Multi-Period Volatility Predictions: A Comparative Study Using MIDAS Regressions

Transcription

1 Multi-Period Volatility Predictions: A Comparative Study Using MIDAS Regressions Douglas G. Santos Flávio A. Ziegelmann Abstract We explore comparative results in the context of multi-period volatility predictions, focusing on the MIDAS approach. First, we compare the MIDAS method with two widely used methods of producing multi-period forecasts: the direct and the iterated approaches. Their relative performances are investigated in a Monte Carlo study and in a first empirical study, where we predict volatility at horizons up to 60 days ahead. The results of the Monte Carlo study indicate that the MIDAS predictions are the most accurate at horizons of 15 days ahead and higher. The iterated forecasts are the best ones for shorter horizons (5 and 10 days ahead). In the first empirical study, using daily returns of the S&P 500 and NASDAQ indexes, the results are not so conclusive, but suggest a better performance for the iterated forecasts. Second, we concentrate on relatively shorter horizons (up to 20 days ahead) to predict future volatility. In the second empirical study, we compare the MIDAS regressions with the HAR regressions and compute forecast combinations of the models. Using intra-daily data of the Bovespa index, we calculate a set of volatility measures to be used as regressors. In this case, the main result is that the combined predictions significantly outperform the individual ones in most cases. All analyses are out-of-sample. Keywords and Phrases: Volatility; Multi-period forecasts; MIDAS regressions; GARCH models; HAR regressions. JEL Classification: C22; C52; C53; G17. Resumo Exploramos resultados comparativos no contexto de previsões de volatilidade multi-período, focando na abordagem MIDAS. Primeiro, comparamos o método MIDAS com dois métodos amplamente utilizados na produção de previsões multi-período: as abordagens direta e i- terada. Suas performances relativas são investigadas em um estudo de Monte Carlo e em um primeiro estudo empírico, onde prevemos volatilidade em horizontes de até 60 dias à frente. Os resultados do estudo de Monte Carlo indicam que as previsões MIDAS são as mais acuradas para horizontes a partir de 15 dias à frente. As previsões iteradas são superiores nos horizontes mais curtos (5 e 10 dias à frente). No primeiro estudo empírico, utilizando retornos diários dos índices S&P 500 e NASDAQ, os resultados não são tão conclusivos, mas sugerem um melhor desempenho das previsões iteradas. Segundo, nos Graduate Program in Economics, Federal University of Rio Grande do Sul, Porto Alegre, RS , Brazil, dgomess@hotmail.com. Department of Statistics and Graduate Programs in Economics and Management, Federal University of Rio Grande do Sul, Porto Alegre, RS , Brazil, flavioaz@mat.ufrgs.br. 1

2 concentramos em horizontes de previsão relativamente mais curtos (até 20 dias à frente). No segundo estudo empírico, comparamos as regressões MIDAS com as regressões HAR e calculamos combinações de previsão dos modelos. Utilizando dados intradiários do índice Bovespa, calculamos um conjunto de medidas de volatilidade a serem usadas como regressores. Neste caso, o resultado principal é que as previsões combinadas superam significativamente as previsões individuais na maior parte dos casos. Todas as análises são fora da amostra. Palavras-chave: Volatilidade; Previsões Multi-período; Regressões MIDAS; Modelos GAR- CH; Regressões HAR. Área de Submissão: ECONOMETRIA. 1 Introduction The literature of financial econometrics exhibits a comprehensive amount of papers in which the focus is on the accuracy of one-period-ahead forecasts of volatility, see for example, Engle (1982), Bollerslev (1986), Andersen and Bollerslev (1998), Hansen and Lunde (2005), among many others. In contrast, studies on long-horizon (multi-period-ahead) volatility predictions exist in small number. In particular, multi-period volatility predictions are quite relevant for risk management, portfolio allocation and regulatory supervision. Amongst the methods to produce multi-period forecasts of volatility, there are the direct, the iterated, and the relatively less explored mixed-data sampling (henceforth MIDAS). Regarding the first approach, the basic idea is to estimate a volatility model, such as a GARCH model, with 5-day, 10-day or k-day returns and then directly calculate the volatility forecasts over the next 5 days, 10 days, and so on. On the other hand, in the iterated approach, a GARCH model, for example, is estimated using daily returns. As a result, the 5-day, 10-day or k-day predictions of the conditional variance are obtained by iterating over the one-period-ahead forecasts for the necessary number of periods. For these methods, see for instance, Bhansali (2002) and Marcellino, Stock and Watson (2006). Alternatively, as a middle ground between the direct and the iterated approaches, the MIDAS models are estimated with daily regressors (squared returns, absolute returns, among others) and directly produce multi-period volatility predictions. MIDAS regression models were introduced by Ghysels, Santa-Clara and Valkanov (2004, 2005). The discussed methods have been widely used in the empirical finance literature; however, there exist few comparative results about their forecasting performance considering multiperiod volatility predictions. Diebold, Hickman, Inoue, and Schuermann (1997), Andersen, Bollerslev, and Lange (1999), and Christoffersen and Diebold (2000) are examples using the direct and iterated methods. In a comprehensive empirical study, Ghysels, Rubia and Valkanov (2009) have found robust results for the MIDAS approach in comparison with the direct, the iterated, and other approaches of obtaining multi-period forecasts of volatility. Ghysels 2

