The volatility of returns from commodity futures: evidence from India
 Isita Mukherjee^{1}Email author and
 Bhaskar Goswami^{1}
Received: 23 September 2016
Accepted: 2 September 2017
Published: 5 September 2017
Abstract
Background
This paper examines the pattern of the volatility of the daily return of select commodity futures in India and explores the extent to which the select commodity futures satisfy the Samuelson hypothesis.
Methods
One commodity future from each group of futures is chosen for the analysis. The select commodities are potato, gold, crude oil, and mentha oil. The data are collected from MCX India over the period 2004–2012. This study uses several econometric techniques for the analysis. The GARCH model is introduced for examining the volatility of commodity futures. One of the key contributions of the paper is the use of the β term of the GARCH model to address the Samuelson hypothesis.
Result
The Samuelson hypothesis, when tested by daily returns and using standard deviation as a crude measure of volatility, is supported for gold futures only, as per the value of β (the GARCH effect). The values of the rolling standard deviation, used as a measure of the trend in the volatility of daily returns, exhibits a decreasing volatility trend for potato futures and an increasing volatility trend for gold futures in all contract cycles. The result of the GARCH (1,1) model suggests the presence of persistent volatility and the prevalence of long memory for the select commodity futures, except potato futures.
Conclusions
The study sheds light on significant characteristics of the daily return volatility of the commodity futures under analysis. The results suggest the existence of a developed market for the gold and crude oil futures (with volatility clustering) and show that the maturity effect is only valid for the gold futures.
Keywords
Commodity futures Daily return Volatility Samuelson hypothesis GARCHBackground
Volatility plays a vital role in derivative pricing, hedging, risk management, and optimal portfolio selection. The concept of volatility relates to the uncertainty or risk about an asset’s value. A higher volatility means that an asset can assume a large range of values, while a lower volatility implies that an asset’s value does not fluctuate dramatically, even though it changes over time. Accurate modeling and forecasting of volatility in asset returns are major issues in financial economics. Derivative markets, particularly commodity futures markets, have become more sophisticated now a day. The futures price depends on the availability of information. A small change in price may have large effects on the trading results across futures markets. Researchers around the world showed increasing interest in the volatility of commodity futures. In the present analysis, an attempt is made to examine the trend and pattern of the volatility of daily returns of few select commodity futures in the Indian context.
As a first step, we examine the characteristics of the commodity futures. In particular, we analyze whether the price variability of a future increases or decreases when the contract approaches maturity. The Samuelson hypothesis for the selected commodity futures is tested. Samuelson (1965) argued that the volatility of the change in futures price increases as the contract approaches maturity. This phenomenon is also called the “Maturity Effect.” The purpose of testing the Samuelson hypothesis is to assess the degree of maturity of Indian commodity futures. From the view point of the Samuelson hypothesis, the prediction of price volatility is very useful for all participants in the futures market, such as hedgers, speculators, and traders. We also address the trend in daily return’s volatility across the contract cycles to decipher the volatility characteristics of the select commodity futures. To this end, we introduce the concept of rolling standard deviation.
We, then, proceed to examine the volatility aspects of the commodity futures. The steps involved in this exercise are the graphical plotting of the daily returns series, followed by its descriptive statistics. The daily returns are tested for stationarity. Then, we explored the GARCH (1, 1) model for the return volatility of the select futures.^{1}
The present paper derives its motivation from the following considerations. First, commodity futures as a financial asset is gaining prominence in the Indian capital market. The uninterrupted transactions in futures contracts from 2004, with a volume of trade surging from Rs 1.29 lakh crore in 2003–2004 to a peak of Rs 181 lakh crore in 2011–2012,^{2} confirms the phenomenal importance of commodity futures. Second, empirically testing the Samuelson hypothesis as an indicator of developed and mature futures market seems necessary for the Indian commodity futures market. One of the key contributions of this paper is to use the GARCH (1,1) process for testing the Samuelson hypothesis on select commodity futures. Testing the Samuelson hypothesis through the β term of the GARCH (1,1) yields meaningful results, as the GARCH (1,1) assumes that the returns are uncorrelated, with zero mean. Moreover, in the GARCH (1,1) process, the present volatility does not depend on past returns, and thereby makes it a suitable methodology to test the Samuelson hypothesis. In this respect, the present analysis aims at filling a gap in the existing literature. Finally, in India, while the volatility issues related to dominant financial assets, such as company shares, have been well researched and documented, only a few studies on commodity futures have been carried out. More specifically, the trend and pattern of the volatility in the daily returns from commodities have been largely ignored in the existing literature. The remainder of this paper is organized as follows. The second section presents the literature. The third section deals with the methodology used in this paper and describes the relevant data sources. The result and discussion of the analysis are carried out in the fourth section. Finally, the fifth section provides our concluding remarks.
Literature review
