Full text of DOI:10.1111/1467-8276.00305

On Jumps and ARCH Effects in
Natural Resource Prices: An Application to
Pacific Northwest Stumpage Prices
Jean-Daniel Saphores, Lynda Khalaf, and Denis Pelletier
Continuous-time models of natural resource prices usually preclude the possibility of large changes
(jumps) resulting from unexpected events. To test for the presence of jumps and/or ARCH effects,
we combine bounds and the Monte Carlo test technique to obtain finite-sample, level-exact p-values.
We apply this methodology to stumpage prices from the Pacific Northwest and find evidence of
jumps and ARCH effects. To assess the impact of neglecting jumps on the decision to harvest old-
growth timber, we develop an autonomous, infinite-horizon stopping model for which we provide a
new method of resolution. Our numerical results show the importance of modeling jumps explicitly.
Key words: conditional heteroscedasticity, jump diffusions, Monte Carlo tests, real options, stumpage
prices.
With the development of real options the-
ory and its applications, there has been
an increasing interest in modeling natural
resource prices with continuous-time stochas-
tic processes, especially the geometric Brow-
nian motion (GBM), to deal with decisions
that have both uncertain and irreversible
characteristics. Examples of papers where the
price of a natural resource follows a GBM
include Pindyck, Brennan and Schwartz, or
Lund (1992). In forestry, the GBM is often
used to model timber prices, as in Morck,
Schwartz, and Stangeland; Clarke and Reed
(1989, 1990); Zinkhan; Thomson; Reed; Con-
rad and Ludwig; or Yin and Newman. While
the usefulness of the geometric Brownian
motion as a theoretical tool is well estab-
lished, its adequacy for deriving practical
decision rules for natural resources has been
questioned on both theoretical and empirical
grounds.
First, Lund (1993) argues that the GBM
is unlikely to be an equilibrium price pro-
cess for exhaustible resources produced by
heterogeneous firms. Moreover, if markets
are sufficiently competitive, we might expect
natural resource prices to exhibit mean-
reversion (Schwartz). In fact, the theoreti-
cal nature of the data generating process for
prices is still an open question given all the
factors contributing to their formation. Here,
although our focus is on the GBM (follow-
ing unit root tests), the methodology we pro-
pose is easily adaptable to mean-reverting
processes.
A second line of inquiry about the ade-
quacy of the GBM (and of pure diffusion
processes in general) for modeling prices
comes from the underlying assumption of
smooth changes. As explained by Merton,
a diffusion process precludes the possibil-
ity of large changes, or jumps, which may
result from the sudden arrival of informa-
tion. In forestry, these jumps can be caused,
for example, by political decisions (a ban
on imported timber), court rulings (logging
restrictions to protect endangered species),
or natural events (fires, diseases, or storms).
If the data generating process of a price
time series is a jump-GBM process, the distri-
bution of the increments of the logarithm of
prices has tails heavier than the normal distri-
bution (the so-called fat tails). This common
feature has been invoked to explain discrep-
ancies between the actual pricing of financial
options and theoretical predictions.
Jean-Daniel Saphores is assistant professor, School of Social
Ecology and Economics Department, University of California
Irvine; Lynda Khalaf is associate professor, Economics Depart-
ment, Université Laval, Québec, Canada. Denis Pelletier is grad-
uate student, Economics Department, Université de Montréal,
Canada.
The authors thank Jean-Thomas Bernard, William Tomek,
and participants at the 1998 annual conference of the Cana-
dian Economic Association and at the 2000 annual conference of
the Canadian Association of Agricultural Economists for helpful
suggestions. Peter Berck and an anonymous referee made valu-
able suggestions that greatly improved this article. Finally, the
authors thank Debra Warren from the Forest Service in Port-
land, OR, for providing some of the stumpage price data. The
authors are, of course, responsible for all remaining errors.
Amer. J. Agr. Econ. 84(2) (May 2002): 387–400
Copyright 2002 American Agricultural Economics Association

388
May 2002
Amer. J. Agr. Econ.
