Which types of commodity price information are more useful for predicting US stock market volatility?

Economic Modelling xxx (xxxx) xxx

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Economic Modelling journal homepage: www.journals.elsevier.com/economic-modelling

Which types of commodity price information are more useful for predicting US stock market volatility? Chao Liang a, Feng Ma a, Ziyang Li b, *, Yan Li a a b

School of Economics & Management, Southwest Jiaotong University, Chengdu, China Business School, Sichuan University, Chengdu, China

A R T I C L E I N F O

A B S T R A C T

JEL classification: C22 C53 C58 G17

This study aims to investigate which types of commodity price information are more useful for predicting US stock market realized volatility (RV) in a data-rich word. The standard predictive regression framework and monthly RV data are used to explore the RV predictability of commodity futures for the next-month RV on S&P 500 spot index. We utilize principal component analysis (PCA) and factor analysis (FA) to extract the common factors for each type and all types of commodity futures. Our results indicate that the futures volatility information of grains and softs has a significant predictive ability in forecasting the RV of the S&P 500. In addition, the FA method can yield better forecasts than the PCA and average methods in most cases. Further analysis shows that the volatility information of grains and softs exhibits higher informativeness during recessions and pre-crises. Finally, the forecasts of the five combination methods and different out-of-sample periods confirm our results are robust.

Keywords: Commodity futures volatility Stock market volatility Factor analysis Principal component analysis

1. Introduction Stock market volatility is inextricably linked to asset pricing, portfolios, derivative hedging strategies, and financial risk management (see, e.g., Bollerslev and Mikkelsen, 1996; Christensen and Nielsen, 2007; Christensen et al., 2010). Therefore, volatility forecasting is of great significance to both academics, policymakers, and practitioners (see, e.g., Koutmos and Booth, 1995; Balaban et al., 2006; Narayan and Narayan, 2007; Wei et al., 2010; Mittnik et al., 2015; Wang et al., 2016; Ma et al., 2017; Ma et al., 2019; Wei et al., 2019). However, it is difficult to improve the accuracy of such predictions, and thus, volatility forecasting remains an area requiring arduous and meaningful study. Various studies have proven that futures market volatility is closely related to spot market volatility (see, e.g., Antoniou and Foster, 1992; Antoniou and Holmes, 1995; Crain and Lee, 1996; Board et al., 2001; Pindyck, 2001; Bae et al., 2004; Basak and Pavlova, 2016; Christoffersen et al., 2019). Meanwhile, financial markets exhibit strong links such as contagion, integration, interdependence, co-movement, return and volatility spillovers (Narayan and Narayan, 2010; Narayan and Sharma, 2011; Diebold and Yilmaz, 2012; Cipollini et al., 2015). More specifically, Pindyck (2001) utilize the data for the petroleum complex from the past two decades to illustrate the link between futures and spot prices, which is a six-month crude oil futures contract that should

underestimate the six-month spot price by approximately 3–4.5 percent. Bae et al. (2004) shows that Korean futures trading increases the spot price volatility and market efficiency, but has a volatility spillover effect on stocks that do not trade. Basak and Pavlova (2016) document that all commodity futures prices, volatility and correlation are related to financialization but are more important for index futures than for nonindex futures. The stock-commodity correlations are also becoming stronger. Financial markets serve as a channel for transmitting external impacts to commodity spot prices. Christoffersen et al. (2019) reveal important characteristics of commodity futures prices and volatility dynamics, and find the factor structure of daily commodity returns is much weaker than the volatility factor structure. Our motivation is straightforward. The commodity futures and stock spots are documented as having strong links such as interdependence, contagion, market comovement, volatility and return spillovers. Following Degiannakis and Filis (2017), they use volatility information from four different asset classes (i.e., Stocks, Forex, Commodities and Macro) to predict the RV of crude oil, suggesting cross-market volatility flows have additional useful information that can significantly improve the predictive accuracy of crude oil RV. In other words, there are information channels between the four different asset classes and the crude oil market. However, there are relatively few relevant studies on forecasting US stock market realized volatility using various types of commodity

* Corresponding author. Sichuan University, 24 South Section 1, Ring Road No.1, Chengdu, Sichuan, Zip code: 610065, China. E-mail addresses: [email protected] (C. Liang), [email protected] (F. Ma), [email protected] (Z. Li), [email protected] (Y. Li). https://doi.org/10.1016/j.econmod.2020.03.022 Received 1 August 2019; Received in revised form 10 March 2020; Accepted 17 March 2020 Available online xxxx 0264-9993/© 2020 Elsevier B.V. All rights reserved.

Please cite this article as: Liang, C. et al., Which types of commodity price information are more useful for predicting US stock market volatility?, Economic Modelling, https://doi.org/10.1016/j.econmod.2020.03.022