Many researchers, such as W. R. Anderson (1985), examined the Samuelson hypothesis using selected agricultural futures contracts and found support for wheat, oat, soybeans, and soybeans meal futures. Bessembinder et al. (1996) provided a new framework for the maturity effect, the ‘BCSS hypothesis’ (based on Bessembinder, Coughenour, Seguin and Smoller). This hypothesis is an extension of the Samuelson hypothesis. The authors found that the Samuelson hypothesis is more likely to hold for those commodities whose price changes can be reversed in future. Black and Tonks (2000) investigated the pattern of the volatility of commodity futures prices over time and revealed the conditions which support the Samuelson hypothesis. Allen and Cruickshank (2000) analyzed the Samuelson hypothesis for selected commodity futures of three different futures markets in three different countries. They performed a regression analysis complemented by ARCH models, and the result suggests that the Samuelson hypothesis holds in the case of maximum selected contracts. Floros and Vougas (2006) investigated the Samuelson hypothesis in the context of the Greek stock index futures market and examined the maturity effect through linear regressions and GARCH models. The result of the study suggests that volatility depends on time to maturity and gives a stronger support to the Samuelson hypothesis compared to linear regressions. Duong and Kalev (2008) examined the Samuelson hypothesis for 336 selected commodities from five futures exchanges observed between 1996 and 2003. Using the JonckheereTerpstra Test, OLS regressions with realized volatility, and various GARCH models, the authors find mixed evidence concerning the support for the Samuelson hypothesis. Even though many studies investigated the Samuelson hypothesis, very few contributions analyzed it in the context of the Indian commodity futures market.
Notable exceptions are Verma and Kumar (2010), who examined the application of the Samuelson hypothesis and BCSS hypothesis in the Indian commodity futures market. Gupta and Rajib (2012) also examined this issue for eight commodities, and they concluded that the Samuelson hypothesis does not hold for the majority of the considered commodity contracts.
Numerous studies investigate the volatility of futures prices worldwide.
Locke and Sarkar (1996) examined the changes in market liquidity following changes in price volatility. The results of the study suggest that market makers are most hurt by volatility in the case of inactive contracts. Richter and Sorensen (2002) analyzed a volatility model for soybean futures and options using panel data. The study suggests the existence of a seasonal pattern in convenience yields and volatility, in line with the storage theory. Chang et al. (2012) examined a long memory volatility model for 16 agricultural commodity futures. The empirical results are obtained using unit root tests, GARCH, EGARCH, APARCH, FIGARCH, FIEGARCH, and FIAPARCH model. Manera et al. (2013) examined the effect of different types of speculation on the volatility of commodity futures prices. The authors selected four energy and seven nonenergy commodity futures observed over the period 1986–2010. Using GARCH models, the study suggests that speculation affects the volatility of returns, and longterm speculation has a negative impact, whereas short term speculation has a positive effect. Christoffersen et al. (2014) analyzed the stylized facts of volatility in the postfinancialization period using data of 750 million futures observed between 2004 and 2013.
Two strands in the existing literature focused on volatility in the Indian commodity futures market. First, the literature is largely dominated by spot price volatility and its spillover effect on future price volatility, that is, the price discovery mechanism of the futures market. Brajesh and Kumar (2009 ) examined the relationship between future trading activity and spot price volatility for different commodity groups, such as agricultural, metal, precious metal, and energy commodities in the perspective of the Indian commodity derivatives market. P. Srinivasan (2012) examined the price discovery process and volatility spillovers in Indian spotfutures commodity markets and the result points to dominant volatility spillovers from spot to futures market. Sehgal et al. (2012) examined the futures trading activity on spot price volatility of seven agricultural commodities and found that unexpected futures trading has strong correlation on spot volatility. Chauhan et al. (2013) analyzed the market efficiency of the Indian commodity market. They found that for guar seed, the volatility in futures prices influences the volatility in spot prices and the opposite result holds for chana. The work by Chakrabarti and Rajvanshi (2013) also explored the determinants of return volatility of select commodity futures in the Indian context. Sendhil et al. (2013) examined the efficiency of commodity futures through price discovery, transmission, and the extent of volatility in four agricultural commodities and found persistence volatility in spot market. Kumar et al. (2014) examined the price discovery and volatility spillovers in the Indian spotfutures commodity market. Gupta and Varma (2015) reviewed the impact of futures trading on spot markets of rubber in India and observed bidirectional flow of volatility between spot and futures market. Vivek Rajvanshi (2015) presented a comparative study on the performance of range and returnbased volatility estimators for crude oil commodity futures. Malhotra and Sharma (2016) investigated the information transmission process between the spot and futures market and found that bidirectional volatility spillovers exists between the spot and futures market.
Second, a few studies specifically focus on the volatility of commodity futures. Kumar and Singh (2008) examined the volatility clustering and asymmetric nature of Indian commodity and stock market using S&P CNX Nifty for the stock market, and gold and soybean for the commodity futures market. Kumar and Pandey (2010) examined the relationship between volatility and trading activity for different categories of Indian commodity derivatives. They find a positive and significant correlation between volatility and trading volume for all commodities, no significant relationship between volatility and open interest, and an asymmetric relationship between trading volume and open interest. Kumar and Pandey (2011) examined the cross market linkages of Indian commodity futures with futures markets outside India. However, all these studies focus on the price volatility of commodity futures. In contrast with the abovementioned studies on the Indian commodity futures market, the present study attempts to examine the return volatility of select commodity futures as financial assets.^{3}
Methods
List of traded contracts (in volume and value) of commodity futures in MCX, India
Bullions  Energy  Agricultural products  Oil & oil related products  