Fat tails may, however, also result from reject the unit root model.3 Since all four of
other processes. A tractable and popular
alternative is a process with time varying
parameters, such as ARCH (Engle). Another
possibility is a GBM with both jumps and
ARCH errors. Since the implications of these
models may differ, it is important to have
a statistical methodology for assessing the
significance of jumps and/or ARCH effects
on price volatility that is reliable with small
samples (a common occurrence in natural
resource economics.) To date, however, no
finite-sample level-exact test for ARCH in
the presence of jumps and for jumps in the
presence of ARCH seems available.1
In this context, this article makes several
contributions. First, building on earlier work
by Khalaf, Saphores, and Bilodeau, we pro-
pose a methodology based on the Monte
Carlo (MC) test technique (Dufour 1995)
to obtain finite-sample, level-exact tests for
ARCH effects in the presence of jumps, and
for jumps in the presence of ARCH effects.2
To deal with nuisance parameters, we derive
exact bound cutoff points to make sure rejec-
tions (which provide evidence in favor of
ARCH and/or jumps) are conclusive.
our time series exhibit fat tails, we investigate
ARCH and mixture distributions as proba-
ble causes and find evidence of jumps and
ARCH effects.
To investigate the empirical impact of
jumps, we reconsider the classical tree-cutting
problem when stumpage prices for old-
growth forest follow a GBM with jumps.
We formulate this autonomous, infinite-
horizon stopping problem in continuous time
using the theory of real options (Dixit and
Pindyck). The resulting Bellman equation is
complicated because it contains both an inte-
gral and derivatives of an unknown func-
tion. To solve it, we propose a new approach,
based on an extension of a Galerkin proce-
dure (Delves and Mohamed), which simply
requires solving a system of linear equations.
We find that ignoring jumps when they are
present can lead to significantly suboptimal
decisions. The study of the impact of ARCH
effects is left for future work because in con-
tinuous time they lead to complex stochastic
variability models (Duan).
This article is organized as follows. In the
next section, we present our econometric
framework. We then describe the data and
report our empirical results. Following the
next section, we develop a simple stopping
problem to assess the impact of neglecting
jumps on the decision to harvest timber. The
last section presents our conclusions.
We then apply this methodology to four
quarterly stumpage price time series from
public forests in the Pacific Northwest region.
Our testing strategy is particularly relevant
here because stumpage price data sets are
usually fairly small. To assess the adequacy of
the GBM, we conduct the well-known Perron
(1989) unit root test and find that we cannot
Models and Tests for Jumps and
Arch Effects
1 Deriving valid p-values for no-jump likelihood ratio tests,
with or without ARCH, is an econometric challenge often over-
looked in empirical work for at least two reasons. First, the rate
of arrival of jumps is on the boundary of its domain, and sec-
ond, there are unidentified nuisance parameters under the null
hypothesis. The former may cause the limiting distribution of
the LR test statistic to be discontinuous (Brorsen and Yang),
and the latter may cause it to be nonstandard. These prob-
lems are compounded in small samples. In addition, standard
ARCH tests (such as Engle’s 1982 test) may not be appropriate
in the presence of jumps. Indeed, the jump process parameters
(which intervene under the null and the alternative hypothe-
ses) are identifiable if the rate of arrival of jumps is restricted
to be strictly positive. This implies, however, that the nuisance
parameters’ space includes a locally almost unidentified region
(Dufour 1997), which may seriously distort test sizes. In practice,
this means that for the tests of interest here, one must seriously
guard against spurious rejections.
Let the random variable St denote a price
(e.g., a stumpage price) at time t. If St follows
a GBM with trend ꢄ and variance parame-
ter ꢅ,
(1) dSt = ꢄSt dt +ꢅSt dz
then xt ≡ Ln St+1ꢃ − Ln Stꢃ is normally dis-
tributed with variance ꢅ2 and mean ꢆ = ꢄ −
0ꢇ5ꢅ2.