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futures volatility flows. Inspired by this lack of studies in this area, our main aim in this paper is to investigate which types of commodity price realized volatility information are more useful for predicting US stock market RV. Following the literature of Diebold et al. (2017), we study six categories of commodities, totally including nineteen sub-indices. We utilized the standard predictive regression framework and monthly realized volatility data to explore the RV predictability of commodity for the next-month RV on S&P 500 over the sample from February 1991 to February 2019. First, we obtain nineteen individual forecasting models by adding the RV of the commodity futures into the benchmark model to examine the predictive ability of each type of commodity futures volatility information in forecasting stock market RV. Second, we utilize the PCA and MA methods to extract the common factors for each type and all types of commodity futures, respectively. Meanwhile, we use a simple average method to combine the individual prediction models of each type and all types. Our prediction results suggest that the futures volatility information of grains and softs has a significant predictive ability for predicting the realized volatility of the S&P 500. The FA method can yield better forecasts than the PCA and average methods in most cases. In addition, we explore the predictability during expansions, recessions, precrises, and postcrises. Our results are consistent with those of Christoffersen et al. (2019) that the grains and softs volatilities exhibit higher informativeness during recessions and precrises. Finally, the empirical results of using five popular combination methods proposed by Rapach et al. (2010) and alternative out-of-sample evaluation period provide consistent evidence, greatly reducing concerns regarding data mining. Our paper contributes the literature from two perspectives. First, this paper is related to a great number of volatility interdependence literature. Such as, Diebold et al. (2017) characterize connectedness in nineteen vital commodity return volatilities based on variance decompositions from high-dimensional vector autoregressions. The study of Christoffersen et al. (2019) use fifteen commodity futures contracts to uncover the common factors in commodity volatility that relate to stock market futures volatility as well as to the business cycle. However, our paper differs these existing studies by focusing on US stock market spot realized volatility based on information of nineteen commodity futures volatilities. Obviously, our paper contributes to the second literature that uses two prevailing methods, namely, PCA and FA, to extract the common factors for each type and all types of commodity futures, respectively. The futures volatility information of grains and softs exhibits preeminent predictability for predicting the monthly RV of the S&P 500. Furthermore, the FA method can yield better forecasts than the PAC and average methods in most cases. The remainder of the paper is organized as follows. We offer the methodology and evaluation method in Section 2. Section 3 describes our data. We report all empirical results in Section 4. In Section 5, we detail the extension and robustness checks. The last Section concludes this research.

post variance, RV includes less noise and is a better measure than the squared monthly returns. In addition, RV is widely used in researches on stock volatility predictions (see, e.g., Christensen and Prabhala, 1998; Corsi et al., 2008; Degiannakis and Filis, 2017; Ma et al., 2019; Zhang et al., 2020). The standard predictive regression framework and monthly RV data are used to explore the RV predictive ability of commodity for the nextmonth RV on S&P 500 spots. Mathematically, the definition of the benchmark model can be written as follows: RVtþ1 ¼ α þ βRVt þ εtþ1 ;

where εtþ1 denotes the error term. To explore the information content of nineteen commodity futures, we extend the benchmark model by adding the RV of nineteen commodity futures as an additional predictor: RVtþ1 ¼ α þ βRVt þ δRVt;i þ εtþ1 ; i ¼ 1; 2; :::; 19;

M RVtþ1 ¼ α þ βRVt þ δCFt;i þ εtþ1 ; i ¼ 1; 2; :::; 7; M ¼ PCA; FA;

c ave;tþ1 ¼ RV

XN RV c k;tþ1 ; k¼1 N

(5)

where d RV ave;tþ1 indicates the average forecast for the US stock market volatility in month tþ1. For example, the grains type includes four commodity futures; thus, N is equal to four. d RV k;tþ1 indicates the individual prediction produced by the kth extension model that utilizes the volatility information from the kth grains commodity futures in month tþ1. 2.2. Forecast evaluation To assess the predictive quality of the extension model, following the studies of Campbell and Thompson (2008), Wang et al. (2018), Liang et al. (2019a), and Zhang et al. (2020), we utilize out-of-sample R2 (R2oos ) test. The R2oos test reflects the percent reduction in the mean squared forecast error (MSFE) of the extension model compared to the MSFE of the benchmark model, which is denoted as follows:

We first construct the monthly RV for all variables used in this study. Consistent with the work of Andersen et al. (2001), Wang et al. (2018), and among others, the definition of the monthly RV can be expressed as follows, r2 ; j¼1 t;j

(4)

M where CFt;i represents the ith common factor of the tth month, and M represents the method of extracting the common factors (i.e., PCA or FA). For example, when i is equal to 7, M is PCA, which refers to the common factor of all commodity futures extracted using the PCA method. Furthermore, we employ a simple average method to combine the individual prediction models to generate combination forecasts. Similarly, we can obtain seven combination models according to Bloomberg’s classification. Statistically, the simple average model can be calculated as

2.1. Benchmark model and extension models

XMt

(3)

where RVt,i represents the RV of the ith commodity futures. Therefore, we can obtain nineteen individual prediction models. In addition, to investigate which types of commodity price information are more useful for predicting US stock market volatility, we use two prevailing methods of PCA and FA, to extract the common factors for each type and all types of commodity futures, respectively. According to Bloomberg’s classification, there are six types of commodity futures. Therefore, we can extract the seven common factors for the six types of commodity futures and all nineteen commodity futures using the PCA and FA methods, respectively. The specification of the model incorporating the common factors as an additional predictor can be expressed as

2. Methodology

RVt ¼

(2)

(1)

Pq c mþk Þ2 ðRVmþk  RV ; R2oos ¼ 1  Pq k¼1 2 c k¼1 ðRVmþk  RV mþk;bench Þ

where RVt represents the RV of the tth month, Mt denotes the total number of trading days in the tth month, and rt,j represents the return of the S&P 500 index in the jth trading day of the tth month. Following the studies of Andersen et al. (2001) and Andersen et al. (2003), relative to ex

(6)

RV mþk and d RV mþk;bench are the S&P 500 RV where RVmþk is actual RV, d predictions of the extension models and the benchmark model in month m þ k, and m and q indicate the length of the first in-sample and the 2

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Table 1 Descriptive statistics of regression variables. Variables

Mean

Std.dev.