Year  Gold^{a}  Silver  Platinum  Crude Oil^{a}  Natural Gas  Gasoline  Potato^{a}  Kapas  Pepper  Mentha Oil^{a}  Mustard seed  RBD Palmolien  
2004  Traded Contracts(in lots)  632,843  138,977 5  NA  NA  NA  NA  NA  1715 8  6406  NA  678  59 
Total Value(in lakhs)  394,070 4.99  449,984 5.34  NA  NA  NA  NA  NA  1197 5.84  4750. 96  NA  1268.4 9  25.43  
2005  Traded Contracts(in lots)  260,040 7  584,476 5  NA  515,781 1  NA  NA  NA  1144 81  1027 08  87,369 3  3826  16,945 
Total Value(in lakhs)  175,513 30.18  196,148 49.81  NA  137,708 85.61  NA  NA  NA  7778 8.06  6997 9.86  18,607 59.84  6708.5 6  23,099.3 5  
2006  Traded Contracts(in lots)  995,735 1  949,854 4  NA  446,653 8  19,537 56  NA  3979 08  3815 73  4431 7  24,282 30  17,073  3759 
Total Value(in lakhs)  760,489 1  506,073 78.39  NA  130,325 62.32  32,625 19.14  NA  7299 85.37  2781 05.65  4612 0.65  55,089 50.19  27,798. 17  13,706.0 5  
2007  Traded Contracts(in lots)  140,242 17  918,327 3  NA  139,388 13  17,327 59  NA  4227 23  2112 87  1924 81  81,674 9  0  NA 
Total Value(in lakhs)  171,474 191.96  515,680 68.06  NA  421,132 66.31  25,869 80.10  NA  7654 16.31  1801 29.77  2509 52.27  16,257 74.77  0.00  NA  
2008  Traded Contracts(in lots)  140,242 17  109,726 76  3790  205,070 01  74,750 6  NA  3900 9  7162 3  2430  42,502 3  NA  NA 
Total Value(in lakhs)  171,474 191.96  704,073 59.66  5133 4.27  859,472 48.64  30,021 86.10  NA  6356 6.98  6958 7.30  3566. 32  84,761 1.43  NA  NA  
2009  Traded Contracts(in lots)  121,449 67  115,555 01  1291 1  410,928 21  11,124 491  5494 2  8996 9  6544 7  4180 3  50,604 2  NA  NA 
Total Value(in lakhs)  184,997 191.41  828,910 95.67  6110 9.94  121,020 964.66  27,497 924.35  2059 35.14  2428 96.24  8029 2.97  4859 4.66  10,114 10.54  NA  NA  
2010  Traded Contracts(in lots)  120,522 25  164,405 33  221  415,370 53  11,176 937  842  4918 52  8759 6  NA  15,710 93  NA  NA 
Total Value(in lakhs)  219,874 783.77  159,664 842.35  1302 .21  150,743 390.24  27,919 327.69  3511. 97  7572 78.05  1168 88.73  NA  50,139 65.14  NA  NA  
2011  Traded Contracts(in lots)  126,557 60  244,345 44  210  547,536 58  98,821 19  20  4671 50  1941 38  NA  15,689 17  NA  NA 
Total Value(in lakhs)  314,713 353.71  408,239 010.89  1325 .36  242,044 737.34  23,293 743.77  107.5 9  8096 64.01  3507 01.82  NA  66,592 52.09  NA  NA  
2012  Traded Contracts(in lots)  102,876 09  172,845 29  21  577,902 29  27,886 670  20  2740 47  3033 23  NA  22,891 39  NA  NA 
Total Value(in lakhs)  305,672 442.56  297,774 497.73  139. 66  289,229 240.48  54,440 421.01  123.6 7  8034 93.25  5644 01.58  NA  12,470 449.09  NA  NA 
In the commodity futures exchanges, trading takes place for 1month, 2month, and 3month contract expiry cycles. However, in India, the 4month, 5month, and up to 1year contract expiry cycles exist, in some cases, and we treat them as unusual exceptions. We only focus on the 1month (near), 2month (nextnear), and 3month (far) expiry cycles for futures. All futures contracts expire on the last Thursday of the month.

