Volatility forecasting using high frequency data: The role of after-hours information and leverage effects

Resources Policy 54 (2017) 58–70

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Volatility forecasting using high frequency data: The role of after-hours information and leverage effects

MARK

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Xuehong Zhua,b, Hongwei Zhanga,b, , Meirui Zhonga,b a b

School of Business, Central South University, Changsha 410083, China Institute of Metal Resources Strategy, Central South University, Changsha 410083, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Non-ferrous metals futures Volatility forecasting After-hours information Leverage effects Volatility clustering

This investigation extends the HAR model to include the role of after-hours information and leverage effects to forecast daily volatility of the Chinese non-ferrous metals futures market. Furthermore, volatility clustering in the residuals of the volatility model is investigated. In addition to the usual leverage effects, the findings indicated new insights into additional leverage effects, which are captured by negative overnight returns and negative lunch-break returns. Moreover, after-hours information has a highly in-sample explanatory and there is no risk–return trade-off in the Chinese non-ferrous metals futures market. One-step ahead forecasts are investigated and the results indicated that the introduction of after-hours information and leverage effects in the HAR model exhibit better predictive power. Finally, the results are robust for various sampling frequencies. Our findings have important significance for investors and policy makers and will elucidate further research directions.

1. Introduction With the development of China's economy in recent years, the demand for non-ferrous metal commodities, such as copper and aluminum, has greatly increased, as they are significant sources of raw materials for industry. Correspondingly, the price of these commodities has had a large impact on the extraction, processing, and manufacturing sectors. However, the increase of uncertainty factors, such as the exchange rate, import and export policies, and the existence of speculators, has lead to great fluctuations in the price of non-ferrous metals. This makes volatility modeling and forecasting of the Chinese non-ferrous metals futures market very significant, as it can assist investors in making portfolio allocation decisions and affect value-at-risk management decisions made by financial traders. Most literatures on forecasting the volatility of metal futures market applied GARCH-type models which were sampled at daily to monthly frequencies (Arouri et al., 2013; Li and Li, 2015; Watkins and Mcaleer, 2008). Other volatility forecasting models commonly used to forecast the future volatility of the financial markets are stochastic volatility (SV)-type models (Baum and Zerilli, 2016; Larsson and Nossman, 2011), autoregressive (AR) models (Sadorsky and Mckenzie, 2008), ARMA models (Xu and Ouenniche, 2012), and the jump-diffusion models (Askari and Krichene, 2008). However, all these models used low-frequency transaction data and can not accurately measure the

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whole-day volatility information. Therefore, they all have their own defects. Using the high-frequency data, Andersen and Bollerslev (1998) first proposed the realized volatility (RV) as the more accurate measure of the integrated variances. Andersen et al. (2003), Andersen et al. (2004) found that simple reduced form models with realized volatility (RV) is significantly better than the popular GARCH and stochastic volatility (SV) models on forecasting future volatility in the financial markets. Corsi (2009) developed the heterogeneous autoregressive (HAR) model of realized volatility and found that the predictive power of HAR model strongly outperforms the GARCH and the ARFIMA-RV models. Therefore, HAR models are receiving more attention from investors. In a majority of the literature, these models used intra-day data that had been observed during normal trading hours. However, the overnight return data collected during break hours is also very important for volatility estimation and forecasting and this information directly affects investor risk. In addition, Wang et al. (2015) demonstrated that lunch-break information may have a large long-run impact on the volatility of the Chinese non-ferrous metals futures market. Therefore, this investigation contributes to the study of this issue by considering the role of after-hours information. In addition, the volatility clustering inherent in the residuals of the HAR models, which is considered in the work of Corsi et al. (2008), Todorova (2015), and Ciarreta and Zarraga (2016), is investigated. We test whether modeling the conditional heteroskedasticity of the realized volatility innovations

Corresponding author at: School of Business, Central South University, Changsha 410083, China. E-mail addresses: [email protected] (X. Zhu), [email protected] (H. Zhang).

http://dx.doi.org/10.1016/j.resourpol.2017.09.006 Received 12 February 2016; Received in revised form 25 August 2017; Accepted 11 September 2017 0301-4207/ © 2017 Elsevier Ltd. All rights reserved.

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forecasting, considering the most recent history of the non-ferrous metals futures market. This investigation also considers the role of after-hours information on volatility forecasting by introducing (negative) overnight returns and (negative) lunch-break returns. Then we test whether after-hours information during non-trading hours has a significant impact on future volatility in the Chinese non-ferrous metals futures market. Moreover additional leverage effects, which are captured by the negative lunchbreak returns and negative overnight returns, are investigated, in addition to the traditional leverage effects. Following Corsi et al. (2008) and Todorova (2015), we account for the volatility of realized volatility by including a GARCH specification. This innovation considers the fact that the residuals of the HAR model reveal significant conditional heteroskedasticity and volatility clustering. Although the HAR model is broadly used in stock and energy markets, research that applies the model to forecast industrial metal volatility is relatively scarce. Most volatility forecasts for metal markets apply GARCH-type models and are based on daily or monthly frequencies. The most recent study by Li and Li (2015) applied GARCH models to forecast copper futures volatility under model uncertainty. Bentes (2015) employed three volatility models of the GARCH family to examine the volatility behavior of gold returns. In contrast, we employ the HAR model, based on intraday data, to forecast the volatility of copper and aluminum futures, which is much more accurate than the GARCH model based solely on daily or monthly frequencies (Wen et al., 2016). The HAR model, first proposed by Corsi (2004), can capture the long-memory feature of realized volatility. The long-memory pattern is captured by aggregating the volatility over different periods; that is, daily for short-term traders, weekly for medium-term, and monthly for long-term. The rest of this investigation is arranged as follows: Section 2 provides the volatility estimation, the data, and the summary statistics. Section 3 describes the volatility models. In Section 4, the results of insample analysis of the extended HAR-LA and HAR-LA-GARCH models are given. Section 5 evaluates the out-of-sample forecasting performances of extended HAR models. In Section 6, we present the robustness test. Finally, the conclusions are discussed in Section 7.

