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Article Excerpt
1. INTRODUCTION
The recent finance and economic literature has recognized the importance of long-range dependence in analyzing time-series data. The recognition of dependence between distant observations has important bearing on modeling and forecasting variables such as inflation, and asset returns, among others. Long memory can undercut standard statistical inferences and usefulness of linear models as accurate tools to forecast the behavior of economic and financial processes (Beran, 1992).
This study is motivated by three considerations. First, to our knowledge, no study has examined long memory in the strategic crude oil and refined product markets (e.g., gasoline and heating oil) despite the importance of these markets and the need to model and forecast their returns more accurately. (1) Many empirical findings suggest that numerous economic and financial data show strong evidence of long memory (see, for example, Booth et al. (1982), Helms et al. (1984), and Lo and MacKinaly (1988)). Those early studies use the Rescale range (R/S) statistic and the variance ratio tests. Later, Lo (1991) uses a modified rescaled range (R/S) statistic and finds no evidence of long memory in the monthly and daily returns on the Center for Research in Security Prices (CRSP) stock indexes. Ding et al. (1993) investigate the long memory properties in several transformations of returns including absolute and squared returns. They find substantial evidence of long memory in absolute returns and squared returns for long lags. More recently, Lobato and Savin (1997) find no evidence of long memory in daily Standard and Poor 500 returns but find strong evidence of long memory in the squared return. They also find that the evidence is even stronger for the absolute returns. (2) None of those studies examine long memory in the petroleum markets whether in terms of crude oil or its refined products. This paper will fill this gap in the petroleum markets by examining long memory in the spot and futures prices of oil, gasoline and heating oil. Second, we investigate the time series properties of petroleum returns whether they show the strictly white noise process or not. Third, recent studies show that there are difficulties in distinguishing between the long memory and structural change processes, and we have taken this issue into consideration because the petroleum markets have weathered many economic and geopolitical crises and weather events. We thus evaluate the possibility of spurious long memory in the petroleum returns in light of structural changes.
The contributions of this paper confirm that the proxies of volatility such as the absolute and squared returns and conditional volatility of petroleum spot and futures prices provide the strongest evidence of long memory. This is consistent with the findings of many studies done on developed countries' stock markets such as Ding et al (1993) and Lobato and Savin (1997). Comparing the results across the petroleum futures absolute returns, the semiparametric results demonstrate that crude oil futures have stronger long memory than gasoline and heating oil. The results also show that gasoline spot and futures absolute returns are more resilient to shocks than their heating oil counterparts. The findings are also consistent across the different methods for the squared returns with the exception of the Modified S/R statistic for the NYMEX heating oil spot, NHOS. When we apply the FIGARCH model to petroleum volatility, most of the results show evidence of long memory, also with the exception of heating oil. The results for the FIGARCH model also demonstrate that the gasoline spots and futures contracts are more resilient to volatility shocks than those for crude oil, confirming the futures results for the absolute returns. Moreover, concurrently with the findings of studies on the stock markets, the petroleum (simple) returns show weak evidence of long memory properties. (3) The results in terms of structural changes demonstrate that: 1) adjusting for structural changes still shows stationary, long memory properties; and 2) structural changes have only a partial effect of reducing the persistence of absolute and squared returns.
We use two forecasting evaluation criteria, namely, the root mean squared error (RMSE) and Diebold and Mariano, hereafter DM, (1995) to compare the forecasting abilities of the ARMA and autoregressive fractionally integrated moving-average (ARFIMA) models. The resulting forecasts suggest that the ARFIMA model, which satisfies the LM property, performs significantly better than the ARMA model which has a short-run process. The paper is organized as follows: After this introduction, section 2 describes the data and analyzes the descriptive statistics for the petroleum price returns. Section 3 describes the estimation and inference on the long memory parameter. Section 4 applies several long memory methodologies to the petroleum price (simple) returns, absolute returns and squared returns, and the FIGARCH model to petroleum volatility. Section 5 compares the empirical results of forecasting abilities between the two competing models, ARFIMA and ARMA. Section 6 concludes.
2. DATA DESCRIPTION
The data include daily time series for the spot and futures prices of crude oil and of two refined products over the period 1/2/1986-7/19/2005. (4) The futures contracts include one-month, two-month and three-month maturities. The crude oil is the US West Texas Intermediate (WTI), while the refined products are the gasoline and heating oil traded and consumed in the United States.
We used the spot and futures prices to calculate the return measures for the oil and refined product prices. The rate of return, which is a continuously compounded, single-period return of that petroleum price from t-1 to t, calculated as a log-differenced price, [r.sub.t] = (log([P.sub.t]) - log([P.sub.t-1])) x 100, where [P.sub.t] is the spot or futures price for a crude oil or a refined product. The daily descriptive statistics for all the return series are given in Table 1. The US spot and futures prices for crude oil averaged at an annualized rate of return ranging between 4.3 and 4.8 percent, based on a yearly business calendar of 260 days.
In terms of US refined products, spot heating oil yielded on average the highest annualized return, reaching about 7.5 percent, followed by spot gasoline which retuned about 6.4 percent. Meanwhile, the one-month futures heating oil yielded an average annualized return of about 4 percent, the lowest among all refined products under consideration. Heating oil spot prices which process less future news than futures prices are relatively more sensitive to changes in inventories and occurrence of spikes due to hurricanes and low temperature conditions. In fact, heating oil spot return is the most volatile of all the crude and refined products considered, having an average annualized volatility of 6.99 compared to 6.77 for spot WTI crude oil and 6.68 for spot gasoline. Gasoline prices are less sensitive to weather conditions and are also watched daily by the US consumers which bring some sort of continuous check on these prices.
It is also interesting to note that the return volatility decreases with the increased maturity length of the futures contracts for all crude oil prices and for the refined products, with the exception of NYMEX two-month gasoline futures. (5) Again this pattern of volatility reflects futures contract liquidity and the ability of futures petroleum markets to process new information better than the spot markets.
All of the displayed returns have non-symmetric distributions as shown by the skewness and primarily kurtosis statistics. NYMEX spot heating oil return...
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