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Revista Contabilidade & Finanças
On-line version ISSN 1808-057X
Rev. contab. finanç. vol.25 no.65 São Paulo May/Aug. 2014
https://doi.org/10.1590/S1519-70772014000200008
ARTICLES
Volatility and Return Forecasting with High‐Frequency and GARCH Models: Evidence for the Brazilian Market^{*}
^{1}Ph.D. Student, Departamentof Management, Pontifícia Universidade Católica do Rio de Janeiro. E-mail: flavio.val@uol.com.br
^{2}Ph.D., Departamentof Management, Pontifícia Universidade Católica do Rio de Janeiro. E-mail: figueiredo@iag.puc-rio.br
^{3}Ph.D., Departamentof Management, Pontifícia Universidade Católica do Rio de Janeiro. E-mail: klotzle@iag.puc-rio.br
Based on studies developed over recent years about the use of high-frequency data for estimating volatility, this article implements the Heterogeneous Autoregressive (HAR) model developed by Andersen, Bollerslev, and Diebold (2007) and Corsi (2009), and the Component (2-Comp)model developed by Maheu and McCurdy (2007) and compare them with the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) family models in order to estimate volatility and returns. During the period analyzed, the models using intraday data obtained better returns forecasts of the assets assessed, both in and out-of-sample, thus confirming these models possess important information for a variety of economic agents.
Key words: Realized volatility; Volatility estimation; Intraday return; HAR
Com base em estudos desenvolvidos em anos recentes sobre o uso de dados de alta frequência para a estimação da volatilidade, este artigo implementa o modelo Autorregressivo Heterogêneo (HAR)desenvolvido por Andersen, Bollerslev, e Diebold (2007) e Corsi (2009), e o modelo Componente (2-Comp) desenvolvido por Maheu e McCurdy (2007) e os compara com a família de modelos Autorregressivos com Heteroscedasticidade Generalizados (GARCH)para estimar a volatilidade e os retornos. Durante o período analisado, os modelos que usam dados intraday obtiveram melhores previsões de retornos dos ativos avaliados, tanto dentro como fora da amostra, confirmando assim que esses modelos possuem informações importantes para uma série de agentes econômicos.
Palavras-Chave: Volatilidade realizada; Estimativa de volatilidade; Retorno intraday; HAR
1 INTRODUCTION
High-frequency data is the result of observations made available over short periods of time. For financial historical series, this could be described as observations that were made availableat a daily frequency or even a shorter period of time, when a number of data bases supplying negotiation by negotiation information regarding financial assets, already existed.
The availability of trader data-bases and the calculation advances have made this data increasingly accessible to researchers and traders and havegenerated an enormous growth in the empirical research in finance.
This development has opened the way for a vast array of empirical applications, in particular onliquid financial markets, dealing large volumes and frequency of negotiations and low transaction costs. Among these applications, research applied to the estimation, forecast and comparison of volatility of returns on financial assets with different frequencies stand out.
In addition, high frequency data is also being widely used to study questions related to the market microstructure, such as: the behavior of participants within a specific market, price dynamics and how they affect transactions and offers for purchase and sale of a particular asset, competition between related markets and real time modeling of the market dynamics.
This article contributes to the literature studying the efficacy of returns estimations produced by volatility models of high-frequency data. Two bivariate returns and realized volatility models were proposed, and their contribution improving returns forecasts. It is worth pointing out that the empirical evidence suggests that the quadratic variation forecasters, based on high-frequency data, are better forecasters than the standard volatility estimation models. Therefore, the results presented here are an important aid for better volatility estimations and pricing of financial assets.In practical terms, models implemented herein can be used to validate and to refine intraday price and return models. Thus, they can be useful in intraday investment strategies, in long-short strategies and in risk management, for instance to calculate different conditional volatilities in order to compare and to improve Value at Risk methodologies.
The article is organized as follows. The next section is a brief overview of the relevant literature. Section 3 describes the data used for constructing daily returns of the estimation of daily realized volatility (RV). In this estimation, the RV adjustments stand outto remove the effects of the market microstructure. Section 4 describes the methodology and estimates return and RV models based on intraday data, and the reference models based on the daily returns. In Section 5 the empirical results are presented and validated throughthe use of the intraday conditional variance to estimate the Capital Asset Pricing Model. And lastly, Section 6 underlines the conclusions of this study.
2 BRIEF OVERVIEW OF THE RELEVANT LITERATURE
Analysis of high-frequency data poses new challenges for researchers, since this data possesses unique characteristics, not present in data-bases presenting lower frequencies.
Since ^{Hsieh (1991)} presented one of the first variance estimations of daily returns taken from intraday returns of the S&P500 shareholder index, progress was made in a number of different research areas. Among other seminal articleswhich deal with the unique properties and characteristics of the distribution of intraday data, it is possible to quote: ^{Zhou (1996)}, who used ultra-high-frequency data (tick by tick) relevant to the currency exchange markets in order to explain the negative autocorrelation of the first order of returns and to estimate volatility for high-frequency data, ^{Goodhart and O'Hara (1997)} which highlight the effects of market structure on the interpretation and analysis of the data, the effects of intra-day seasonal and the effects of time-varying volatility, and ^{Andersen and Bollerslev (1997}, ^{1998a)} who analyzed the behavior of intraday volatility, the volatility shocks due to macroeconomic pronouncements and the long-term persistence in the temporal series of realized volatility, also on the currency exchange market.
Other important works, such as that of ^{Andersen and Bollerslev (199b)8}, ^{Andersen, Bollerslev, Diebold, and Ebens (2001a)}, ^{Andersen, Bollerslev, Diebold, and Labys (2001b)}, ^{Barndorff-Nielsen and Shephard (2002)} and ^{Meddahi (2002)} established the theoretical and empirical properties of the quadratic variation estimation for a large class of stochastic processes in finance, thus making empirical research feasible with a new class of estimators, among which realized volatility is included.