3 et al. (2009) use a data set consisting of daily returns of the US stock market to forecast the volatility at horizons up to 60 trading days. Their overall findings indicate that at relatively short horizons of 5 and 10 days ahead, the iterated predictions are quite precise. Nevertheless, at horizons of 10 days ahead and longer, the MIDAS forecasts are significantly better than the ones obtained by the other approaches. On the other hand, predicting relatively shorter horizons (one day, and from one week to four weeks), Ghysels, Santa-Clara and Valkanov (2006) examine the forecasting ability of a set of daily predictors of future volatility (measured as increments in quadratic variation) in the context of MIDAS regressions. Using intra-daily data of the US stock market, they consider the predictors: daily squared returns, daily absolute returns, daily range, daily realized variance and daily realized power (i.e., sum of intra-daily absolute returns). In particular, Realized Power Variation (RPV) is a volatility measure proposed by Barndorff-Nielsen and Shephard (2003, 2004). Ghysels, Santa-Clara and Valkanov (2006) find that daily realized power is the best overall predictor of future volatility. In addition, this empirical evidence was supported by theoretical and further empirical work by Forsberg and Ghysels (2007). In an extensive empirical study, Forsberg and Ghysels (2007) consider volatility measures such as Realized Variance (RV), Realized Power Variation (RPV), among others, to be used as regressors in MIDAS models and in the Heterogeneous Autoregressive (HAR) model of Corsi (2009). The HAR model can be seen as a MIDAS regression (for a particular specification), as explained in Ghysels, Sinko, and Valkanov (2007). Forsberg and Ghysels (2007) show that RPV is the most preferred regressor to predict future increments in quadratic variation at different forecasting horizons (one day, and from one week to four weeks). Some of the favorable arguments to RPV are related to higher persistence, predictability and less measurement noise in comparison with RV. Furthermore, comparing the MIDAS regressions with the HAR regressions (involving the same regressors), they find a similar predictive performance. In this paper, our objective is to explore comparative results in the context of multi-period volatility predictions, focusing on the MIDAS approach. To accomplish that, we carry out three numerical applications that differ mostly in relation to the prediction horizons, the forecasting approaches, the regressors and the data we analyze. First, we compare the MIDAS method with two widely used methods of producing multiperiod forecasts of volatility: the direct and the iterated approaches. In particular, the direct and the iterated forecasts of volatility are calculated using a set of GARCH models. The relative performance of the forecasting approaches is investigated in a Monte Carlo study and in a first empirical study, where we compute out-of-sample predictions of volatility for horizons ranging from 5 days to 60 days ahead. In the Monte Carlo study, we evaluate two specifications based on squared returns for each forecasting method. Further, we consider a Data Generating Process (DGP) in which, depending on the assumptions over the innovations, we are able to test the robustness of the methods. The overall findings are quite favorable to the MIDAS 3

4 approach, where we find that the MIDAS predictions are the most accurate at horizons of 15 days ahead and higher. On the other hand, the iterated forecasts are the most precise for shorter horizons (5 and 10 days ahead). In the first empirical study, we consider specifications involving squared returns and (or) absolute returns for each forecasting approach. Our data set consists of daily returns of the S&P 500 and NASDAQ stock indexes, a different data set from the one used in Ghysels et al. (2009). Regarding the results, the iterated predictions dominate in terms of Mean Squared Error (MSE) in almost all forecasting horizons. However, in most cases, the three methods produce forecasts that are not statistically different based on the modified Diebold and Mariano (1995) test of Harvey et al. (1997). Second, we concentrate on relatively shorter horizons to forecast the volatility, more precisely, at horizons of 1 day, and from 5 days to 20 days ahead. In a second empirical study, we compare the MIDAS forecasting regressions with a related approach: the HAR regressions. Additionally, we compute simple forecast combinations of the MIDAS and HAR models. In particular, the combination weights are obtained using linear regression (see Granger and Ramanathan, 1984). For applications in the context of volatility forecasting using forecast combinations, see e.g., Donaldson and Kamstra (1996), Becker and Clements (2008), McAleer and Medeiros (2008), Patton and Sheppard (2009), among others. Given the increasing importance of the Brazilian stock market, we explore empirical results for the Bovespa stock index (Ibovespa). Using intra-daily data (15-minute returns) of the Ibovespa, we calculate a set of daily volatility measures to be used as regressors, such as daily realized variance (RV), daily realized power (RPV), daily squared returns and daily absolute returns. Moreover, we predict future realized variance as well as its square root and log transformations. Summarizing the main results, we find a slight advantage in forecasting performance for the MIDAS models with RP V -based regressors in relation to the HAR models (with the same regressors). Furthermore, we find robust results in favor of the forecast combinations of the MIDAS and HAR models when compared to the individual predictions of the models. The forecast combinations significantly improve the predictions in most cases, indicating a viable way of producing volatility forecasts. Lastly, take notice that all of the reported results are related to the out-of-sample performance of the forecasting approaches. The remainder of the paper is organized as follows: section 2 covers the methodologies that are used in the three numerical applications, focusing on the MIDAS approach; section 3 includes the numerical applications, a Monte Carlo study and two empirical studies; and, finally, section 4 concludes. 2 Methodologies In this section, in order to facilitate the exposition, we will separate the presentation of the two methodologies that are employed in three numerical applications. The first methodology 4