The contract starts on July 30, 2010, and expires on October 20, 2010.

Nominal return for 1month contract = ln(closing price on October 20)ln(opening price on October 1); (October 1 is the Friday following the last Thursday of September, with 1 month to expiry, approximately.).

Nominal return for 2month contract = ln(closing price on October 20)ln(opening price on August 27); (August 27 is the Friday following the last Thursday of August, with 2 months to expiry, approximately).

Nominal return for 3month contract = ln(closing price on October 20) ln(opening price on July 30).
Here, the daily return on futures is calculated as the value of the continuously compounded rate of the return multiplied by 100. As such, the Log return of the price series = ln(Ft /Ft1) *100, where Ft and Ft1 are the closing prices on day t and (t1) of a futures contract. The standard deviation of the daily return is also calculated for all the three categories of contract cycles.
Returns of near, next near and far month contracts of mentha oil maturing on 31st December, 2010
Col.1  Col.2  Col.3  Col.4  Col.5  Col.6  Col.7 

Contract/expiry month  Near month  Returns  Next near month  Returns  Far month  Returns 
31Dec10  1Dec10  0.018  1Nov10  0.039  1Oct10  0.012 
31Dec10  2Dec10  0.017  2Nov10  0.039  4Oct10  −0.012 
31Dec10  3Dec10  −0.009  3Nov10  0.003  5Oct10  0.016 
31Dec10  4Dec10  0.005  4Nov10  −0.023  6Oct10  0.015 
31Dec10  6Dec10  −0.030  5Nov10  0.018  7Oct10  0.039 
31Dec10  7Dec10  −0.012  6Nov10  0.032  8Oct10  0.035 
31Dec10  8Dec10  −0.019  8Nov10  −0.018  9Oct10  −0.004 
31Dec10  9Dec10  −0.041  9Nov10  −0.029  11Oct10  −0.013 
31Dec10  10Dec10  −0.028  10Nov10  −0.041  12Oct10  0.012 
31Dec10  11Dec10  −0.041  11Nov10  −0.024  13Oct10  0.039 
31Dec10  13Dec10  0.005  12Nov10  0.003  14Oct10  −0.017 
31Dec10  14Dec10  0.032  13Nov10  0.022  15Oct10  −0.029 
31Dec10  15Dec10  0.012  15Nov10  0.007  16Oct10  0.036 
31Dec10  16Dec10  −0.006  16Nov10  0.011  18Oct10  −0.007 
31Dec10  17Dec10  −0.035  17Nov10  0.000  19Oct10  −0.008 
31Dec10  18Dec10  0.002  18Nov10  −0.004  20Oct10  −0.004 
31Dec10  20Dec10  0.035  19Nov10  −0.030  21Oct10  0.038 
31Dec10  21Dec10  −0.012  20Nov10  0.023  22Oct10  0.039 
31Dec10  22Dec10  −0.011  22Nov10  0.015  23Oct10  −0.006 
31Dec10  23Dec10  0.023  23Nov10  −0.033  25Oct10  0.039 
31Dec10  24Dec10  0.000  24Nov10  0.001  26Oct10  0.039 
31Dec10  27Dec10  0.013  25Nov10  0.011  27Oct10  0.011 
31Dec10  28Dec10  0.008  26Nov10  −0.005  28Oct10  −0.004 
31Dec10  29Dec10  0.009  27Nov10  0.014  29Oct10  0.006 
31Dec10  30Dec10  0.024  29Nov10  0.011  30Oct10  0.011 
31Dec10  31Dec10  0.100  30Nov10  −0.005  –  – 
Std. dev  0.029  0.022  0.021 
We also introduce the concept of 25day moving standard deviation (also known as the rolling standard deviation) as a measure of the trend in the volatility of the daily returns.