^{Andersen and Benzoni (2008)} relate the empirical applications derived from the measurements constructed from high-frequency data, highlighting at least four large research areas (i) volatility forecasting, with emphasis on research focused on improving performance of this forecast, on the relevant literature related to the detection of jumps and on investigating problems related to microstructure in the performance of forecasting; (ii) implications in the distribution of returns under the conditions of non-arbitrage; (iii) multivariate measures of the quadratic variation and (iv) realized volatility, specification and estimation models.
Within these sub-areas of research, this article focuses on the improvement of performance of volatility forecasting, in which special attention has been given to the properties of temporal series and the enhancement of estimation procedures, namely, using realized volatility.
Following are some of the articles which stand out in this sub-area of research.
^{Andersen et al. (2001a)} estimate the daily realized volatility of a number of shares on the Dow Jones Industrial Average - DJIA index. The authors obtain results which affirm that the unconditional distribution of the realized variance and covariance are highly asymmetric towards the right while the realized standard logarithmic deviation and the correlations are approximately Gaussian, as is the returns distribution scaled by the realized standard deviations.
^{Andersen, Bollerslev, Diebold, and Labys (2003)} offer a general structure for using intraday high-frequency data to measure, model and forecast the volatilities and returns distributions at a daily frequency or over lower periods.
^{Ghysels, Santa-Clara, and Valkanov (2005)} introduce a new estimator which forecasts the monthly varianceusing the past daily squared returns and name it Mixed Data Sampling (MIDAS).
^{Andersen et al. (2007)} affirm that more and more literature confirms gains in the volatility forecast of financial assets using measurements based on high-frequency data. They implement a new volatility measure (bipower variation measure) and corresponding non-parametric tests for jumps. The empirical analysis of exchange rates, shares returns index and rates for bonds suggest that the volatility component due to jumps is very important and less persistent than the continuous component, and that the separation of jump movements from soft movements (continuous) results in a significant improvement in the out-of-sample volatility forecast. In addition to this, many significant jumps are associated with new announcements of macroeconomic events.
^{Maheu and McCurdy (2007)} propose a flexible and parsimonious model of the combined dynamic of the return and the market risk to forecast the time-varying market equity premium. This volatility model allows its components to have different decay ratios, generating average returns forecasts and allowing variance targeting.
^{Corsi (2009)} proposes an additive volatility model of components defined in different time horizons. This model possesses components which are auto-regressive in realized volatility and is named the Heterogeneous Autoregressive Model of Realized Volatility - HAR-LOG(RV). Easy to implement, the simulated results show that this model manages to reproduce the principle characteristics of returns on financial assets (long memory, fat tails and self-similarity). In addition, empirical results show excellent forecast performance.
Few articles have studied the benefits of incorporating RV into returns distribution. Among them, we have ^{Andersen et al. (2003)} and ^{Giot and Laurent (2004)}, who consider the value of RV for estimations and to calculate the Value at Risk, comparing the performance of one ARCH model, which uses daily returns, with the performance of a model based on the daily realized volatility - which uses intraday returns - in shares index portfolios and exchange rates. These approaches separate the dynamics of returns and volatilities and assume that RV is a sufficient measure to represent the conditional variance of returns. ^{Ghysels et al. (2005)} point out that the high-frequency volatility measures identify the tradeoff between risk and return at lower frequencies.
Among the studies applied to the Brazilian market, ^{Moreira and Lemgruber (2004)} evaluate the use of high frequency data in volatility forecasting and in VaR using GARCH models in daily and intraday horizons. Among the results, they highlighted that the intraday data can bring significant improvements to the one-day VaR. The most noteworthy article in the Brazilian market is that developed by ^{Wink Junior and Valls Pereira (2012)} which, in a pioneering manner, choose the optimal intraday time interval, deal with the question of noise generated by the market microstructure and implement two recent models, which use high-frequency data to estimate and forecast the volatility of fiverepresentative shares of the Bovespa Index.
3 DATA AND ESTIMATION OF REALIZED VOLATILITY
In this study, the prices negotiated for the PETR4 and VALE5 shares were used, the two most liquid shares on the Brazilian share market. The prices of the negotiations of these shares were obtained directly from BM&FBOVESPA.
The sample data covers the period through December 1^{st}, 2009 and March 23, 2012 for both shares.
After removing errors from the negotiation data, a 5 minute grid was constructed inside the negotiation timeframe of the electronic auction, finding the price negotiated equal to or afterwards closer to each interval on the grid. From this grid, 5 minute continually composed returns were constructed (log returns). These returns were multiplied by 100, and denoted as r_{t,i} = 1,..., I ,where I is the number of intraday returns on the day t. For this 5 minute grid, the average is I=83 for each day of negotiation. This routine generated, respectively, 47,334 and 47,322 five minute returns for the PETR4 and VALE5 over the 573 days in which the shares were negotiated.
The increment of the quadratic variationis a natural measure of the ex-post variance within a specific period of time. The realized variance, also known as realized volatility is one of the most popular quadratic variation estimators, calculated as the sum of squared returns over a specific period of time.
Thus, given the intraday returns r_{t,i} = 1,..., I, a non- adjusted daily estimator of the RV is
However, in the presence of market microstructure dynamics, the RV can be distorted and is an inconsistent estimator for quadratic variation (^{Bandi & Russell, 2004)}. Therefore, the daily realized volatility was adjusted using the moving average method, used by ^{Andersen et al. (2001a)} and later generalized by ^{Hansen, Large, and Lunde (2008)} and also implemented in the Brazilian shares market by ^{Wink Junior and Valls Pereira (2012)}.
Thus, in the event the intraday return of an asset follows a moving average process of the order (MA(q)) given by r_{t,m} = ε_{t} - θ_{1}ε_{t-1,m} -... - θ_{q}ε_{t-q,m}, ^{Hansen et al. (2008)} show that, considering some hypotheses, the estimator that corrects the non-adjusted RV bias, based on the MA(q) process, is given by:
So in order to have no gap between interday and intraday volatility measures, the daily returns r_{t} used in the GARCH family models, were calculated by the logarithmic difference of the last price of the day and the last price of the previous day, both captured in the 5 minute grid. These returns were also multiplied by 100.