5 covers three fundamentally distinct methods of obtaining multi-period forecasts of volatility: the direct, the iterated and the MIDAS approaches. We will focus on the MIDAS method because it is relatively less explored. The first methodology will be used in a Monte Carlo study and in a first empirical study. The second methodology comprises the MIDAS and HAR regressions as well as forecast combinations of these models. This methodology is applied to multi-period predictions of volatility in a second empirical study. A final subsection covers the way we evaluate the accuracy of the predictions in each study. 2.1 Direct, Iterated and MIDAS Methods To present the first methodology, we partially follow the notation used in Ghysels, Rubia and Valkanov (2009). Firstly, consider D daily returns, d = 1, 2,..., D and multi-period returns, at periods of k days, t = 1, 2,..., T k, where T k = [D/k] and [ ] is the integer operator. For example, in our numerical applications we have D = 7500 daily observations, with which we can calculate non-overlapping returns of 5 days T 5 = 1500, 10 days T 10 = 750, and so on. 1 Secondly, define the daily return as r d = ln P d ln P d 1 and the k-period, non-overlapping, continuously compounded return as Rt k = ln P d ln P d k. Regarding the information sets, the daily returns set at time t is I t = {r t, r t 1, r t 2,..., r 0 } and the set of k-period returns is It k = {Rt k, Rt 1 k k, Rt 2 k k,..., R0 k}. Moreover, I T and IT k denote the information sets based on the entire history of returns, where IT k I T. Thirdly, denote the volatility forecasts according to the method as, V M (i, ii, iii) for MI- DAS, V D (i, ii, iii) for direct and V I (i, ii, iii) for iterated. Also, i is the starting period of the forecast, ii is the forecast horizon and iii represents the related information set. For instance, V I (t, k, I t ) and V M (t, k, I t ) are, respectively, iterated and MIDAS conditional forecasts using daily data to provide k-period ahead forecasts at time t. Lastly, V T (t, k, I t ) denotes the true conditional variance given past daily data Direct Approach The direct method is carried out using the multi-period returns Rt k to directly produce the multi-period prediction of the conditional variance as a single step forecast. The predictions can be computed, for instance, by means of a GARCH(p, q) model. According to Ghysels, Rubia and Valkanov (2009), estimates obtained by V D (T, k, IT k ) would be potentially unbiased, but at the same time, less efficient than the ones obtained by a method that uses the information set I T. In the applications, we use a set of different models of the GARCH family, estimated by MLE 2, for both methods: the direct and the iterated. Furthermore, in the empirical study we 1 To avoid cumbersome notation, we will henceforth drop the subscript from T. 2 We use R codes, especially the package fgarch. 5

6 select the order of the GARCH(p, q) models according to the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and other standard statistics of goodness-of-fit Iterated Approach The iterated method is employed using the daily returns r t to produce the forecasts of volatility V I (T, k, I T ), where r t I T. Therefore, iterated predictions of the daily volatility, k periods ahead, are performed. In this way, assuming that the conditional covariances are zero, V I (T, k, I T ) = k j=1 V D(T + j, 1, I T ). To summarize the presentation of the GARCH models considered in this paper, we use the APARCH(p, q) model specification r t = E(r t I t 1 ) + ε t, ε t = σ t z t, z t D(0, 1) p σt δ = ω + α i ( ε t i γ i ε t i ) δ + i=1 q β j σt j, δ (2.1) where δ > 0, 1 < γ i < 1, and D(0, 1) is the probability density function (pdf) of the iid process z t with zero mean and unit variance. This model was introduced by Ding, Granger, and Engle (1993) and adds the flexibility of a varying exponent with an asymmetry coefficient to take the leverage effect into account. A stationary solution exists if ω > 0, and i α iκ i + j β j < 1, where κ i = E( z + γ i z) δ. Notice that, if γ 0 and (or) δ 2, the κ i depend on the assumptions made on the innovation process. Additionally, if E(r t I t 1 ) 0 we use an ARMA(p, q) model to remove the serial correlation of the returns series. The APARCH model nests the GARCH models that we employ as special cases: ARCH model of Engle (1982) when δ = 2, γ i = 0, and β j = 0. GARCH model of Bollerslev (1986) when δ = 2, and γ i = 0. TS-GARCH model of Taylor (1986) and Schwert (1989) when δ = 1, and γ i = 0. GJR model of Glosten, Jagannathan, and Runkle (1993) when δ = 2. TGARCH model of Zakoian (1994) when δ = 1. The iterated approach improves upon the direct method, regarding efficiency, once we are using daily data to estimate the forecasting model. On the other hand, since the forecasts are iterated and aggregated, small errors related to model misspecification will potentially be amplified (Ghysels et al., 2009). j=1 6

7 2.1.3 MIDAS Approach The last approach, the mixed-data sampling, is to use the daily returns r t to directly calculate a multi-period ahead forecast of the volatility. This technique was introduced by Ghysels et al. (2004). To present the method, consider a MIDAS forecasting regression: where Ṽ k t+1 k j=1 r2 t+j j max 1 Ṽt+1 k = µ k + φ k b k (j, θ)rt j 2 + ε k,t, (2.2) j=0 is a measure of (future) volatility such as realized variance, Ṽ k t+1 = RV t+1 k and b k(j, θ) is a parsimonious weighting function parameterized by only a few parameters, e.g., θ = [θ 1 ; θ 2 ]. In this paper, the parameters µ k, φ k, and θ are estimated by NLS. 3 The regression (2.2) involves data sampled at different frequencies. For example, in our applications, the realized variance (low frequency variable) is measured at horizons from 5 days (one week) to 60 days (three months), whereas the regressors (high frequency variables) are available at daily frequencies. Hence, assuming, e.g., j max = 60 means that we are using the previous 60 daily squared returns to compute a forecast of next week s volatility (if k = 5). According to Ghysels and Valkanov (2010), the choice of j max can be done conservatively (by taking a large value) and letting the weights die out as determined by the parameter estimation. Normally, daily lags longer than about 50 days do not improve (nor harm) the predictions. The weights placed on the daily regressors are estimated in-sample and used to form the out-of-sample predictions. 4 Additionally, following Ghysels et al. (2006), we also estimate the equation (2.2), in the empirical study, using daily absolute returns r t j as regressors. One of the most important features in a MIDAS regression is the weighting function. The parametric restriction imposed to the lag coefficients b k (j, θ) circumvents the proliferation of parameters associated with the daily regressors. In this way, we consider one of the most used and flexible parameterizations, named the Beta function: b k (j, θ 1, θ 2 ) = j f( j, θ max 1 ; θ 2 ) j max j=1 f( j j, θ max 1 ; θ 2 ), (2.3) where f(z, a, b) = z a 1 (1 z) b 1 /β(a, b) and β(a, b) is based on the Gamma function, β(a, b) = Γ(a)Γ(b)/Γ(a + b). This scheme guarantees that the weights are positive, ensuring that the forecasted volatility is also positive. Furthermore, the weights add up to one, which allows the identification of the scale parameter φ k. The functional form in equation (2.3) can provide a 3 We use R codes to estimate MIDAS regressions. 4 For more details, see Ghysels, Santa-Clara and Valkanov (2005, 2006), Ghysels, Rubia and Valkanov (2009) and Andreou, Ghysels and Kourtellos (2010). 7

