This paper investigates the trend in the monthly real price of oil between 1990 and 2008 with a generalized autoregressive conditional heteroskedasticity (GARCH) model. Trend and volatility are estimated jointly with the maximum likelihood estimation. There is long persistence in the variance of oil price shocks, and a GARCH unit root (GUR) test can potentially yield a significant power gain relative to the augmented Dickey-Fuller (ADF) test. After allowing for nonlinearity, the evidence supports a deterministic trend in the price of oil. The deterministic trend implies that influence of a price shock is transitory and policy efforts to restore a predictable price after a shock would be unwarranted in the long run.Oil Price, Volatility, Trend, GARCH, Fourier Form

Jing, Li

Thompson, Henry

English

Research Papers in Economics

1 
 
A Note on the Oil Price Trend and GARCH Shocks 
 
Jing Li*   
South Dakota State University 
 
Henry Thompson 
Auburn University 
 
ABSTRACT 
This paper investigates the trend in the monthly real price of oil between 1990 and 2008 with a 
generalized autoregressive conditional heteroskedasticity (GARCH) model.  Trend and volatility are 
estimated jointly with the maximum likelihood estimation.  There is long persistence in the variance 
of oil price shocks, and a GARCH unit root (GUR) test can potentially yield a significant power gain 
relative to the augmented Dickey-Fuller (ADF) test. After allowing for nonlinearity, the evidence 
supports a deterministic trend in the price of oil. The deterministic trend implies that influence of a 
price shock is transitory and policy efforts to restore a predictable price after a shock would be 
unwarranted in the long run.  
 
Keywords: Oil Price, Volatility, Trend, GARCH, Fourier Form 
JEL Classifications: Q31, C12, C53 
_____________ 
Thanks go to four anonymous referees and the editor of this journal for very helpful suggestions.  
* Corresponding Author, Department of Economics, South Dakota State University, Brookings SD 
57007, jing.li@sdstate.edu, Phone 605-688-4848, Fax 605.688.6386 
Henry Thompson, Economics, Comer Hall, Auburn University AL 36849, thomph1@auburn.edu, 
Phone 334-844-2910 
 2 
 
A Note on the Oil Price Trend and GARCH Shocks 
1. INTRODUCTION 
From the perspective of both oil supply and demand, it is critical to understand price 
behavior following a shock.  If the price were mean reverting or trend reverting, shocks would 
dissipate and policy efforts to restore price following a shock would be unwarranted.  If, however, 
there were no price reversion in a random walk or a stochastic trend, policy intervention would be 
wise to overcome the permanent effect of a price shock.  
For instance, price controls imposed by the US government following the oil price shock of 
the early 1970s would have been warranted if price were not to revert to its long term trend.  The 
same can be said for the OPEC production quotas of the early 1980s.  If price were a deterministic 
trend, however, the shocks would have had no permanent effects and these policies would have 
been redundant. 
The present paper examines the evidence of a deterministic trend in the monthly real price 
of oil from 1990 to 2008, a period of relatively stable market structure with no such overwhelming 
breaks in the price series.  Results from a GARCH unit root test that simultaneously estimates trend 
and variance are compared with well known difference stationarity tests.  The present results have 
direct implications for energy policy. 
2. GARCH AND FOURIER METHODOLOGY 
Suppose the price of oil   follows an AR(1) process  .  If   then 
 as   and the effect of a shock diminishes with time.  The series is then 
stationary and mean reverting following a shock.  In contrast, if   the effect of a shock never 
dies and the series is a nonstationary random walk with no reverting behavior.  3 
 