The contract starts on July 30, 2010, and expires on October 20, 2010.

We consider the first 25 days starting from July 30, 2010 to calculate the standard deviation.

For the next period, the initial day (July 30, 2010) is left out and 1 day is added to the end of the period (August 24, 2010) so that the 25 days begin from July 31, 2010, and end on August 24, 2010. The standard deviation is calculated for these 25 days.

The above process is repeated for the entire length of the contract cycles to obtain the rolling standard deviation for the concerned futures.

In this example, 25days are considered as the average number of trading days per month (leaving aside Sundays and other holidays). Therefore, the total annual trading days for commodity futures is 305 days.
We then proceed to plot graphically the daily returns series over time so that volatility clustering can be verified.
Descriptive statistics
To analyze the characteristics of the daily return series of the commodity futures market during the study period, the descriptive statistics show the mean (X), standard deviation (σ), Skewness (S), Kurtosis (K), and JarqueBera statistics results.
Test for stationarity
Test for heteroskedasticity
The presence of heteroskedasticity in asset returns has been well documented in the existing literature. If the error variance is not constant (heteroskedastic), then, the OLS estimation is inefficient. The tendency of volatility clustering in financial data can be well captured by a Generalized Autoregressive Conditional Heteroskedastic (GARCH) model. Therefore, we modeled the timevarying conditional variance in our study as a GARCH process.
The ARCH model as developed by Engle (1982) is an extensively used timeseries models in the financerelated research. The ARCH model suggests that the variance of residuals depends on the squared error terms from the past periods. The residual terms are conditionally normally distributed and serially uncorrelated. A generalization of this model is the GARCH specification. Bollerslev (1986) extended the ARCH model based on the assumption that forecasts of the timevarying variance depend on the lagged variance of the variable under consideration. The GARCH specification is consistent with the return distribution of most financial assets, which is leptokurtic and it allows long memory in the variance of the conditional return distribution.
The Generalized Arch Model (GARCH)
The GARCH model (Bollerslev 1986) assumes that the volatility at time t is not only affected by q past squared returns but also by p lags of past estimated volatility. The specification of a GARCH (1, 1) is given as:
Wald test
The Wald test estimates the test statistic by computing the unrestricted regression equation, without imposing any coefficient restrictions, as specified by the null hypothesis. The Wald statistic (under the null hypothesis) measures how the unrestricted estimates satisfy the restrictions. If the restrictions are valid, then, the unrestricted estimates should fulfill the restrictions.
Under the null hypothesis H˳, the Wald statistic has an asymptotic χ^{ 2 } (q) distribution, where q is the number of restrictions.
The result of the above tests is derived using Eviews 7.
Result and discussion
Volatility of daily returns
Commodity futures  1 month contract  2 month contract  3 month contract  

