1 DEPARTMENT OF ECONOMICS ISSN DISCUSSION PAPER 20/14 Unit root properties of natural gas spot and futures prices: The relevance of heteroskedasticity in high frequency data Vinod Mishra and Russell Smyth Abstract We apply a unit root test that allows for heteroskedasticity and two structural breaks to daily data for natural gas spot and futures prices. In contrast to previous findings, the main result is that natural gas spot and futures prices are mean reverting. More generally, the results emphasize the need to allow for heteroskedasticity when applying unit root tests to energy spot and futures prices with high frequency data, such as daily data. Keywords: Natural gas spot and futures prices, unit root, heteroskedasticity Department of Economics, Monash University. Department of Economics, Monash University. Address for correspondence: Russell Smyth, Department of Economics, Monash University 3800 Australia. Telephone: We thank Paresh Narayan for providing the GAUSS codes for the Narayan and Popp (2010) and Narayan and Liu (2013) unit root tests Vinod Mishra and Russell Smyth All rights reserved. No part of this paper may be reproduced in any form, or stored in a retrieval system, without the prior written permission of the author. 1
2 1. Introduction There is a large literature which tests for a unit root in spot and futures energy prices (Barros et al., 2014; Elder & Serletis, 2008; Ghoshray & Johnson, 2010; Lee et al., 2006; Lee & Lee, 2009; Presno et al., 2014; Maslyuk & Smyth, 2008; Ozdemir et al., 2013; Sadorsky, 1999; Serletis, 1992). The results from existing studies have been mixed. Some studies have concluded that spot and/or futures energy prices are stationary (Elder & Serletis, 2008; Lee et al., 2006; Lee & Lee, 2009; Sadorsky, 1999; Serletis 1992). Other studies have concluded that spot and/or futures energy prices are non-stationary or persistent or find mixed evidence that spot and future prices are stationarity (Barros et al., 2014; Ghoshroy & Johnson, 2010; Maslyuk & Smyth 2008; Ozdemir et al., 2013; Pindyck, 1999; Presno et al. 2014). The issue of whether spot and future energy prices contain a unit root has several important implications for forecasting, investors and the transmission of non-stationarities throughout the economy. If energy prices are mean reverting, it is possible to forecast energy prices into the future, but if energy prices exhibit a random walk it is not possible to forecast future prices based on past prices. If energy prices contain a unit root, the market is weak form efficient, but if energy prices are mean reverting, it is possible for investors to make a profit through predicting price movements based on historical data. If energy prices contain a unit root, non-stationarities will be transmitted throughout the economy given the close links between movements in energy prices and the real economy. When testing for a unit root in spot and future prices, researchers typically employ high frequency data (such as daily or weekly data). High frequency financial data is subject to heteroskedasticity. Failure to account for heteroskedasticity in the data lowers the power to reject the unit root null hypothesis (Narayan & Liu, 2013).
3 In this paper we test for a unit root in natural gas spot and futures prices using daily data. Our main motivation is to illustrate the importance of accommodating heteroskedasticity when testing for a unit root in high frequency spot and futures energy price data. We focus on natural gas to illustrate our point, given that there is much evidence in the existing literature that has employed unit root tests that natural gas prices follow a random walk or exhibit high persistence, even after allowing for structural breaks (see eg. Barros et al., 2014; Ferreira et al., 2005; Lee & Lee, 2009; Pindyck, 1999; Presno et al. 2014). For the purposes of benchmarking, we begin with conventional unit root and stationarity tests - Augmented Dickey Fuller (ADF) (Dickey & Fuller, 1979), Phillips-Perron (1988) and KPSS (Kwiatkowski et al., 1992) in addition to two well-known unit root tests with two structural breaks; namely, Lee and Strazicich (2003) and Narayan and Popp (2010). 1 To illustrate the importance of accommodating heteroskedasticity in addition to structural breaks, we employ the Narayan and Liu (2013) unit root test Method 2.1. Data The data consisted of daily observations for spot and future prices for natural gas for all trading days between 01 January 1997 and 03 March 2014 (the latest date available at the time of this study) collected from the U.S. Energy Information Administration (EIA) 1 The Lee and Strazicich (2003) and Narayan and Popp (2010) tests have been shown to have better size and power properties, and to estimate the break dates more accurately, than alternative two break unit root tests, such as Lumsdaine and Papell (1997) (see Lee & Strazicich, 2003; Narayan & Popp, 2013). 2 Other applications of the Narayan and Liu (2013) test (or earlier working paper versions) in the commodities/energy area are Mishra and Smyth (2014), Narayan and Liu (2011), Salisu and Fasanya (2013) and Salisu and Mobolaji (2013). Of these, Salisu & Fasanya 2013 is the only previous study to apply Narayan and Liu (2013) to energy prices (oil prices). But, they do not consider energy futures prices. There are no previous applications of Narayan and Liu (2013) to natural gas spot or futures prices.