can improve forecasting performance. The focus of this study is to investigate the role of after-hours information, volatility of realized volatility and asymmetry in the non-ferrous metals futures market. Our research is motivated by the work of Wang et al. (2015), who extended the HAR-RV model to enable it to forecast volatility by considering the role of lunch-break returns, overnight returns, trading volume, and leverage effects. However, the paper did not account for the volatility of realized volatility, which was considered in Corsi et al. (2008), and this aspect is considered to be one of the innovations in this investigation. Therefore, we extend the HAR model by incorporating a GARCH specification to forecast the volatility of Chinese non-ferrous metals futures. This investigation is also related to the recent work of Todorova (2015), who captured the dynamics of realized volatility and leverage effects and constructed the HAR-L-GARCH model to analyze the volatility of the London Metals Exchange (LME) non-ferrous metal market. However, in contrast to the present work, Todorova (2015) made no attempt to consider the impact of after-hours information and the additional asymmetry effects with respect to overnight returns and lunch-break returns. Previous studies focused primarily on mature futures markets, such as the London Metals Exchange (LME) and the Chicago Mercantile Exchange (CME), rather than emerging markets. In contrast, this investigation explores the Shanghai Futures Exchange (SHFE), which represents a typical Chinese commodity futures market and has been the most dramatically expanding market in the world during the past three decades. In fact, the trading volume of the SHFE in 2013 was 624 million hands and the SHFE rose to the top spot once again in terms of trading volume after three years, according to the volume ranking list developed by the global commodity futures exchange in 2013. The 2013 vol of the SHFE increased by 80% from the previous year. According to data from the World Federation of Exchanges (WFE), the 2015 trading volume growth rate of China's commodities exchanges was the first in the global market. The volume of commodity contract trades in the SHFE soared to more than 1 billion copies and increased by 25% in 2015. The trading volume of the CME in 2015, by contrast, grew by only 17%, while the LME trading volume fell by 4%. This paper focuses on copper and aluminum futures traded on the Shanghai Futures Exchange (SHFE) in China, because they are the most actively traded commodity futures in China. For example, in 2013, the trading volume of copper futures on the SHFE was 33,460 billion RMB Yuan (approximately US $5577 billion), which accounted for 27.69% of the total futures trading volume on the SHFE, and was 3 times the trading volume of copper futures on the LME (Liu and An, 2014).1 The volume growth in the China Commodity Exchange in recent years has dwarfed other competitors such as LME and CME. Furthermore, the SHFE copper futures price was lifted to the status of one of the authoritative quotes provided by the three major pricing centers of the global copper market. However, Chinese futures markets are still relatively immature, less information efficiency, more volatile, and less liquid than mature futures markets (Liu and An, 2014). Compared with the LME or the CME in US and European markets, the SHFE displays strong regional characteristics and has significant structural and institutional distinctions and, therefore, exhibits unique volatility and risk characteristics. Thus, from an empirical perspective, the SHFE is an interesting case for research. This investigation makes several contributions to the existing literature. First, in contrast to stock and energy markets, volatility forecasting in non-ferrous metals futures is considerably limited. Although non-ferrous metal commodities play a very significant role in national economies, there exists only 45 refereed publications from 1980 to 2002 concerning the price of industrial metals (Watkins and Mcaleer, 2004). Moreover, this investigation covers the period from July 1, 2010 to December 1, 2015, and hence may be more significant for volatility

1

2. Volatility estimation and data An appropriate volatility measure series is important for high-frequency volatility forecasting. Realized volatility is widely used as an appropriate realized measure in many papers. However, realized volatility only reflects volatility during trading hours, while ignoring the overnight information produced during non-trading hours. Therefore, a more proper proxy for daily volatility is required for forecasting accuracy. 2.1. Intraday integrated variance estimation In the course of exploring the volatility of non-ferrous metals futures, we used realized volatility as the measure of daily quadratic variation. By dividing a trading day into M periods of time, the resultant continuous intraday returns based on intraday metals futures’ quotations, pt , j / M , can be written as:

rt , j = 100(pt ,/j / M − pt ,(j − 1)/ M ) (j = 1, 2, 3, …, M ),

(1)

with the first index t denoting the day of observation t = 1,2,……, T. The realized volatility, which is estimated by the sum of squared intraday returns: M

RVt =

∑ rt2,j j=1

(2)

The realized volatility can be used as an estimator of intraday integrated variance. However, both the data frequency and market

Sources: www.shfe.com.cn and www.dce.com.cn.

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Fig. 1. Closing prices over the period of July 1, 2010–December 31, 2015.

2.2. Daily integrated variance estimation

microstructure noise influence the accuracy and precision of this calculation. High-frequency returns are subject to market microstructure noise and bid-ask bounce effects, which distort the return properties. With the premise of no microstructure noise, RV converges to the daily volatility as n→∞ (Barndorff-Nielsen, 2002). However, with an increase in sampling frequency, market microstructure noise rapidly grows. Therefore, the determination of an optimal sample frequency to balance the market microstructure noise and efficiency is key.2 The majority of previous research investigations have employed a 5-min sample interval, which is regarded as a reasonable sampling frequency to minimize the bias between RV arising from market microstructure noise due to high frequency and efficiency loss of RV due to low frequency. Therefore, we run parallel to most of the literature to facilitate a comparison, and we used the 5-min frequency in this analysis as a base for copper and aluminum futures. Furthermore, in the final robust test, a comparison of different frequencies was investigated. To alleviate the effects of market microstructure noise, five-minute return data were used in this investigation. However, Zhang et al. (2012) found that this was not enough to solve the problem. In practice, the realized volatility, RVt , calculated from Eq. (2) may be not a robust estimator due to autocorrelation that occurs in high-frequency returns, which is caused by market microstructure noise (Bandi and Russell, 2006; Hansen and Lunde, 2006). Hansen and Lunde (2006) studied market microstructure noise in high-frequency data systematically and showed that a kernel-based estimator can effectively capture the effects of autocorrelation on high frequency returns. Therefore, we focused on the kernel-based approach proposed by Hansen and Lunde (2006) and made the following adjustment: M

RVt =

q

∑ rt2,j + ∑ ⎛1 − ⎜

j=1

i=1

⎝

i ⎞ ⎟ q + 1⎠

The true value of daily volatility is the goal of prediction. However, realized volatility only reflects the volatility during trading hours and neglects the information of the overnight period, when the market is closed. Therefore, a consistent proxy for the daily integrated variance estimation is needed for the entire day. We followed the approach of Blair et al. (2001), Gong et al. (2014), Todorova (2015), and Zhu et al. (2017), which are widely used in the literature, and simply added the squared overnight returns to the realized volatility as the measure of daily volatility. 3 Therefore, the daily volatility of a trading day can be given as: M

RVtON =

j=1

(4)

rt2, on

is the squared overnight return, which reflects the difference where between opening price and the closing price from the previous day. rt , on is defined as rt , on = 100(pt , o − pt − 1, c ) , where pt , on is the opening price at time t and pt − 1, c is the closing price at time t-1. 2.3. Data The empirical research sample used in this investigation covered the period from July 1, 2010 to December 31, 2015; a period of 1338 trading days in total for three-month copper and aluminum futures on the Shanghai Futures Exchange, which consisted of five-minute frequency price data and daily price data. Currently, the three-month futures contract has the largest trading volume and has the greatest influence on the spot price. Within the sample interval, the Shanghai Futures Exchange trading hours are from 9:00 a.m. to 11:30 a.m. and from 13:30 p.m. to 15:00 p.m., which comprises a total of 47 fiveminute intervals every day, resulting in M = 47. The transaction prices

M−i

∑ rt,j rt,j+i. j=1

∑ rt2,j + rt2,on,

(3)

This estimator can always provide positive values for realized volatility (Barndorff-Nielsen et al., 2008), which makes it very convenient for calculation. Following Todorova and Souček (2014), we set the width of the lag window to 30 min, and the number of autocovariances to q = 6.

3 In addition to the broadly used method in Eq. (4), an alternative method is to estimate the volatility of a whole day by scaling the realized volatility for the trading hours by a factor (Hansen and Lunde, 2005). However, this method only scale the resulting value upward and actually ignores the overnight period. The third method to measure the daily volatility is to optimally weight the squared overnight return and the sum of squared intraday returns, which is expressed as RVtHL = w1 rt2, on + w 2 RVt . However, the existence of outliers in aluminum futures’ high-frequency trade data leads to high variations of overnight returns of the given data set, which makes the case that w1* becomes negative (− 1.5523). Therefore, for the instance of daily integrated variance estimation, we regard the widely used RVtON as the proxy for daily volatility.