8 wide variety of shapes for different values of θ 1 and θ 2. For instance, for θ 1 = 1 and θ 2 > 1 one has a slowly decaying pattern typical of volatility filters. We also employ this restricted Beta function (i.e., with θ 1 = 1), leaving only one parameter to determine the shape. The speed with which the weights decay controls the effective number of daily regressors used to estimate the volatility. The specification (2.3) was introduced in Ghysels, Santa-Clara, and Valkanov (2004) and further explored in Ghysels, Sinko, and Valkanov (2007). Finally, MIDAS predictions will be denoted by V M (T, k, I T ). 2.2 MIDAS and HAR Regressions To describe the methodology used in the second empirical study, we begin with the definition of the daily return as r t,t 1 = ln P t ln P t 1, where the time index t refers to daily sampling. The intra-daily return is then denoted by r (m) t,j = ln P t,j/m ln P t,(j 1)/m, where m is the number of observations within a trading day. For instance, in our study, we have prices observed every 15 minutes of the Bovespa stock index and this corresponds to m = 28. Our objective is to predict a measure of volatility over a future horizon of k days, V t+k,t. As a main measure of volatility for the period t to t + k, we consider the increments in the quadratic variation of the return process. The quadratic variation is not directly observed but can be measured with some discretization error. 5 One such measure would be the sum of intra-daily squared returns, named the Realized Variance (RV), RV (m) t+k,t = km j=1 (r (m) t,j )2. (2.4) Another class of empirical processes related to volatility is Realized Power Variation (RPV), introduced by Barndorff-Nielsen and Shephard (2003, 2004), which can be defined for our purposes 6 as RP V (m) m t+1,t = r (m) t,j, (2.5) for example, over one day (k = 1). Recent studies in the context of MIDAS regressions as, for instance, Ghysels et al. (2006), Forsberg and Ghysels (2007) and Chung et al. (2008) have shown robust results in favor of RPV as a predictor of future volatility (measured by increments in quadratic variation). j=1 Some of the favorable arguments to RPV are related to higher persistence, predictability and less measurement noise in comparison with RV. To simplify the notation, we will henceforth drop the superscript m. 5 For a review on the subject, see for example, Andersen et al. (2001), Barndorff-Nielsen and Shephard (2002a,b), Andersen et al. (2007), Forsberg and Ghysels (2007), among others. 6 Note that the asymptotic theory establishes (when m ) the need of a m normalization, as shown by Barndorff-Nielsen and Shephard (2003, 2004). We do not deal with asymptotic analysis in the context of this paper. 8

9 The multi-period realized variances will be constructed the same way as in Andersen et al. (2007) and Forsberg and Ghysels (2007). The multi-period measures will be defined by the normalized sum of the one-period realized variances, as follows: RV t+k,t = k 1 (RV t+1,t + RV t+2,t RV t+k,t+k 1 ), (2.6) for k = 5, 10, 15 and 20, denoting realized variances from one week to four weeks, respectively. Additionally, in contrast to Section 2.1, the multi-period measures are no longer computed without overlap. To predict the realized variance over a future horizon of k days, we first consider the MIDAS-RV model: jmax 1 RV t+k,t = µ k + φ k j=0 b k (j, θ 1, θ 2 )X t j,t j 1 + ε t+k, (2.7) where X t j,t j 1 denotes a daily regressor (calculated using 15-minute returns) from t j 1 to t j. In MIDAS regressions the selection of X t j,t j 1 corresponds to choosing the best predictor of future quadratic variation from various possible measures of past fluctuations in returns (Ghysels and Valkanov, 2010). Hence, we explore the forecasting ability of the following daily predictors: RV t,t 1, RP V t,t 1, RP V 2 t,t 1, r2 t,t 1 and r t,t 1. The use of RP V 2 t,t 1 is suggested in Forsberg and Ghysels (2007) to be on the same scale as RV t,t 1. Also, we employ the restricted Beta function 7 (i.e., with θ 1 = 1) to parameterize the lag coefficients b k (j, θ) (weights), with j max = Besides predicting RV, as in equation (2.7), we also forecast RV in standard deviation and log form. Henceforth, to facilitate, a specific regression will be denoted by its dependent variable and regressor as, for example, MIDAS-RV-RV t,t 1. 9 This way, we have MIDAS-RV 1/2 -X, with X = RV 1/2 t,t 1, RP V t,t 1, and r t,t 1 as well as MIDAS-ln RV -X, with X = ln RV t,t 1, ln RP V t,t 1, and ln r t,t 1. The second type of regressions that we undertake is based on the Heterogeneous Autoregressive (HAR-RV) model of Corsi (2009), which is an extension of the Heterogeneous ARCH (HARCH) model of Müller et al. (1997). 10 and Ghysels (2007), the HAR-RV model can be written as Following Andersen et al. (2007) and Forsberg RV t+k,t = β 0 + β D RV t,t 1 + β W RV t,t 5 + β M RV t,t 20 + ε t+k, (2.8) 7 Presented in Section In fact, we use 60 daily lags for all applications with MIDAS models (other lag lengths were tested, yielding basically the same results). 9 We adopt the same simplified notation used in Forsberg and Ghysels (2007). 10 The motivation behind the HARCH model, and consequently behind the HAR-RV model, is based on the existence of different types of market participants with distinct investment horizons (Heterogeneous Market Hypothesis). 9

