Forecast evaluation in daily commodities futures markets

Transcription

1 Int. J. Financial Markets and Derivatives, Vol. 1, No., Forecast evaluation in daily commodities futures markets Periklis Gogas* Department of International Economic Relations and Development, Democritus University of Thrace, Komotini Campus, Greece *Corresponding author Apostolos Serletis Department of Economics, University of Calgary, Calgary, Alberta, TN 1N4, Canada Abstract: In this paper, we use recent advances in the financial econometrics literature to model the time-varying conditional variance in five energy markets crude oil, gasoline, heating oil, propane, and natural gas using daily data over the period from January 3, 1994 to September 3, 008. We estimate autoregressive conditional heteroscedasticity (ARCH) and generalised ARCH (GARCH) models using a variety of error densities (the normal, Student-t, and generalised error distribution) and diagnostic checks. We use the models to perform static and dynamic forecasts over different horizons and compare their performance to that of a random walk model. Keywords: energy markets; forecasting; autoregressive conditional heteroscedasticity; derivatives; ARCH. Reference to this paper should be made as follows: Gogas, P. and Serletis, A. (010) Forecast evaluation in daily commodities futures markets, Int. J. Financial Markets and Derivatives, Vol. 1, No., pp Biographical notes: Periklis Gogas is a Faculty member at the Department of International Economic Relations and Development of the Democritus University of Thrace. He teaches macroeconomics at the undergraduate level and international economics, international finance, and international banking and finance at the graduate level. He is a Financial Consultant for the Gerson Lehrman Group, Austin, Texas. He also works on special research projects for the Greek Ministry of Finance and the European Union. He has published his research and acts as a referee for journals like the J. of Banking and Finance, Applied Financial Economics, J. of Economic Studies, J. of Macroeconomic Dynamics, etc. Apostolos Serletis is a Faculty member of the Department of Economics at the University of Calgary. His research interests include: macroeconomics, monetary and financial economics and non-linear and complex dynamics. He teaches macroeconomics and banking and finance courses in the undergraduate, master s and PhD levels. He has published over 150 research papers in journals including: J. of Econometrics, J. of Applied Econometrics, Journal of Macroeconomics, J. of Economic Dynamics and Control, J. of Money Credit Copyright 010 Inderscience Enterprises Ltd.

2 156 P. Gogas and A. Serletis and Banking, etc. He is the Associate Editor of Macroeconomic Dynamics. He has published several books including: Barro and Serletis, Macroeconomics: A Modern Approach (009), Mishkin and Serletis, The Economics of Money, Banking, and Financial Markets (008), Serletis, The Demand for Money: Theoretical and Empirical Approaches (007). 1 Introduction Recently, economists have been creating new models and tools that can capture important non-linearities in economic and financial data. There have been, e.g., exciting advances in dynamical systems theory, non-linear time-series analysis, and stochastic volatility models. One reason for the interest in non-linear methods is what one might call the forecasting paradox the fact that linear models produce invariably good in-sample fits, but usually fail miserably at out-of-sample prediction. One is therefore tempted to explore means by which apparent dependencies in the residuals of linear models (that are inconsistent with a linear data generator) can be exploited to produce better forecasts. Recent leading-edge research has applied Engle s (198) autoregressive conditional heteroscedastic (ARCH) model and Bollerslev s (1986) and Baillie and Bollersev s (1989) generalised ARCH (GARCH) model to estimate time-varying variances in commodity prices. In this paper, we follow these recent advances in the financial econometrics literature and conduct a thorough investigation to properly identify the type of heteroscedasticity in the data generation process of five energy prices crude oil, gasoline, heating oil, propane, and natural gas. This is of major importance in forecasting, since these models allow the conditional variance to depend on elements of the information set. In doing so, we use a variety of error densities, including the normal, the Student-t distribution, and the generalised error distribution (GED), as well as a comprehensive set of diagnostic checks. The remainder of the paper is organised as follows. The next section describes the data and examines the univariate time series properties of the crude oil, gasoline, heating oil, propane, and natural gas price series, using the augmented Dickey-Fuller (ADF) and the Phillips-Peron (PP) unit root testing procedures. Sections 3 and 4 model the changing volatility of energy price changes, by specifying parametric ARCH-type models for volatility and Section 5 uses the best fitted models to perform static and dynamic forecasts, over different forecast horizons, and to compare the forecasting performance of the ARCH-type models to that of a random walk. The final section summarises the paper. The data and stochastic trends The data used in this paper consist of daily futures prices on five energy commodities crude oil, gasoline, heating oil, propane, and natural gas. The sample period is from January 3, 1994 to September 3, 008, for a total of 3,676 observations. Prices are in US dollars per barrel in the case of crude oil, per gallon in the case of gasoline, heating oil, and propane, and per million British thermal units (MMBTU) in the case of natural gas. Figure 1 to Figure 5 plots the logged levels and the logarithmic first differences of the series.