2 Hansen and Lunde (2006) had a thorough discussion of this issue. Furthermore, Bandi and Russell (2008) presented a method for computing the optimal sample frequency. The optimal frequency processes for copper and aluminum futures markets were calculated using the method of Bandi and Russell (2008), and the averages were found to be 7.1531 and 7.2360, respectively.

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under consideration was steeper than that of the LME metal market. The larger skewness and kurtosis of the SHFE reflected that our dataset was more asymmetrical and contained a greater degree of right skew due to the presence of outliers in the dataset. This reflects the fact that the Chinese non-ferrous metals futures market is more volatile and vulnerable to the influence of financial factors caused by international speculative capital or changes in exchange rates. These results also reflect an immature development of the Chinese metal futures market. This is consistent with Liu and An’s findings that Chinese futures markets are typically more volatile, less liquid, and less information efficiency than mature markets, such as those in the US and Europe (Liu and An, 2011, 2014). There are many financial factors that are increasingly exerting more influence on international metal prices, such as the increased information transmission efficiency between China’s commodity markets and international commodity markets, changes in the exchange rate of major international currencies, such as the US dollar (Harri et al., 2009; Wen et al., 2017), international capital speculation (Andreasson et al., 2016; Huchet and Fam, 2016; Li et al., 2015; Manera et al., 2016; Mellios et al., 2016), the shock of interest rates (Akram, 2009; Frankel, 2008), and the linkage of oil price (Baffes, 2007; Sari et al., 2010; Soytas et al., 2009). All of these financial factors exert more influence on international metal prices than simply the impact of basic supply and demand. To analyze the characteristic daily volatility of copper and aluminum futures markets, we plotted Figs. 2 and 3 to present the daily realized volatility, daily returns, and overnight returns during the break hours of the entire sample. The figures clearly indicate volatility clustering in each of time series (Figs. 2 and 3).

Table 1 Descriptive statistical analysis of copper and aluminum futures daily realized volatility and its logarithmic form. Copper futures

Aluminum futures

Statistics

RVt

ln(RVt)

RVt

ln(RVt)

Mean Std. Dev. Skewness Kurtosis Min Max Obs

1.4015 3.2919 8.2445 90.2797 0.0368 48.2921 1338

− 0.4037 1.0811 0.5607 3.7522 − 3.3033 3.8773 1338

0.4518 1.0935 8.7146 101.1988 0.0111 16.6647 1338

− 1.5638 1.1091 0.5161 3.6329 − 4.4983 2.8133 1338

Ling-Box Q-statistics Copper futures

Aluminum futures

Lags

RVt

ln(RVt)

RVt

ln(RVt)

5

382.5525 (0.0000) 561.4481 (0.0000) 729.9853 (0.0000) 838.9284 (0.0000)

1297.4874 (0.0000) 2144.5687 (0.0000) 2720.9226 (0.0000) 3204.8948 (0.0000)

643.0296 (0.0000) 711.3901 (0.0000) 754.2258 (0.0000) 789.6203 (0.0000)

1123.1896 (0.0000) 1794.0118 (0.0000) 2238.3607 (0.0000) 2620.5843 (0.0000)

10 15 20

Note: P values are shown in parentheses.

were obtained from CSMAR (http://www.gtarsc.com/). Fig. 1 shows the price series of copper and aluminum futures, which clearly illustrates that the price trend of copper futures resembles that of aluminum futures. The price trend plot illustrates that at the beginning of the sample period, the price level exhibited an upward trend, while the price experienced a persistent decline from the second half of 2011, despite occasional modest increases over the sample period. That may have been caused by an increase in the metal supply due to unsustainable rapid expansion of metal production at the outset, while the demand was decreasing. Table 1 shows the descriptive statistics of daily volatility, RVt , defined in Eq. (4) for copper and aluminum futures and their logarithmic form. The sample mean indicates that the volatility of copper is significantly higher than the volatility of aluminum futures. Correspondingly, copper futures prices are more volatile than those of aluminum futures, which is indicated by the standard deviations. The standard deviations also reflect some time-varying nature. Moreover, daily volatility for the two metals futures both exhibit positive skewness and excess kurtosis. Ljung-Box Q statistics of the daily volatility, RVt , reveal that they both have strong sequence autocorrelation. A comparison of the skewness and kurtosis of the daily volatility and the logarithmic form indicates that the distribution of the logarithmic form of daily volatility is close to a normal distribution, which is beneficial for statistical purposes. Therefore, we used the logarithmic form of daily realized volatility in the volatility equation. Todorova (2015) studied the course of realized volatility in the LME non-ferrous metal market using high frequency data. Compared with the LME metal market, realized volatility on the SHFE revealed some commonalities about their distributional characteristics. For example, aluminum futures prices are less volatile than copper futures prices, and the realized volatility of the SHFE also exhibited a significant timevarying pattern. Furthermore, the realized volatility of copper and aluminum futures showed positive skewness, excess kurtosis, and high autocorrelations. However, significant differences in volatility properties were observed. For example, the standard deviations and the difference between the minimum and maximum of realized volatility were significantly larger than those found for the LME metal market. Furthermore, the reported kurtosis, shown in Table 1, was significantly greater than that of the LME metal market, which indicated that the distributional pattern of realized volatility of the SHFE in the period

3. Volatility model In the following sections, we present the high frequency volatility forecasting models, which include the standard heterogeneous autoregressive (HAR) model introduced by Corsi (2009) and its extended models. The extended HAR models were constructed by considering the leverage effects, the impact of after-hours information, and the volatility of realized volatility. 3.1. HAR-LA model Based on the Heterogeneous Market Hypothesis proposed by Uller et al. (1993), Corsi proposed the HAR model. Generally, participants in financial markets trade at different frequencies. Short-term participants will be easily influenced by both short-term and long-term volatility, but not vice versa. Hence the volatility over longer time periods has a strong influence on the volatility over short time periods. Therefore, the HAR model aggregates the volatility over different periods, on a daily, weekly, and monthly basis, to capture the long memory feature. The HAR model, which can capture long-range dependence, can be seen as a lagged autoregressive process. Since it is concise, easy to estimate, and a well fitted long-term memory feature of RVt , this model is widely used by the majority of scholars in this field. In view of the volatility clustering inherent in the residuals of HAR models, Corsi et al. (2008) accounted for the volatility of realized volatility by including a GARCH-type extension of the HAR model and he constructed the HAR-GARCH model. Corsi and Renò (2010) extended the HAR model by considering the fact that volatility tends to increase more after a negative shock than after a positive shock of the same magnitude and proposed the HAR-L model. Moreover, Wang et al. (2015) went one step further and considered the impact of lunch-break returns, overnight returns and leverage effects on volatility forecasting and found leverage effects had a significant role in volatility forecasting, not only with respect to daily returns, but also with respect to lunch-break returns and overnight returns. Therefore, following the research of Wang et al. (2015), in addition to the usual leverage effects, we took into account the additional leverage effects with respect to 61

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Fig. 2. Daily realized volatility, daily returns and overnight returns for copper futures.