4 website. 3 This is a total of 4294 observations. The spot prices are the Henry Hub natural gas spot prices and the futures prices are for the 1, 2, 3 and 4-month futures contracts. All prices are expressed in dollars per million Btu. All the analysis was performed on the natural log of the series. The time series for the natural gas spot and futures price series are graphed in Figures 1 and 2. Table 1 presents descriptive statistics for daily spot and futures prices for natural gas for 1-4 month contracts over the sample period. [Insert Table 1 and Figures 1 & 2 here] 2.2. Unit root tests We do not reproduce the method for the ADF, Phillips-Perron and KPSS tests, nor the unit root tests with structural breaks proposed by Lee and Strazicich (2003) or Narayan and Popp (2010), given that these tests are well known in the literature. We outline the method in Narayan and Liu (2013) because it is relatively new and still in working paper form. High frequency data is often characterised by time varying volatility, also known as ARCH/GARCH effects. Narayan and Liu (2013) argued that structural break unit root tests based on standard linear models (independently and identically distributed (iid) innovations) are inappropriate if the data has significant ARCH effects. Narayan and Liu (2013) propose a unit root test with two endogenous structural breaks that specifically include a GARCH(1,1) model in the data generating process. This test allows for two structural breaks in the intercept of the time series and jointly estimates the autoregressive parameter (for an ADF based unit root test) and the GARCH parameter using maximum likelihood estimation. The data generating process is specified as: 3 (last accessed March 15, 2014).
5 Here, are break dummy coefficients. follows the first order GARCH (1,1) model of the form: Here, is a sequence of iid random variables with zero mean and unit variance. The unknown break dates ( ) are estimated using a sequential procedure. Initially, the location of the first break date is determined using the maximum absolute t-value of the break dummy coefficient, as follows: Next, imposing the first break estimate as given, a second break date can be searched: The 5 per cent critical values based on 50, 000 replications for N = 500 are given in Narayan and Liu (2013). 4 It is to be noted that the critical values depend on the relative location of the two breaks and the value of the two GARCH parameters. 3. Results Table 2 presents the results for the ADF, Phillips-Perron and KPSS tests with an intercept and with an intercept and trend. Overall, there is strong evidence that natural gas spot and futures prices contain a unit root. Table 3 presents the results for the Lee and Strazicich (2003) lagrange multiplier (LM) unit root test with two breaks in the intercept and two breaks in the intercept and trend. In each case, the Lee and Strazicich (2003) unit root test concludes that 4 These are also reproduced at the bottom of Table 5 below.
6 the spot and futures prices contain a unit root. Table 4 presents the results for the Narayan and Popp (2010) unit root test with two breaks in the intercept and two breaks in the intercept and trend. The unit root null cannot be rejected for any series at the 5 per cent level or better. [Insert Tables 2-4 here] Each of these tests ignore the possible existence of heteroskedasticity in the data. The last column of Table 1 reports the result of the ARCH LM test to examine whether there is heteroskedasticity in the spot and futures prices series. In order to conduct this test, we first filtered the each series using an AR(1) model, then used the residuals to run an ARCH LM test. The null hypothesis of no arch effect was rejected at the 5 per cent level; indicating the presence of significant time varying volatility in the natural gas spot and futures price series. As the data was of daily frequency, the choice of filtering it with an AR(1) seems the obvious choice; however, as a robustness check we also repeated the ARCH LM test after filtering the data using an AR(5) model (to capture weekly seasonality) and an AR(24) model (to capture monthly seasonality). The results were found to be robust to choice of different lags for filtering the series. We also computed the Durbin-Watson statistics after each filtering and it ranged between , suggesting that the filtered data is independent of autocorrelations in each case. As the models with different AR lags produced the same results, we only report the most parsimonious AR(1) model in the paper. Table 5 reports the results of the Narayan and Liu (2013) test with two structural breaks in the intercept. The Narayan and Liu (2013) test suggests that natural gas spot and futures prices are mean reverting in each case at the 5 per cent level. [Insert Table 5 here]
7 Most of the breaks across the three tests occur in 1998, 1999, 2000, 2001 or The breaks in the 4-month futures contract with the Narayan and Popp (2010) test occur in Figure 3 plots the time series for natural has spot prices with breaks in the intercept showing the breaks identified by the three unit root tests with structural breaks which we employ. 