Skew.

Kurt.

Max.

Min.

Jbstat

ADFstat

RV Crude Oil Heating Oil Natural Gas Unleaded Gasoline Gold Silver Aluminum Copper Nickel Zinc Live Cattle Lean Hogs Corn Soybeans Soybean Oil Wheat Coffee Cotton Sugar

0.003 0.009 0.008 0.016 0.008 0.002 0.007 0.003 0.005 0.009 0.006 0.002 0.005 0.005 0.004 0.004 0.006 0.011 0.005 0.008

0.005 0.009 0.006 0.013 0.008 0.002 0.008 0.003 0.007 0.010 0.006 0.001 0.004 0.005 0.004 0.003 0.005 0.012 0.004 0.006

7.098 3.656 2.520 2.373 4.047 3.358 4.040 2.596 7.000 4.797 3.102 4.363 3.381 2.400 2.697 3.715 2.490 4.604 2.505 1.946

68.631 19.650 8.813 8.436 24.160 14.551 21.847 8.623 73.348 36.016 15.509 38.623 17.544 9.383 11.334 24.078 8.606 26.862 8.324 6.144

0.057 0.082 0.044 0.104 0.072 0.020 0.069 0.019 0.094 0.105 0.058 0.017 0.037 0.037 0.032 0.036 0.038 0.104 0.028 0.044

0.000 0.001 0.001 0.001 0.001 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.001 0.000 0.000 0.001 0.001 0.002 0.001 0.001

67174.618 *** 6015.784 *** 1412.000 *** 1283.092 *** 8882.435 *** 3516.884 *** 7425.736 *** 1388.522 *** 76250.353 *** 19001.225 *** 3817.734 *** 21438.047 *** 4837.678 *** 1520.804 *** 2156.747 *** 8684.622 *** 1354.499 *** 11035.628 *** 1293.600 *** 724.400 ***

5.380 6.094 4.160 3.864 5.670 5.007 6.950 3.256 5.708 3.905 2.258 7.489 3.297 2.678 8.666 4.284 2.963 4.246 3.330 3.954

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

Notes: Jbstat and ADFstat indicate statistical tests for normal distribution and stationarity. The entire sample contains 337 observations with a time span from February 1991 to February 2019. *** denotes significant at the 1% level.

length of the out-of-sample period, respectively. If the value of R2OOS is greater than 0, indicating that the prediction model of interest can yield superior result than the benchmark model. The significance of R2OOS are calculated by the MSFE-adjusted statistic based on the Clark and West (2007) test. Mathematically, the Clark and West (2007) statistic is expressed as follows:

addition, the monthly RV for each variable is right-skewed and leptokurtic. The Jbstat further demonstrates that these monthly RVs are not normally distributed at the 1% significance level. Moreover, the ADFstat suggests that all the time series are stationary.

c t;bench Þ2  ðRVt  RV c t;model Þ2 þ ð RV c t;bench  RV c t;model Þ2 ; ft ¼ ðRVt  RV

In this study, we only focus on the out-of-sample predictions and ignore the in-sample estimation results. The major reason for this is that market participants, policymakers, and researchers have shown more interest in the out-of-sample prediction accuracy of the prediction model, and they are more concerned about the use of this prediction model in future practical applications. Based on these facts, we put more emphasis on assessing the out-of-sample predictive quality. We employ a prevailing approach of rolling window to yield the out-of-sample predictions. The fixed length of the rolling window contains 48 observations.

4. Empirical results

(7)

where d RV t;bench and d RV t;model denote the S&P 500 RV predictions of the benchmark model and the extension model of interest on month t, respectively. 3. Data In this study, we use the price information of nineteen commodity futures to explore the forecasting power for forecasting the S&P 500 RV. Following the approach of Diebold et al. (2017), we study six categories of commodities (i.e., energy commodities, precious metals, industrial metals, livestock commodities, grains, and softs), including the nineteen subindices of the Bloomberg Commodity Price Index.1 It is worth noting that the category label is not ours; instead, it is the standard for industry participants. The nineteen subindices of the commodity price index specifically include four energy commodities (crude oil, heating oil, natural gas, and unleaded gasoline), two precious metals (gold and silver), four industrial metals (aluminum, copper, nickel, and zinc), two livestock commodities (live cattle and lean hogs), four grains (corn, soybeans, soybean oil, and wheat), and three softs (coffee, cotton, and sugar). We collect daily price data for all 19 commodity indices from Bloomberg. We can obtain the daily price data of the S&P 500 index from Yahoo! Finance.2 The whole sample period containing 337 observations spans February 1991 to February 2019. Table 1 reports the descriptive statistics of the monthly RV for the nineteen commodity futures and S&P 500 that are used in this study. The first column of Table 1 shows all twenty variables used. The last two columns show the Jarque-Bera statistic (Jbstat) and the augmented Dickey–Fuller statistic (ADFstat) of each series. Based on the sample mean, we find that gold and live cattle have the lowest volatility, while natural gas has the largest volatility among the other variables. In

1 2

4.1. Correlation analysis We report the Spearman correlation coefficient between these realized volatilities to alleviate the concern that the RV of S&P 500 is highly correlated with the realized volatility of 19 community futures due to volatility spillover effect in Table 2. First, we observe a positive correlation between the RV of S&P 500 and the realized volatility of 19 community futures. Second, we observe that the realized volatility of most of our commodity futures is weakly correlated with the realized volatility of S&P 500, because the correlation coefficient is relatively small. Only three commodity futures RV (i.e., crude oil, heating oil, and unleaded gasoline) and S&P 500 RV have a correlation coefficient exceeding 0.4. Finally, it is obvious that the correlation between commodity RVs in the same category is relatively high, but the correlation with other types of commodity futures RV is low. 4.2. Out-of-sample results and analysis The forecasts of the nineteen individual models can be generated by adding the monthly RVs of the nineteen commodity futures into the benchmark model. Table 3a reports the R2oos results of the nineteen individual models used, as well as the statistical significance of the R2oos value.3 We find that the five individual prediction models containing five

Based on Bloomberg. finance.yahoo.com.