2 To define an exact test (or p-value), consider a test prob-
lem pertaining to a parametric model (i.e., the data generating
process is determined up to a finite number of unknown real
parameters ꢀ ∈ ꢁ). Let ꢁ0 refer to the subspace of ꢁ com-
patible with the null hypothesis H0, which we suppose (with-
out loss of generality) corresponds to a test statistic with critical
region S ≥ c. To have an ꢂ-level test, c must be chosen so that
supPꢀ S ≥ c ꢀ ꢀ ∈ ꢁ0ꢃ ≤ ꢂ. This test has size ꢂ if and only if
supPꢀ S ≥ c ꢀ ꢀ ∈ ꢁ0ꢃ = ꢂ. Size control is usually very difficult
to achieve.
3 Note that the performance of unit root tests (including Per-
ron’s test) in the presence of jumps has not been formally
assessed. In addition, it is well known that ARCH and/or breaks-
in-variance can lead to serious under- or over-rejections in
these tests (Kim and Schmidt). Since the processes we consider
involve heteroscedasticity with related characteristics; this ques-
tion deserves further consideration; yet it is beyond the scope of
this article.

Saphores, Khalaf, and Pelletier
Jumps and ARCH in Natural Resource Prices
389
ꢂ
ꢃ
ꢄ
T
To allow for discontinuities in St, consider
ꢁ
1−ꢊ
xt −ꢆ
(7) LJump-GBM
=
ln
'
the mixed jump-diffusion process,
ꢅ
ꢅ
t=1
ꢃ
ꢄꢅ
(2) dSt = ꢄSt dt +ꢅSt dz+St dq
ꢊ
xt −ꢆ−ꢋ
+ √
'
√
where ꢄ dt is the expected change in St dur-
ing dt when there is no jump, dz is an incre-
ment of a standard Wiener process, and dq is
a discrete increment in St due to a jump. We
assume that dq and dz are independent. If a
jump occurs at ꢈ, then dq = Yꢈ −1, where Yꢈ
is the percentage change in Sꢈ (Merton), and
0 otherwise. More formally, Yꢈ is the ratio of
Sꢈ after a jump divided by Sꢈ before a jump:
Yꢈ ≡ Sꢈ+/Sꢈ− with
ꢅ2 +ꢌ2
1
ꢅ2 +ꢌ2
ꢆ
ꢇ
ꢄ
ꢃ
T
ꢁ
xt −ꢆ
(8) LARCH
=
ln ꢀ
'
ꢀ
ht
ht
t=1
ꢆ
ꢃ
ꢄ
T
ꢁ
1−ꢊ
xt −ꢆ
(9) LJump-ARCH
=
ln
ꢀ
'
ꢀ
ꢀ
ht
ht
t=1
ꢇ
ꢃ
ꢄ
ꢊ
xt −ꢆ−ꢋ
+ ꢀ
'
ꢇ
ht +ꢌ2
ht +ꢌ2
S
ꢈ+ = lim St and
S
ꢈ− = lim Stꢇ
t→ꢈꢉt>ꢈ
t→ꢈꢉt<ꢈ
For both LARCH and LJump-ARCHꢉht is defined
by equation (5).
Following Ball and Torous, we assume that
the arrival of jumps has a Bernoulli distribu-
tion with arrival rate ꢊ and identically inde-
pendently distributed lognormal jump sizes
ln Ytꢃ ∼ N ꢋꢉꢌ2ꢃꢃꢍ ꢋ is thus the mean of the
logarithm of jump sizes and ꢌ2 is the corre-
sponding variance. With the jump process,
To obtain evidence on ARCH/jump effects,
we conduct four likelihood ratio-based (LR)
tests. For ease of exposition, we introduce
the notation Hij , where index i refers to the
ꢁ
absence (when i = Aꢉ%1 = 0) or presence
(when i = Aꢉ%1 > 0) of ARCH effects, and
ꢁ
index j refers to the absence (when j = J,
(3) xt = ꢆ+ꢅꢎt +ln Ytꢃntꢇ
ꢊ = 0) or presence (when j = Jꢉꢊ > 0) of
jumps. If LR Hij ꢉHklꢃ denotes the LR statis-
tic for testing the null, Hij , against the alter-
native Hkl, hypothesis, then
In the above, ꢎt ∼ N 0ꢉ1ꢃ and nt is a
Bernoulli random variable. It equals one
when a jump occurs (with probability ꢊ) in
the interval !t −1ꢍ t", and zero otherwise.