4. Discussion When we use conventional unit root tests with up to two structural breaks we find that natural gas spot and futures prices contain a unit root. When we employ a unit root test that simultaneously accommodates structural breaks and heteroskedasticity, we find that natural gas spot and futures prices are mean reverting. Our results highlight the importance of accommodating heteroskedasticity and suggest that just allowing for structural breaks is not sufficient to reject the unit root null in natural gas spot and futures prices. Our results are consistent with previous studies that have found natural gas prices contain a unit root when applying tests that do not allow for heteroskedasticity (Ferreira et al., 2005; Lee & Lee, 2009; Pindyck, 1999; Presno et al. 2014). It is accommodating heteroskedasticity that is driving the results. More generally, the results point to the need to accommodate heteroskedasticity when testing for a unit root in high frequency energy spot and futures price data. The break dates can be linked to global economic events that have affected global energy and financial markets and are similar to those identified in previous studies (see eg. Maslyuk & Smyth, 2008; Salisu & Fasanya, 2013). The break dates in 1998 and 1999 coincide with the Asian financial crisis. The break dates in 2001 coincide with the 9/11 terrorist attacks on the World Trade Centre in New York. The breaks in 2003 could be linked to the outbreak of the Gulf War in Iraq, which commenced in March The breaks in 2009 coincide with the global financial crisis. 5. Conclusion and policy implications
8 The main conclusion is that when one employs a unit root test that simultaneously allows for heteroskedasticity and multiple structural breaks, natural gas spot and futures prices are mean reverting. While this result is specifically for natural gas spot and futures prices, more broadly it illustrates the importance of accommodating heteroskedasticity when testing for a unit root in high frequency energy spot and futures prices data. This main result for natural gas prices has the following implications. First, the weak form of the efficient market hypothesis is rejected and there is potential for investors to make profits through trading in natural gas futures using technical analysis. Moreover, it is possible to forecast natural gas prices based on historical data. The major policy implication here is that in circumstances in which investors can analyse price movements to make profits, there is greater justification (and need) to regulate natural gas markets, relative to the situation if natural gas markets were weak form efficient. Second, that natural gas prices are mean reverting suggests that shocks to energy prices due to events such as 9/11 or the global financial crisis will only result in temporary deviations from the long-run growth path and that natural gas prices will return to their long run equilibrium. It follows that given natural gas prices are stationary, non-stationarities will not be transmitted from the energy market to the real economy. If non-stationarities are not transmitted to the real economy, economic theories, such as real business cycles, are not affected by shocks occurring in energy markets. This, in turn, has implications for the g v m f macroeconomic policies. If there is no transmission of hysteresis from energy prices to the goods and labour markets, ceteris paribus, there is less need for Keynesian demand management policies because output will be mean reverting.
9 References Barros, C.P., Gil-Alana, L.A., Payne, J.E. (2014). Long range dependence and breaks in energy prices. Energy Sources, Part B: Economics, Planning and Policy, 9, Dickey, D.A. and Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (366): Elder, J. and Serletis, A. (2008). Long memory in energy futures prices. Review of Financial Economics, 17, Ferreira, P., Soares, I., Araujo, M. (2005). Liberalization, consumption, heterogeneity and the dynamics of energy prices. Energy Policy, 33, Ghoshray, A. and Johnson, B. (2010). Trends in world energy prices. Energy Economics, 32, Kwiatkowski, D., Phillips, PCB., Schmidt, P. and Shin, P. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, Lee, C-C. and Lee, J.D. (2009) Energy Prices, multiple structural breaks and efficient market hypothesis, Applied Energy, 86, Lee, J., List, J.A. and Strazicich, M.C. (2006). Non-renewable resource prices: Deterministic or stochastic trends. Journal of Environmental Economics and Management, 51, Lee, J. and Strazicich, M. C. (2003) Minimum Lagrange multiplier unit root test with two structural breaks, Review of Economics and Statistics, 85, Lumsdaine, R. and Papell, D. (1997) Multiple trend breaks and the unit root hypothesis, Review of Economics and Statistics, 79, Maslyuk, S. and Smyth, R. (2008) Unit root properties of crude oil spot and futures prices, Energy Policy, 36, Mishra, V. and Smyth, R. (2014) Is monthly US natural gas consumption stationary? New evidence from a GARCH unit root test with structural breaks, Energy Policy, in press. Narayan, PK. and Liu, R. (2011) Are shocks to commodity prices persistent? Applied Energy, 88, Narayan, PK. and Liu, R. (2013). New evidence on the weak-form efficient market hypothesis. Working Paper, Centre for Financial Econometrics, Deakin University Narayan, PK. and Popp, S. (2010). A new unit root test with two structural breaks in level and slope at unknown time. Journal of Applied Statistics, 37, Narayan, PK. and Popp, S. (2013). Size and power properties of structural break unit root tests. Applied Economics, 45, Ozdemir, Z.A., Gokmenoglu, K. and Ekinci, C. (2013). Persistence in crude oil spot and futures prices. Energy, 59,
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11 Tables and Figures Table 1: Descriptive statistics of natural gas spot and futures prices during the sample period Series Observations Mean Std. Dev. Min. Max. Skewness ARCH(1) LM Test Spot Price *** Futures 1M *** Futures 2M *** Futures 3M *** Futures 4M *** Notes: Sample consisted of daily data on natural gas spot and futures prices for all the trading days for the period 07 January March All the prices are expressed in US Dollars per Million Btu. All further analysis was conducted using the natural log of the series. Results for the ARCH LM test were filtered using a AR(1) model, given the data is daily, and the residuals were tested for ARCH effects.