Figure 1 shows that the realized volatilities of the shares analyzedpossess significant serial autocorrelation.
Table 1 shows the descriptive statistics for the daily returns and for the estimated daily RV using the 5 minute grid. There is a certain bias in the non-adjusted RV. Following the analysis of the daily RV correlograms and the criterion adopted by ^{Maheu and McCurdy (2011)} to remove this bias, an MA process with 8 gaps (q=8) appears necessary for the PETR4 returns, while q=11 is appropriate for the VALE5 returns. From here on, RV_{t} = RV_{t, MAq} will be used with q=8 and q=11 for the estimations, respectively, of PETR4 and VALE5.
Mean | Variance | Skewness | Kurtosis | Minimum | Maximum | ||
---|---|---|---|---|---|---|---|
PETR4 | r_{t} | -0.090 | 3.121 | -0.404 | 1.359 | -7.596 | 5.328 |
RV_{u} | 3.155 | 11.124 | 4.580 | 28.561 | 0.417 | 32.743 | |
RV_{ma1} | 0.629 | 0.443 | 4.580 | 28.561 | 0.083 | 6.532 | |
RV_{ma2} | 0.023 | 0.001 | 4.580 | 28.561 | 0.003 | 0.237 | |
RV_{ma3} | 0.083 | 0.008 | 4.580 | 28.561 | 0.011 | 0.863 | |
RV_{ma4} | 0.083 | 0.008 | 4.580 | 28.561 | 0.011 | 0.863 | |
RV_{ma4} | 0.330 | 0.121 | 4.580 | 28.561 | 0.044 | 3.420 | |
RV_{ma5} | 1.031 | 1.188 | 4.580 | 28.561 | 0.136 | 10.699 | |
RV_{ma6} | 1.746 | 3.405 | 4.580 | 28.561 | 0.231 | 18.115 | |
RV_{ma7} | 2.223 | 5.521 | 4.580 | 28.561 | 0.294 | 23.067 | |
RV_{ma8} | 3.083 | 10.624 | 4.580 | 28.561 | 0.408 | 31.999 | |
RV_{ma9} | 3.645 | 14.844 | 4.580 | 28.561 | 0.482 | 37.823 | |
RV_{ma10} | 0.083 | 0.008 | 4.580 | 28.561 | 0.011 | 0.863 | |
RV_{ma10} | 3.951 | 17.443 | 4.580 | 28.561 | 0.522 | 41.001 | |
VALE5 | r_{t} | -0.011 | 3.015 | -0.364 | 3.009 | -9.958 | 5.815 |
RV_{u} | 2.866 | 14.783 | 6.344 | 56.496 | 0.303 | 46.845 | |
RV_{ma1} | 0.967 | 1.684 | 6.344 | 56.496 | 0.102 | 15.812 | |
RV_{ma2} | 0.166 | 0.050 | 6.344 | 56.496 | 0.018 | 2.717 | |
RV_{ma3} | 0.000 | 0.000 | 6.344 | 56.496 | 0.000 | 0.001 | |
RV_{ma4} | 0.204 | 0.075 | 6.344 | 56.496 | 0.022 | 3.328 | |
RV_{ma5} | 0.499 | 0.448 | 6.344 | 56.496 | 0.053 | 8.155 | |
RV_{ma6} | 0.646 | 0.750 | 6.344 | 56.496 | 0.068 | 10.555 | |
RV_{ma7} | 0.699 | 0.879 | 6.344 | 56.496 | 0.074 | 11.421 | |
RV_{ma8} | 1.568 | 4.427 | 6.344 | 56.496 | 0.166 | 25.635 | |
RV_{ma9} | 1.646 | 4.878 | 6.344 | 56.496 | 0.174 | 26.908 | |
RV_{ma10} | 1.830 | 6.029 | 6.344 | 56.496 | 0.193 | 29.917 | |
RV_{ma11} | 3.016 | 16.374 | 6.344 | 56.496 | 0.319 | 49.302 | |
RV_{ma12} | 3.468 | 21.650 | 6.344 | 56.496 | 0.367 | 56.691 |
4 METHODOLOGY
In this article, bivariate models were proposed based on two alternative ways in which the RV is related to the conditional variance of returns and the GARCH family models are the reference for performance analysis of intraday models.
As in ^{Maheu and McCurdy (2011)}, two functional forms were proposed for the bivariate models of the returns and the RV. The first model uses the heterogeneous autoregressive (HAR) function of the lagged log(RV) (^{Corsi, 2009}; ^{Andersen, Bollerslev, & Diebold, 2007)}. The second model allows the components of the log(RV) to have different decay ratios (^{Maheu and McCurdy, 2007)}.
A way to connect the RV to the returns variance was also considered, imposing the restriction that the conditional variance of the daily returns be equal to the conditional expectation of the daily RV.
As with EGARCH and TGARCH models, bivariate models allow the so called leverage effect, or asymmetries, of the negative innovations versus the positive innovations of the returns.
One way to confirm that the intraday information contributes to the improvement of the estimations of the returns distributions is to compare the estimates of the bivariate models of the return and log(RV) specified in the estimates of the GARCH family models:
The main comparison method implementeduses the root mean squared errors and the Modified ^{Diebold and Mariano (1995)} test, based on the work of ^{Harvey, Leybourne, and Newbold (1997)}. Intuitively, better estimated models will have less forecast errors and, if compared with the other inferior performance models, will present statistical differences in their errors. Therefore, to assess the models implemented in this article, we focused on the relative accuracy of those models when estimating the in-sample and the out-of-sample returns.
One important aspect of the approach used is the possibility to directly compare traditional volatility specifications, such as GARCH family models, with the bivariate returnsmodels and RV, because the implemented models possess one common criterion - returns forecast. The average and the statistical test of these forecast errors allow us to investigate the relative contribution of the RV in the forecasts.
