Breaks, Trends, and Unit Roots in Energy Prices: A New View from a Long-run Perspective

Transcription

1 Breaks, Trends, and Unit Roots in Energy Prices: A New View from a Long-run Perspective CFES Area Studies Working Paper, No.1 Wendong Shi 1 and Guoqing Zhao 1,2 1 School of Economics, Renmin University of China, Beijing , China 2 Corresponding author. Tel: +86 (10) Center for Far Eastern Studies, University of Toyama 3190 Gofuku, Toyama, Toyama April 2017

2 Breaks, Trends, and Unit Roots in Energy Prices: A New View from a Long-run Perspective Wendong Shi 1, Guoqing Zhao 1 Abstract This paper investigates the trend properties in the annually price series for coal, natural gas, and petroleum over the period 1880 to In particular, we examine the presence of breaks in trend using a sequential procedure proposed by Kejriwal and Perron (2010), which offers consistent results regardless of whether the noise component is integrated or stationary. We then test for unit root according to the occurrences of breaks using various model specifications. For price series that has no breaks in the trend, we directly perform the Ng-Perron (2001) unit root test. For prices with at least one breaks, we employ the procedure of Carrion-i-Silvestre et al. (2009) to allow for an arbitrary number of breaks with unknown break dates under both the null and alternative hypotheses. We further investigate their trend properties by estimating deterministic trend with an integrated or stationary noise component, using the Perron-Yabu trend estimate. We find evidence against unit root and rising deterministic trend for all the three price series; multiple breaks in trends are observed for coal and petroleum. Keywords: structural break; unit root; trend; energy price. JEL Classification: C4, C5, Q4. This study was supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China, #12XNK School of Economics, Renmin University of China, Beijing , China Corresponding author. Tel: +86 (10) ; address: zhaogq@ruc.edu.cn. 1

3 1. Introduction Much attention and intense debate have been attracted to the trend properties of non-renewable energy resource prices. One complicated issue in modeling the prices is whether or not they can be classified as trend-stationary or difference-stationary processes. Adopting different models might lead to contradict conclusion about how energy prices evolve through the time passage. In particular, Berck and Roberts (1996) explore the trend properties of eleven non-renewable natural resources prices. They predict rising price trends using a trend-stationary model, but only find weak evidence for the increasing prices when a difference-stationary model is used. Therefore, they conclude that the prediction of natural resource prices depends strongly on which time series methods are employed. They also perform the Lagrange-multipliers (LM) tests on the prices and find that the null hypothesis of a unit root cannot be rejected. Following Berck and Roberts (1996), many studies are conducted on examining the trend properties of resource prices, but the results are controversial. For example, Ahrens and Sharma (1997) analyze the prices of eleven resource commodities and find that five of them exhibit stochastic trends, while the others are stationary around a deterministic trend. Lee et al. (2006) find evidence against the unit root hypothesis, and demonstrate that natural resource prices appear to be stationary around deterministic trends with structural breaks. More recently, Ghoshray and Johnson (2010) perform unit root tests allowing for structural breaks on energy resource prices. They find evidence of stationarity, and argue that the spurious unit roots in previous literature might be induced by breaks in trend. Despite extensive studies on the trend properties of energy resources, there still exist critical limitations in testing for unit roots and estimating stochastic trends. First, the conventional unit root tests without allowing for structural breaks perform poorly if the trend function is subject to changes in level and/or slope. Most of the previous empirical studies on resource prices exclude the presence of breaks in the unit root tests, which might lead to inconsistent results when breaks actually occur. In particular, Perron (1989) show that a shift in the trend function might bias the sum of autoregressive coefficients towards unit. When there is a shift in the trend function, the null hypothesis of a unit root is hardly rejected; besides, a spurious unit root might arise from white noise disturbances around the trend. Second, among the unit root tests allowing for a break (or breaks), most appear to be restrictive in determining number and/or time of the break(s). For example, Perron (1989, 1990) proposes alternative unit root tests, which allow for a one-time break under both the null and alternative hypotheses. His work largely improves the conventional approach, but assumes a prior knowledge on the break time, which is controversial (Christiano, 1992). Lee and Strazicich (2004, 2003) then develop minimum LM unit root tests with one or two breaks. Later, Kim and Perron (2009) extend Perron s (1989, 1990) work to the case of one break with an unknown break date. More recently, Carrion-i-Silvestre et al. (2009) improve the 2

