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Theoretical Economics Letters, 2018, 8, 1599-1613 http://www.scirp.org/journal/tel
ISSN Online: 2162-2086 ISSN Print: 2162-2078
Modelling and Forecasting Unbiased Extreme Value Volatility Estimator: A Study Based on EUR/USD Exchange Rate
Indian Institute of Management, Kashipur, India
Abstract The paper provides a framework to model and forecast volatility of EUR/USD exchange rate based on the unbiased AddRS estimator as proposed by Kumar and Maheswaran . The framework is based on the heterogeneous autore- gressive (HAR) model to capture the heterogeneity in a market and to account for long memory in data. The results indicate that the framework based on the unbiased extreme value volatility estimator generates more accurate forecasts of daily volatility in comparison to alternative volatility models.
Keywords Volatility Modeling, Volatility Forecasting, Forecast Evaluation, Economic Significance Analysis, Bias-Corrected Extreme Value Estimator
The volatility of the financial market has implications towards asset pricing, portfolio and fund management and in risk measurement and management. Moreover, more accurate prediction of volatility is important for option valua- tion, in implementing successful trading strategies and in construction of the optimal hedge using futures. Another important application of volatility is in the estimation and forecasting of Value-at-Risk and expected shortfall.
There are various ways to estimate daily volatility and it depends on the kind of data used. The squared return and the absolute return are the very basic esti- mates of daily volatility based on close-to-close prices. However, these estimates of daily volatility are highly inefficient in nature . The GARCH class of mod- els uses the squared return as a measure of unconditional volatility in order to generate conditional volatility estimates and forecasts. The daily realized volatil-
How to cite this paper: Kumar, D. (2018) Modelling and Forecasting Unbiased Ex- treme Value Volatility Estimator: A Study Based on EUR/USD Exchange Rate. Theo- retical Economics Letters, 8, 1599-1613. https://doi.org/10.4236/tel.2018.89102 Received: March 9, 2018 Accepted: June 10, 2018 Published: June 13, 2018 Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/
DOI: 10.4236/tel.2018.89102 Jun. 13, 2018 1599 Theoretical Economics Letters
http://www.scirp.org/journal/tel https://doi.org/10.4236/tel.2018.89102 http://www.scirp.org https://doi.org/10.4236/tel.2018.89102 http://creativecommons.org/licenses/by/4.0/
ity is yet another popular estimate of daily volatility and is estimated by using intraday high-frequency data. The realized volatility is estimated by taking sum of squared high-frequency intraday returns. However, the intraday high-frequency data are suffered from various market microstructure issues, which makes its es- timation complex. These market microstructure issues can make the realized volatility estimator highly biased. Moreover, high-frequency data are difficult to obtain and are usually expensive and sometime may not be available or may be available only for a shorter duration for various tradable assets. The high-frequency data demands substantial computational resources and time which may not be of interest to the practicing quant .
The third popular way of estimating daily volatility makes use of the daily opening, high, low and closing prices. These include the method of moments es- timators       and maximum likelihood (ML) estimators   . These volatility estimators are highly efficient when compared with the close-to-close return based volatility estimators. Out of all these, the Rogers and Satchell  (RS) volatility estimator stands out as it is unbiased regardless of the drift. Kumar and Maheswaran  highlight the presence of downward bias in the RS estimator and propose a bias corrected version of the RS estimator, re- ferred as the AddRS estimator.
The main contribution of the paper is to propose the use of heterogeneous autoregressive (HAR) model for the AddRS estimator. The model is named as HAR-AddRS model. The HAR-AddRS model can capture the long memory characteristics and heterogeneity in the markets. We also examine the statistical and distributional properties of the AddRS and Log (AddRS) estimator and ob- tain similar inference as given in the findings of Kumar and Maheswaran . We also implement the HAR-AddRS-GARCH model to account for the hete- roskedasticity in the residuals of the HAR-AddRS model. We evaluate the fore- casting performance of the AddRS based models using the error statistics ap- proach and the superior predictive ability (SPA) approach and compare the re- sults with the corresponding results based on alternative returns based and range based volatility models. Our findings indicate that the AddRS based models per- form much better than the alternative models in forecasting realized volatility.
The remainder of this paper is organized as follows. Section 2 presents the li- terature review. Section 3 describes the data, methodology and preliminary analysis. Section 4 reports our empirical findings, discussion and policy implica- tions. Section 5 concludes with a summary of our main findings.
2. Literature Review
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) fam- ily of models and stochastic volatility models are highly popular in capturing the dynamics of the return based volatility  . The GARCH family of models has origin due to the seminal study by Engle  and Bollerslev . Since then, these have become the standard models for estimating and forecasting volatility.
DOI: 10.4236/tel.2018.89102 1600 Theoretical Economics Letters
Subsequently, various extensions of GARCH models have been proposed in the literature. Engle and Bollerslev  propose the Integrated GARCH (IGARCH) model to capture the impact of a shock on the future volatility over an infinite horizon. Baillie, Bollerslev  propose the fractionally integrated GARCH (FIGARCH) model to allow for fractional orders I(d) of integration, where 0 < d < 1. The EGARCH model , GJR-GARCH model  and APARCH model  are popular asymmetric models from the GARCH family. Bollerslev and Ole Mikkelsen  fractionally integrated exponential GARCH (FIEGARCH) model and Tse  fractionally integrated asymmetric power ARCH (FIAPARCH) model help us incorporate long memory effect with asymmetry in volatility. The literature related to stochastic volatility models started with the work of Clark  and Taylor . These models assume that the evolution of volatility over time is an independent stochastic process and also that past re- turns do not impact future volatility.
Alizadeh, Brandt  propose the use of two-factor stochastic volatility model to model range based volatility measures and find that the range based volatility models are highly efficient than the returns based counterparts. Chou  pro- poses the Conditional Autoregressive Range Model (CARR) to model the dy- namics in the range-based volatility measures. Chou  also provides the ex- tension of the CARR model which include the exogenous variables and named that model as CARRX model to predict volatilities. Chou  finds that the CARR model better predicts the volatility in comparison to the GARCH based models. Brandt and Jones  suggest the use of exponential generalized auto- regressive conditional heteroskedasticity (EGARCH) model with trading range to generate more accurate forecasts of the trading range. Chen, Gerlach  provide another extension of the CARR model which include the threshold con- ditional heteroskedastic autoregressive model. Chen, Gerlach  find the pres- ence of significant threshold non-linearity in the data. Chiang and Wang  propose the logarithm conditional autoregressive range based model with log- normal distribution to capture the smooth transition in the range process. Li and Hong  suggest the use of auto-regression model for range based volatility. Chan, Lam  propose the CARR model based on the geometric framework and named that model as the conditional autoregressive geometric process range (CARGPR) model to allows for flexible trend patterns, threshold effects, leverage effects, and long-memory dynamics in financial time series. Kumar and Maheswaran  propose the use of ARFIMA based model to generate forecasts based on AddRS estimator. All these studies imply the fact that the forecasts of volatility based on range based volatility estimator are more accurate when compared with the forecasts based on the returns based volatility models. Kumar  incorporate the impact of structural breaks in the CARR model while mod- elling and predicting the RS estimator. Kumar  incorporate the impact of structural breaks in modelling the Yang and Zhang  volatility estimator. In another study, Kumar  incorporate the impact of structural breaks in the CARR model to capture the dynamics of the range based volatility estimator.
DOI: 10.4236/tel.2018.89102 1601 Theoretical Economics Letters
Kumar  make use of ARFIMA-AddRS model to model and to generate fore- casts of volatility of ene