4.1 Bivariate Returns Models and Realized Variance.
In this subsection, two combined specifications of the daily return and the RV were implemented. These bivariate models are differentiated by their alternative conditions over the RV dynamic. In each case, restrictions between the equations connect the returns variance to the RV specification.
The corollary one of ^{Andersen et al. (2003)} shows that, in realistic empirical conditions, the conditional expectation of the quadratic variation (QV_{t}) is equal to the conditional variance of the returns, i.e., E_{t-1} (QV_{t}) = Var_{t-1} (r_{t})≡ σ_{t}^{2}. If the RV is a non-biased estimator of quadratic variation, it follows that the conditional variance of the returns can be linked to RV as σ_{t}^{2} = E_{t-1} (RV_{t}), where the combination of information is defined as σ_{t-1} ≡ {r_{t-1}, RV_{t-1}, r_{t-2}, RV_{t-2}, ... r_{1}, RV_{1}}. Supposing that RV possesses a log-normal distribution, the restriction takes on the form:
4.1.1 Heterogeneousauto-regressive specification: HAR Model.
The first model implemented possesses a bivariate specification for the daily returns and RV in which the conditional returns are driven by normal innovations and the dynamic of log (RV_{t}) is captured by a heterogeneous autoregressive function (HAR) of the lagged log (RV_{t}). ^{Corsi (2009)} and ^{Andersen et al. (2007)} use the HAR functions aiming to capture the dependence of long-term memory parsimoniously. Motivated by these studies, we defined:
For example, (RV_{t-22,22}) is estimated calculating the average of log (RV) for the last 22 days, i.e., from t-22 to t-1, log(RV_{t-5,5} ) considers the average of the last five days.
This takes the specification of the daily returns and RV with the dynamic of log(RV_{t} ) being modeled as an asymmetrical HAR function of the past log(RV)
^{1}. This bivariate system is summed up as follows:
This bivariate specification of the daily returns and RV imposes the restriction equation that relates the conditional variance of daily returns with the conditional expectation of daily RV, as shown in (4.5).
Since the data-baseanalyzed in this article is that of shares returns, it is important to allow asymmetrical effects into the volatility. To facilitate comparisons with the EGARCH reference model, the parameterization in equation (4.8) includes the asymmetric term γe_{t-1} associated with the innovations of the e_{t-1} returns. The impact coefficient for negative innovations of the returns will be γ. Typically, < 0, which means the negative innovations of returns imply larger conditional variance for the next period.
4.1.2 Component-Log(RV) Specification: 2-Comp Model.
This Bivariate specification for daily returns and RV possesses conditional returns guided by normal innovations, but the dynamic of log(VR) is captured by two components (2-Comp) with a different decay ratio, as shown in ^{Maheu and McCurdy (2007)}. In particular, this bivariate system can be represented in the equation:
Again, a restriction was imposed by equation (4.5) which related the conditional variance of the daily returns with the conditional expectation of the daily RV. For this specification, the dynamic of daily log(RV) was parameterized by equations (4.10) and (4.11), substituting the HAR function in equation (4.8).
The average of the return time series was estimated by ARMA (p, q) models, using the R software function auto.arima. All other estimates were made using the software eviews 7.1. Aiming to combine parsimony and robustness of these estimates, we established the maximum lag length (p+q) of 4 and automatic lag length selection using the Schwarz Information Criterion (BIC). Thus, for the average of the series we have an AR (1) modeling the daily and weekly series of VALE5 and PETR4 and an ARMA (2,2) for weekly and monthly series of both assets.
The bivariate systems were estimated in two steps. Initially the equations of means were estimated and then the return innovations were modeled by different volatility models. Volatility forecasts were made by a sequence of one-step ahead forecasts, using the current values for lagged depended variable and return forecasts considering r_{i,t} = f_{i,t} + e_{i,t}, e_{i,t} = σ_{i,t} u_{t} , u_{t} ~ NID(0,1) where f_{i,t} is the estimated equation for the mean process for asset i in time t, e_{i,t} is the return innovation for the asset i in time t and α_{i,t} is the estimated conditional standard deviation for the asset i in time t.
Aiming to do a practical exercise for the applicability of the models presented here, we use the conditional variance estimated by the monthly 2-Comp model, which returned the lowest error predictions among analyzed models, in the Capital Asset Pricing Model-CAPM estimation. The CAPM developed by ^{Sharpe (1964)}, ^{Treynor (1961)}, ^{Lintner (1965)} and ^{Black, Jensen, and Scholes (1972)} has become in recent decades the most widespread model for determination of asset prices (^{Barros, Famá, & Silveira, 2002}). This model states that assets are priced compatible with a trade-off between non-diversifiable risk and expectations of return.
The CAPM can be formally presented as E(r_{i}) = r_{f} + β_{i} (E(r_{m}) - r_{f}) where E(r_{i}) is the expected return on asset i over a single time-period, r_{f} is the riskless rate of interest rate over the period, E(r_{m}) is the expected return on the market over the period, and identifies the exposure of asset i to the market.
To estimate β we used: (i) the statistical covariance Cov (r_{m}, r_{i}) of PETR4 and VALE5 regarding BOVA11 in a moving window of 22 days and (ii) the conditional variance σ^{2}(r_{m}) estimated by the 2-Comp model in this same time frame. The Interbank Certificate of Deposits - CDI was selected to represent the risk free interest rate risk, following the work of ^{Barros, Famá, and Silveira (2002)}.
To represent the market portfolio, we choose the exchange traded fund BOVA11, considering that its expected return for the next period is a function of the preceding period return. Among the qualities of BOVA11 we have: (i) it is effectively traded in an active market, enabling the extraction of realized volatility and its use in intraday volatility models, (ii) it has an average correlation of more than 99% with the Ibovespa^{2} in the analyzed period, (iii) it is an asset with increasing liquidity, with average daily turnover of R$44.1mi in 2011, and by far the most traded ETF in the Brazilian market.