3

3

The values of R2oos in all tables in this study are expressed as percentages.

Notes: This table reports the Spearman correlation coefficient between these realized volatilities. RV represents the monthly realized volatility of S&P 500 spots, C1 to C19 indicate the monthly realized volatility of 19 commodity futures.

0.319 0.254 0.272 0.099 0.249 0.216 0.128 0.153 0.119 0.229 0.217 0.127 0.320 0.235 0.146 0.220 0.316 0.032 0.323 1.000 0.135 0.090 0.086 0.162 0.075 0.071 0.046 0.133 0.013 0.048 0.155 0.038 0.078 0.026 0.149 0.151 0.058 1.000 0.017 0.032

0.215 0.226 0.243 0.079 0.262 0.203 0.184 0.205 0.162 0.286 0.191 0.210 0.237 0.352 0.382 0.403 0.411 0.017 1.000 0.323

C17

0.229 0.251 0.181 0.037 0.228 0.318 0.290 0.288 0.331 0.367 0.412 0.343 0.301 0.717 0.541 0.376 1.000 0.058 0.411 0.316

C15

0.181 0.266 0.310 0.084 0.334 0.274 0.294 0.192 0.268 0.291 0.266 0.202 0.199 0.473 0.682 1.000 0.376 0.151 0.403 0.220 0.174 0.222 0.213 0.065 0.231 0.317 0.260 0.187 0.287 0.232 0.221 0.202 0.169 0.719 1.000 0.682 0.541 0.149 0.382 0.146 0.212 0.212 0.156 0.036 0.186 0.408 0.380 0.319 0.422 0.345 0.410 0.239 0.113 1.000 0.719 0.473 0.717 0.026 0.352 0.235 0.222 0.320 0.313 0.129 0.353 0.089 0.011 0.010 0.136 0.295 0.198 0.443 1.000 0.113 0.169 0.199 0.301 0.078 0.237 0.320

C14 C13 C12 C11

0.047 0.303 0.285 0.037 0.298 0.183 0.243 0.113 0.226 0.341 0.269 1.000 0.443 0.239 0.202 0.202 0.343 0.038 0.210 0.127 0.143 0.196 0.200 0.106 0.293 0.458 0.508 0.603 0.561 0.622 1.000 0.269 0.198 0.410 0.221 0.266 0.412 0.155 0.191 0.217

C10 C9

0.257 0.341 0.383 0.289 0.446 0.444 0.358 0.464 0.541 1.000 0.622 0.341 0.295 0.345 0.232 0.291 0.367 0.048 0.286 0.229 0.184 0.300 0.298 0.190 0.334 0.367 0.479 0.514 1.000 0.541 0.561 0.226 0.136 0.422 0.287 0.268 0.331 0.013 0.162 0.119

C8 C7

0.104 0.108 0.104 0.090 0.160 0.343 0.453 1.000 0.514 0.464 0.603 0.113 0.010 0.319 0.187 0.192 0.288 0.133 0.205 0.153 0.097 0.182 0.126 0.054 0.188 0.643 1.000 0.453 0.479 0.358 0.508 0.243 0.011 0.380 0.260 0.294 0.290 0.046 0.184 0.128 0.309 0.297 0.284 0.235 0.361 1.000 0.643 0.343 0.367 0.444 0.458 0.183 0.089 0.408 0.317 0.274 0.318 0.071 0.203 0.216 0.419 0.870 0.890 0.390 1.000 0.361 0.188 0.160 0.334 0.446 0.293 0.298 0.353 0.186 0.231 0.334 0.228 0.075 0.262 0.249

C6 C5 C4 C3

0.263 0.295 0.385 1.000 0.390 0.235 0.054 0.090 0.190 0.289 0.106 0.037 0.129 0.036 0.065 0.084 0.037 0.162 0.079 0.099 0.448 0.892 1.000 0.385 0.890 0.284 0.126 0.104 0.298 0.383 0.200 0.285 0.313 0.156 0.213 0.310 0.181 0.086 0.243 0.272 0.477 1.000 0.892 0.295 0.870 0.297 0.182 0.108 0.300 0.341 0.196 0.303 0.320 0.212 0.222 0.266 0.251 0.090 0.226 0.254 1.000 0.477 0.448 0.263 0.419 0.309 0.097 0.104 0.184 0.257 0.143 0.047 0.222 0.212 0.174 0.181 0.229 0.135 0.215 0.319

C2 C1 RV

RV C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19

Table 2 Spearman correlation coefficient.

