Volatility GARCH models with the ordered weighted average (OWA) operators
Introduction
One of the most studied volatility models is the autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH (GARCH) models [18], [7], [31], [22]. The ARCH-GARCH models and their extensions have been used in the stock market [48], oil price [56], derivates markets [4], the exchange rate [16], [10] and others.
GARCH models reflect the non-linear dependence of the conditional variance in time series, estimating a conditional variance. However, these models can present structural problems when the data is imprecise and with uncertainty. Hung [27], [28] shows fuzzy adaptation in GARCH modeling to obtain favorable results in the estimation and forecast. Thavaneswaran, Appadoo, & Paseka [45] use fuzzy numbers to moderate the complex data in GARCH modeling. D'Urso, Giovanni, & Massari [13] proposed different fuzzy clustering models for classifying heteroskedastic time series usefulness in volatility.
The ordered weighted averaging (OWA) operator [50] provides an aggregation mean, which lies between the maximum and minimum. The OWA operator has been used for better results in complex and uncertain scenarios [55], [30]. Additionally, this methodology has been combined with other proposals to deal with data and risks such as linear regression [53], [29], variance [6], [34] and covariance [41], [39].
An interesting extension of the OWA operator is the induced ordered weighted averaging (IOWA) operator [54], which works with variables influenced by other variables. Yager [52] proposes the generalized ordered weighted aggregation operator (GOWA), which provides a generalization of the OWA operator by combining it with the generalized mean operator. These two operators are combined in one formulation called induced generalized (IGOWA) operator developed by Merigó and Gil-Lafuente [40].
This work proposes the use of OWA operators with multiple linear regressions to estimate a GARCH volatility model. The main advantage of this new methodology is to add to the GARCH models a component (OWA operator) for periods of great uncertainty. This component has the characteristic of adding additional information from the studied scenarios to underestimate or overestimate the parameters and obtain a forecast that is close to reality.
The remainder of this paper is organized as follows. Section 2 gives some background about the OWA operator, the regression and ARCH-GACH models. Section 3 develops OWA operators in linear regression, where LR-IOWA, MLR-IOWA, LR-IGOWA and MLR-IGOWA are proposed. Section 4 proves OWA operators in ARCH-GARCH models, we call them OWA-ARCH, IOWA-ARCH, GOWA-ARCH, IGOWA-ARCH, OWA-GARCH, IOWA-GARCH, GOWA-GARCH, IGOWA-GARCH. Section 5 presents an application in volatility exchange rate MX/US. Section 6 summarizes the main results and gives some future works.
Section snippets
Preliminaries
In this section, a review of the OWA operators, linear regression and ARCH-GARCH models is presented to analyze the main properties.
OWA operators in linear regression
The main limitation in linear regression and ordinary least squares is to consider absolute linearity when the data tend to be non-linear [47], [42]. IOWA operator in linear regression uses a non-linear method where IOWA means, variance and covariance IOWA are employed to estimate the parameters in a simple linear regression (LR-IOWA) and multiple linear regression (MLR-IOWA). In the case of simple linear regression, it is defined as follows: Definition 9 Is an LR-IOWA of dimension n if we have a mapping IOWA
An ARCH model with OWA operators
Estimating and forecasting volatility with ARCH-GARCH models implies that we consider a given behavior in the financial series. However, many of the data tend to present non-normality and atypical behaviors caused by impression on the variables [1], [49], [2]. The use of OWA operators in the estimation of ARCH-GARCH parameters allows us to analyze a series of scenarios from maximum to minimum where the above characteristics can appear. We present an ARCH Model estimated with simple linear
Conclusions
The volatility forecast is a controversial issue due to the statistical behavior in the financial time series. Features such as data accuracy and high uncertainty make volatility difficult to predict. In this sense, two of the most used models are the autoregressive conditional heteroskedasticity (ARCH) and generalized ARCH (GARCH).
This paper proposes OWA aggregation operators' use in conjunction with the ordinary least squares estimator for the volatility forecast with ARCH-GARCH models. We
CRediT authorship contribution statement
Martha Flores-Sosa: Conceptualization, Validation, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Supervision. Ezequiel Avilés-Ochoa: Validation, Formal analysis, Resources, Data curation, Writing - original draft. José M. Merigó: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Supervision. Ronald R. Yager: Methodology, Validation, Writing - original draft.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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