5 EMPIRICAL RESULTS
In this chapter, we will present the estimated results of models for the 1, 5 and 22 day time models.
Table 2 presents estimates of the GARCH family models. The Schwarz Information Criterion (BIC) indicates that asymetric model sare as well adjusted to the data as the estimated GARCH model, which was confirmed in nearly all the in sample forecasts of these models. Based on fitted EGARCH and TGARCH models, aside from the monthly estimate of PETR4, all leverage effect coefficientsare significant at the 10% level, confirming the asymmetrical impact between the positive and negative returns of assets.
| |||||||||||
Horizon | Share/ Parameter | ρ_{1} | ρ_{2} | θ_{1} | θ_{2} | Adj. R^{2} | ω | β | α | γ | BIC |
1 day | PETR4 | 0.159*** | 0.883* | 0.07** | 3.774 | ||||||
0.090 | 0.060 | 0.030 | |||||||||
VALE5 | 0.104** | 0.011 | 0.165** | 0.835* | 0.096* | 3.668 | |||||
0.042 | 0.072 | 0.057 | 0.034 | ||||||||
5 days | PETR4 | 0.815* | 0.661 | 0.243 | 0.849* | 0.099** | 4.325 | ||||
0.025 | 0.153 | 0.068 | 0.046 | ||||||||
VALE5 | 1.299* | -0.664* | -0.664* | 0.779* | 0.707 | 0.211*** | 0.856* | 0.092** | 4.222 | ||
0.042 | 0.040 | 0.034 | 0.029 | 0.118 | 0.057 | 0.039 | |||||
22 days | PETR4 | 0.908* | 0.978* | -0.041 | 0.889 | 0.285 | 0.892* | 0.053** | 4.483 | ||
0.017 | 0.043 | 0.044 | 0.215 | 0.062 | 0.029 | ||||||
VALE5 | 0.228** | 0.638* | 0.907* | 0.276* | 0.905 | 0.019 | 0.948* | 0.047* | 4.325 | ||
0.119 | 0.114 | 0.118 | 0.043 | 0.032 | 0.021 | 0.018 | |||||
| |||||||||||
Horizon | Share/ Parameter | ρ_{1} | ρ_{2} | θ_{1} | θ_{2} | Adj. R^{2} | ω | β | α | γ | BIC |
1 day | PETR4 | -0.051 | 0.9405* | 0.115** | -0.138* | 3.778 | |||||
0.044 | 0.033 | 58.000 | 0.041 | ||||||||
VALE5 | 0.104** | 0.011 | -0.005 | 0.904* | 0.101** | -0.183* | 3.637 | ||||
0.042 | 0.032 | 0.022 | 0.044 | 0.037 | |||||||
5 days | PETR4 | 0.815* | 0.661 | -0.070 | 0.971* | 0.1314** | -0.118* | 4.325 | |||
0.025 | 0.044 | 0.022 | 0.059 | 0.034 | |||||||
VALE5 | 1.299* | -0.664* | -0.664* | 0.779* | 0.707 | -0.032 | 0.937* | 0.145** | -0.091* | 4.221 | |
0.042 | 0.040 | 0.034 | 0.029 | 0.039 | 0.026 | 0.059 | 0.036 | ||||
22 days | PETR4 | 0.908* | 0.978* | -0.041 | 0.889 | 0.613 | 0.496*** | 0.245** | -0.078 | 4.499 | |
0.017 | 0.043 | 0.044 | 0.429 | 0.287 | 0.111 | 0.073 | |||||
VALE5 | 0.228** | 0.638* | 0.907* | 0.276* | 0.905 | -0.069* | 1.003* | 0.078* | -0.045** | 4.329 | |
0.119 | 0.114 | 0.118 | 0.043 | 0.021 | 0.005 | 0.028 | 0.020 | ||||
| |||||||||||
Horizon | Share/ Parameter | ρ_{1} | ρ_{2} | θ_{1} | θ_{2} | Adj. R^{2} | ω | β | α | γ | BIC |
1 day | PETR4 | 0.155** | 0.858* | -0.003 | 0.152** | 3.769 | |||||
0.081 | 0.051 | 0.027 | 0.060 | ||||||||
VALE5 | 0.104** | 0.011 | 0.142* | 0.879* | -0.041** | 0.216* | 3.629 | ||||
0.042 | 0.036 | 0.030 | 0.017 | 0.046 | |||||||
5 days | PETR4 | 0.815* | 0.661 | 0.125 | 0.902* | -0.015 | 0.164** | 4.316 | |||
0.025 | -0.015 | 0.048 | 0.032 | 0.054 | |||||||
VALE5 | 1.299* | -0.664* | -0.664* | 0.779* | 0.707 | 0.196** | 0.888* | 0.002 | 0.123** | 4.213 | |
0.042 | 0.040 | 0.034 | 0.029 | 0.079 | 0.041 | 0.031 | 0.051 | ||||
22 days | PETR4 | 0.908* | 0.978* | -0.041 | 0.889 | 0.215 | 0.913* | 0.031 | 0.026 | 4.497 | |
0.017 | 0.043 | 0.044 | 0.157 | 0.048 | 0.031 | 0.039 | |||||
VALE5 | 0.228** | 0.638* | 0.907* | 0.276* | 0.905 | -0.027*** | 0.979* | -0.013 | 0.073* | 4.328 | |
0.119 | 0.114 | 0.118 | 0.043 | 0.016 | 0.012 | 0.016 | 0.022 |
Note: *, ** and *** stand for rejection of the null hypothesis at the1%, 5% and 10% significance levels, respectively.
Both asymmetric models provide similar leverage effects for the estimated time horizons. Considering a standardizes shock of 2 standard deviation, the leverage effect for EGARCH(1,1,1) can be estimated (^{Tsay, 2010}) as , and for TGARCH(1,1,1) as .