An alternative to the GBM with jumps
that can also produce fat tails is a GBM
with ARCH(1) errors, with or without jumps.
ꢈ
ꢊ
ꢉ
ꢉ
(10) LR Hij ꢉHklꢃ = 2 LH −LH
kl
ij
ꢉ
ꢉ
where LH and LH are respectively the max-
imum of ijthe log-kllikelihood function under
the null and the alternative hypotheses. We
With the notation above, it can be written
ꢀ
(4) xt = ꢆ+ htet +ln Ytꢃntꢇ
test for jumps in the GBM LR H ꢉ HAJ ꢃꢃ
ꢁꢁ
AJ
ꢁ
In equation (4) et ∼ N 0ꢉ1ꢃ and the condi-
andinthejump-ARCH LR HAJꢁꢉHAJ ꢃꢃmod-
tional variance, ht, is defined by
els. We also test for ARCH in the jump-ARCH
(5) ht = %0 +%1 xt−1 −ꢆꢃ2ꢇ
Duan shows that the discrete model
described by (4) and (5) converges to a
stochastic volatility model as the frequency
of observations goes to infinity.
We estimate the parameters of these mod-
els by numerical maximization of the likeli-
hood function of the parameter vector given
the observations xt ≡ Ln St+1ꢃ − Ln Stꢃꢉ t =
1ꢉꢇꢇꢇ ꢉT , after taking out seasonal effects
and the impacts of a definition change (see
below). Let ' designate the density of the
standard normal distribution. Then the log-
likelihood functions for the GBM, jump-
GBM, ARCH, and jump-ARCH models are
given respectively by
model (LR HAJ ꢉHAJ ꢃ), and we apply Engle’s
ꢁ
no-ARCH test, denoted by LM H ꢉH ꢃ, to
ꢁꢁ
AJ
ꢁ
AJ
(4) and (5). LM H ꢍH ꢃ is the Lagrange
ꢁꢁ
ꢁ
multiplier (LM) test foArJ ARCH(1) effects
over a GBM. It equals TR2 from the regres-
sion of xt2 on a constant and xt2−1, where R2
is the coefficient of determination and T is
the sample size in terms of xt. The asymptotic
distribution of LM H ꢍH ꢃ is 02 1ꢃ.
AJ
ꢁꢁ
AJ
ꢁ
AJ
It is well known that Engle’s LM test
tends to under-reject if 02 1ꢃ critical points
are used (see Dufour et al. and references
therein). The no-jump tests LR HAJꢁꢉHAJ ꢃ
and LR H ꢉHAJ ꢃ may suffer from even
ꢁꢁ
AJ
ꢁ
more serious problems: as observed in
Brorsen and Yang, when the no-jump
hypothesis is imposed, ꢊ (the rate of arrival
of jumps) lies on the boundary of the param-
eter space and the nuisance parameters ꢋ
ꢂ
ꢃ
ꢄꢅ
T
ꢁ
1
ꢅ
xt −ꢆ
(6) LGBM
=
ln
'
ꢅ
t=1

390
May 2002
Amer. J. Agr. Econ.
and ꢌ (respectively the mean and the stan- Note that
dard deviation of the logarithm of jumps)
ꢉ
RN LR0ꢃ−1
are not identified. Therefore, the standard
1−
02-approximation to the null distribution of
N
LR does not obtain, even asymptotically,
and the statistic’s limiting null distribution
is nonstandard (Davies 1977, 1987; Hansen).
Finally, when testing for ARCH in the pres-
ence of jumps, the nuisance parameters ꢊ,
ꢋ, and ꢌ are “estimable” under the null and
the alternative hypothesis if the restriction
ꢊ > 0 is imposed. Although this justifies the
use of standard asymptotic cutoff points, it
may conceal important distributional prob-
lems because the relevant nuisance param-
eter space includes a locally almost uniden-
tified (LAU) region as ꢊ approaches the
would often be used in a standard bootstrap
as it relies only on asymptotic arguments.