3 Forecast evaluation in daily commodities futures markets 157 First, we test for stochastic trends in the autoregressive representation of the logged returns on the futures prices, using two alternative unit root testing procedures, in an attempt to deal with the fact that the series may not be very informative about the existence or not of a unit root. In particular, we use the ADF test [see Dickey and Fuller (1981) for more details] and the non-parametric Z( t ) test of Phillips and Perron (1988). a Figure 1 Crude oil prices and returns (in basis points) (see online version for colours) Figure Heating oil prices and returns (in basis points) (see online version for colours)

4 158 P. Gogas and A. Serletis Figure 3 Gasoline prices and returns (in basis points) (see online version for colours) Figure 4 Natural gas prices and returns (in basis points) (see online version for colours)

5 Forecast evaluation in daily commodities futures markets 159 Figure 5 Propane prices and returns (in basis points) (see online version for colours) The ADF test is conducted using the following regression equation l t t j t j t j= 1 Δ log z = a + at+ a log z + β Δ log z + ε (1) where z t is the series under consideration and l is selected large enough such that ε t is white noise. The alternative non-parametric Z( t a ) test involves estimating (1) with l = 0 and then transforming the test statistic to correct for serial correlation in its asymptotic distribution. As discussed in Pantula et al. (1994), the Z( t a ) test is robust to a wide variety of serial correlation and time-dependent heteroskedasticity. Table 1 Unit root test result Variable l t t j t j t j= 1 Regression: Δ log z = α0 + α1t + α log z 1 + β Δ log z + ε ADF p-values KPSS LM-statistic Intercept Intercept and trend Intercept Intercept and trend Decision Crude oil I(0) Heating oil I(0) Gasoline I(0) Natural gas I(0) Propane I(0) Note: KPSS test critical values at 1% and 5% levels are and 0.463, respectively, with an intercept and 0.16 and with an intercept and trend

6 160 P. Gogas and A. Serletis The selection of the optimal lag length in the ADF test is done using the Schwartz information criterion (SIC), considering values of l from one to 9. The p-values, reported in Table 1, show that the null hypothesis of a unit root is rejected with probability p for both tests using both an intercept in the estimated function or an intercept and a trend. Thus, we conclude that the series are stationary [or integrated of order zero, I(0)] in the terminology of Engle and Granger (1987). 3 The GARCH model specification In conventional econometric models, stochastic variables are assumed to have a constant variance (and are called homoscedastic, as opposed to heteroscedastic). Many macroeconomic and financial variables, however, exhibit clusters of volatility and tranquility (i.e., serial dependence in the higher conditional moments). In such circumstances, the homoscedasticity assumption is inappropriate. Having concluded that the logged first differences energy futures prices are stationary, we use the following model for purposes of forecasting these prices r 5 t i t i k kt t i= 1 k= Δ log z = φ Δ log z + d D + ε () In equation (), D kt are day of the week dummy variables, r is the order of the autoregression, and φ and d are unknown parameters to be estimated. We used both the SIC and the Akaike information criterion (AIC) to optimally determine the value of r in equation (), by estimating several models with r = 1 to r = 50. However, as the AIC tends to overparameterise the model while the SIC tends to select the true model as the sample size increases (and if the true model is included in the choices), we follow the SIC in selecting the optimal lag length of the autoregression, r. The results are reported in Table. Both visual inspection and the use of the Q(36) statistic for residual serial correlation (as seen in the last two columns of Table ) suggest that the residuals of the autoregressive model with the order of the autoregression, r, chosen as above are not serially correlated. However, the Q (36) statistic, which represents the Q-statistic for the squared residuals and is designed to pick non-linearities and the presence of heteroscedasticity, is highly significant providing evidence for the presence of conditional heteroscedasticity in the error term. For this reason in order to capture the heteroscedasticity in the error term we estimate the autoregressive AR(r) model () for each series assuming that ε t is IN(0, follows, σ t ) with q p 5 = w0 + ai t i + j t j + d kdkt i= 1 j= 1 k= σ t following a GARCH (p, q) process as σ ε β σ, (3) or an EGARCH(p, q) process as follows, 5 = 0 + q + + log + p εt i εt i t w ai i j t j dkdkt = 1 σt i σ i t i j= 1 k= logσ γ β σ see, e.g., Bollerslev (1986) and Nelson (1978), respectively, for more details. (4)

