for ARCH/GARCH models (see, for instance, , , ,  and ) and also for EGARCH models (see, for instance, ).. Therefore, we consider here a simulation study to investigate the finite sample performance of the estimator. Since it is expected that, the better the fit, the better the forecasting, we also investigate the fitted models’ forecasting performance.
The paper is organized as follows: Section 2 presents the formal definition of FIEGARCH process and its theoretical properties. We give a recurrence formula to obtain the coefficients in the power series expansion of the polynomial that describes the volatility and we show their asymptotic properties. The autocovariance and spectral density functions of the processes and are also presented and analyzed. The asymmetry and kurtosis measures of any stationary FIEGARCH process are also presented. Section 3 presents the theoretical results regarding the forecasting. Section 4 presents a Monte Carlo simulation study including the generation of FIEGARCH time series, estimation of the model parameters and the forecasting based on the fitted model. Section 5 presents the analysis of an observed time series and the comparison of the forecasting performance for different ARCH-type and radial basis models. Section 6 concludes the paper.
2 FIEGARCH Process
In this section we present the Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedastic process (FIEGARCH). This class of processes, introduced by , describes not only the volatility varying on time and the volatility clusters (known as ARCH/GARCH effects) but also the volatility long-range dependence and its asymmetry.
Here, we present some results related to the existence, stationarity and ergodicity for these processes. We analyze the autocorrelation and the spectral density functions decay for both and processes. Conditions for the existence of a series expansion for the polynomial that describes the volatility are given and a recurrence formula to calculate the coefficients of this expansion is presented. We also discuss the coefficients asymptotic behavior. We observe that if is a FIEGARCH process then is an ARFIMA process and we prove that, under mild conditions, is an ARFIMA process with correlated innovations. We present the expression for the kurtosis and the asymmetry measures for any stationary FIEGARCH process.
Throughout the paper, given , means that , for some , as ; means that , as ; means that , as . We also say that , as , if for any , there exists such that , for all . Also, given any set , corresponds to the set and is the indicator function defined as , if , and 0, otherwise.
From now on, let be the operator defined by its Maclaurin series expansion as,
where is the gamma function, is the backward shift operator defined by , for all , and the coefficients are such that and , for all .
Note that expression (3) is valid only for non-integer values of . When , is merely the difference operator iterated times. Also, one observe that, upon replacing by , the operator has the same binomial expansion as the polynomial given in (3), that is
where , for all . Moreover, , as (see ). Therefore, , as goes to infinity.
Suppose that is a sequence of independent and identically distributed (i.i.d.) random variables, with zero mean and variance equal to one. Let and be the polynomials of order and defined, respectively, by
with . We assume that , if , and that and have no common roots. These conditions assure that the operator is well defined.
Let be the stochastic process defined as
where and is defined by
Then is a Fractionally Integrated EGARCH process, denoted by FIEGARCH.
Figure 1 presents samples from FIEGARCH processes, with observations, considering two different underlying distributions. To obtain these samples we consider Definition 1 and two different distributions for . For this simulation we set , , , and . These are the parameter values of the FIEGARCH model fitted to the Bovespa index log-returns in Section 5. Figures 1 (a) - (c) consider and show, respectively, the time series , the conditional variance and the logarithm of the conditional variance . Figures 1 (d) - (e) show the same time series as in Figures 1 (a) - (c) when the distribution for is the Generalized Error Distribution (GED), with tail-thickness parameter .
For practical purpose, it is important to observe that slightly different definitions of FIEGARCH processes are found in the literature. Usually it is easy to show that, under certain conditions, the different definitions are equivalent to (2). For instance,  defines the conditional variance of a FIEGARCH process through the equation
This is the definition considered, for instance, in the software S-Plus (see ) and it is equivalent to (2) whenever , , , , and for all . This equivalence is mentioned in  and a detailed proof is provided in . In  only the case and is considered and is defined as
where is a Gaussian white noise process with variance equal to one. This is the definition considered, for instance, in the G@RCH package version 4.0 of . Notice that, by setting , and , (9) is equivalent to (7) if and only if the equality holds.
where and are real, nonstochastic, scalar sequences for which the process is well defined, is white noise process with variance not necessarily equal to one and is any measurable function. In particular, Theorems 1 and 2 below, which are stated and proved in , assume that is given by (10) (the notation was adapted to reflect the one used in this work), with and as in Definition 1. Although (10) is more general than (7), in practice the applicability of the model is somewhat limited given that the parameter estimation is far more complicated when compared to the model (7).