Fig. 3. Daily realized volatility, daily returns and overnight returns for aluminum futures.

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Fig. 4. Residual analysis of the HAR-LA model for realized volatility in copper futures market. Shown are the time series of the residuals, the sample autocorrelation functions (acf) and partial autocorrelation functions (pacf) of the squared residuals.

lunch-break return, respectively We adopted the logarithmic transform of volatility instead of volatility itself in the volatility Eq. (5), since the distributions of realized logarithmic variance often appear approximately normal. Moreover, the logarithmic form avoids the non-negative constraints in the volatility equation. Many papers adopted this approach and proved that the logarithmic transformation of volatility can achieve excellent performance when used for volatility forecasting (Andersen et al., 2008; Corsi et al., 2008; Huang et al., 2013; Todorova, 2015).

after-hours information captured by negative lunch-break returns and negative overnight returns. Furthermore, the overnight returns and lunch-break returns may also have an impact on the following daily volatility. Therefore, the extended HAR model (i.e. HAR-LA model) is presented in the following form:

ln(RVt + 1) = β0 + βc ln(RVt ) + βw ln(RVt − 5, t ) + βm ln(RVt − 22, t ) + γd rt− + γw rt5 − + γm rt22 − + γ0 rt , on + γ1 rt−, on + λ 0 rt , lunch + λ1 rt−, lunch + εt + 1, (5)

3.2. HAR-LA-GARCH model

where RVt − i, t = (RVt + RVt − 1+⋯+RVt − (i − 1) )/ i, (i = 5, 22) mean the weekly and monthly sample path variation respectively. rt , on and rt , lunch are the overnight returns and lunch-break returns, respectively. 1 and rt22 −= rt− = min(rt , 0),rt5 − = min 5 (rt − 4 + rt − 3+⋯+rt ), 0

(

(

1

In empirical studies, the residuals of HAR models may exhibit volatility clustering (Corsi et al., 2008). Figs. 4 and 5 show the residual analyses for the copper and aluminum futures markets, respectively. The time series plots of residuals and the sample autocorrelation and partial autocorrelation functions of the squared residuals of both HARLA models for copper and aluminum futures exhibit volatility clustering. Meanwhile, the Ljung-Box Q-statistics for the squared and

)

)

min 22 (rt − 21 + rt − 20+⋯+rt ), 0 are defined as the usual leverage effects comprising negative shocks over the last day, last week and last month daily returns respectively. and rt−, on = min(rt , on, 0) rt−, lunch = min(rt , lunch, 0) are the negative overnight return and negative

Fig. 5. Residual analysis of the HAR-LA model for realized volatility in aluminum futures market. Shown are the time series of the residuals, the sample autocorrelation functions (acf) and partial autocorrelation functions (pacf) of the squared residuals.

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Table 2 Estimation results from the standard HAR model and its extended models (HAR-GARCH, HAR-L, HAR-L-GARCH, HAR-LA and HAR-LA-GARCH models) for copper futures. Variables

HAR

HAR-GARCH

HAR-L

HAR-L-GARCH

HAR-LA

HAR-LA-GARCH

βD

0.0658* (0.0346) 0.416*** (0.0561) 0.309*** (0.0529)

0.0452 (0.0363) 0.421*** (0.0576) 0.328*** (0.0507)

0.00713 (0.0345) 0.298*** (0.0573) 0.408*** (0.0539) − 0.0358 (0.0371) − 0.427*** (0.0904) − 0.333* (0.170)

0.00333 (0.0381) 0.308*** (0.0586) 0.414*** (0.0515) − 0.0292 (0.0367) − 0.410*** (0.0879) − 0.343** (0.166)

0.0729* (0.0425) 0.293*** (0.0573) 0.394*** (0.0542) − 0.0635 (0.0445) − 0.420*** (0.0897) − 0.316* (0.167) − 0.131** (0.0610) 0.314*** (0.101) 0.220 (0.215) − 0.425 (0.313)

0.0642 (0.0475) 0.298*** (0.0584) 0.403*** (0.0514) − 0.0626 (0.0452) − 0.412*** (0.0907) − 0.323* (0.169) − 0.123 (0.0804) 0.310** (0.128) 0.259 (0.281) − 0.435 (0.429) 0.0265* (0.0160) 0.845*** (0.114)

βW βM rt−

rt5 − rt22 −

rt , on rt−, on rt , lunch rt−, lunch α

0.0587*** (0.0191) 0.771*** (0.0878)

β R-squared

0.339

0.0330* (0.0173) 0.816*** (0.116) 0.368

0.374

Notes: This table provides the parameter estimation results for HAR and its extended realized volatility models for the whole sample period from July 1, 2010 through December 31, 2015 for copper futures. The numbers in the parentheses are robust standard error. The asterisks *, ** and *** denote rejections of null hypothesis at 10%, 5% and 1% significance levels, respectively.

daily volatility for the futures of the two metals. It is particularly noteworthy that the coefficients of weekly and monthly volatility components are significantly larger than those of daily components, which is in line with the results found by Wang et al. (2015). Moreover, the monthly volatility component has a stronger impact on future volatility than the weekly component, which implies that past long-term volatility plays a predominant role in the future volatility of copper and aluminum futures. This implies that there is long memory in the Chinese nonferrous metal futures market and it illustrates that the volatility of metal futures is mainly determined by the trade behavior of longterm investors. Next, we analyzed the leverage effects on the Chinese non-ferrous metals futures market. The leverage effects reflect volatilities that respond differently to bad news and good news. In particular, a negative market shock typically exerts a stronger impact on volatilities than does a positive shock of the same magnitude. A profound understanding of this asymmetric effect is crucial to investors, as it can be utilized to help accurately forecast future volatilities and is critical to risk management and asset pricing. In contrast to the results reported by Todorova (2015), the coefficient estimates associated with the one day lagged daily negative returns for copper and aluminum futures were generally insignificant, albeit negative in most cases while the coefficients related to the one week lagged negative returns were negative, as expected, and significant at levels lower than 1%. These results indicate that there is some evidence of significant mid-term leverage effects in the volatility of copper and aluminum futures. Further, the parameters of the monthly negative past returns for copper futures are statistical and economic significant because they imply that long-term leverage effects exist in the copper futures market. This implies that investors are more prone to react to negative news in comparison to positive news in the medium and long term. Therefore, mid-and-long term leverage effects lead to a remarkable increase in the explanatory power, revealing a persistent leverage effect. In other words, leverage effects play an important role in forecasting the future volatility of the Chinese non-ferrous metals futures market. Finally, it should be noted that the coefficients of the GARCH

absolute residuals (available upon request) indicate strong conditional heteroskedasticity. Hence, following Corsi et al. (2008) and Todorova (2015), we went one step further and took the volatility of realized volatility into account by combining a GARCH specification extension of our HAR-LA model. To keep the model parsimonious, we adopted the GARCH(1,1) specification for the conditional variance of logarithmic realized volatility. Hence the proposed model is represented as:

ln(RVt + 1) = β0 + βC ln(RVt ) + βW ln(RVt − 5, t ) + βM ln(RVt − 22, t ) + γd rt− + γw rt5 − + γm rt22 − + γ0 rt , on + γ1 rt−, on + λ 0 rt , lunch + λ1 rt−, lunch + ht + 1 = ω + ut + 1 =