4 previous procedure to allow for an arbitrary number of breaks with unknown break dates under both the null and alternative hypotheses. However, all the above tests require the number of break(s) to be given at the initial step. For a comprehensive review of unit root tests, please refer to the recent work by Xiao (2014). Third, the conventional trend estimates generally assume the noise component to be stationary, i.e. I(0), which is impropriate for many economic time series including energy resources prices. As widely documented (e.g., Nelson and Plosser, 1982), price series might have a noise component with an autoregressive unit root, i.e. I(1), or a root that is close to one, and the prior knowledge about this is usually unavailable. Having an I(1) noise or noise with a root close to one could result in poor performance of estimation. In particular, when the noise component is I(1), the least square estimate of the linear trend slope is no longer asymptotically efficient; when the noise component has a root close to one, the limiting normal distribution is a poor approximation in finite sample sizes. Empirical studies treading noise component as I(0) or I(1) could lead to inconsistent results. For example, as shown by Berck and Roberts (1996), the trend-stationary model predicts rising trends in resource prices, while the difference-stationary model does not. Finally, similar to the trend estimates, many structural break tests are based on the assumption of I(0) noise, but prior knowledge about whether the noise component is actually I(0) or I(1) is generally not available. For example, Lee et al. (2010) study the structural breaks in daily oil price using the structural break model of Bai and Perron (2003). Their procedure allows one to test multiple structural changes in linear models, only under the condition of assuming the noise component to be stationary. Supposition of I(0) noise might result in questionable testing results when the noise component is actually I(1). Sometimes, structural break tests are performed using either first-differenced data or growth rates, where the noise component is assumed to be I(1); the test results could be very poor, e.g., giving different limit distribution, if the noise component is actually I(0). Motivated by the above considerations, we apply a set of robust procedures to determine the number and time of breaks (if there exist any), test the unit root hypothesis with allowing for breaks, and estimate the trend function according to the estimated breaks. This paper contributes to the literature in several ways. First, we employ a sequential procedure, proposed by Kejriwal and Perron (2010), to test the number and time of breaks in trend regardless of whether the noise component is I(0) or I(1). Their tests have several advantages over the previous ones, such as maintaining the correct size in finite samples and being more powerful than the LM tests. Second, after determining the number of breaks, we use the unit root test proposed by Carrion-i-Silvestre et al. (2009) for the price series with multiple breaks, allowing for breaks in both the null and alternative hypothesis; for the price series with no breaks, the Ng- Perron unit root test is applied. Third, when estimating the deterministic trend of energy prices, we used 3

5 the robust test proposed by Perron and Yabu (2009a), which is valid with either I(1) or I(0) noise. The Perron-Yabu test, based on a Feasible Quasi Generalized Least Squares (GLS) method, gives the same limit distribution regardless of whether the noise component is I(0) or I(1). Their procedure is also shown to have better size and power properties than the commonly used methods. The rest of this paper is organized as follows: Section 2 introduces the empirical methodology; Section 3 describes the data utilized, followed by the empirical results; Section 4 provides conclusions and policy implications. 2. Method 2.1 Estimating deterministic trend with an integrated or stationary noise component Perron and Yabu (2009a) analyze the problem of hypothesis testing on the slope coefficient of a linear trend model, without knowing whether the noise component is integrated or stationary. Their test is based on a Feasible GLS procedure with an estimate of the sum of the autoregressive coefficients truncated to be one. For a more detailed description of their models, please refer to Perron and Yabu (2009a), or Estrada and Perron (2012). We use the following data generating process: y t = x t Ψ + u t (1) u t = αu t 1 + e t for t = 1,, T, e t ~i. i. d. (0, σ 2 ), x t = (1, t) are deterministic components, and Ψ = (μ, β) is model specific. We assume the initial condition u 0 to be bounded in probability. We allow the noise component to be either I(1) or I(0) by assuming that α 1. The estimating procedure is as follows: 1. Run the OLS regression on the first equation in (1) to obtain the residuals u t; 2. Estimate the second equation in (1) using u t to obtain α ; 3. Use α to get the Roy and Fuller (2001) biased corrected estimate α M; 4. Apply the truncation α MS = { α M if α M 1 > T if α M 1 T 0.5; 5. Run the GLS regression using α MS to obtain the estimates of the trend coefficients and variance of residuals. 2.2 A sequential procedure to determine the number of breaks in trend with an integrated or stationary noise component Based on the previous work of Perron and Yabu (2009a and 2009b), Kejriwal and Perron (2010) propose a sequential procedure to consistently test the number of breaks regardless of whether the noises 4