Notice that the function can be rewritten as
This expression clearly shows the asymmetry in response to positive and negative returns. Also, it is easy to see that is non-linear if and the asymmetry is due to the values of . While the parameter , also known in the literature as leverage parameter, shows the return’s sign effect, the parameter denotes the return’s magnitude effect. Therefore, the model is able to capture the fact that a negative return usually results in higher volatility than a positive one. Proposition 1 below presents the properties of the stochastic process . Although the proof is straightforward, these properties are extremely important to prove the results stated in the sequel.
Let be a sequence of i.i.d. random variables, with . Let be defined by (8) and assume that and are not both equal to zero. Then is a strictly stationary and ergodic process. If , then is also weakly stationary with zero mean (therefore a white noise process) and variance given by
See . ∎
Theorem 1 below provides a criterion for stationarity and ergodicity of EGARCH (FIEGARCH) processes. As pointed out by , the stationarity and ergodicity criterion in Theorem 1 is exactly the same as for a general linear process with finite variance innovations. Obviously, different definitions of in (10) will lead to different conditions for the criterion in Theorem 1 to hold. In  it is stated that, in many applications, an ARMA process provides a parsimonious parametrization for . In this case, is defined as , , where and are the polynomials given in (5), leading to an EGARCH process. For this model, the criterion in Theorem 1 holds whenever the roots of are outside the closed disk . We shall later discuss the condition for the criterion in Theorem 1 to hold when is defined by (7), leading to a FIEGARCH process.
Define , and by
where and are real, nonstochastic, scalar sequences, and assume that and do not both equal zero. Then , and are strictly stationary and ergodic and is covariance stationary if and only if . If , then almost surely. If , then for , , and .
See theorem 2.1 in . ∎
Define by (12)-(13), and assume that and do not both equal zero. Let be i.i.d. GED with mean zero, variance one, and tail-thickness parameter , and let . Then and possess finite, time-invariant moments of arbitrary order. Further, if , conditioning information at time 0 drops out of the forecast th moments of and , as :
where denotes the limit in probability.
See theorem 2.2 in . ∎
From now on, let be the polynomial defined by
where and are defined in (5). Since it is assumed that has no roots in the closed disk , and also and have no common roots, the function is analytic in the open disc ( if , in the closed disk ). Therefore, it has a unique power series representation and (7) can be rewritten, equivalently, as
Notice that, with this definition we obtain a particular case of parametrization (10).
Theorem 3 below gives the convergence order of the coefficients , as goes to infinity. This theorem is important for two reasons. First, it provides an approximation for , as , and this result plays an important role when choosing the truncation point in the series representation for simulation purposes. Second, and most important, the asymptotic representation provided in this theorem plays the key role to establish the necessary condition for square summability of . More specifically, from Theorem 3 one concludes that if and only if and whenever .
Let be the polynomial defined by (14). Then, for all , the coefficients satisfy
Consequently, , as goes to infinity.
Denote by . Since has no roots in the closed disk , one has
From (18), one has
In particular, .
Moreover, since , as , it follows that for all , there exists , such that, for a given and for all , for all and . Hence, for sufficiently large,
Notice that, since , as , one can choose such that , for all and . Consequently,
However, and , as . So, we have
It follows that and , as . Hence, , as , which concludes the proof. ∎
Let be defined by (14). Consequently,
The applicability of Theorem 1 to long memory models was briefly mentioned (without going into details) in . Corollary 1 below is a direct application of Theorem 3 and provides a simple condition for the criterion in Theorem 1 to hold when is defined by (7), which leads to a long memory model whenever .
Let be a FIEGARCH process, given in Definition 1. If , is stationary (weakly and strictly), ergodic and the random variable is almost surely finite, for all . Moreover, and are strictly stationary and ergodic processes.
The square summability of implies that the process is stationary (weakly and strictly), ergodic and the random variable is almost surely finite, for all (see Theorem 1). Now, since is a white noise (Proposition 1), it follows immediately that is an ARFIMA process (for details on ARFIMA processes see, for instance, , ). This result is very useful, not only for forecasting purposes (see Section 3) but also, to conclude the following properties
if , the autocorrelation function of the process is such that
where , and the spectral density function of the process is such that