Two of the test criteria we use, (LR H
ꢃ and LM H ꢉH ꢃ), are pivotal (their
distribution under H0 does not depend on
nuisance parameters). For LR H ꢉHAJ ꢃ,
the null hypothesis sets ꢊ (the rate of arrival
of jumps) to zero so neither ꢋ nor ꢌ (the
mean and standard deviation of the logarithm
of jumps) is identifiable. However, since the
MC p-value calculated as described above
by drawing from the no-jump DGP depends
neither on ꢋ nor on ꢌ, the null distribution
ꢉ
ꢁꢁ
AJ
H
ꢁ
AJ
ꢁꢁ
AJ
ꢁ
AJ
ꢁꢁ
AJ
ꢁ
zero boundary. As demonstrated in Dufour of LR H ꢉHAJ ꢃ is nuisance parameter-free.
(1997), severe size distortions may then occur The same argument holds for LM H ꢉH
ꢁꢁ
ꢁ
AJ
ꢃ
ꢁ
ꢁꢁ
and for LR H ꢉHAJ ꢃ, a test statisAtiJc used
AJ
with standard critical points, even if identi-
fying restrictions are imposed. The challenge
for all tests considered is thus how best to
approximate the statistics’ finite sample dis-
tribution under the null hypotheses.
To circumvent the unidentified nuisance
parameter problem and obtain improved p-
values in finite samples, we combine bounds
and the MC test technique (Dufour 1995),
which is closely related to the parametric
bootstrap. Whereas a standard parametric
bootstrap does not, in principle, guarantee
size or level control for finite T (sample size)
or N (replications), the MC technique pro-
vides a randomized version of a test that con-
trols its size provided this test’s null distribu-
tion can be simulated.
ꢁꢁ
below. ApplyiAnJg the MC test procedure
to LR H ꢉH ꢃ and LM H ꢉH ꢃ thus
ꢁꢁ
ꢁ
ꢁꢁ
AJ
ꢁ
AJ
AJ
yields exact sAizJe p-values.4 To emphasize
the pivotal test property, the associated MC
p-value is labeled PMC (“P” stands for
pivotal).
However, our other two test statistics
(LR HAJꢁꢉH ꢃ and LR HAJ ꢉHAJ ꢃ) are not
ꢁ
pivotal. The AAJ RCH parameter %1 for the for-
mer and jump parameters ꢊꢉꢋ, and ꢌ for
the latter intervene as (identified) nuisance
parameters. A standard parametric bootstrap
would rely on point estimates of the nuisance
parameters to generate a p-value. It would be
unlikely to yield reliable results because of
potential convergence failure due to bound-
ary problems for LR HAJꢁꢉHAJ ꢃ (ꢊ is on the
frontier of its parameter space under H0),
and to the presence of the LAU nuisance
Let us first describe how the MC test tech-
nique may be implemented for a right-tailed
LR test when there are no nuisance parame-
ters under H0:
parameter ꢊ for LR HAJ ꢉHAJ ꢃ, which may
ꢁ
cause spurious rejections in finite samples.
By contrast, the MC method described
above can be modified to still guarantee
level control: in this case, the MC p-value
is defined as the largest simulated p-value
over the relevant nuisance parameter space.
For details on the validity and the imple-
mentation of the MC method in the pres-
ence of nuisance parameters, see Dufour
(1995). However, this approach is likely to
be computationally demanding. This diffi-
culty may be avoided, however, if we can
1. Using the observed sample, calculate the
LR statistic LR0.
2. Using draws from the null data generat-
ing process (DGP), generate N simulated
samples.
3. For simulated sample nꢉ1 ≤ n ≤ N, com-
pute the LR statistic LRn.
ꢉ
4. In LR0ꢉꢇꢇꢇ ꢉLRN , find the rank RN LR0ꢃ
of the observed statistic LR0.