6 component is I(0) or I(1). Their test has the correct size in finite samples and is more powerful than the LM tests. First, we modified the model in equation (1) to test for a one-time change in the trend function slope. The model is specified with x t = (1, t, DT t ) and Ψ = (μ 0, β 0, β 1 ) where DT t = 1(t > T 1 )(t T 1 ), the break date T 1 = [λt] for some λ (0,1), with [ ] denoting the largest integer that is not more than λt and 1( ) being the indicator function. The hypothesis to test is β 1 = 0. Then, the estimating procedure is as follows: 1. For any given break date, follow the procedure 1-5 in section 2.1 and construct the standard Waldstatistic W; 2. Since the break date is assumed to be unknown, we repeated the above step for all the permissible break dates, in order to construct the exp function of the Wald test: ExpW = log [T 1 exp ( 1 2 W(λ)) ] where Λ = {λ; ε λ 1 ε} for some ε > 0. In line with the literature, we set ε = Λ Then, conditional on the presence of one break, we perform the test on the two segments defined by the break date. If the maximum of the tests is significant, we conclude that there exist two breaks instead of one. In other words, we reject the null hypothesis that there is no additional break in the first stage, if the following exp function is sufficiently large: where ExpW i is the Perron-Yabu test in segment i. ExpW(2 1) = max 1 i 2 {ExpWi } The above procedure can be extended to the case with k breaks by using the following exp function for k+1 segments: ExpW(k + 1 k) = max 1 i k+1 {ExpWi } 2.3 Unit root tests with multiple breaks on unknown break dates under both the null and alternative hypotheses We modified the model in equation (1) to test unit root allowing for m-time changes in the level and slope of trend function. The model is specified with x t = (1, t, DU 1t, DT 1t,, DU mt, DT mt ) and Ψ = (μ 0, β 10, β 11,, β m0, β m1 ). Here, DU it = I(t > T i ), DT it = 1(t > T i )(t T i ) (i = 1,2,, m) with break date T i = [λ i T] for some λ i (0,1) (i = 1,2,, m), where [ ] denotes the largest integer that is not more than λ i T and 1( ) is the indicator function. The hypothesis to test is α = 0. The estimating procedure is as follows: 5

7 1. Estimate the break fraction λ i and the regression parameters by minimizing the sum of squared residuals from the quasi-differenced regression as in Harris et al. (2009). The sum of squared residuals at these estimates is denoted by S(α(λ ), λ ) with α(λ ) = 1 c(λ ) T. 2. The feasible point optimal statistic is given by P T gls (λ ) = S(α(λ ), λ ) α(λ )S(1, λ ) s 2 (λ ) where s 2 (λ ) is an autoregressive estimate of spectral density of v t at frequency zero. 3. The M-class of test is given by with MZ α gls (λ ) = (T 1 y T2 s 2 (λ )) (2T 2 MSB α gls (λ ) = (T 2 T t=2 2 y t 1 1/2 ) MZ t gls (λ ) = (T 1 y T2 s 2 (λ )) (4s 2 (λ )T 2 MP T gls (λ ) = [c 2 (λ )T 2 T 2 y t 1 t=2 T t=2 2 y t 1 s 2 (λ ) T t=2 2 y t 1 1 ) 1/2 ) + (1 c(λ )) T 1y T2 ] s 2 (λ ) y t = y t Ψ x t (λ ), x t (λ ) = (1, t, DU 1t (λ ), DT 1t (λ ),, DU mt (λ ), DT mt (λ )) and Ψ being the OLS estimate obtained from the quasi-differenced regression. 3. Data and Empirical Results The data consist of annual price series for coal, natural gas, and petroleum over the period 1880 to We construct the sample by updating the data in Berck and Roberts (1996) 2 to 2012, with coal, natural gas, and petroleum beginning in 1880, 1919, and 1900, respectively. All prices are deflated by the United States Producer Price Index, and expressed in the natural logarithm. A descriptive statistics is provided in Table 1. We utilize the data with low frequency, instead of using the high frequent data such as daily energy prices. As in previous studies on the trends of natural resources prices (e.g., Ahrens and Sharma, 1997; Lee et. al, 2006), we focus on the long run behavior of the prices, for which employing lowfrequent data is more suitable because many volatile movements are smoothed out. In other words, the high frequency data is less likely to contain any significant trends due to the sharp movements in it. 2 Data are available at 6