S.D(%)  Rolling s.d  S.D(%)  Rolling s.d  S.D(%)  Rolling s.d  
Potato  1.96  Decreasing  2.19  Decreasing  3.79  Decreasing 
Crude oil  1.86  Constant  1.71  Marginal increase  2.4  Decreasing 
Mentha oil  2.56  Constant  2.16  Decreasing  4.65  Constant 
Gold  1.06  Marginal increase  0.97  Increasing  1.01  Increasing 
Table 3 also presents the rolling standard deviation of the four commodity futures for all the three types of contract cycles.
For potato futures, there is a decreasing trend in volatility for near, next near, and far month contracts, with near contract exhibiting the least declining trend in volatility, and far month contract showing the maximum declining trend in volatility.
For crude oil and mentha oil futures, the near month volatility trend of daily returns is almost constant, and the magnitude of rolling standard deviation (volatility trend) is the highest for the far month contract.
For gold futures, the trend in volatility is increasing for all types of contract (1month, 2month, and 3month). Moreover, this rise in the trend in volatility is greater for the 1month contract, suggesting that the gold futures trend is more volatile as the contract approaches the maturity date.
Simple Statistics for all three types of contracts for four commodities
Mean  Median  Maximum  Minimum  Std. Dev.  Skewness  Kurtosis  JarqueBera  Probability  

Contracts  
Potato  Near  0.000978  0.002409  0.285775  −0.406582  0.019603  −3.660024  181.8045  1,802,719  0.000000 
Next Near  0.00158  0.002358  0.264099  −0.437677  0.021901  −4.900318  182.2342  1,440,545  0.000000  
Far  0.000579  0.002266  0.47847  −0.557147  0.037906  −2.459749  121.0494  449,624.1  0.000000  
Mentha oil  Near  0.00275  0.000226  0.150534  −0.094422  0.025606  1.582898  11.14921  6391.611  0.000000 
Next Near  −3.34E05  0.000000  0.101813  −0.410464  0.021692  −4.171503  77.02822  461,557  0.000000  
Far  −0.002403  0.000000  0.669725  −0.591267  0.046594  −3.343164  83.00441  525,838.3  0.000000  
Crude oil  Near  −0.000249  0.000621  0.088606  −0.094389  0.018635  −0.228756  5.63786  628.0626  0.000000 
Next Near  −0.000128  0.000763  0.084266  −0.09662  0.017191  −0.276003  5.628188  634.6635  0.000000  
Far  −0.000574  0.000362  0.33352  −0.242462  0.024035  −0.12481  48.82034  180,300.2  0.000000  
Gold  Near  0.000515  0.000677  0.081194  −0.064016  0.010629  −0.235882  9.932995  2690.098  0.000000 
Next Near  0.000429  0.000797  0.051936  −0.065173  0.009777  −0.649765  8.615752  1582.362  0.000000  
Far  0.000434  0.000511  0.08112  −0.08509  0.010107  −1.026436  17.93436  12,290.4  0.000000 
Correlogram test (upto 24 lags)
Commodity futures →  Potato  Mentha oil  Crude oil  Gold  

Contract types →  Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far 
Residuals are serially correlated  No  No  Yes  Yes  Yes  No  No  No  No  No  No  No 
Result of unit root test
Commodity futures →  Potato  Mentha oil  Crude oil  Gold  

Contract types →  Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far 
ADF Test Statistic  −28.242  −31.121  −27.875  −41.075  −45.046  −5.682  −43.527  −43.770  −43.860  −37.804  −32.710  −34.323 
Prob.  0.000  0.000  0.000  0.000  0.0001  0.000  0.000  0.0001  0.0001  0.000  0.000  0.000 
Philips Perron Test Statistic  −36.475  −31.120  −27.875  −41.179  −44.800  −44.315  −43.525  −43.781  −43.860  −37.821  −32.692  −34.331 
Prob.  0.000  0.000  0.000  0.000  0.0001  0.0001  0.000  0.0001  0.0001  0.000  0.000  0.000 
Test critical value  
1%  −3.434  −3.436  −3.438  −3.433  −3.433  −3.433  −3.433  −3.433  −3.433  −3.435  −3.435  −3.435 
5%  −2.863  −2.864  −2.865  −2.862  −2.862  −2.862  −2.862  −2.862  −2.862  −2.863  −2.863  −2.863 
10%  −2.567  −2.568  −2.568  −2.567  −2.567  −2.567  −2.567  −2.567  −2.567  −2.567  −2.568  −2.567 
Both the test statistics reported in Table 6 reject the null hypothesis at the 1% significance level, with the critical value of −3.43 for both the ADF and PP tests. These results confirm that the series are stationary.
The graphs of daily returns confirm the absence of a clustering effect for potato futures and menthe oil futures. Only the 3 month contracts for menthe oil futures exhibits a small clustering effect for some periods. The graphs of crude oil and gold futures for all types of contracts show that the daily return series exhibits a clustering effect or volatility.
Result of ARCHLM test for residuals
Potato  Mentha oil  Crude oil  Gold  

Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far  
Obs Rsquared  0.604  0.046  0.010  4.45 8  30.52 2  0.037  78.95 5  64.85 4  0.099  48.72 8  26.39 4  142.6 915 
Prob. ChiSquare  0.437  0.831  0.919  0.03 5  0.000  0.847  0.000  0.000  0.752  0.000  0.000  0.000 
Estimated result of GARCH (1,1) Model
Commodity futures→  Mentha oil  Crude Oil  Gold  

Contract types →  Next Near  Far  Near  Next Near  Far  Near  Next Near  Far 
Coefficients↓  
Mean  
μ (constant)  −4.30E − 05^{c}  −0.002299^{b}  8.02E05^{c}  0.000215^{c}  0.00044^{c}  0.000251^{c}  0.000436^{c}  −5.84E − 05^{b} 
Variance  
ω (constant)  5.01E − 06^{b}  0.000315^{b}  2.34E06^{b}  1.72E06^{b}  5.14E05^{b}  1.60E06^{b}  1.35E06^{b}  4.82E06^{b} 
α (arch effect)  0.099648^{b}  0.003595^{b}  0.032133^{b}  0.030668^{b}  0.00364^{b}  0.071647^{b}  0.084121^{b}  0.217631^{b} 
β (garch effect)  0.891336^{b}  0.856938^{b}  0.960178^{b}  0.962725^{b}  0.9111^{b}  0.917732^{b}  0.903237^{b}  0.775297^{b} 
α + β  0.990984  0.853343  0.992311  0.993393  0.90746  0.989379  0.987358  0.992928 
Log likelihood  5376.339  3241.84  5582.369  5772.936  4778.326  4315.175  3862.321  4295.068 
Akaike info. Criterion (AIC)  −5.384801  −3.307962  −5.30673  −5.464648  −4.6343  −6.45236  −6.75537  −6.61537 
Schwarz info. Criterion (SIC)  −5.370770  −3.293707  −5.29329  −5.451254  −4.62063  −6.43291  −6.73330  −6.59545 
Residual Diagnostics for GARCH (1, 1):ARCHLM (1) test for heteroskedasticity  
Obs^{a} Rsquared  1.204135  0.025106  0.264945  0.057879  0.003309  2.001935  0.12552  0.617633 
Prob. ChiSquare(1)  0.2725  0.8741  0.6067  0.8099  0.9541  0.1571  0.7231  0.4319 
Wald Test  
Fstatistic  1433.702  1191.443  1497.929  1524.253  2935.317  10.09822  9.188416  3.26E34^{a} 
Probability  0.000  0.000  0.000  0.000  0.000  0.0015  0.0025  1.000 
The constant (ω), ARCH term (α), and GARCH term (β) are statistically significant at the 1% level. In the variance equation, the estimated β coefficient is considerably greater than the α coefficient, which implies that the volatility is more sensitive to its lagged values. The result suggests that the volatility is persistent. Moreover, the β term is greater for the near month contract cycles for gold futures, which confirms the validity of the Samuelson hypothesis. The sum of these coefficients (α and β) is close to unity, which indicates that a shock will persist for many future periods, suggesting the prevalence of long memory. However, the Wald test indicates the acceptance of the null hypothesis that α + β = 1 for far month contract cycles of gold futures only.
To check the robustness of the GARCH (1,1) model, we employed the ARCHLM test (Engle 1982) to verify the presence of any further ARCH effect. As shown in the Table 7, the ARCH LM test statistic for the GARCH (1,1) model does not show any additional ARCH effect in the residuals of the model, which implies that the variance equation is well specified for the select commodity futures.
As a result, we can conclude that, among the select commodity futures, the clustering effect is present in the volatility of daily returns for crude oil and gold commodity futures in all contract cycles. Mentha oil futures also present a clustering effect in far month contracts.
Conclusions
This paper addresses the volatility of four select commodity futures: potato, mentha oil, crude oil, and gold. All the three types of contract cycles (near month, next near month, and far month) are considered for volatility analysis. The conventional approach based on standard deviation as a measure of volatility is considered to test the Samuelson hypothesis. To further corroborate the findings, the βterm of the GARCH (1,1) is also used to verify the Samuelson hypothesis. The results suggest that the Samuelson hypothesis does not hold for the select commodity futures in the Indian context, except for the gold futures. These results are in line with the findings of Gupta and Rajib (2012) and suggest that the Indian gold futures market is as developed as in the advanced countries.
The trend in the volatility of daily returns is captured by the concept of rolling standard deviation. The volatility trends in crude oil and mentha oil futures highlight the significance of the available information as the far month volatility is higher than the near month volatility. The fluctuations in the world markets for oil commodities have a lagged impact on the domestic market. Finally, the objective of futures market in terms of price discovery and hedging against future risks seems to be satisfied for potato futures. To test the presence of a unit root in the daily return series, we performed the ADF and PP tests. The results confirmed the stationarity of the daily return series for all the commodity futures.