5. Reject the null hypothesis at level ꢂ if
ꢉ
RN LR0ꢃ ≥ N + 1ꢃ 1 − ꢂꢃ + 1. A MC p-
value may be obtained from:
4 LR H ꢉH ꢃ, LR H ꢉH ꢃ, and LM H ꢉH ꢃ can also
ꢁꢁ
ꢁ
ꢁꢁ
ꢁꢁ
AJ
ꢁ
AJ
be seen tAoJ be ApJivotal beAcaJ use under H0, xt follows a normal
distribution for which there is invariance to location and scale
(ꢆ and ꢅ).
AJ
ꢉ
RN LR0ꢃ−1
pˆN LR0ꢃ = 1−
ꢇ
N +1

Saphores, Khalaf, and Pelletier
Jumps and ARCH in Natural Resource Prices
391
find a pivotal statistic that bounds our LR (skewness and kurtosis) tests, (iii) the Ljung–
test statistic. This is the approach we fol- Box tests, and (iv) the Lo and McKinlay vari-
low here. Indeed, both the null distribu- ance ratio tests (Campbell, Lo, and McKin-
tions of LR HAJ ꢉHAJ ꢃ and LR HAJꢁꢉHAJ ꢃ lay, chapter 2).
ꢁ
are bounded by the null distribution of the
pivotal statistic LR H ꢉH ꢃ: since H
⊆
ꢁꢁ
ꢁꢁ
AJ
AJ
H
and Hꢁꢁ ⊆ H , both LARJ HAJ ꢉHAJ ꢃ and
ꢁ
ꢁ
ꢁ
AJ
AJ
AJ
Application to Forestry Prices
LR HAJꢁꢉHAJ ꢃ are smaller than or equal to
LR H ꢉHAJ ꢃ. Thus, if we use the cutoff
ꢁꢁ
pointsAaJssociated with LR H ꢉHAJ ꢃ, we are
Stumpage Data
ꢁꢁ
AJ
sure that rejections are conclusive.5 These
bounding cutoff points must be obtained by
simulation here since the null distribution of
Quarterly stumpage “cut prices” from 1973
to the first quarter of 1997 for Douglas
Fir, Ponderosa and Jeffrey Pines, Western
Hemlock, and True Firs were provided by
the USDA Pacific Northwest Research Sta-
tion, in Portland, Oregon. The “cut price”
is the high-bid price adjusted for rates actu-
ally paid for timber, when the logs are scaled
after harvest; it thus represents the current
value of harvested timber in the marketplace.
We deflate these data using the wholesale
price deflators from the Bureau of Economic
Analysis. The National Forest Service also
publishes “sold-stumpage prices,” which are
three-month averages of high-bid prices for
the right to harvest timber, but they are avail-
able only as an average for all tree species.
Because of the increased difficulty of log-
ging during the winter months and the cost
of storing logs that could be harvested dur-
ing the summer months, there are seasonal
variations in stumpage prices (Sohngen and
Haynes). Fall and winter stumpage prices
tend to be higher, while summer stumpage
prices tend to be lower than the yearly aver-
age. In addition, since 1984, stumpage prices
have included estimated purchaser credit for
road construction (Haynes and Warren). To
account for these seasonal effects and for the
1984 definitional change, we follow David-
son and MacKinnon (1993, chapter 19). We
perform a preliminary maximum likelihood
estimation (MLE) for all maintained models
using a definition change dummy and three
seasonal dummies. The residuals from the
preliminary MLE yield a seasonally adjusted
series upon which we perform our diagnos-
tic tests and to which we fit our models. This
is numerically equivalent to adding the dum-
mies to the models analyzed.
LR H ꢍHAJ ꢃ is nonstandard. The p-values
ꢁꢁ
thus oAbJtained are labeled BMC (for bounds
MC) while the parametric bootstrap p-value
obtained in the same context are referred to
as local MC (LMC) p-values.
To illustrate the implementation of this
procedure, consider the case of LR HAJꢁ
,
HAJ ꢃ:
1. Using the observed sample, estimate (4)
and (5) with and without jumps to get HAJ
and HAJꢁ respectively, then calculate the
ꢉ
ꢉ
likelihood ratio LR0 = 2!LH −LH ꢁ".