12 Table 2: Results of traditional unit root and stationarity tests Intercept Intercept and Trend Series ADF Test PP Test KPSS Test ADF Test PP Test KPSS Test Spot Price *** *** Futures 1M *** *** Futures 2M *** *** Futures 3M *** *** Futures 4M *** *** Notes: ADF= Augmented Dickey Fuller; PP= Phillips Perron; KPSS is the Kwiatkowski et al., 1992) stationarity test. *** denotes significant at 1 per cent level.
13 Table 3: Results for Lee and Strazicich (2003) LM unit root test with two structural breaks Break in Intercept Break in Intercept and Trend Series Test statistic TB1 TB2 Test statistic TB1 TB2 Spot Price Feb Nov May Jul 2008 Futures 1M Sep Feb Oct Nov 2005 Futures 2M Jan Feb Apr Oct 2005 Futures 3M Feb May Apr Aug 2008 Futures 4M Feb Jun Apr Aug 2008 Critical values for unit root test Break in Intercept only (Model AA) 1% 5% 10% Break in Intercept and trend (Model CC) λ λ 1 1% 5% 10% 1% 5% 10% 1% 5% 10% Notes: TB 1 and TB 2 are the dates of the structural breaks. λ j denotes the location of the breaks. The LM unit root test for model AA is invariant to the location of the breaks; however, this invariance does not hold for model CC, for which the null distribution of the LM test depends on the relative location of the breaks.
14 Table 4: Results for Narayan and Popp (2010) unit root test with two structural breaks. Break in Intercept Break in Intercept and Trend Series Test statistic TB1 TB2 Test statistic TB1 TB2 Spot Price Dec Feb Dec Feb 2003 Futures 1M Sep Sep Sep Sep 2001 Futures 2M Sep Aug Sep Aug 2009 Futures 3M Dec Jul Dec Jul 2009 Futures 4M Jul Jun Jul Jun 2012 Critical values for unit root test 1% 5% 10% Model M1 (Break in Intercept only) Model M2 (Break in Intercept and Trend) Notes: The critical values are taken from Narayan and Popp (2010).
15 Table 5: Results for Narayan and Liu (2013) test with two structural breaks in the intercept. Series Test statistic TB1 TB2 Spot Price -8.51** 24 Feb Jul 2008 Futures 1M -4.44** 19 Mar July 2008 Futures 2M -3.97** 19 Mar July 2008 Futures 3M -3.75** 19 Mar July 2008 Futures 4M -3.81** 01 Mar July 2008 Notes: The 5% Critical values for Narayan and Liu (2013) test are provided in Table below. ** indicates rejection of the null hypothesis of a unit root at the 5% level of significance. T 5% C V f N = 500 GARCH m [α β] [0.05,0.90] [0.45,0.50] [0.90,0.05] t1/t
16 Spot Price ($/MBtu) Figure 1: Time series plot of daily natural gas spot prices. Henry Hub Natural Gas Spot Price t
17 Futures Contract 3 ($/MBtu) Futures Contract 4 ($/MBtu) Futures Contract 1 ($/MBtu) Futures Contract 2 ($/MBtu) Figure 2: Time series plot of daily natural gas futures prices for 1 4 months contracts. Natural Gas Futures Contract 1 Natural Gas Futures Contract t t Natural Gas Futures Contract 3 Natural Gas Futures Contract t t
18 ln(spot Price ($/MBtu)) Figure 3: Time series plot of daily natural gas spot price with breaks in the intercept according to the Lee and Strazicich (2003), Narayan and Popp (2010) and Narayan and Liu (2013) tests Natural Gas Spot Price Narayan and Popp (2010) Breaks Lee and Strazicich (2003) Breaks Narayan and Liu (2013) Breaks t
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