7 Forecast evaluation in daily commodities futures markets 161 Table Optimal AR lag specifications and serial correlation and heteroscedasticity tests r Δ log z = ϕ Δ log z + d D + ε t i t 1 k kt t i= 1 k= Series SIC lag (r) selection Q(36) p-values Q (36) p-values Crude oil Heating oil Gasoline Natural gas Propane In equations (3) and (4) above, p, q [1, ] such that eight different conditional heteroskedasticity specifications are estimated for each series. The lagged values of the error term, ε t 1, i = 1,, q, in equations (3) and (4) represent news in the market about volatility in the previous period, while the lagged values of the conditional variance, σ t j, j = 1,, p, are lagged forecasted variances. Thus, this period s variance prediction is formed as a weighted average of a long term average (the constant, w 0 ), the forecasted variance from previous periods, and information about volatility observed in earlier periods. This variance modelling is consistent with the volatility clustering observed in the returns of the five series (see Figure 1 to Figure 5). In the first column of Table 3 we reproduce the optimal AR lag chosen in Table and in the second column we report the conditional heteroskedasticity model that is selected by the SIC for each of the five series, conditional on the chosen AR lag. As can be see, the GARCH (1, ) model is the optimal specification for crude oil and gasoline, the GARCH (1, 1) model for heating oil, the EGARCH (1, 1) model for natural gas, and the GARCH (, ) model for propane. To check the robustness of these results, we also estimated the following simple random walk model for each of the five series 5 Δ log z = d D + ε (5) t k kt t k = where D kt are the day of the week dummy variables, as before. The residuals and squared residuals of each of these random walk models indicate the presence of autocorrelation and heteroskedasticity. In fact, we formally tested for serial correlation and heteroscedasticity, using the Q(36) and Q (36) statistics, and strongly rejected the null hypotheses of no serial correlation and no heteroscedasticity. In order to best capture these dependencies in the error term we then estimated eight random walk models for each one of the five series assuming that ε t is IN(0, σ t ) with σ t following both a GARCH (p, q) and an EGARCH (p, q) process as in equations (3) and (4). As expected, the selected specifications of the conditional variance functions, based on the SIC, are the ones shown in the last column of Table 3.

8 16 P. Gogas and A. Serletis Table 3 Optimal AR lag and conditional variance specifications r Δ log z = ϕ Δ log z + d D + ε t i t 1 k kt t i= 1 k= with equations (3) or (4) Series SIC lag (r) selection SIC lag (q, p) selection, conditional on r Crude oil GARCH (1, ) Heating oil 1 GARCH (1, ) Gasoline 6 GARCH (1, ) Natural gas 9 EGARCH (1, 1) Propane 1 GARCH (1, ) 5 4 Alternative distributional assumptions The models estimated and selected in the previous section use the normal distribution as the density function for the error term. Now, we explore different error distributions in an attempt to improve the fit of the models. In particular, in addition to the normal distribution we use the Student-t distribution, used by Bollerslev (1987), and the GED, used by Nelson (1978), for both the autoregressive and the random walk models for each of the five energy price series. The Student-t distribution is given by n+ 1 n z f( z) = n π Γ Γ 1+ n.5( n+ 1) where n is the degree of freedom and Γ( ) is the gamma function. This distribution is normalised to have unit variance and becomes the standard normal distribution when n. The density of a GED random variable normalised to have a mean of zero and a variance of one is given by 1 v vexp z/ λ f( z) =, (1+ 1/ v) λ Γ(1/ v) where < z <, 0 < v <, Γ( ) is the gamma function, and ( / v) Γ(1/ v) λ Γ(3/ v) 1/ Above, v is a tail-thickness parameter. When v =, z has a standard normal distribution. For v <, the distribution of z has thicker tails than the normal (e.g., when v = 1, z has a double exponential distribution). For v >, the distribution of z has thinner tails than the normal (e.g., for v = 1, z is uniformly distributed on the interval [ 3 1/, 3 1/ ].