10 where the explanatory variables are basically the lagged realized variance over a day, a week, and four weeks (i.e., over different interval sizes). 11 Besides being simple and parsimonious, the model is capable of capturing the long memory behavior of the realized variance (for more details, see Corsi, 2009). Additionally, the HAR-RV model can be seen as a MIDAS-RV regression with step-functions (one of the possible weighting functions for MIDAS models); see, e.g., Ghysels, Sinko, and Valkanov (2007) and Ghysels and Valkanov (2010). In our empirical study, we analyze different regressors and transformations of the dependent variable RV t+k,t for the HAR-RV model. Therefore, we consider HAR-RV-X, with X = RV, RP V and RP V 2, also HAR-RV 1/2 -X, with X = RV 1/2 and RP V as well as HAR-ln RV -X, with X = ln RV and ln RP V. Finally, the measures RP V t,t 5 and RP V t,t 20 are computed similarly to equation (2.6). As a final investigation, we also take into account simple forecast combinations of MIDAS and HAR regressions. The idea of combining different forecasts of the same quantity, with the objective of improving predictive accuracy, was first proposed by Bates and Granger (1969). In particular, suppose we have a pair of multi-period forecasts f (1) (2) t+k,t and f t+k,t representing, for instance, MIDAS-RV-RV t,t 1 and HAR-RV-RV predictions of RV t+k,t, respectively. Using the method proposed in Granger and Ramanathan (1984), we can combine the pair of predictions as follows f (c) t+k,t = β 0 + β 1 f (1) t+k,t + β 2f (2) t+k,t, (2.9) where the weights (β 0, β 1, β 2 ) are obtained from the standard least squares regression, RV t+k,t = β 0 + β 1 f (1) t+k,t + β 2f (2) t+k,t + ε t+k. (2.10) This general formulation, including an intercept and with unconstrained weights, allows for the possibility of biased forecasts (for reviews on the subject, see Clemen, 1989; Diebold and Lopez, 1996; Hendry and Clements, 2004; Timmermann, 2006; among others). applications in volatility forecasting using forecast combinations, see for instance, Donaldson and Kamstra (1996), Becker and Clements (2008), McAleer and Medeiros (2008), and Patton and Sheppard (2009). In our study, we only combine individual forecasts obtained by the same regressors. Hence, using our previous notation, we will consider forecast combinations of MIDAS+HAR-RV-X, with X = RV, RP V and RP V 2, also MIDAS+HAR-RV 1/2 -X, with X = RV 1/2 and RP V as well as MIDAS+HAR-ln RV -X, with X = ln RV, and ln RP V. 11 Originally, the HAR-RV model is specified using the lagged RV over a month (i.e., RV t,t 22). To carry out comparisons between MIDAS and HAR regressions, we use four weeks (i.e., k = 20) as in Forsberg and Ghysels (2007). Also, the parameters of the HAR-RV model are estimated by OLS. For 10

11 2.3 Comparing the Predictions We discuss here how the predictions are going to be compared according to the methodology used in the different applications. First, we consider the evaluation measures used in the Monte Carlo study and in the first empirical study. Next, the evaluation measures related to the second empirical study are presented Evaluation Measures: Direct, Iterated and MIDAS Methods In the Monte Carlo study we compare the forecasts V D (T, k, I k T ), V I(T, k, I T ) and V M (T, k, I T ) with the true multi-period volatility V T (T, k, I T ), specified by the Data Generating Process (DGP). Particularly, we compute V T (T, k, I T ) = k j=1 V T (T +j, 1, I T ), which is an aggregation of the true one-period volatilities. However, in the real data analysis, we have to use a proxy for V T (T, k, I T ) since the true k-period volatility is unobservable. Andersen and Bollerslev (1998), Andersen et al. (2001), amongst many others, advocate the use of realized variance RVT k +1, instead of squared returns, because of its better statistical properties. Nonetheless, realized variance is calculated using intra-daily returns. Since we do not have intra-daily data for our sample period, we compute the multi-period measures with daily returns. As a result, our estimates of RV k T +1 will be a noisy proxy of V T (T, k, I T ), especially for shorter horizons of k days. To evaluate the out-of-sample predictions, we make use of two loss functions. This way, let us define the out-of-sample forecast error, for instance, for the MIDAS approach as where k denotes the forecasting horizon. e k M,T +1 V T (T, k, I T ) V M (T, k, I T ), (2.11) We have defined a robust loss function for the Monte Carlo simulations to deal with the occurrence of outliers. Given t = T 1,..., T n final observations (forecasting sample) of each generated series, we compute the Median Absolute Error (MdAE) as MdAE k M = median( e k M,t ). (2.12) On the other hand, using RVT k +1 in the empirical study, we calculate the feasible out-ofsample forecast error as and the Mean Squared Error (MSE) at the k-horizon u k M,T +1 = RV k T +1 V M (T, k, I T ), (2.13) MSE k M = mean[(u k M,t) 2 ]. (2.14) The use of MSE for the empirical study is related to the work of Patton (2011), who classifies the measure as belonging to a class of loss functions that are robust to the presence of noise in 11