8 To capture the trend properties of energy resource prices, we first test for the presence and the number of breaks in the trend function using the procedure proposed by Perron and Yabu (2009), as well as Kejriwal and Perron (2010), both of which yield consistent results regardless of whether the noise component is stationary or not. As presented in Table2, the test statistics ExpW(1 0) is the Perron-Yabu test of none versus one break, while ExpW(2 1) is the Kejriwal-Perron sequential test of one versus two breaks, and so forth. The testing results indicate four, zero, and three breaks for coal, natural gas, and petroleum, respectively. For the coal price series, four breaks are detected in the year 1914, 1943, 1973, and 1998; for the petroleum price series, the breaks occur in 1927, 1970, and The break dates were reasonably coincident with historical events related to price fluctuations. For example, a structural break in the year 1998 was detected for petroleum with statistical significance at the 1% level, reflecting a series of oil embargo by OPEC and non-opec countries in response to the East Asian economic crisis in the late 1990s; prices of coal and petroleum shared a similar break point in the early 1970s, which was largely related to the 1973 oil crisis caused by American involvement in the Yom Kippur War and cut in oil supply of OPEC countries. After determining the number of breaks in each energy price series, we apply the Ng-Perron unit root test (the no break M-test) to the natural gas price, and use the test proposed by Carrion-i-Silvestre et al. (2009) (the M-test) for coal and petroleum. Table 3 presents the results for the tests. For all the three price series, the null hypothesis of a unit root is rejected, indicating that they can be classified as trendstationary processes. Since the test of Carrion-i-Silvestre et al. (2009) allows for an arbitrary number of breaks with unknown break dates under both the null and alternative hypotheses, our results avoid the spurious rejection problems that exist in most literature. To further investigate the evolution of energy prices during this period, we applied the trend estimate proposed by Perron and Yabu (2009b) to the whole sample, as well as to subsamples constructed according to the estimated structural break date. Results for the Perron-Yabu trend estimate are included in Table 4. For all the energy price series, the estimates for the constant term μ are statistically significant at the 1% level, either for the full sample or the subsamples, indicating breaks in the level of the trend function. The estimates for β is insignificant except for coal during , petroleum over and Hence, no evidence for rising trend is found. This finding contradicts some earlier studies. In particular, Slade (1982) finds significant rising trends for 11 out of 12 natural resource commodities using a quadratic model, and gets mixed conclusion from a linear model. However, our results re-verify the conclusions in many recent studies, such as Berck and Roberts (1996), and Ghoshray et al. (2013). 7

9 Results in tables 4 are further investigated in Figure 1(a)-(c), with the plot of the actual energy price and fitted values from a linear time trend regression. The dashed lines indicate the fitted values for the full sample without allowing for breaks. For coal and petroleum, both with multiple breaks, the dash lines appear to be poor fits. However, when we divided the sample according to the break date and fit each regimes, the actual values could be fitted well. Figure 1 confirm the findings in Table 3 and 4: (1) the three price series are trend stationary; (2) breaks in trend exist for coal and petroleum, but not for natural gas; (3) for the whole period, no rising (or decreasing) deterministic trend is found; (4) signs of the estimated trend coefficients are different. For example, for petroleum price, trend during differs from that in ; the U-shaped trend during this period is consistent with the 1980 oil gluts and OPEC s supply cut starting in Conclusions and Policy Implications This paper applies a new set of robust econometric methods to examine the trend properties in the prices of coal, natural gas, and petroleum over We initially test for number and time of breaks in trend using Kejriwal and Perron s (2010) sequential procedure, which is robust to whether the noise component is I(0) or I(1). After confirming the presence and number of breaks for coal and petroleum, we conduct the unit root test proposed by Carrion-i-Silvestre et al. (2009), allowing for multiple breaks in both the null and alternative hypotheses; for natural gas, we use the Ng-Perron test with no breaks. Further, we perform the Perron-Yabu trend estimate on each regime delineated by the estimated break time. Combining the findings in unit root tests and trend estimation, we conclude that the price series are trend stationary without significant rising deterministic trend; structural breaks exist in the prices for coal and petroleum while not for natural gas. Our results further support the findings in previous studies (e.g., Berck and Roberts, 1996; Lee, et al., 2006; Ghoshray and Johnson, 2010) in the sense that we find evidence against unit roots and rising deterministic trend. Results in this paper provide important policy implications. In particular, based on the results of break test and trend estimate in Table 2 and Figure 1, we observe that the estimated break dates and trends coincide with historical events related to energy supply or demand. For example, a break date in 1979 is observed for petroleum, which might be partly caused by the National Energy Act of This act, including Energy Tax Act, promotes fuel efficiency and renewable energy, and therefore cause negative shocks in demand of conventional resources. Breaks in 1998 are observed for both coal and petroleum, which might be result from the series of OPEC supply cuts. However, the impacts of such policies are 8

10 temporary in the sense that they are ineffective in the long run. According to Table 4, there exists no evidence of positive or negative trends for coal, natural gas, and petroleum in long run path. 9