9 Forecast evaluation in daily commodities futures markets 163 We use the SIC to determine the best overall model and present the results in Table 4. For the autoregressive models, the Student-t distribution provides the best fit for the crude oil and gasoline series, while the GED is the best error representation for heating oil, natural gas, and propane. For the random walk models the GED is selected in the case of heating oil, while for the remaining four series we select the Student-t distribution. It is to be noted that the results are robust to the use of the AIC in selecting the best distribution for the error in both the autoregressive and random walk models. In Figure 6 to Figure 10 we plot the conditional variances of the energy futures returns implied by our estimated, best-fitted autoregressive models. Table 4 Selection of best overall model r 5 t ϕi log t 1 k kt ε t i= 1 k= Δ log z = Δ z + d D + with equations (3) or (4) Series AR lag Conditional variance Autoregressive model Random walk model (r = 0) Crude oil GARCH (1, ) Student-t Student-t Heating oil 1 GARCH (1, 1) GED GED Gasoline 6 GARCH (1,) Student-t Student-t Natural oil 9 EGARCH (1, 1) GED Student-t Propane 1 GARCH (, ) GED Student-t Figure 6 Crude oil conditional variance (see online version for colours)

10 164 P. Gogas and A. Serletis Figure 7 Heating oil conditional variance (see online version for colours) Figure 8 Gasoline conditional variance (see online version for colours)

11 Forecast evaluation in daily commodities futures markets 165 Figure 9 Natural gas conditional variance (see online version for colours) Figure 10 Propane conditional variance (see online version for colours)

12 166 P. Gogas and A. Serletis 5 Forecasting and model comparison We have selected for each of the five series and for each of the autoregressive and random walk specifications the best model in terms of modelling the conditional variance and the distribution of the error term. Next, we use these models to produce in-sample static and dynamic forecasts of energy futures returns at forecast horizons of one week, two weeks, and one month ahead. As we use daily data this means that we use the models to forecast the next five, ten, and days, respectively. To access the quality of the static and dynamic forecasts and to formally compare them we calculate the mean error (ME), mean absolute error (MAE), and root mean squared error (RMSE) statistics. These statistics are calculated using the following formulas 1 = F ME et+ f ; F f = 1 1 = F MAE et+ f F f = 1 ; 1 = F RMSE t f, F e + f = 1 * t+ f t+ f t+ f where e = y y, with y t+f being the actual value of the series at period t + f and * t+ f y being the forecast for y t+f. F is the forecast window, in our case for one week, two weeks, and one month ahead forecasts, F = [5, 10, ]. Based on the ME, MAE, and RMSE statistics and in the case of static forecasts, at the one week ahead forecast window the autoregressive model outperforms the random walk model for all series with the exception of propane. At the two weeks ahead forecast horizon the results are mixed as the autoregressive model is selected for gasoline and natural gas and the random walk model for crude oil, heating oil, and propane. When the one month ahead static forecasts are considered the autoregressive model dominates the random walk model for all series except for propane. In the case of dynamic forecasts, at the one week ahead forecast horizon the autoregressive model is selected for all series with the exception of propane, and at two weeks ahead forecasts the random walk model dominates for all series except the natural gas. Finally at one month ahead dynamic forecasts the statistics select the autoregressive model for crude oil, heating oil, and natural gas, and the random walk model for the gasoline and propane. These results for both the static and dynamic forecasts are summarised in Table 5. It is interesting to note that for natural gas the best forecasting model for all three forecast horizons and for both static and dynamic forecasts is the autoregressive model with r = 9 in equation (), an EGARCH (1, 1) specification for the variance function, and the GED distribution for the errors. In the case of propane and for all forecast horizons, and for both static and dynamic forecasts, the random walk model is selected with a GARCH (, ) specification for the variance function and the Student-t distribution for the errors.