12 the volatility proxy; thus, producing a consistent ranking of the forecasts even in the absence of the true volatility. In addition to the direct comparison of the MSEs of the competing methods, we apply the modified Diebold and Mariano (1995) test of Harvey et al. (1997) to the series of predictions. The test of equal forecast accuracy for two competing forecasting methods, f 1 and f 2, can be summarized by the following null hypothesis: H 0 : E[(u k f 1,t )2 (u k f 2,t )2 ] = 0. (2.15) Evaluation Measures: MIDAS and HAR Regressions In the second empirical study, we evaluate multi-period predictions of the realized variance (and its transformations). Further, we analyze the forecasting ability of a set of regressors for MIDAS and HAR regressions as well as forecast combinations of the models. To carry out comparisons, we calculate the MSE for all predictions and apply the modified Diebold- Mariano test of equal forecast accuracy. Firstly, let us denote RV t+k,t as the actual value of the realized variance for the horizon of k days. Also, let RV t+k,t denote the out-of-sample prediction of the RV. Hence, the MSE is given by MSE = N 1 N i=1 (RV i+k,i RV i+k,i ) 2, (2.16) where N is the number of out-of-sample predictions. Secondly, when modeling the transformed measures of RV t+k,t, i.e., RV 1/2 t+k,t and ln RV t+k,t, we transform the predicted values of the dependent variables to realized variance before calculating the MSE. This allows us to compare the regressions with distinct dependent variables. In particular, when we model the realized standard deviation, RV 1/2 t+k,t, we undo the transformation of the predicted value as follows MSE = N 1 N i=1 ( ( ) 1/2 2 2 RV i+k,i RV i+k,i). (2.17) Finally, when modeling the log of the realized variance, ln RV t+k,t, we undo the transformation of the predicted value as follows MSE = N 1 N i=1 ( RV i+k,i exp(ln RV i+k,i )) 2. (2.18) Note, as pointed out by Forsberg and Ghysels (2007), that realized variance forecasts obtained by undoing the transformations are not going to be unbiased (see also Granger and Newbold, 1976). This way, we only compare regressions with different response variables through MSEs. Using the modified Diebold-Mariano test, we compare the predictions obtained by al- 12

13 ternative regressors against the ones generated by realized variance (benchmark regressor). Moreover, we explore differences in forecast accuracy for MIDAS models and MIDAS+HAR combinations against the HAR model (benchmark model) using the same regressors. 3 Numerical Applications We carry out three studies in which the objective is to predict the volatility over a future horizon of k days. In particular, all the forecasts are out-of-sample. Firstly, we compare the multi-period predictions of volatility produced by the MIDAS, the direct and the iterated methods for horizons from 5 to 60 days ahead. We begin with an investigation of the forecasting performance of the aforementioned methods in a Monte Carlo study. Next, in a first empirical study, we analyze the same distinct approaches using the S&P 500 and NAS- DAQ daily returns series. Secondly, we focus on relatively shorter horizons to predict the volatility, where we consider the horizons of 1 day, and from 1 week to 4 weeks ahead. In a second empirical study, we compare the MIDAS regressions with the HAR regressions and calculate simple forecast combinations of these models. Using 15-minute intra-day returns of the Bovespa index, we compute a set of daily volatility measures to be used as regressors. Lastly, besides predicting the level of volatility, we also forecast specifications of the log and the square root of the volatility. 3.1 Monte Carlo Study: Direct, Iterated and MIDAS Methods In this section, we simulate observations from an ARCH(2) model, defined as follows: r t = σ t z t, σ 2 t = r 2 t r 2 t 2, (3.1) where z t N(0, 1) for one set of simulations, and z t t 6 for the other set. For both distributions of z t, we replicate 1000 series and each one has 7500 observations. Also, for the estimation procedures, we consider the conditional mean as known and equal to zero. We compute, for each series, non-overlapping k-period returns R k t, and realized variances RV k t for the horizons k = 5, 10, 15, 20, 30 and 60. Recall that the daily returns r t are used to produce the iterated GARCH and MIDAS forecasts, whereas the returns R k t are used in the direct GARCH approach. Regarding the models, we compare the predictions of six models based on squared returns. Two MIDAS regressions, one using the unrestricted Beta function (V M1 ) and the other based on the restricted Beta function (V M2 ). Two direct GARCH models: a GARCH(1,1), (V D1 ) and a GJR(1,1), (V D2 ). Finally, an iterated GARCH(1,1), (V I1 ) and an iterated GJR(1,1), (V I2 ). In particular, the first 6000 observations are used to estimate the parameters of the 13

14 models. We left 1500 final observations to compute the out-of-sample forecasts. Moreover, we re-estimate the models after each k-period ahead prediction. Figure 1 shows the box plots of the Median Absolute Errors (MdAEs) of the six models using three forecasting methods. In addition, Table 1 presents some statistics of these simulations for Gaussian errors. Considering the results, the MIDAS regressions present the best performance, in terms of bias, for longer prediction horizons (k 15), with slightly better results for the V M2 forecasts. In contrast, the iterated GARCH models are superior to the others for relatively short horizons (5 and 10 days ahead). Also, the iterated method shows less variability for the out-of-sample predictions in relation to the MIDAS method. Lastly, the direct GARCH models exhibit the worst performance. Specifying a distribution with heavier tails than the Normal for z t, the results are even better for the MIDAS models. Figure 2 and Table 2 include the results of the simulations for Student-t errors. In terms of bias, the MIDAS models provide the most accurate predictions in almost all horizons (k 10). Besides that, the models present less variability in their predictions for longer horizons, especially the MIDAS model specified with the restricted Beta function (V M2 forecasts). The iterated GARCH models are the best ones, considering the bias, for the first horizon (k = 5); however, they are the worst ones for longer horizons (k 20). Additionally, the direct GARCH method presents more variability in its forecasts when compared to the other methods, for all horizons. 3.2 Empirical Study 1: Direct, Iterated and MIDAS Methods In this empirical study, we have 7500 daily returns of the S&P 500 and NASDAQ stock indexes from August 31, 1981 to May 20, A few descriptive statistics of the daily returns are provided in Table 3. Similarly to the Monte Carlo experiment, we use the first 6000 observations to estimate the parameters of the models. This will generate [1500/k] outof-sample predictions for each forecasting horizon of k days. Furthermore, we estimate four MIDAS regressions using squared returns (V M1s and V M2s ) and absolute returns (V M1a and V M2a ) as regressors. In these regressions, the unrestricted Beta function is used for (V M1s, V M1a ), whereas the restricted version of the Beta function is used for (V M2s, V M2a ). For the direct and iterated approaches, we always estimate a GARCH(1,1) model as benchmark (V D1 and V I1 ), and another specification of the GARCH family (GJR, TS-GARCH or TGARCH) for V D2 and V I2. The GARCH type models are selected by information criteria and other usual goodness-of-fit measures. 13 Finally, note that the model for V D2 forecasts may be distinct for each forecasting horizon, given that the time series of k-period returns R k t changes according to each k. 12 The data were obtained from Yahoo! Finance. 13 Also, we specify two distributions for the errors, the Normal and the Student-t. 14