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C16

C18

C19

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Table 3a The R2oos test results of the nineteen individual models. Classification

Forecasting models

R2oos

MSFE-adjusted

Energy commodities

Crude Oil Heating Oil Natural Gas Unleaded Gasoline Gold Silver Aluminum Copper Nickel Zinc Live Cattle Lean Hogs Corn Soybeans Soybean Oil Wheat Coffee Cotton Sugar

39.571 2.565 2.379 6.642 0.045 10.642 0.588 2.297 5.888 1.846 1.168 14.733 5.431 8.372 * 4.195 * 2.969 0.135 1.778 * 0.138

1.314 1.042 1.046 1.361 0.939 0.894 1.037 0.008 1.102 0.195 0.261 1.279 1.120 1.475 1.615 0.898 0.830 1.351 0.519

Precious metals Industrial metals

Livestock commodities Grains

Softs

Notes: This table presents the R2oos results for the nineteen individual models we use. The fixed length of the rolling window contains 48 observations. * denotes significant at the 10% level.

commodity futures (i.e., gold, corn, soybeans, soybean oil, and cotton), respectively, can generate positive R2oos values. However, the R2oos values for the two individual models containing gold and corn are insignificant. Obviously, the model containing soybean monthly RV can generate the largest R2oos value of 8.372% and is significant at the 10% level, which means that the monthly RV of soybean exhibits the best informativeness. In addition, the two models containing the monthly RV of soybean oil and cotton can generate R2oos values of 4.195% and 1.778%, respectively, and are significant at the 10% level. To explore which types of commodity price information are more effective in predicting the RV of the S&P 500, we utilize the PCA and FA methods to extract the common factors for each type and all types of commodity futures, respectively. Meanwhile, we use a simple average method to combine the individual prediction models of each type and all types. Therefore, we obtain seven PCA models, seven FA models, and seven average models. Table 3b reports the R2oos results of three sets of forecasting models. We can observe several important findings. First, the R2oos value is significantly positive for four prediction models, FA-Grains, FA-Softs, AVE-Grains, and AVE-Softs. In addition, the FA method exhibits better performance in extracting common factors than the PCA method. We can see that the PCA models can generate positive R2oos values, but they are not significant. Furthermore, the common factor of grains can generate more accurate forecasts than that of other commodities. More precisely, the FA-Grains and AVE-Grains models can generate R2oos values of 8.932% and 7.084%, respectively, while the FA-Softs and AVE-Softs models can generate R2oos values of 1.712% and 0.982%, respectively. 4.3. Predictability during expansions and recessions According to the studies of Neely et al. (2014), Feng et al. (2017), and Chatziantoniou et al. (2019), among others, the business cycle may affect the predictability of the market, and the predictability of the stock market will produce different results during different business cycles. Therefore, based on the determination of the NBER, we further analyze the predictive performance of these predictors during different business cycles of expansions and recessions. Panel A and panel B of Table 4a report the R2oos results for the nineteen individual models during expansions and recessions, respectively. First, we find that all nineteen individual models have negative R2oos values during the expansion state. Second, during the recession state, the nine individual prediction models can generate positive R2oos , four of which are 4

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Table 3b The R2oos test results of the PCA, FA, and average models. Forecasting models

R2oos

MSFE-adjusted

Panel A: PCA method

R2oos

MSFE-adjusted

Panel B: FA method 9.070 11.147 2.342 13.822 4.322 0.234 0.519

PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

Forecasting models

1.561 0.927 0.131 1.274 1.247 0.722 1.003

R2oos

MSFE-adjusted

7.879 2.373 1.743 7.099 7.084 * 0.982 * 0.933

1.519 0.946 0.328 1.317 1.350 1.479 0.808

Panel C: Average method 6.572 7.708 2.340 5.856 8.932 * 1.712 * 4.646

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Forecasting models

1.424 1.018 0.033 1.053 1.505 1.513 1.250

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Notes: This table presents the R2oos results of the three sets of models, including seven PCA models, seven FA models, and seven average models. The fixed length of the rolling window contains 48 observations. * denotes significance at the 10% level.

recessions. Consistent with our expectations, all PCA, FA, and average models generate negative R2oos values during expansions. In contrast to Table 3b, during recessions, four models (i.e., FA-Grains, FA-Softs, AVEGrains, and AVE-Softs) can produce higher R2oos values, implying that these four models have better predictive power during recessions. Moreover, the PCA-ALL model can generate an R2oos value of 1.166% and is significant at the 10% level during recessions; that is, the common factor of all commodity futures RV is effective for forecasting the RV of the S&P 500.

Table 4a The R2oos test results of individual models during expansions and recessions. Classification

Forecasting models

R2oos

Panel A: Expansion

Panel B: Recession

Energy commodities

Crude Oil Heating Oil Natural Gas Unleaded Gasoline Gold Silver Aluminum Copper Nickel Zinc Live Cattle Lean Hogs Corn Soybeans Soybean Oil Wheat Coffee Cotton Sugar

3.190 5.674 3.295 1.003

0.429 0.432 0.712 0.855

47.057 1.932 2.193 7.811

1.320 0.950 0.932 1.449

35.415 80.184 11.645 14.555 8.502 9.043 3.228 10.575 3.221 15.658 7.145 2.892 0.709 9.380 5.723

0.226 0.421 0.092 0.346 0.346 0.207 0.325 1.288 0.833 0.110 0.431 0.653 1.002 0.681 0.324

7.298 3.565 1.663 * 0.206 5.354 0.375 0.722 15.598 7.206 13.295 * 6.523 * 2.966 0.033 4.059 ** 1.003

1.001 0.798 1.292 0.518 1.046 0.092 0.155 1.228 1.033 1.510 1.613 0.879 0.026 1.857 0.640