αut2

ht + 1 εt + 1,

+ βht ,

ht + 1 εt + 1

εt + 1 Ω ∼ (0, 1),

(6)

where Ωt is the σ -field generated by all the information available up to time t. The residual ht + 1 εt + 1 follows a conditional density with timevarying variance. We label Eq. (6) a HAR-LA-GARCH model. 4. In-sample analysis For comparison, the full in-sample estimation results of the standard HAR model of Corsi (2009) and its extended models (HAR-GARCH, HAR-L, HAR-L-GARCH, HAR-LA and HAR-LA-GARCH models) for copper and aluminum futures daily volatilities are shown in Tables 2 and 3. The left columns report the OLS estimate results with NeweyWest standard errors, and the right reports the estimate results with the incorporation of a GARCH specification. 4.1. Long memory, leverage effects, and volatility clustering It is apparent that almost all the coefficients of the three lagged volatility components are positive and highly significant for copper and aluminum futures, as expected. This implies a strong heterogeneity and dynamic dependencies in realized volatility. The daily, weekly, and monthly components all have consistent, heavy impact on the future 64

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Table 3 Estimation results from the standard HAR model and its extended models (HAR-GARCH, HAR-L, HAR-L-GARCH, HAR-LA and HAR-LA-GARCH models) for aluminum futures. Variables

HAR

HAR-GARCH

HAR-L

HAR-L-GARCH

HAR-LA

HAR-LA-GARCH

βD

0.244*** (0.0331) 0.274*** (0.0549) 0.258*** (0.0556)

0.239*** (0.0333) 0.271*** (0.0525) 0.280*** (0.0485)

0.213*** (0.0342) 0.244*** (0.0547) 0.268*** (0.0557) − 0.0379 (0.0662) − 0.528*** (0.170) − 0.321 (0.275)

0.214*** (0.0344) 0.240*** (0.0533) 0.287*** (0.0491) − 0.00146 (0.0718) − 0.485*** (0.179) − 0.457 (0.316)

0.284*** (0.0400) 0.227*** (0.0542) 0.244*** (0.0559) − 0.0173 (0.0770) − 0.595*** (0.174) − 0.147 (0.286) − 0.459*** (0.139) 0.892*** (0.223) 1.162** (0.452) − 1.982*** (0.732)

0.287*** (0.0402) 0.223*** (0.0519) 0.257*** (0.0476) 0.0293 (0.0767) − 0.534*** (0.173) − 0.285 (0.311) − 0.480*** (0.132) 0.910*** (0.207) 0.997** (0.461) − 1.930*** (0.653) 0.00639** (0.00320) 0.992*** (0.00480)

βW βM rt−

rt5 − rt22 −

rt , on rt−, on rt , lunch rt−, lunch α

0.0113** (0.00528) 0.983*** (0.00989)

β R-squared

0.316

0.0140** (0.00660) 0.977*** (0.0141) 0.327

0.342

Notes: This table provides the parameter estimation results for HAR and its extended realized volatility models for the whole sample period from July 1, 2010 through December 31, 2015 for aluminum futures. The numbers in the parentheses are robust standard error. The asterisks ** and *** denote rejections of null hypothesis at 5% and 1% significance levels, respectively.

innovations (α and β , respectively) are generally significant, which reflects volatility clustering and time-varying volatility of the realized volatility for copper and aluminum futures. Moreover, compared with the ARCH coefficients, the GARCH coefficients are relatively large and positive, which implies that previous periods’ volatility has a strong positive impact on the current volatility of the realized volatility. This also confirms that volatility clustering of realized volatility occurs in the Chinese non-ferrous metals futures market. Furthermore, the sum of the parameter estimates of the GARCH variance equation for aluminum is very close to 1, which indicates the high persistence of the volatility of realized volatility for aluminum.

Besides the leverage effects with respect to daily returns and overnight returns, a remarkable finding is that there exists significant leverage effects with respect to lunch-break returns in the aluminum futures market, which is in line with the results found by Wang et al. (2015). This novel finding suggests that the market might be more sensitive to price declines that occurred in the previous lunch-break period. Another possible interpretation may be that if there is any information released in the lunch-break period, particularly bad news, it could subsequently be incorporated into the price through trading activities during the following afternoon and overnight periods. This would then influence the volatility of the futures market on the following day. Therefore, including the after-hours information may improve the forecast ability of a volatility model. The impact of after-hour information, especially the overnight information, on volatility in Chinese non-ferrous metal futures market may be related to local features of the SHFE. For example, Chinese futures markets trade from 9:00 a.m. to 3:00 p.m. (Beijing time (BT)), while the Chicago Mercantile Exchange (CME) trades from 11:30 p.m. to 3:15 a.m. (BT), and the London Metal Exchange (LME) trades from 5:45 p.m. to 1:00 a.m. (BT). Therefore, trading of non-ferrous metals continues in other exchanges after the SHFE closes and the trading information released during the overnight period in international developed markets, especially US and European markets, such as LME or CME, would certainly influence the Chinese non-ferrous metal futures market. Trading activities in these international futures markets represent a significant component of the non-trading information in the corresponding Chinese markets (Liu and An, 2014). Along with the internationalization of China's commodity market, the linkage of the commodity futures market and the international market increasingly close, and therefore the price fluctuations are more frequent and violent. The policy of the SHFE to correct overnight due to trading in the bigger exchanges is conducive to allow domestic investors and risk managers to timely digest the international market information, and this contributes to risk management throughout the domestic market. The copper and aluminum futures markets have different characteristics with respect to the roles of usual leverage effects and after-

4.2. The impact of after-hours information The particularly interesting finding is the role of after-hours information outside the trading hours, which includes the overnight and lunch-break periods, on volatility forecasting. The parameters of overnight returns for both models are of strong significance, which reveals that overnight information has an important role in volatility forecasting. This result agrees with the findings of Todorova and Souček (2014) and Wang et al. (2015). An explanation for this result be that overnight returns contain new information such as monetary policy, dividend news, and new IPOs, which will interfere with investor sentiment and further impact trader decisions on the following day. Another possible explanation is that traders tend to trade based on demands for liquidity right after regular trading hours end (Jayawardena et al., 2015). Therefore, overnight information has important influence on the following day’s volatility. Moreover, additional leverage effects, captured by negative overnight returns, are found. This new finding reveals that the futures market may be more sensitive to bad news that has been reported in the previous overnight period. In contrast for the aluminum futures market, lunch-break period information also plays a significant role in volatility forecasting. This implies that lunch- break hour information acts as an additional volatility measurement and might have a strong influence on future volatility in the aluminum futures market. 65

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Table 4 OLS estimates of the regression of daily returns on daily variance forecasts Vˆt obtained with different models. Risk-return regression: rt = c + βVˆt + εt Copper

Aluminum

Model

Slope βˆ

t statistics

R2

Slope βˆ

t statistics

R2

HAR HAR-GARCH HAR-L HAR-L-GARCH HAR-LA HAR-LA-GARCH

0.0336 0.0409 − 0.2122*** − 0.1950*** − 0.1837*** − 0.1730***

0.33 0.41 − 2.95 − 2.63 − 3.23 − 2.63

0.0004 0.0006 0.0331 0.0277 0.0285 0.0235

− − − − − −

− − − − − −

0.0042 0.0040 0.0366 0.0242 0.0416 0.0344

0.1905 0.1792 0.4284** 0.3596* 0.4382*** 0.4105***

0.86 0.84 2.45 1.95 3.67 3.30

Notes: The asterisks *, ** and *** denote rejections of null hypothesis at 10%, 5% and 1% significance levels, respectively.