15 Firstly, for the S&P 500 series, the iterated forecasts V I2 are calculated using a GJR(1,1) model with Student-t distribution for the errors. 14 In particular, for both iterated models we specify for the conditional mean equation an ARMA(1,2) model. Additionally, the direct predictions V D2 are computed by means of a GJR(1,1) model for (k 20) and using a TGARCH(1,1) model for k = 30, 60. These models are estimated with Gaussian errors and, for the first prediction horizon (k = 5), we use an AR(1) model for the mean equation. The comparative results of the multi-period forecasts are presented in Table 4. The main evidence is that the iterated predictions V I2 present the smallest Mean Squared Errors (MSEs) for all horizons. The MIDAS method is the second best method for the horizons k = 5, 10, considering the V M1s and V M2s predictions. In contrast, the direct method is the second best for longer horizons (k 15), more precisely, the V D2 forecasts. Besides that, using the iterated predictions V I2 as the benchmark, we apply the modified Diebold-Mariano test to find out significant differences in forecast accuracy. Based on the Diebold-Mariano test, the forecasts are not statistically different in almost all cases. The exceptions are the direct predictions V D1, for the horizons k = 5, 10, that have inferior performance in relation to the benchmark. Secondly, for the NASDAQ series, the iterated predictions V I1 are computed using an ARMA(1,2)-GARCH(1,1) model with Gaussian distribution for the errors, whereas the V I2 forecasts are calculated by means of an ARMA(1,2)-GJR(1,1) model with Student-t errors. Furthermore, the direct forecasts V D2 are obtained by an AR(1)-GJR(1,1) model for the horizons (k = 5, 10) and by means of an AR(1)-TGARCH(1,1) model for k = 15. For longer horizons (k 20), they are calculated using a TS-GARCH(1,1) model. Also, these models are estimated with Student-t errors for (k 15) and with Gaussian errors for (k 20). Lastly, the GARCH(1,1) model used to obtain the V D1 predictions is estimated with Gaussian errors and, for the horizons (k 15), we include an AR(1) model for the mean equation. Table 5 includes the out-of-sample forecasting results for the NASDAQ series. Once again, the iterated forecasts V I2 present the smallest MSEs, except for the horizon of ten days ahead in which the direct forecasts V D2 are the most accurate. Regarding the methods, the MIDAS method is the second best for all horizons, excluding k = 10. Also, the direct method has the worst performance. Finally, applying the modified Diebold-Mariano test, there is a significant difference in performance only for the horizon of five days ahead, in which the direct predictions V D1 are less precise than the iterated forecasts V I2 (the benchmark). 3.3 Empirical Study 2: MIDAS and HAR Regressions In this section, we use MIDAS and HAR regressions to predict the volatility in the Brazilian stock market. In addition, we compute simple forecast combinations of the MIDAS and HAR models. Our data set consists of 15-minute intra-day returns of the Bovespa stock index 14 The GARCH(1,1) model (V I1 predictions) is also estimated with Student-t errors. 15

16 (Ibovespa) from May 14, 2004 to October 08, The entire sample contains 1330 trading days with 28 observations per day for a total of 37, minute returns (trading days of 7 hours). From this sample, we calculate daily realized variance (RV), daily realized power (RPV), daily squared returns and daily absolute returns. 16 A few descriptive statistics for the daily volatility measures of the Ibovespa are reported in Table 6. The objective here is to predict the future volatility for horizons of k days. In particular, we consider the out-of-sample predictions for horizons of one day, one week, two, three and four weeks, i.e., for k = 1, 5, 10, 15 and 20. These horizons are quite relevant for risk management and option pricing routines. We use the first 1110 daily observations to estimate the parameters of the models and the remaining 220 daily observations to compute the outof-sample forecasts. Also, the models are re-estimated after each k-period ahead prediction. In addition, the number of out-of-sample forecasts will be different across forecast horizons. We have 220, 216, 211, 206, 201 predictions for the horizons of k = 1, 5, 10, 15 and 20 days, respectively. Finally, we predict the realized variance RV t+k,t, and the RV in standard deviation and log form. To simplify the exposition of our main findings, we report the results according to the response variable in the subsections below Predicting RV We begin with the results for the out-of-sample forecasts of the realized variance RV t+k,t. In Table 7 we note, for the MIDAS RV regressions, that RP Vt,t 1 2 as regressor produces the smallest Mean Squared Errors (MSEs) for the horizons of one day, three and four weeks. In particular, the transformation of RPV to the same scale as RV is necessary and improves the forecasting ability of the predictor. The second best predictor is RV t,t 1. However, the forecasts of the MIDAS models that use RP Vt,t 1 2 and RV t,t 1 as regressors are not statistically different according to the modified Diebold-Mariano (DM) test. The MIDAS forecasts produced by the remaining regressors are significantly worse than the ones generated by RV t,t 1. Considering the HAR RV regressions in Table 8, the use of RV as regressor produces the best predictions in MSE terms. Once again, forecasts that involve RV and RP V 2 as regressors are not statistically different according to the modified DM test. For the MIDAS+HAR RV forecast combinations in Table 9, the best results in terms of MSE come from combining individual forecasts considering RP V as the regressor. Note that the intercept term included in the combination scheme, see equation (2.10), corrects the bias of the forecasts related to the difference in scale between RPV and RV. The combined predictions involving RP V as 15 The data were obtained from CMA trade solution. We thank Osvaldo C. Silva Filho for kindly providing the data. 16 As in Ghysels et al. (2006) and Forsberg and Ghysels (2007), our daily volatility measures are not adjusted for microstructure noise. In this regard, see also Ghysels and Sinko (2006) and Ghysels et al. (2007). 16