AJ
AJ
2. Generate N simulated samples drawing
from the null DGP ((4) and (5) without
jumps).
3. For simulated sample nꢉ1 ≤ n ≤ N, esti-
mate (4) and (5) first without ARCH
nor jumps and then with ARCH and
jumps; compute the likelihood ratio LRn =
ꢉ
ꢉ
2!LH −LHꢁꢁ".
AJ
AJ
ꢉ
4. In LR0ꢉꢇꢇꢇ ꢉLRN , find the rank RN LR0ꢃ
of LR0.
5. The BMC p-value is
ꢉ
RN LR0ꢃ−1
1−
ꢇ
N +1
TheapplicationtoLR HAJ ꢉHAJ ꢃisstraight-
ꢁ
forward. This conservative approach prevents
spurious rejections of ARCH effects in the
presence of jumps (LR HAJ ꢉHAJ ꢃ) and of
ꢁ
jumps in the presence of ARCH effects
(LR HAJꢁꢍHAJ ꢃ), even with small samples.
We also conduct commonly used random
walk diagnostic tests: (i) Perron’s (1989,
1993) unit root test,6 (ii) the Jarque–Bera
Results
5 This is the basic reasoning behind the Durbin–Watson auto-
correlation bounds test.
Table 1 gives a summary of diagnostic statis-
tics for the logarithm of stumpage prices
Ln Stꢃ (Perron’s test) and their change
Ln St+1ꢃ − Ln Stꢃ (Jarque–Bera, Ljung–Box,
6 We use Perron’s test instead of more “standard” tests like
the Dickey–Fuller unit root test to account for a 1984 definition
change that may create a structural break in our data (see below
and notes pertaining to table 1).

392
May 2002
Amer. J. Agr. Econ.
Table 1. Summary Statistics
5%
Douglas
Fir
Ponderosa and
Jeffrey Pines
Western
Hemlock
Statistic
Critical Points
True Firs
Perron (1989)
Skewness
Kurtosis
−3ꢇ93
0ꢇ479
3ꢇ52
−3ꢇ55
0ꢇ728
−2ꢇ44
−0ꢇ006
4ꢇ698
−4ꢇ15
−1ꢇ276
10ꢇ686
262ꢇ357
−3ꢇ79
−0ꢇ684
4ꢇ710
8ꢇ907
Jarque–Bera
Autocorrelations
Lag 1
5ꢇ99
148ꢇ052
11ꢇ527
19ꢇ195
−0ꢇ255
0ꢇ018
−0ꢇ105
−0ꢇ040
7ꢇ75
−0ꢇ119
−0ꢇ007
0ꢇ058
−0ꢇ420
0ꢇ049
−0ꢇ341
−0ꢇ056
0ꢇ016
Lag 2
Lag 3
−0ꢇ062
Lag 4
−0ꢇ190
0ꢇ111
−0ꢇ184
Ljung–Box
Variance ratios
Z∗ 2ꢃ
9ꢇ49
5ꢇ44
19ꢇ36
15ꢇ32
1ꢇ96
1ꢇ96
−1ꢇ302
−1ꢇ232
−0ꢇ793
−0ꢇ618
−2ꢇ021
−1ꢇ814
−2ꢇ126
−1ꢇ972
Z∗ 4ꢃ
Notes: Each stumpage price sample (S ꢉꢇꢇꢇ ꢉS ) has 97 observations. Except for Perron’s test, diagnostic tests are applied to changes in the logarithm of
1
T
stumpage prices (Ln S
ꢃ−Ln S ꢃ) after removing seasonal effects and the impacts of a definition change.
1+1
t
Since standard unit root tests (e.g., Dickey–Fuller) are not reliable in the presence of structural breaks, we apply the unit root test proposed by Perron
(1989, Model B) when there is an exogenous break point occurring at a known date t . We first de-trend each series by regressing Ln S ꢃ on a time trend
B
t
∗
∗
∗
e
t
∗
and the structural change dummy variable DT = t − T . Let e denote the residuals from this regression. We then regress
− e ꢃ on a constant,
t−1
B
t
t
∗
∗
the seasonal dummies, and e . The standard t statistic associated with e
t−1
yields a valid unit root test criterion, provided cutoff points form Perron
t−1
(1993, table I) are used. Perron shows that these cutoff points are typically farther in the tails than the corresponding Dickey–Fuller tests critical points.