Precious metals Industrial metals

Livestock commodities Grains

Softs

MSFEadjusted

R2oos

MSFEadjusted

4.4. Predictability during pre- and postcrises It is also of interest to explore the predictive power of all used models during pre- and postcrises. We also employ the rolling window method that covers 48 months to generate the forecasts. Then, we select the outof-sample predictions from January 2007 to December 2009 as the precrisis forecasting performance and select the out-of-sample predictions from January 2010 to December 2012 as the postcrisis forecasting performance. Since we focus on the predictability of the RV of the S&P 500, we calculate the R2oos statistics and compare them to determine the precrisis/postcrisis volatility effect. Panel A of Table 5a presents the R2oos statistics calculated for all nineteen individual prediction models during the precrisis period. The R2oos statistics of the four models (i.e., aluminum, soybeans, soybean oil, and cotton) are positive and significant at the 10% level during precrises. In particular, the model containing the monthly RV of soybeans can generate the largest R2oos value of 11.324%. Panel B of Table 5a reports the R2oos statistics calculated for all nineteen individual prediction models during the postcrisis period. We find that only the lean hogs model has a positive R2oos value of 1.248% and is significant at the 10% level. Tables 5b and 5c report the predictive performance of the PCA, FA, and average models during precrises and postcrises, respectively. As expected, the precrisis forecasts are superior to the postcrisis forecasts. More precisely, during precrises, five models—the PCA-ALL, FA-Grains, FA-Softs, AVE-Grains, and AVE-Softs models—have positive R2oos values and are significant at the 10% level. The FA method can produce more

Notes: This table presents the R2oos test results of the nineteen individual models during expansions and recessions. Business cycle expansion and recession are as determined by the NBER. The fixed length of the rolling window contains 48 observations. * and ** denote significant at the 10% and 5% levels, respectively.

significant. In addition, compared to Table 3a, we find that the model containing the monthly RV of aluminum generates an R2oos value of 1.663% and is significant at the 10% level. Furthermore, the three individual models containing the monthly RV of soybeans, soybean oil, and cotton can generate more predictive accuracy. In particular, the model containing the monthly RV of soybeans exhibits the highest R2oos value, at 13.295%. Tables 4b and 4c report the predictive performance of the PCA, FA, and average models by calculating the R2oos values during expansions and Table 4b The R2oos test results of the PCA, FA, and average models during expansions. Forecasting models

R2oos

MSFE-adjusted

Forecasting models

2.996 85.465 10.791 10.018 4.211 5.961 2.700

1.498 0.419 0.388 1.216 0.824 0.331 0.146

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Panel A: PCA method PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

R2oos

MSFE-adjusted

Forecasting models

2.556 80.296 12.807 6.635 12.239 8.003 0.929

0.305 0.352 0.209 1.098 0.246 0.419 1.165

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Panel B: FA method

R2oos

MSFE-adjusted

2.096 42.274 7.433 4.872 3.137 2.741 0.822

0.345 0.077 0.201 0.933 0.456 0.012 0.083

Panel C: Average method

Notes: This table presents the R2oos test results of the three sets of models during expansions, including seven PCA models, seven FA models, and seven average models. Business cycle expansion and recession are as determined by the NBER. The fixed length of the rolling window contains 48 observations. 5

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Table 4c Out-of-sample R2 test results of the PCA, FA, and average models during recessions. Forecasting models

R2oos

MSFE-adjusted

Forecasting models

10.321 4.037 0.618 14.615 6.071 1.486 1.166 *

1.526 0.837 0.326 1.220 1.199 0.895 1.368

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Panel A: PCA method PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

R2oos

MSFE-adjusted

Forecasting models

7.407 7.130 0.203 5.693 13.271 * 3.690 ** 5.785

1.434 0.977 0.408 0.946 1.519 1.922 1.159

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Panel B: FA method

R2oos

MSFE-adjusted

9.074 5.783 0.582 7.549 9.188 * 1.738 ** 1.293

1.533 0.943 0.356 1.262 1.324 1.644 0.798

Panel C: Average method

Notes: This table presents the R2oos test results of the three sets of models during recessions, including seven PCA models, seven FA models, and seven average models. Business cycle expansion and recession are as determined by the NBER. The fixed length of the rolling window contains 48 observations. * and ** denote significant at the 10% and 5% levels, respectively.

Table 5a The R2oos test results of individual models during pre- and postcrises. Classification

R2oos

Forecasting models

MSFE-adjusted

Panel A: Precrisis Energy commodities

46.890 1.987 2.239 7.800 7.285 3.656 1.519 * 0.048 5.342 0.380 0.862 15.761 6.497 11.324 * 5.779 * 3.083 0.031 4.035 * 0.976

Crude Oil Heating Oil Natural Gas Unleaded Gasoline Gold Silver Aluminum Copper Nickel Zinc Live Cattle Lean Hogs Corn Soybeans Soybean Oil Wheat Coffee Cotton Sugar

Precious metals Industrial metals

Livestock commodities Grains

Softs

R2oos

MSFE-adjusted

Panel B: Postcrisis 1.317 0.977 0.908 1.444 1.070 0.828 1.527 0.466 1.043 0.077 0.230 1.236 0.986 1.441 1.516 0.876 0.123 1.867 0.668

2.668 9.192 5.041 0.334 82.634 203.749 28.428 37.393 14.183 15.446 2.871 1.248 * 0.383 12.744 4.077 1.718 1.133 16.975 11.826

1.274 1.042 0.838 0.937 0.462 0.051 0.784 0.789 0.628 0.282 0.308 1.997 1.134 0.431 0.623 0.846 0.898 0.573 0.951

Notes: This table presents the R2oos test results of the nineteen individual models during pre- and postcrises. The fixed length of the rolling window contains 48 observations. The predictions during precrises include the period January 2007 to December 2009, and the predictions during postcrises include the period January 2010 to December 2012. * denotes significant at the 10% level.