13 Forecast evaluation in daily commodities futures markets 167 The autoregressive model is selected for both crude oil and heating oil and for both static and dynamic forecasts of one week and one month ahead forecast horizons. In the case of crude oil, however, as can be seen in Table 4, r = in equation (), a GARCH (1, ) specification is chosen for the variance function, and the Student-t distribution for the errors whereas in the case of heating oil, r = 1 in equation (), a GARCH (1, 1) specification is chosen for the variance function, and the GED distribution for the errors. In the case of medium range forecasts for both crude oil and heating oil the random walk model is selected. Finally, for gasoline, the autoregressive model is selected in the case of static forecasts for all forecast horizons and in the case of dynamic forecasts for the one week ahead horizon, with r = in equation (), a GARCH (1, ) specification for the variance function, and the Student-t distribution for the errors. In the case of dynamic forecasts for two weeks and one month forecast horizons the random walk model is selected, again with r = in equation (), a GARCH (1, ) specification for the variance function, and the Student-t distribution for the errors. Table 5 Best forecast model Series One week Two weeks One month A Static forecasts Crude oil AR RW AR Heating oil AR RW AR Gasoline AR AR AR Natural oil AR AR AR Propane RW RW RW B Dynamic forecasts Crude oil AR RW AR Heating oil AR RW AR Gasoline AR RW RW Natural oil AR AR AR Propane RW RW RW 6 Conclusions This paper provides a study of daily price changes of five energy products crude oil, gasoline, heating oil, propane, and natural gas using data over the period from January 3, 1994 to September 3, 008. We have implemented GARCH and EGARCH models and used a variety of error densities and diagnostic checks and found that these models can remove all heteroscedasticity in energy futures returns in all five energy markets. This is of major importance in forecasting, since these models allow the conditional variance to depend on elements of the information set. The contribution of the paper is its use of models of changing volatility and alternative distributional assumptions to properly identify the type of heteroscedasticity in the data-generation processes. This is of major importance in forecasting. Instead of using volatility measures that are based on the assumption of constant volatility, one can use

14 168 P. Gogas and A. Serletis these models to extract volatility estimates form the data. As Diebold and Watson (1996, p.453) put it forecasting is re-emerging as an exciting and vital research area, fuelled not only by its tremendous practical importance, as always, but also by recent advances in both analytic methods and computational methods. The new methods and models, however, are very different from those of twenty five years ago. References Baillie, R.T. and Bollerslev, T. (1989) The message in daily exchange rates: a conditional-variance tale, Journal of Business and Economic Statistics, Vol. 7, pp Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, Vol. 31, pp Bollerslev, T. (1987) A conditional heteroskedastic time series model for speculative prices and rates of return, Review of Economics and Statistics, Vol. 9, pp Dickey, D.A. and Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, Vol. 49, pp Diebold, F.X. and Watson, M.W. (1996) Introduction: econometric forecasting, Journal of Applied Econometrics, Vol. 11, pp Engle, R.F. (198) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica, Vol. 50, pp Engle, R.F. and Granger, C.W. (1987) Cointegration and error correction: representation, estimation and testing, Econometrica, Vol. 55, pp Nelson, D. (1978) Conditional heteroscedasticity in asset returns: a new approach, Unpublished PhD Dissertation, Department of Economics, Massachusetts Institute of Technology. Pantula, S.G., Gonsalez-Farias, G. and Fuller, W.A. (1994) A comparison of unit-root test criteria, Journal of Business and Economics Statistics, Vol. 1, pp Phillips, P. and Perron, P. (1988) Testing for a unit root in time series regression, Biometrica, Vol. 75, pp