futures. Thus, there is no risk-return trade-off in the Chinese non-ferrous metal futures market. This is consistent with Gong et al. (2017)’s findings on investigating the risk-return trade-off for crude oil futures.

hours information on volatility forecasting. In addition to the usual weekly leverage effects, there are also monthly leverage effects that occur in the copper futures market, which are different with the leverage effects that occur in the aluminum futures market. This implies that the copper futures market may be more sensitive to long-term investor trading behaviors and the market mechanism is relatively more mature. It is also apparent that after-hours information will not have the same effect in the two different futures markets. For aluminum futures, not only does overnight information have an important role on volatility forecasting, but so lunch-break information also plays a significant role in volatility forecasting. Moreover, additional leverage effects captured by negative lunch-break returns also exist in the aluminum futures market. This implies that the aluminum futures market may possess greater fluidity than copper futures market, and thus traders in the aluminum futures market may have higher degree of sensitivity to market changes, even with regard to the small bits of information released during the lunch-break period.

5. Out-of-sample forecasting performance In addition to modeling the volatility of the Chinese non-ferrous metals futures market, volatility forecasting is potentially a useful application for hedging, derivative pricing, and performance evaluation. In this section, we considered the one-day-ahead out-of-sample volatility forecasting performance of the extended HAR models for non-ferrous metals futures with and without considering the volatility of realized volatility, which was a innovation to consider the conditional heteroskedasticity of the realized volatility. Further, we evaluated the one-day-ahead forecasting performance of the HAR models with and without considering the leverage effects, not only with respect to the daily returns, but also with respect to after-hours period returns. Therefore, we compared the logarithmic variance for the one-dayahead forecast evaluation of the six volatility models, which included HAR, HAR-L, HAR-LA, HAR-GARCH, HAR-L-GARCH and HAR-LAGARCH, to assess which model was more accurate volatility forecasting. The first subsection below discusses the model evaluation methods and the subsequent section reports on the results from the outof-sample forecasts.

4.3. Investigating the risk-return trade-off Portfolio allocation is decided according to the return and risk of investments, and volatility is generally used to measure risk. In general, given the return, a portfolio with low volatility is preferred. Therefore, understanding the relationship between risk and return is essential for portfolio allocation and risk-management. This can help investors to make choices given the uncertainty in the metal market. The volatility forecasts developed in this study can be utilized to investigate the presence of the risk–return trade-off in the Chinese non-ferrous metals futures market. The majority of studies investigating risk-return trade-off use lowfrequency data to measure risk. Instead, we used high-frequency transaction data to measure the volatility, which can more accurately measure the risks because high-frequency transaction data contains far more information than the low-frequency transaction data (Andersen et al., 2007; Gong et al., 2017). Following the methods presented by Corsi and Reno (2012), we used the high-frequency volatility forecasts of (1) the standard HAR model (HAR), (2) the HAR model with GARCH specification (HAR-GARCH), (3) the HAR model with leverage effects (HAR-L), (4) the HAR model with leverage effects and GARCH specification (HAR-L-GARCH) (5) the HAR model with leverage effects and after-hours information (HAR-LA), and (6) the HAR model with leverage effects, after-hour information and GARCH specification (HAR-LA-GARCH) to measure the risk. The regression results of the return on the variance forecasts obtained via OLS with Newey-West standard errors are displayed in Table 4. We found that volatility forecasts had a significant impact on returns in most cases, even if we had a very low R2, as it is common in this kind of applications. The results indicated a contemporaneous relationship between risk and return that was negative and statistically significant, with the exception of the HAR and HAR-GARCH for copper

5.1. Model evaluation methods To assess the accuracy of the extended HAR models forecasts, the loss functions were adopted to evaluate the volatility forecasting performance. Loss functions adopted in this section include the Mean Absolute Error (MAE) and Mean Square Error (MSE), which are defined as N

MAE = N−1 ∑ |(log(RVt + 1) − log(Vt + 1, M )|, i=1

(7)

N

MSE = N−1 ∑ (log(RVt + 1) − log(Vt + 1, M ))2 , i=1

(8)

where RVt + 1 and Vt + 1, M denote the actual and one-day-ahead forecasts of daily volatility from model M, respectively. N is the number of sample days observed. To determine whether the difference in the loss functions were statistically significant, we implemented the Diebold and Mariano test. The Diebold and Mariano test, as applied here, is a pairwise comparison of the forecasts from each of the extended HAR models to the forecasts from the standard HAR model. It uses t-statistics or p-values to measure the significance of the difference of the loss function of the two volatility forecasts. The null hypothesis was expressed as follows:

H0 = E [L (i)] − E [L (j )] = E [L (i) − L (j )] = E [dL (i, j )] = 0 66

(9)

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log(RVt + 1) = α + β log(Vt + 1, M ) + μt + 1 ,

Table 5 Loss function value of out-of-sample prediction for copper and aluminum. Cu Model

MAE

MSE

R2

MAE

MSE

R2

HAR HAR-GARCH

0.7034 0.7061*** [0.0044] 0.6799* [0.0601] 0.6816* [0.0814] 0.6785* [0.0542] 0.6791* [0.0646]

0.7717 0.7745 [0.1399] 0.7288** [0.0463] 0.7312* [0.0637] 0.7263** [0.0353] 0.7268** [0.0419]

0.1715 0.1684

0.6779 0.6750 [0.1254] 0.6782 [0.9860] 0.6722 [0.6964] 0.6764 [0.9188] 0.6689 [0.6391]

0.7338 0.7272* [0.0959] 0.7283 [0.8809] 0.7173 [0.6274] 0.7194 [0.6421] 0.7136 [0.5298]

0.3203 0.3236

HAR-L HAR-L-GARCH HAR-LA HAR-LA-GARCH

If the model is correctly established, then E (Vt + 1, M ) = RVt + 1. α and β are expected to be zero and one, respectively, with a high statistical significance. Andersen and Bollerslev (1998) found that the coefficients are easily subject to a standard errors-in-variables problem, which reduces the explanatory strength. Nonetheless, they argued that the R2 of the M-Z regression can still be used to evaluate the performance of volatility forecasts.