Table 5b The R2oos test results of the PCA, FA, and average models during precrises. Forecasting models

R2oos

MSFE-adjusted

Panel A: PCA method PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

Forecasting models

R2oos

MSFE-adjusted

Panel B: FA method 10.318 4.140 0.679 14.715 5.462 1.508 1.184 *

1.522 0.870 0.326 1.226 1.177 0.956 1.391

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Forecasting models

R2oos

MSFE-adjusted

9.060 5.921 0.596 7.675 8.531 * 1.772 * 1.260

1.524 0.999 0.225 1.280 1.280 1.692 0.787

Panel C: Average method 7.404 7.259 0.286 5.653 11.685 * 3.711 * 5.667

1.431 1.028 0.426 0.943 1.455 1.956 1.147

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Notes: This table presents the R2oos test results of the three sets of models during precrises, including seven PCA models, seven FA models, and seven average models. The fixed length of the rolling window contains 48 observations. The predictions during precrises include the period January 2007 to December 2009. * denotes significant at the 10% level.

the US economy, and both their production and export volume rank first in the world. In addition, the FA method can yield better forecasts than the FA and average methods in most cases. The main reason is that the FA method is not a choice of original variables but a recombination based on the information of the original variables to determine the common factors affecting the variables and simplify the data. Further analysis shows that the price information for four commodities (i.e., aluminum, soybeans, soybean oil, and cotton) and two types (i.e., grains and softs) exhibit higher informativeness during recessions and precrises, which is

accurate predictions than the PCA and average methods, which is consistent with the previous analysis. We find that only the R2oos statistic of the PCA-Live model is positive and significant. In summary, we find that the three individual models containing the monthly RV of soybeans, soybean oil, and cotton have significant predictability for the RV of the S&P 500 spot. The price information of two commodities (i.e., grains and softs) exhibits better out-of-sample performance. The possible reason for this is that soybeans and cotton are the main agricultural products of the US, occupying an important position in 6

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Table 5c Out-of-sample R2 test results of the PCA, FA, and average models during postcrises. Forecasting models

R2oos

MSFE-adjusted

Forecasting models

3.291 215.962 24.960 1.205 * 3.648 15.224 8.055

1.059 0.126 0.628 1.886 0.886 0.890 0.685

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Panel A: PCA method

R2oos

MSFE-adjusted

Forecasting models

2.757 199.022 28.957 0.911 9.545 15.140 0.309

1.213 0.132 0.672 0.079 0.541 0.557 0.449

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Panel B: FA method

PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

R2oos

MSFE-adjusted

3.671 104.855 19.945 0.427 0.446 5.834 2.660

1.054 0.348 0.682 0.089 0.736 0.209 0.591

Panel C: Average method

Notes: This table presents the R2oos results of the three sets of models during postcrises, including seven PCA models, seven FA models, and seven average models. The fixed length of the rolling window contains 48 observations. The predictions during postcrises include the period January 2010 to December 2012. * denotes significant at the 10% level. Table 6 The R2oos test results of combination models. R2oos

Forecasting models

Mean Median Trimmed DMSPE (1) DMSPE (0.9)

MSFEadjusted

R2oos

Table 7a The R2oos test results of individual models with different forecasting window. MSFEadjusted

R2oos

MSFEadjusted

Panel A: 48 observations

Panel B: 36 observations

Panel C: 60 observations

1.032 0.394 1.220 1.090 1.074

0.235 0.077 0.476 0.262 0.186

1.326 0.064 1.271 1.340 1.354

0.864 0.859 1.027 0.907 0.879

0.476 0.034 0.584 0.492 0.436

1.078 0.096 1.171 1.096 1.100

1.198 1.194 1.030 1.478 0.843 0.599 1.308 0.499 1.275 0.371 0.424 1.293 0.803 1.369 1.565 0.880 0.853 1.412 1.230

Notes: This table presents the R2oos test results of the nineteen individual models at different forecasting window sizes. The fixed length of the rolling window contains 36 observations. * denotes significant at the 10% level.

We report the predictive performance for five combination models in Table 6. The forecasts in the three panels are produced using a rolling window, with each window covering 48 months, 36 months, and 60 months. Obviously, all five combination models can generate positive R2oos statistics but are insignificant when we use the rolling window that covers 48 months. Moreover, the R2oos statistics of the combination models are also positive and insignificant, except for those of the median model. Overall, the performance of the five combination models for the nineteen individual models is not ideal, implying that all volatility information of the nineteen commodity futures is not effective in predicting the RV of the S&P 500.