The following Table 3 compare these asymmetric models assuming that e_{t-1} = ± 2σ_{t-1} so that u_{t-1} = ± 2. The impact of a negative shock of size 2 standard deviation is more powerful for daily returns. Thus, the leverage effect is reduced when time horizons increase.
Model | Share | 1 day | 5 days | 22 days |
---|---|---|---|---|
EGARCH | PETR4 | 1.74 | 1.60 | 1.37 |
VALE5 | 2.21 | 1.51 | 1.31 | |
TGARCH | PETR4 | 1.72 | 1.78 | 1.10 |
VALE5 | 2.08 | 1.44 | 1.20 |
For instance, for VALE5 1 day horizon, the impact of a 2 standard deviation negative shock is about 121% and 108% higher, respectively for EGARCH and TGARCH models, than that of a positive shock of the same magnitude, while for 22 days horizon these impacts are reduced for 31% and 20%.
Tables 4 and 5, following, show the estimations of the HAR-Log(RV) and 2-Comp models respectively, for the 1, 5 and 22 day time horizons.
| ||||||||||||||
Horizon | Share/ Parameter | ρ_{1} | ρ_{2} | θ_{1} | θ_{2} | Adj. R^{2} | ω | ø_{1} | ø_{2} | ø_{3} | γ | η | Adj. R^{2}_{j} | Adj. R^{2} |
1 day | PETR4 | - | - | - | - | - | 0.156** | 0.290* | 0.345* | 0.093 | -0.038** | -0.021 | 0.3354 | 0.3310 |
- | - | - | - | - | 0.064 | 0.060 | 0.091 | 0.090 | 0.017 | 0.026 | ||||
VALE5 | 0.104** | - | - | - | 0.011 | 0.057 | 0.312* | 0.220** | 0.267* | -0.058* | -0.018 | 0.3891 | 0.3753 | |
0.042 | - | - | - | 0.055 | 0.060 | 0.089 | 0.088 | 0.019 | 0.027 | |||||
5 days | PETR4 | 0.815* | 0.661 | 0.053* | 0.113* | 0.855* | -0.016 | -0.011* | -0.001 | 0.9049 | 0.9040 | |||
0.025 | 0.019 | 0.018 | 0.028 | 0.028 | 0.004 | 0.008 | ||||||||
VALE5 | 1.299* | -0.664* | -0.664* | 0.779* | 0.707 | 0.037** | 0.138* | 0.807* | 0.022 | -0.002 | -0.007 | 0.8991 | 0.8990 | |
0.042 | 0.040 | 0.034 | 0.029 | 0.019 | 0.020 | 0.030 | 0.030 | 0.005 | 0.009 | |||||
22 days | PETR4 | 0.908* | 0.978* | -0.041 | 0.889 | 0.004 | 0.018* | 0.039* | 0.944* | -0.001 | 0.001 | 0.9859 | 0.9859 | |
0.017 | 0.043 | 0.044 | 0.006 | 0.006 | 0.009 | 0.008 | 0.001 | 0.002 | ||||||
VALE5 | 0.228** | 0.638* | 0.907* | 0.276* | 0.905 | 0.005 | 0.014* | 0.0036* | 0.949* | -0.004* | -0.002 | 0.9891 | 0.9888 | |
0.119 | 0.114 | 0.118 | 0.043 | 0.005 | 0.005 | 0.008 | 0.008 | 0.001 | 0.002 |
Note: *, ** and *** stand for rejection of the null hypothesis at the1%, 5% and 10% significance levels, respectively.
| |||||||||||||
Horizon | Share/ Parameter | ρ_{1} | ρ_{2} | θ_{1} | θ_{2} | Adj. R^{2} | ω | ø_{1} | ø_{2} | α_{1} | α_{2} | γ | η |
1 day | PETR4 | 0.071* | 0.544* | 0.242* | 0.784 | 0.001 | -0.017** | -0.007 | |||||
0.023 | 0.093 | 0.064 | 0.007 | 0.011 | |||||||||
VALE5 | 0.104** | 0.011 | 0.034 | 0.516* | 0.345* | 0.408 | 0.904 | -0.027* | -0.006 | ||||
0.042 | 0.022 | 0.076 | 0.107 | 0.008 | 0.011 | ||||||||
5 days | PETR4 | 0.815* | 0.661 | 0.027* | 1.505* | -0.579* | 0.003 | 0.354 | -0.005* | -0.002 | |||
0.025 | 0.007 | 0.122 | 0.125 | 0.002 | 0.003 | ||||||||
VALE5 | 1.299* | -0.664* | -0.664* | 0.779* | 0.707 | 0.024* | -0.507* | 1.432* | 0.349 | 0.018 | -0.003 | -0.004 | |
0.042 | 0.040 | 0.034 | 0.029 | 0.007 | 0.138 | 0.135 | 0.002 | 0.004 | |||||
22 days | PETR4 | 0.908* | 0.978* | -0.041 | 0.889 | 0.009* | -0.222* | 1.199* | 0.770 | 0.001 | -0.001 | -0.001 | |
0.017 | 0.043 | 0.044 | 0.002 | 0.026 | 0.025 | 0.001 | 0.001 | ||||||
VALE5 | 0.228** | 0.638* | 0.907* | 0.276* | 0.905 | 0.006* | -0.307* | 1.291* | 0.628 | 0.001 | -0.002* | -0.001 | |
0.119 | 0.114 | 0.118 | 0.043 | 0.002 | 0.034 | 0.033 | 0.000 | 0.001 |
Note: *, ** and *** stand for rejection of the null hypothesis at the1%, 5% and 10% significance levels, respectively.
The results found in Table 4 are in agreement with those presented by ^{Andersen et al. (2007)} and ^{Wink Junior and Valls Pereira (2012)}. The statistically significant estimates of the coefficients of the daily (ø_{1}), weekly (ø_{2}) and monthly (ø_{3}) volatility components confirm the presence of high persistence in the volatility^{3}. The relative weight of the daily volatility component decreases from the daily regressions to the weekly and monthly component while the monthly component tends to be relatively more important in the regressions over longer periods^{4}.