Figure 1 presents the deflated monthly spot price of West Texas Intermediate Oil between 
January 1990 and February 2008 from the Federal Reserve Economic Data.  The base year is 1990 
and CPI is used for deflating nominal oil price. The issue is whether the price trend is deterministic 
or stochastic.   
The present study focuses on monthly prices since most trading is done on a monthly basis.  
Also, the general autoregressive conditional heteroskedasticity( GARCH) effect becomes stronger 
and the GARCH unit root (GUR) test brings more power gain for higher frequency data.  Preliminary 
analysis indicates that annual data lead to a weak GARCH effect and to difficulty in convergence of 
the maximum likelihood estimator (MLE) algorithm.  The dollar price is chosen following convention 
in the oil market and given the lack of significant swings in the trade weighted dollar.   
Results of recent studies based on annual data are mixed.  Berck and Roberts (1996) and 
Ahrens and Sharma (1997) find stochastic trends across a menu of unit root tests. Allowing for 
multiple breaks,  Lee, List, and Strazicich (2006) report a deterministic trend.  The present paper 
improves upon the previous methodologies in two ways.   
First, there is evidence that monthly price shocks follow the GARCH process proposed by 
Bollerslev (1986) regardless of the specification of deterministic terms in mean regressions.  Trend 
and volatility are then investigated jointly with an MLE-based GARCH unit root test.  Seo (1999) 
shows that this GUR test utilizes information in the volatility and therefore enjoys a power gain 
relative to the augmented Dickey Fuller (ADF) test.   
Another innovation of the present paper is the treatment of possible structural breaks.  
Breaks are accounted for first by dummy variables following Ahrens and Sharma (1997) and Lee, List, 
and Strazicich (2006).  The dummy variable approach, however, is restrictive given the assumptions 
about the maximum number and the functional form of breaks.  The present paper employs the 4 
 
Fourier form of Becker, Enders, and Hurn (2004) and Becker, Enders, and Lee (2006) that 
approximates breaks, or general nonlinear deterministic terms, with robustness.  
3. MODEL SPECIFICATION FOR THE PRICE OF OIL 
Consider solving the dynamic problem of extracting oil over n-periods subject to a total 
reserve constraint 
  ( )+ ,     
where   is the amount of oil extracted,   is the discount rate,  is consumer surplus, 
 is constant marginal cost,   is the Lagrangian multiplier, and   is oil reserves.  First order 
conditions imply  ( ) from which the price of oil   
satisfies the first-order difference equation   
There are various ways to add a stochastic element to price, and both stochastic and 
deterministic trends can be accommodated by this extraction model.  Consider first adding the 
stationary process   in the difference equation  .  Price is then a 
random walk when   = 0, and this random walk has a drift term if the mean of   is not zero.  The 
trend accumulated by the drift term is stochastic.   
Alternatively, solve the difference equation to find  .  We have a trend 
stationary price of oil after adding a stationary process to the solution  .  In 
this case, the trend is deterministic and nonlinear.   
Statistically we use the following models to test the type of trend.  
                                    (1) 
                                  (2) 
  (3) 
                             (4) 5 
 
Note the linear and quadratic trends are included in (1) and (2), respectively.  Let   and  be the 
two unknown break dates.  The dummy break variables in (3) are specified as 
  when   and   otherwise.  Lee and Strazicich (2001) shows that the ADF 
type unit root tests that allow for endogenous structural breaks have the drawback of spurious 
rejection of the null hypothesis. By contrast the LM test proposed by Lee and Strazicich (2003) does 
not have that drawback, and therefore the LM test is used here. The Fourier form in (4) employs   
and   terms to approximate instantaneous or gradual breaks in deterministic terms.  The Fourier 
form may also describe a general nonlinear model without the necessity of breaks
1. 
          The break dates  and the frequency  are estimated with data dependent methods.  The two 
break dates are estimated by the values that minimize the LM statistics of Lee and Strazicich (2003). 
There is a possible efficient estimator of the break date that minimizes the residual sum squares 
(RSS) weighted by the shock variance
2. The estimated frequency minimizes RSS.  In general, the 
distribution for the unit root test with unknown breaks or unknown Fourier frequency is different 
from that with known breaks or frequency.  Critical values for the tests are simulated if unavailable 
in the literature.  
The primary goal is to test the null hypothesis   in (1) through (4).  The GUR test 
explicitly takes into account information in the variance of  .  In contrast, ADF tests do not include 
any effect of variance.  The conditional variance of  is specified in the following GARCH process 
                                   (5) 
where   denotes the conditional variance of .  Hillebrand (2005) shows long persistence in 
variance may be caused by breaks in the variance, a complexity left for future research since the 
related GUR test is underdeveloped.  
                                                           