Al

0.2188 0.2170 0.2246 0.2245

0.3087 0.2882

5.2. Out-of-sample forecasts

0.3144

To evaluate the out-of-sample forecasting performance of the different models, we computed a one-day-ahead out-of-sample rolling forecast from August 14, 2014 to December 31, 2015, which resulted in 338 daily forecasts. The volatility forecasts were obtained with a rolling fixed-window approach, including 1000 latest realized volatility observations. Table 5 shows the results of the MAE, MSE, and theR2 statistic from the M-Z regressions of real next-day realized volatility on the one-dayahead forecasts for the six volatility models. In addition, the corresponding p-values of the Diebold and Mariano test are reported in square brackets. According to the forecasting results for copper futures, we found that all values of MAE and MSE criterion of the extended HAR models with leverage effects and after-hours information (HAR-L, HARL-GARCH, HAR-LA, and HAR-LA-GARCH models) are lower than the standard HAR model. Moreover, the corresponding p-values of the Diebold and Mariano test, reported in square brackets, indicated that the evaluation results had statistically significant difference when compared to the standard HAR model, which showed that leverage effects and after-hours information can significantly improve out-ofsample forecasting power. Comparing the relative performance of these models, we observed that the extended HAR-LA model exhibit best forecasting power for copper futures. With regards to aluminum futures, GARCH structure in the innovations may be preferred, although the corresponding p-values of the Diebold and Mariano test indicated

0.3378

Note: The value of square brackets is p-value of the Diebold and Mariano test, which is a pairwise comparison of the forecasts from each of the five extended HAR models to the forecasts from the standard HAR model. The asterisks *, ** and *** denote rejections of null hypothesis at 10%, 5% and 1% significance levels for the Diebold and Mariano test, respectively. The results for the best forecasts per criterion are printed bold.

where L (i) is a loss function for the model i and dL (i, j ) is the loss differential equation between two models i and j, for a given loss functions. The test statistic is expressed as

tL (i, j ) =

DL (i, j ) V (DL (i, j ))

(10) m

where DL (i, j ) = 1/ m ∑t = 1 dL (i, j ) , V (DL (i, j )) is the Newey-West variance estimator, and the test statistic is asymptotically distributed as a standardized normal. For ease of comparison and display, we only show the p-values of the Diebold and Mariano (2002) test. In order to further analyze the relative performance of extended HAR models, MincerZarnowitz (M-Z) regressions were also taken into consideration. M-Z regressions, proposed by Mincer and Zarnowitz (1969), are widely used to evaluate sample prediction. Its regression equation is: Table 6 Model estimation for copper futures. HAR-LA

HAR-LA-GARCH

Frequency

1 min

5 min

10 min

15 min

1 min

5 min

10 min

15 min

βD

0.187*** (0.0473) 0.248*** (0.0538) 0.357*** (0.0503) − 0.0522 (0.0408) − 0.413*** (0.0818) − 0.307** (0.151) − 0.215*** (0.0585) 0.474*** (0.0976) 0.162 (0.190) − 0.364 (0.287)

0.0729* (0.0425) 0.293*** (0.0573) 0.394*** (0.0542) − 0.0635 (0.0445) − 0.420*** (0.0897) − 0.316* (0.167) − 0.131** (0.0610) 0.314*** (0.101) 0.220 (0.215) − 0.425 (0.313)

0.0435 (0.0402) 0.279*** (0.0591) 0.426*** (0.0570) − 0.0458 (0.0480) − 0.430*** (0.0947) − 0.317* (0.176) − 0.104 (0.0644) 0.243** (0.104) 0.306 (0.221) − 0.541* (0.323)

0.0227 (0.0401) 0.261*** (0.0596) 0.453*** (0.0587) − 0.0388 (0.0493) − 0.452*** (0.0990) − 0.312* (0.183) − 0.0703 (0.0676) 0.180* (0.108) 0.393* (0.226) − 0.669** (0.334)

0.182*** (0.0501) 0.254*** (0.0548) 0.366*** (0.0467) − 0.0506 (0.0406) − 0.397*** (0.0834) − 0.305** (0.154) − 0.211*** (0.0768) 0.478*** (0.127) 0.208 (0.260) − 0.382 (0.387) 0.0364** (0.0167) 0.807*** (0.105)

0.0642 (0.0475) 0.298*** (0.0584) 0.403*** (0.0514) − 0.0626 (0.0452) − 0.412*** (0.0907) − 0.323* (0.169) − 0.123 (0.0804) 0.310** (0.128) 0.259 (0.281) − 0.435 (0.429) 0.0265* (0.0160) 0.845*** (0.114)

0.0363 (0.0465) 0.282*** (0.0604) 0.433*** (0.0545) − 0.0458 (0.0470) − 0.426*** (0.0950) − 0.315* (0.177) − 0.0954 (0.0806) 0.234* (0.129) 0.327 (0.304) − 0.539 (0.459) 0.0240 (0.0175) 0.790*** (0.219)

0.0156 (0.0451) 0.264*** (0.0628) 0.455*** (0.0570) − 0.0416 (0.0503) − 0.447*** (0.0984) − 0.299 (0.185) − 0.0584 (0.0819) 0.167 (0.134) 0.419 (0.323) − 0.679 (0.480) 0.0329* (0.0197) 0.649* (0.345)

0.415

0.374

0.349

0.331

βW βM

rd rw rm

rt , on rt−, on rt , lunch rt−, lunch α β R-squared

Notes: This table provides the parameter estimation results for our extended realized volatility models for the whole sample period from July 1, 2010 through December 31, 2015. The numbers in the parentheses are standard error. The asterisks *, ** and *** denote rejections of null hypothesis at 10%, 5% and 1% significance levels, respectively.

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Table 7 Model estimation for aluminum futures. HAR-LA

HAR-LA-GARCH

Frequency

1 min

5 min

10 min

15 min

1 min

5 min

10 min

15 min

βD

0.365*** (0.0409) 0.199*** (0.0482) 0.230*** (0.0451) − 0.0510 (0.0663) − 0.535*** (0.137) − 0.532*** (0.117) 1.052*** (0.183) 1.133*** (0.393) − 1.830*** (0.581)

0.285*** (0.0400) 0.225*** (0.0518) 0.248*** (0.0497) − 0.0189 (0.0767) − 0.632*** (0.158) − 0.466*** (0.133) 0.910*** (0.205) 1.151** (0.451) − 1.955*** (0.665)

0.236*** (0.0396) 0.246*** (0.0539) 0.252*** (0.0524) 0.000629 (0.0831) − 0.654*** (0.170) − 0.406*** (0.141) 0.760*** (0.217) 1.019** (0.484) − 1.890*** (0.714)

0.195*** (0.0395) 0.258*** (0.0552) 0.261*** (0.0540) − 0.0144 (0.0863) − 0.671*** (0.176) − 0.348** (0.145) 0.664*** (0.222) 1.052** (0.500) − 1.956*** (0.735)

0.372*** (0.0403) 0.197*** (0.0509) 0.248*** (0.0452) − 0.0107 (0.0660) − 0.519*** (0.134) − 0.589*** (0.119) 1.141*** (0.186) 0.895** (0.418) − 1.599*** (0.588) 0.0190*** (0.00682) 0.965*** (0.0155)

0.291*** (0.0399) 0.221*** (0.0517) 0.263*** (0.0472) 0.0245 (0.0763) − 0.612*** (0.154) − 0.494*** (0.130) 0.946*** (0.201) 0.983** (0.459) − 1.878*** (0.643) 0.00577* (0.00304) 0.993*** (0.00458)

0.242*** (0.0410) 0.244*** (0.0545) 0.268*** (0.0504) 0.0438 (0.0843) − 0.627*** (0.168) − 0.432*** (0.144) 0.803*** (0.220) 0.846* (0.490) − 1.760** (0.690) 0.00887** (0.00445) 0.987*** (0.00799)

0.195*** (0.0410) 0.258*** (0.0539) 0.262*** (0.0496) − 0.0184 (0.0893) − 0.667*** (0.182) − 0.349** (0.144) 0.668*** (0.217) 1.069** (0.515) − 1.981*** (0.700) − 0.00390 (0.0138) − 0.903** (0.447)

0.404

0.342

0.304

0.285

βW βM

rd rw

rt , on rt−, on rt , lunch rt−, lunch α β R-squared

Notes: This table provides the parameter estimation results for our extended realized volatility models for the whole sample period from July 1, 2010 through December 31, 2015. The numbers in the parentheses are standard error. The asterisks *, ** and *** denote rejections of null hypothesis at 10%, 5% and 1% significance levels, respectively.