In addition to this, when comparing the adjusted R^{2} of the HAR which includes the asymmetrical γ component with the same standard HAR statistic (last column of Table 3), there is noted little improvement in the models' estimation, indicating that the HAR coefficients was already capturing some of the asymmetric dynamic of the asset returns.
However, the results presented in Table 5 show that the estimated 2-Comp model was able to efficiently capture the different volatility dynamics, with the persistent coefficients α_{1} and α_{2} clearly differentiated by each time horizon (see Figure 2). In addition to this, the minor α coefficient in each equation shows less persistent effect, being more influenced by the more recent RV observations.
In-sample | ||||||
Horizon | Model | GARCH | EGARCH | TGARCH | HAR-log(RV) | 2-Comp |
1 day | PETR4 | 2.15 | 2.16 | 2.16 | 2.19 | 1.91 |
5 days | 2.88 | 2.89 | 2.87 | 2.56 | 2.34 | |
22 days | 3.13 | 3.13 | 3.13 | 2.72 | 2.49 | |
1 day | VALE5 | 2.14 | 2.12 | 2.14 | 2.17 | 1.89 |
5 days | 2.76 | 2.74 | 2.76 | 2.49 | 2.26 | |
22 days | 3.04 | 3.04 | 3.03 | 2.69 | 2.44 | |
Out-of-sample | ||||||
Horizon | Model | GARCH | EGARCH | HAR-RV | HAR-log(RV) | 2-Comp |
1 day | PETR4 | 3.07 | 2.97 | 3.15 | 3.02 | 2.59 |
5 days | 4.55 | 4.31 | 4.55 | 3.85 | 3.44 | |
22 days | 4.19 | 4.00 | 4.20 | 3.80 | 3.42 | |
1 day | VALE5 | 3.14 | 3.06 | 3.37 | 3.19 | 2.63 |
5 days | 3.85 | 3.80 | 3.98 | 3.58 | 3.09 | |
22 days | 4.40 | 4.32 | 4.47 | 3.91 | 3.49 |
Thus, as in the HAR model that was used, the asymmetric γ components of the 2-Comp model are negative but also relatively very small, indicating that these models without γ are also able to partially capture the asymmetric balance of the returns.
Figure 2 presents the graphs of the historical s_{1,t} and s_{2,t} series, generated by the decline of factors α_{1} and α_{2} (graph 2.1) and realized and estimated Log(RV) (graph 2.2) for VALE5 in the one day time horizon.
In order to assess the accuracy of the models used, they were assessed for the forecast of the following 1, 5 and 22 day time horizons, in and out-of-sample, using the root mean squared error (RMSQ) measure. The Modified Diebold Mariano test was used^{5} to estimate the statistical differences between the models.
For the in-sample period, data collected between 01/07/2010 and 7/29/2011 was considered (388 observations) and for the out-of-sample, data between 08/01/2011 and 03/21/2012 (160 observations).
Table 6 presents the RMSQ of the forecasts in the three defined time horizons. The 2-Comp model returned better forecasts for all time horizons, and HAR returned the second best forecast for the 5 and 22 day horizons. But are these results statistically better than the GARCH family models? Table 7 tries to answer these and other questions.
7.1 In-sample | 7.2 Out-of-sample | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 day horizon | 1 day horizon | ||||||||||||
GARCH | EGARCH | TGARCH | HAR-RV (log) | 2-Comp | GARCH | EGARCH | TGARCH | HAR-VR (log) | 2-Comp | ||||
GARCH | PETR4 | 100.0% | 94.4% | 96.9% | 67.2% | 3.8% | GARCH | PETR4 | 100.0% | 8.7% | 3.3% | 11.6% | 0.0% |
EGARCH | 100.0% | 99.4% | 81.0% | 0.4% | EGARCH | 100.0% | 0.0% | 9.9% | 0.0% | ||||
TGARCH | 100.0% | 77.0% | 0.8% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% | ||||||||||
GARCH | VALE5 | 100.0% | 82.1% | 96.3% | 74.6% | 0.2% | GARCH | VALE5 | 100.0% | 7.5% | 0.0% | 2.5% | 0.0% |
EGARCH | 100.0% | 91.6% | 52.8% | 0.4% | EGARCH | 100.0% | 0.0% | 0.0% | 0.0% | ||||
TGARCH | 100.0% | 74.0.0% | 0.0% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV(log) | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% | ||||||||||
5 days horizon | 5 days horizon | ||||||||||||
GARCH | PETR4 | 100.0% | 85.0% | 95.4% | 0.0% | 0.0% | GARCH | PETR4 | 100.0% | 0.0% | 57.4% | 0.0% | 0.0% |
EGARCH | 100.0% | 93.2% | 0.0% | 0.0% | EGARCH | 100.0% | 0.0% | 0.0% | 0.0% | ||||
TGARCH | 100.0% | 0.0% | 0.0% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV(log) | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% | ||||||||||
GARCH | VALE5 | 100.0% | 80.3% | 93.2% | 0.0% | 0.0% | GARCH | VALE5 | 100.0% | 32.6% | 0.0% | 0.0% | 0.0% |
EGARCH | 100.0% | 93.0% | 0.0% | 0.0% | EGARCH | 100.0% | 0.0% | 0.0% | 0.0% | ||||
TGARCH | 100.0% | 0.0% | 0.0% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV(log) | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% | ||||||||||
22 days horizon | 22 days horizon | ||||||||||||
GARCH | PETR4 | 100.0% | 95.1% | 99.4% | 0.0% | 0.0% | GARCH | PETR4 | 100.0% | 0.0% | 91.7% | 0.0% | 0.0% |
EGARCH | 100.0% | 97.3% | 0.0% | 0.0% | EGARCH | 100.0% | 0.0% | 0.0% | 0.0% | ||||
TGARCH | 100.0% | 0.0% | 0.0% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV(log) | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% | ||||||||||
GARCH | VALE5 | 100.0% | 93.5% | 89.8% | 0.0% | 0.0% | GARCH | VALE5 | 100.0% | 0.9% | 1.5% | 0.0% | 0.0% |
EGARCH | 100.0% | 97.2% | 0.0% | 0.0% | EGARCH | 100.0% | 0.0% | 0.0% | 0.0% | ||||
TGARCH | 100.0% | 0.0% | 0.0% | TGARCH | 100.0% | 0.0% | 0.0% | ||||||
HAR-RV(log) | 100.0% | 0.0% | HAR-RV(log) | 100.0% | 0.0% | ||||||||
2-Comp | 100.0% | 2-Comp | 100.0% |
Share | Returns | Mean | Median | Maximum | Minimum | Standard Dev. | Skewness | Kurtosis | Jarque-Bera |
---|---|---|---|---|---|---|---|---|---|
PETR4 | Observed | (2.70) | (2.48) | 17.15 | (24.16) | 6.70 | (0.26) | 3.29 | 5.93 |
Expected | (0.85) | (0.27) | 11.99 | (22.31) | 5.32 | (1.11) | 5.38 | 170.93 | |
VALE5 | Observed | 0.50 | 0.42 | 15.95 | (27.30) | 7.60 | (0.46) | 3.59 | 19.42 |
Expected | (1.19) | 0.64 | 19.20 | (46.36) | 10.12 | (1.48) | 7.05 | 407.92 |
Table 7 shows the p-values of the Modified Diebold Mariano Statistical Test.