17 regressor are significantly better than the ones based on RV as regressor for the horizons of three and four weeks. Moreover, as can be seen from the tables, almost all MSEs decrease for all regressors as the forecasting horizon increases. As pointed out by Forsberg and Ghysels (2007), this is an indication that RV computed over longer horizons are easier to forecast as they are smoother series. 17 Lastly, besides investigating the best predictors of future volatility, we are able to evaluate differences in forecast accuracy between the MIDAS RV and HAR RV regressions as well as between the MIDAS+HAR RV forecast combinations and the HAR RV models. The comparative results using the modified DM test are shown in Table 10. First, when comparing the MIDAS RV regressions against the HAR RV regressions, we do not observe differences in forecast accuracy for predictions involving RV as regressor. However, for predictions obtained by using RP V 2 as regressor, the MIDAS RV RP V 2 model provides the most accurate forecasts for two, three and four weeks ahead. Second, the MIDAS+HAR RV forecast combinations generate more precise predictions than the HAR RV regressions for all regressors and almost all horizons Predicting RV 1/2 We also have results for the out-of-sample predictions of the realized variance in standard deviation form RV 1/2 t+k,t. From Table 7 we observe, for the MIDAS RV 1/2 regressions, that RV 1/2 t,t 1 used as regressor is the best predictor considering the MSEs, and daily realized power RP V t,t 1 is the second option. Nevertheless, the forecasts produced by using these regressors are not statistically different. In addition, from Table 8 we see that the best results for the HAR RV 1/2 regressions, in terms of MSE, are obtained using RV 1/2 as the regressor. However, the HAR RV 1/2 forecasts involving RV 1/2 and RP V as regressors are significantly distinct only for the horizon of 10 days. Furthermore, comparing the MIDAS RV 1/2 and HAR RV 1/2 forecasts with the MIDAS RV and HAR RV predictions in terms of MSE, involving the same regressors, we note that the square root transformation is generally beneficial. The results for the MIDAS+HAR RV 1/2 forecast combinations are reported in Table 9, where we can see that the smallest MSEs are related to combined predictions involving RP V as the regressor. In particular, these combined forecasts are statistically different from the ones considering RV 1/2 as predictor for horizons of three and four weeks ahead. Additionally, in Table 10 we compare the forecasting results of the MIDAS RV 1/2 regressions against the HAR RV 1/2 models using the modified DM test. Firstly, the forecasts obtained by using RV 1/2 as regressor are not statistically different. Secondly, the MIDAS-RV 1/2 RP V model 17 Recall that the dependent variable, in this study, is the multi-period normalized realized variance, i.e., RV t+k,t /k, for all models. See equation (2.6). 17

18 provides the most precise forecasts at horizons of two, three and four weeks. Besides that, from Table 10 we also note that the MIDAS+HAR RV 1/2 forecast combinations are significantly better than the HAR RV 1/2 predictions for both the regressors and for most horizons Predicting ln RV Lastly, we report the results for the out-of-sample forecasts of the realized variance in log form, ln RV t+k,t. The results for the MIDAS ln RV regressions are shown in Table 7, where we observe that ln RP V t,t 1 used as regressor generates the best predictions in terms of MSE for most horizons. The forecasts produced by ln RP V t,t 1 and ln RV t,t 1 as regressors are statistically different at horizons of three and four weeks, where the former predictor is the best one. In addition, we note from Table 8 for the HAR ln RV models that ln RP V is the preferred predictor considering the MSEs. Nonetheless, the forecasts obtained by ln RP V and ln RV as regressors are not statistically different. Moreover, comparing the MSEs of the MIDAS ln RV and HAR ln RV predictions with the MSEs of the MIDAS RV and HAR RV forecasts, taking the same regressors, we see that the log transformation is beneficial for most horizons. Regarding the MIDAS+HAR ln RV forecast combinations in Table 9, the combined predictions that involve ln RP V as regressor present the smallest MSEs. Particularly, these combined forecasts are significantly more accurate than the ones involving ln RV as regressor for the horizons of three and four weeks ahead. Furthermore, we investigate differences in forecast accuracy between the MIDAS ln RV regressions and the HAR ln RV models using the modified DM test in Table 10. Firstly, we observe that the HAR ln RV ln RV forecasts significantly outperform the MIDAS ln RV ln RV predictions at horizons of one, two and three weeks. Secondly, the HAR ln RV ln RP V model produces the best predictions for the horizon of one week. In addition, we also compare in Table 10 the MIDAS+HAR ln RV forecast combinations and the HAR ln RV models, where we find that the forecasts based on ln RV as regressor are not statistically different. In contrast, predictions involving ln RP V as regressor are statistically different for the horizons of two and four weeks. The MIDAS+HAR ln RV ln RP V combination provides the best forecasts for these horizons Summary of the Main Findings We start with the best predictors in terms of MSE to forecast the multi-period volatility measures: RV t+k,t, RV 1/2 t+k,t and ln RV t+k,t. First, for the MIDAS regressions the best forecasting results involve RP V -based regressors in most cases. Second, for the HAR regressions the RV - based regressors produce the best predictions in most cases. Finally, the best outcomes for the MIDAS+HAR forecast combinations come from combining individual forecasts considering RP V as regressor. 18