11 References Ahrens, W., Sharma, V., Trends in natural resource commodity prices: deterministic or stochastic? Journal of Environmental Economics and Management 33, Bai, J., Perron, P., Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, Berck, P., Roberts, M., Natural resource prices: will they ever turn up? Journal of Environmental Economics and Management 31, Christiano, L.J., Searching for a break in GNP. Journal of Business and Economic Statistics 10, Estrada, F., Perron, P., Breaks, trends and the attribution of climate change: a time-series analysis. Working paper, Boston University. Estrada, F., Perron, P., Gay C., Martinez, B., A Time series analysis of the 20th century climate simulations produced for the IPCC's fourth assessment report. Plos One, 8(3), e Ghoshray, A., Johnson, B., Trends in world energy prices. Energy Economics 32, Kejriwal, M., Perron, P., A sequential procedure to determine the number of breaks in trend with an integrated or stationary noise component. Journal of Time Series Analysis 31, Kelly, T.D., and Matos, G.R., comps., 2013, Historical statistics for mineral and material commodities in the United States (2013 version): U.S. Geological Survey Data Series 140, accessed [Month day, year], at Kim, D., Perron, P., Unit roots tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses. Journal of Econometrics 148, Lee, J., List, J.A., Strazicich, M.C., Non-renewable resource prices: deterministic or stochastic trends? Journal of Environmental Economics and Management 51, Lee, Y., Hu, H., Chiou, J., Jump dynamics with structural breaks for crude oil prices. Energy Economics 32, Nelson, C.R., Plosser, C., Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10,

12 Ng, S., Perron, P., Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, Noguera, J Oil price: breaks and trends. Energy Economics, Perron, P., The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, Perron, P., Testing for a unit root in a time series with a changing mean. Journal of Business and Economic Statistics 8, Perron, P., Vogelsang, T.J., Erratum: The great crash, the oil price shock and the unit root hypothesis. Econometrica 61, Perron, P., Yabu, T., 2009a. Estimating deterministic trends with an integrated or stationary noise component. Journal of Econometrics 151, Perron, P., Yabu, T., 2009b. Testing for shifts in trend with an integrated or stationary noise component. Journal of Business and Economic Statistics 27, Said, S.E., Dickey, D.A., Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, Slade, M.E., Trends in natural resource commodity prices: An analysis of the time domain. Journal of Environmental Economics and Management 9, Xiao, Z., Unit roots: A selective of the contributions of Peter C. B. Phillips. Econometric Theory / FirstView Article,

13 Appendix Table 1: Descriptive statistics for energy prices Series Period # Obs. Mean Std. dev. Min. Max. Skewness Coal Natural gas Petroleum

14 Table 2: Results for Kejriwal-Perron break test ExpW(1 0) ExpW(2 1) ExpW(3 2) ExpW(4 3) ExpW(5 4) Series: Coal price Test *** *** *** *** Break Date Series: Natural gas price Test Break Date Series: Petroleum price Test * ** * Break Date Notes. *, **, and *** indicate that the statistic is significant at the 10%, 5%, and 1% levels, respectively. 13

15 Table 3: Results for Ng-Perron and Carrion- unit root tests Model pt Mpt Mza Mzt MSB Carrion-i-Silvestre et al. test Coal (4 breaks) *** *** *** Petroleum (3 breaks) *** *** * ** Ng-Perron test Natural gas ( 0 break) *** *** *** Notes. *, **, and *** indicate that the statistic is significant at the 10%, 5%, and 1% levels, respectively. 14

16 Table 4: Results for Perron-Yabu trend estimate μ β K (MAIC) Series: Coal price *** (0.0927) (0.0080) *** (0.0771) (0.0116) *** (0.0723) (0.0166) *** (0.0896) (0.0164) *** (0.1229) (0.0246) *** (0.0734) * (0.0183) 1 Series: Natural gas price *** (0.1299) (0.0134) 11 Series: Petroleum price *** (0.1949) (0.0183) *** (0.2046) (0.0394) *** ** 1 (0.1181) (0.0044) *** (0.2306) (0.0529) *** * 1 (0.2206) (0.0570) Notes. *, **, and *** indicate that the statistic is significant at the 10%, 5%, and 1% levels, respectively. k is the number of lagged differences added to correct for serial autocorrelation. 15

17 Fig.1(a). Time series plot of coal price with 4 breaks Fig.1(b). Time series plot of natural gas price without breaks 16

18 Fig.1(c). Time series plot of petroleum price with 3 breaks 17