Table 8 Out-of-sample forecasting performance of HAR-LA model for copper and aluminum futures. Cu

Al

Frequency

1 min

5 min

10 min

15 min

1 min

5 min

10 min

15 min

MAE

0.593 [0.1429] 0.5565* [0.0644] 0.2713

0.6785* [0.0542] 0.7263** [0.0353] 0.2246

0.7398* [0.0685] 0.8592* [0.0712] 0.1915

0.7822 [0.1457] 0.976 [0.1045] 0.1763

0.5613 [0.7394] 0.5069 [0.8872] 0.3899

0.6764 [0.9188] 0.7194 [0.6421] 0.3144

0.7336 [0.5922] 0.8590 [0.6182] 0.2512

0.7748 [0.5082] 0.9561 [0.6399] 0.2234

MSE

R2

Note: The value of square brackets is p-value of the Diebold and Mariano test, which is a pairwise comparison of the forecasts from HAR-LA model to the forecasts from the HAR model. The asterisks * and ** denote rejections of null hypothesis at 10% and 5% significance levels for the Diebold and Mariano test, respectively.

6. Robustness tests

that the evaluation results had no statistically significant difference with the standard HAR model. The MAE, MSE and the R2 statistic from the M-Z regressions for aluminum futures are more in favor of the HARLA model. To summarize, the analysis of the forecasting performance in copper and aluminum futures suggests that the introduction of leverage effects and after-hours information contain significant incremental information for out-of-sample forecasting the future volatility of the Chinese non-ferrous metal futures market. The GARCH structure in the innovations for copper futures failed to deliver consistent out-of-sample forecasting performance across all loss functions and M-Z regressions. Compared with the innovation of considering the volatility of realized volatility, the role of after-hours information and leverage effects on volatility forecasting may be more significant. This can be explained by a higher parameter uncertainty leading to a deterioration of the forecasting performance due to the need for more parameter estimation, which is consistent with the study of Todorova (2015), who compared the HAR and HAR-GARCH models’ realized forecasting performance for the LME non-ferrous metal market. Both studies could not provide evidence that the HAR-GARCH model could significantly outperform the simple HAR model, even though it may have had a good in-sample fit.

In this subsection, we examine whether our results are robust for different sampling frequencies. This investigation was performed to evaluate how sensitive the models were to sampling frequencies. Tables 6–9 present model estimation results and forecasting performance at 1-min, 5-min, 10-min, and 15-min frequencies, respectively. It is noted that the findings are similar to the findings discussed previously. Further, we compared the volatility forecasting performance at different frequencies to determine the optimal sampling frequency (1 min, 5 min, 10 min, or 15 min). The evidence shows that the forecasting performance of volatility models was the most accurate at the sampling frequency of 1-min frequency for both copper and aluminum futures, since the lost functions (MAE, MSE) are at their minimum at the 1-min sampling frequencies. Meanwhile, adjusting R2 of the M-Z regression was maximal at a 1-min frequency. In short, it shows the results are robust to different frequencies. 7. Conclusions The main purpose of this investigation was to model and forecast the daily realized volatility and apply this model to the Chinese non68

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Table 9 Out-of-sample forecasting performance of HAR-LA-GARCH model for copper and aluminum futures. Cu

Al

Frequency

1 min

5 min

10 min

15 min

1 min

5 min

10 min

15 min

MAE

0.5941 [0.2442] 0.5570* [0.0663] 0.2708

0.6791* [0.0646] 0.7268** [0.0419] 0.2245

0.7429 [0.1233] 0.8614* [0.0879] 0.1896

0.7840 [0.1968] 0.9781 [0.1303] 0.1747

0.5528 [0.7039] 0.4920 [0.4417] 0.4029

0.6689 [0.6391] 0.7136 [0.5298] 0.3378

0.7260 [0.9737] 0.8398 [0.7744] 0.2629

0.7687 [0.8434] 0.9402 [0.8227] 0.2329

MSE

R2

Note: The value of square brackets is p-value of Diebold and Mariano test, which is a pairwise comparison of the forecasts from HAR-LA-GARCH model to the forecasts from the HAR model. The asterisks * and ** denote rejections of null hypothesis at 10% and 5% significance levels for the Diebold and Mariano test, respectively.

accuracy of market volatility forecasting.

ferrous metals futures market. Unlike the usual volatility forecasting model that is used in most of the empirical literature, we investigated whether after-hours information, including the overnight and lunchbreak periods, could play significant role in the volatility forecasting of the Chinese non-ferrous metals futures market. We included the overnight returns and lunch-break returns as additional explanatory variables. Besides the usual leverage effects, additional leverage effects with respect to overnight returns and lunch-break returns were considered. Moreover, the volatility of realized volatility was taken into consideration by including the GARCH specifications for the variance equation. The estimation results of our extended volatility models may be summarized as follows: First, there exist significant dynamic dependencies in the Chinese non-ferrous metals futures market volatility, which is well captured by the HAR models with a daily, weekly, and monthly component. Furthermore, significant weekly leverage effects were found in both markets. Additionally, there was a monthly leverage effect in the copper futures market. The particularly interesting finding was the significant role of after-hours information during the nontrading hours on volatility forecasting. The overnight information had an important influence on the following day’s volatility for both markets, and lunch-break hour information had significant impact on the next day’s volatility for the aluminum futures market. Additional leverage effects, which were captured by the negative overnight returns for both markets and negative lunch-break returns for aluminum futures market were discovered. The additional leverage effects have an important impact on the next day’s volatility and should be considered in the volatility forecasting. Moreover, GARCH specifications for the variance equation indicated the highly persistent volatility of realized volatility and volatility clustering. Second, the one-day-ahead out-of-sample forecasts showed that the extended HAR-LA and HAR-LA-G mode was superior for out-of-sample predictive ability for the daily volatility, although the p-values of the Diebold and Mariano test for aluminum futures indicated no statistically significant difference with the standard HAR model. This research is of great significance to academics, policymakers, and futures markets participants for several reasons. First, it should be noted that after-hours information is a very rich predictor for future volatility of the Chinese non-ferrous metals futures market, and the inclusion of after-hours information in the HAR model showed consistent outperformance. This has great implications for investors who use Shanghai copper and aluminum in their portfolio hedging and trading strategies. Second, our findings are not only important for the future of metal markets performance research, it also may be of interest to those wanting to apply the method to other commodity futures, spot markets, or even derivatives markets. This method considers nontrading period information in volatility forecasting performance, which may lead to new findings when applied to other markets. Finally, realized volatility spillovers of Chinese non-ferrous metals futures markets, an important and interesting research topic considered by Todorova et al. (2014), may also have an impact on volatility forecasting. These discoveries can lead to new research directions for improving the

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