1 We are grateful to a referee for pointing this out. 
2 We thank a referee for suggesting this estimator.  6 
 
Equations (1) through (4) along with (5) are estimated with the maximum likelihood method. 
Let   denote the MLE estimate of  .  The GUR test is computed as 
                                           (6) 
Seo (1999) shows the null distribution of the GUR test is a weighted average of the Dickey Fuller t 
and standard normal distributions.  The weight is controlled by the parameter   that is bounded 
between 0 and 1.  A smaller   implies more weight for the standard normal distribution and more 
power gain from the GUR test.   
4. RESULTS 
Table 1 reports the main findings.  The GUR test is justified by various deterministic terms in 
mean regressions since ARCH tests always indicate autoregressive conditional heteroskedasticity.  
The GUR weight   is close to 0.6 in all cases implying the Dickey-Fuller t distribution only accounts 
for 0.6/(0.6 + (1 – 0.6
2)
1/2) = 43% of the hybrid distribution
3.  The power gain of the GUR test is 
substantial.  
The ADF test, LM test with two breaks, and GUR test reject a stochastic trend at the 5% level 
except for model (1), providing strong evidence against a stochastic trend.  The sharp difference 
between (1) and (2) through (4) illustrates the importance of allowing for the nonlinear specification 
when examining the time series properties of oil price. Nonlinear specification is also emphasized by 
Ahrens and Sharma (1997) and Lee, List and Strazicich (2006).   
Similar results from the ADF/LM tests and GUR test are not evidence against the GUR test.  
Notice that the ADF and LM tests reject the null hypothesis for models (2) through (4). That means 
the power of the ADF and LM tests is so high (for this problem) that it disables the presumably 
superior GUR test to produce qualitatively different results.  
                                                           
3 See Equation (13) of Seo (1999) for details.  7 
 
The BIC for the ADF/LM tests (denoted by BIC
1) and for the GUR test (denoted by BIC
2) are 
reported for model selection.  The quadratic trend model (2) outperforms the linear trend model (1) 
by the BIC criterion as it is better able to capture the upturn in 1999.   Two dummy break variables 
in (3) and the sin term of the Fourier model (4) are significant.  The Fourier model (4) outperforms 
the dummy model (3) by BIC
2.  Both BIC
1 and BIC
2 pick the quadratic trend model (2) as the best 
model.  
The conclusion is that the trend in the price of oil is deterministic.  The trend may be 
quadratic or linear with unspecified breaks, or it may have a general nonlinear functional form. 
 
5. CONCLUSION 
The deterministic trend in the monthly price of oil between 1990 and 2008 suggests policy 
reactions to oil price shocks are unwarranted in the long run since price reverts to its long term 
trend.  Evidence of the deterministic trend also provides support for optimal depletion models of 
Hotelling (1931).  Our main findings are consistent with those of Lee, List, and Strazicich (2006). In 
terms of methodology, the findings in this paper contribute to the growing literature that suggests 
price models can be improved by allowing for a nonlinear specification.  
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
   9 
 
 
 
 
Table 1.  Results  
Model  (1)  (2)  (3)**  (4) 
ARCH(4)  test  14.50*  14.32*  22.82*  23.82* 
  0.55  0.57  0.57  0.58 
ADF test  -0.98  -3.96*  na  -4.09* 
LM test***  na  na  -6.12*  na 
BIC
1  1.34  1.29  1.34  1.34 
GUR Test  -1.18  -3.95*  -6.30*  -5.04* 
BIC
2  1.49  1.44  1.47  1.44 
         
  0.0042*  -0.0430*  -0.0143*  0.3293* 
  na  0.0002*  na  na 
  na  na  1.6482*
  na 
  na  na  0.0112  na 
  na  na  -0.2506  -55.228* 
  na  na  0.1411*  19.596 
*Significant at 5% level. ARCH(4) denotes the test for autoregressive conditional heteroskedasticity 
with 4 lags.  denotes the weight of Dickey-Fuller distribution. BIC
1 denotes the BIC for the ADF or 
LM test, and BIC
2 for the GUR test. 
**The estimated break dates are July 1999 and December 2002. 
***See Lee and Strazicich (2003) for the details about the LM unit root test with two breaks.  
 
   10 
 
References 
 
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Becker, R., Enders, W., and Lee, J. (2006). “A Stationarity Test in the Presence of an Unknown 
Number of Smooth Breaks.” Journal of Time Series Analysis 27: 381-409.  
 
Berck, P. and Roberts, M. (1996). “Natural Resource Prices: Will They Ever Turn Up.” Journal of 
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Bollerslev, R. (1986). “Generalized Autoregressive Conditional Heteroskedasticity.”  Journal of 
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A Note on the Oil Price Trend and GARCH Shocks

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