For volatility modeling, we first considered the graphical representation of volatility clustering along with the descriptive statistics for all contract cycles of each commodity future. We, then, introduced a correlogram to check for serial correlation in the residuals, and, finally, the ARCHLM test was conducted to check for the presence of an ARCH effect. All contract cycles of potato futures did not show any volatility clustering, and the result of the ARCHLM test ruled out any ARCH effects in the daily return series. However, for all types of contract cycles of gold futures, we found unambiguous volatility clustering, and the ARCHLM test results also suggested the presence of an ARCH effect. These results are in line with the findings of Kumar and Singh (2008) for gold futures.
For mentha oil and crude oil futures, the result obtained from the volatility clustering and ARCH LM test was ambiguous for different contract cycles. Although the result of the ARCHLM test implied no ARCH effect for the far month of mentha oil and crude oil futures, a trace of volatility clustering was observed in the daily return graph. Hence, we considered the far month contracts of mentha oil and crude oil futures for the GARCH analysis.
Furthermore, the result of the GARCH (1,1) model shows that three parameters, the constant(ω), ARCH (α) term, and GARCH (β) term, are significant at the 1% level. In the variance equation, the estimated β coefficient is greater than the α coefficient, which implies that the volatility is more sensitive to its lagged values. Hence, the volatility is persistent. The sum of these coefficients (α and β) are close to the unit, which suggests that a shock will persist for many future periods. This is particularly true for gold futures of far month contract, in line with the findings of Kumar and Singh (2008).
The volatility clustering effect shows that the crude oil and gold futures markets are rather similar. The crude oil futures market is largely dependent on the global market conditions, which are highly volatile. The spillover effect of global volatility has an impact on the Indian crude oil futures market. Other significant macroeconomic variables (such as the interest rate, exchange rate, and so on, which are fluctuating in nature) have a significant impact on gold futures market in India. Thus, after examining the Samuelson hypothesis and volatility features, we concluded that, out of the selected commodity futures, gold futures are well developed and organized in the Indian market.
The aim of this paper is to portrait the simplest form of return volatility of the select commodity futures. Therefore, advanced volatility models (like EGARCH, TGARCH, PGARCH) are not considered, although the inclusion of such models would definitely enrich the present study.
The factors affecting the return volatility of commodity futures (like trading volume and open interest) are not under the purview of the present study as that would unnecessarily complicate and shift the focus out of the presented issue.
Identical results hold for gold futures, for which we test the Samuelson hypothesis using the β term of GARCH (1, 1) model as a measure of volatility, as reported in Table 8.
The graph for the next near month contract of menthe oil shows volatility clustering although the JarqueBera value suggests that the residuals are not normally distributed. In addition, the correlogram shows that the residuals are serially correlated. Therefore we perform the ARCHLM test and we observe the presence of ARCH effect.
Although the result of the ARCHLM test implies no ARCH effect for the far month contract of mentha oil and crude oil futures, a trace of volatility clustering is observed in the daily return graph. Hence, we also consider the far month contracts of mentha oil and crude oil futures for the GARCH analysis.
Declarations
Acknowledgements
The authors are indebted to three anonymous referee of this journal for their constructive comments of on the earlier draft of the manuscript. However, the usual disclaimer applies.
Funding
There is no financial assistance received in carrying out this particular research activity.
Availability of data and materials
The dataset is obtained from the publicly available repository, MCX, India website.
Authors’ contributions
BG initiated the thematic concept of the current research while IM carried out the exercise using statistical tools and techniques with the help of EViews 7. Both authors read and approved the final manuscript.
Ethics approval and consent to participate
Not Applicable.
Consent for publication
Not Applicable.
Competing interests
The authors declare that they have no competing interests.
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