In this subsection, we use five combination methods of Rapach et al. (2010) and the forecasts of the nineteen individual models generated above to yield five different combination predictions. As a great deal of the extant literature (Rapach et al., 2010; Zhang et al., 2018, 2019a, 2019c; Liang et al., 2019b; Liu et al., 2019) use the five combination methods and obtain reliable and accurate forecasting results. Mathematically, the combination prediction models can be written as k

51.086 5.024 3.866 15.314 1.915 12.620 0.791 * 1.071 11.883 2.784 1.922 16.238 4.341 2.834 * 6.400 * 4.704 3.787 3.214 * 0.285

Softs

5.1. Combination forecasts

c tþ1 ; ωk;t RV

Crude Oil Heating Oil Natural Gas Unleaded Gasoline Gold Silver Aluminum Copper Nickel Zinc Live Cattle Lean Hogs Corn Soybeans Soybean Oil Wheat Coffee Cotton Sugar

Grains

In this section, we first consider five prevailing combination methods to generate five combination forecasts to explore whether these five combination methods are more effective than the PCA, FA, and average methods. Furthermore, we conduct a series of additional analyses to examine whether our empirical results are robust.

k¼1

MSFE-adjusted

Energy commodities

Livestock commodities

5. Extension and robustness checks

XN

R2oos

Industrial metals

consistent with the results of Christoffersen et al. (2019) that the volatility of commodities in recessions and stock market decline is more closely related to the volatility of other markets.

c

Forecasting models

Precious metals

Notes: This table presents the R2oos test results for five combination models. The fixed lengths of the rolling windows in Panels A, B, and C include 48 observations, 36 observations, and 60 observations, respectively.

c tþ1 ¼ RV

Classification

5.2. Alternative out-of-sample evaluation periods (8) Rossi and Inoue (2012) and Inoue et al. (2017) stress that choosing different window sizes in practical applications may lead to completely different out-of-sample predictions. Therefore, the size of prediction window plays a very important role in the out-the-sample evaluation. We then test the prediction results of the two prediction windows, namely, the fixed length of the rolling window includes 36 and 60 observations, respectively. Table 7a presents the R2oos results of the nineteen individual models with different forecasting window sizes. We find that three individual models (i.e., soybeans, soybean oil, and cotton) have positive R2oos statistics and are significant at the 10% level based on the results in panel A of Table 7a. Table 7b reports the R2oos values of three sets of models, including seven PCA models, seven FA models, and seven average

c where d RV tþ1 represents the combination prediction for the S&P 500 in k

month t þ 1, d RV tþ1 represents the individual RV forecast for the S&P 500 utilizing the monthly RV information from commodity k in month t þ 1, and ωk;t denotes the combining weight for the kth individual prediction. Obviously, five different combined prediction results can be obtained by using five different weight settings. The five combination models are Mean, Median, Trimmed, DMSPE (1), and DMSPE (0.9).4

4 Since these combination methods are already common, we do not report the details of the weight setting. For more introduction, please refer to Rapach et al. (2010) and Zhang et al. (2019b).

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Table 7b The R2oos test results of the PCA, FA, and average models with different forecasting window. Forecasting models

R2oos

MSFE-adjusted

Panel A: PCA method PCA-Energy PCA-Precious PCA-IndMet PCA-Live PCA-Grains PCA-Sofs PCA-ALL

Forecasting models

R2oos

MSFE-adjusted

Panel B: FA method 21.308 11.878 3.924 15.186 0.121 0.676 1.176

1.587 0.678 0.467 1.293 1.204 1.251 0.079

FA-Energy FA-Precious FA-IndMet FA-Live FA-Grains FA-Sofs FA-ALL

Forecasting models

R2oos

MSFE-adjusted

12.910 0.963 2.645 7.806 6.161 * 1.056 0.505

1.460 0.784 0.468 1.346 1.322 1.210 0.609

Panel C: Average method 29.150 1.624 2.619 6.497 5.106 * 1.619 * 0.095

1.379 0.871 0.163 1.021 1.430 1.466 0.936

AVE-Energy AVE-Precious AVE-IndMet AVE-Live AVE-Grains AVE-Sofs AVE-ALL

Notes: This table presents the R2oos test results of the PCA, FA, and average models with different forecasting window. The fixed length of the rolling window contains 36 observations. * denotes significant at the 10% level.

models. There are two major findings. First, three models (i.e., FA-Grains, FA-Softs, and AVE-Grains) have positive R2oos statistics and are significant at the 10% level. Second, the FA method outperforms the PCA and average methods. In conclusion, the empirical results of alternative forecasting window sizes further confirm our results are robust.5

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6. Conclusions In this study, we investigate which types of commodity price information are more useful in predicting the RV of the US stock market. The standard predictive regression framework and monthly RV data are used to explore the RV predictability of commodity futures for the future RV on the S&P 500 spot. First, we examine the predictive power of each commodity futures by incorporating the RV of nineteen commodity futures as an additional predictor into the benchmark model. Second, we utilize PCA and FA to extract the common factors for each type and all types of commodity futures. Meanwhile, we use a simple average method to combine the individual prediction models of each type and all types. Our results provide important evidence that the futures price information of grains and softs has a significant predictive ability for forecasting the RV of the S&P 500 spot. In addition, the factor analysis method can yield better forecasts than the principal component analysis method and average method in most cases. Further analysis shows that the price information of grains and softs exhibits higher informativeness during recessions and precrises, which is consistent with the results of Christoffersen et al. (2019) that the volatility of commodities in a recession and stock market decline is more closely related to the volatility of other markets. In particular, the individual model containing the monthly RV of aluminum exhibits significant predictability during recessions and precrises. Furthermore, we utilize five prevailing combination methods to generate five combination forecasts to confirm that the methods (i.e., PCA, FA, and average methods) we chose are optimal. Finally, the forecasts based on different out-of-sample periods confirm our results are robust. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are grateful to the editor and anonymous referees for insightful comments that significantly improved the paper. This work is supported by the Natural Science Foundation of China [71701170,

5 The predictions are also robust when we use the rolling window covering 60 observations.

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