Regarding the return forecasts in bothin andout of the sample periods, the p-values show that: (i) 2-Comp model provides the best forecasts in the three time horizons; (ii) HAR model has the second best prediction for the 5 and 22 day horizons.
Considering the return forecasts in the sample period: (i) there is no significant difference between the GARCH, EGARCH and HAR models in one day time horizon; (ii) there is no significant difference between the GARCH family models in 5 and 22 day time horizons.
Furthermore, returnforecastsonly inthe out of the sample period show: (i) in one day time horizon: TGARCH not kept the EGARCH performance, vis a vis the similar in sample forecasts of both models; GARCH and EGARCH return better predictions than HARmodel for VALE5; there are no significant forecast differences of HAR, GARCH and EGARCH models for PETR4; (ii) in the 5 day time horizon, the EGARCH model returns better predictions than TGARCH model; (iii) in the 22day time horizon, the EGARCH model returns better predictions than the GARCH and TGARCH models.
Figure 3 represents the graphs for the two best out-of-sample forecasts for the PETR4 and VALE5 shares for the three time horizons. It can be observed that the high-frequency models implemented here present very similar forecasts among themselves. In addition, these forecasts appear to be strongly adherent to the realized returns in all analyzedtime horizons.
Considering CAPM results, Figure 4 shows the behavior of estimated expected returns series and PETR4 and VALE5 return series for the 22 days time horizon. There is a high adherence of the estimated returns to realized returns, with a correlation of 65% and 90%, respectively, for PETR4 VALE5.
The following table summarizes the descriptive statistics for expected and observed monthly returns of PETR4 and VALE5. We can mention among the results that: (i) the distribution of the expected returns of both shares are asymmetric to the left and exhibit more fat tails (more leptokurtic) than the realized returns and (iii) distributions of the realized and the estimated returns are not normal at the 5% level of significance.
6 CONCLUSIONS
This article proposed alternative models which used the daily returns and the RV, relating the RV to the variance of returns. In addition, it sought to explore the possible benefits of using intraday data to obtain better volatility estimates and forecast of returns.
The empirical applications implemented in the returns of PETR4 and VALE5 reveal the importance of the information contained in the intraday returns and the use of log(RV). The results found confirm that: (i) bivariate models which use high-frequency data provide a significant improvement in the forecasts compared with the standard models, from daily data, confirming the results found by ^{Maheu and McCurdy (2011)} on the North-American stock market; (ii) the two bivariate high-frequency models, in a parsimonious and singular manner, obtained success in modeling volatility as presented by ^{Wink Junior and Valls Pereira (2012)}, showing excellent performance in the forecast of the returns and confirming results found in ^{Corsi (2009)}.
These findings can be useful in intraday investment strategies, in long-short strategies and in risk management. HAR and 2-Comp conditional volatilities can be used, for instance, in order to compare and refine the performance of different Value at Risk methodologies.
At the end of this article, we also sought answers to the question: does high-frequency price models offer better return forecasts than the accepted models using closing prices?
It is possible to confirm; yes. The models using high-frequency data implemented here appear to contribute to better volatility and return forecasting. These results were obtained in the in and out-of-sample periods using the root mean squared error and the Modified Diebold Mariano test of the one, five and twenty-two day time horizon forecasts. Nevertheless, the estimation of these models for other financial assets and longer historical series could confirm and validate the results obtained here.
*The authors would like to thank Prof. Dr. Claudio Henrique da Silveira Barbedo from IBMEC-Rio and Banco Central do Brasil for his valuable comments.
^{1}The temporal RV series for the assets used here are stationary according to the unit root test, which rejects the null hypothesis of non-stationarity.
^{2}Bovespa Index (Ibovespa) is the most important indicator of average prices of shares traded on the São Paulo Stock Exchange and it is made up of stocks with the highest trading volume in recent months.
^{3}Only three out of 18 volatility coefficients are not significant to a degree of 10%. These exceptions occur for the monthly volatility components estimated in equations with 1 and 5 days time horizons.
^{4}As noted by ^{Andersen et al. (2007)}, although the structure of the HAR model does not formally possess a long memory, the combination of few volatility components is capable of reproducing a notable smooth fall of the autocorrelation of this volatility, being almost indistinguishablefromthe hyperbolic decay (long memory).
^{5}This statistical test consists of testing the null hypothesis of equality between the quadratic error mean of two forecasts, using the critical values of a t-Student distribution with (n-1) degrees of liberty.
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Received: April 21, 2013; Revised: April 29, 2013; Accepted: March 07, 2014