In connection, see Perron (1989, p. 1378).
Critical points for the skewness and kurtosis tests are taken from D’Agostino and Stephens (1986, table 9.3, p. 379 and table 9.5, p. 385). Under the null
2
hypothesis of normality and relevant regularity conditions, the asymptotic distribution of the Jarque–Bera is 0 2ꢃ.
and variance ratios) after taking out sea- can conclude that the GBM hypothesis seems
soundly rejected.7
sonal effects and the impacts of a definition
Table 2 shows the parameters estimated
by fitting a GBM, a jump-GBM, an ARCH,
and a jump-ARCH to the four series con-
sidered. These parameters were obtained by
numerical maximization of the correspond-
ing log-likelihood functions (see equations
(6) to (9)), using GAUSS. Looking first at
the continuous components of our models,
we note that ꢆ, the trend parameter for
Ln Stꢃ, appears to be close to 0, with the
possible exception of the jump-ARCH for
Ponderosa and Jeffrey Pines, and both jump
models for True Firs. As expected, the inclu-
sion of a jump process reduces the vari-
ance of the continuous process. Looking at
the jump parameters, we observe that the
rate of arrival of jumps (ꢊ) varies from ∼0ꢇ2
(i.e., one jump every 5 quarters on aver-
age) for Douglas Fir to between ∼0ꢇ4 and
∼0ꢇ5 (one jump every 2 to 2.5 quarters on
average) for True Firs.8 We also note that ꢋ
change. First, except for Western Hemlock,
Perron’s test fails to reject the presence of
a unit root at the 5% level. Note, however,
that in this case, Perron’s test is not signif-
icant at 2.5%. This result gives some sup-
port to our choice of GBM-based models,
but it must be qualified given that the per-
formance of unit root tests in the presence of
jumps has not been formally assessed. Sec-
ond, we observe that for three of the series,
logarithmic changes are skewed at 5% (the
exception is Ponderosa and Jeffrey Pines),
and we see evidence of high kurtosis (espe-
cially for Douglas Fir and Western Hemlock).
This is confirmed by the Jarque–Bera statis-
tic, which is significant at 5% for all times
series, a clear sign of fat-tailed distributions.
Third, the Ljung–Box autocorrelation test is
significant at 5% for Western Hemlock and
True Firs, but not for Douglas Fir or Pon-
derosa and Jeffrey Pines. This effect, how-
ever, may be due to the presence of het-
eroscedasticity that distorts the test’s size in
smaller samples (see Jorion, p. 432). Fourth,
the variance ratio tests also reject the ran-
dom walk null for Western Hemlock and
True Firs. As is well known, caution must be
exercised in interpreting decisions from a bat-
7 In a discrete time framework, Haight and Holmes find that
quarterly (average) stumpage prices of Loblolly Pine follow a
GBM, but that nonaveraged quarterly or monthly stumpage
prices follow stationary autoregressive models. They do not,
however, consider the presence of seasonal effects or of jumps
in their data.
8 ꢊ is smaller for Western Hemlock and somewhat larger for
Ponderosa and Jeffrey Pines but we will see below that jumps
tery of diagnostic tests. Yet on the whole, we are not statistically significant for these two series.

Saphores, Khalaf, and Pelletier
Jumps and ARCH in Natural Resource Prices
393
Table 2. Maximum Likelihood Parameters
Parameter
ꢆ
ꢅ
%
%
ꢊ
ꢋ
ꢌ
0
1
Douglas Fir
GBM
−0ꢇ014
0ꢇ038ꢃ
0ꢇ003
0ꢇ029ꢃ
−0ꢇ047
0ꢇ033ꢃ
−0ꢇ014
0ꢇ023ꢃ
0ꢇ370
0ꢇ027ꢃ
0ꢇ191
0ꢇ032ꢃ
Jump-GBM
ARCH
0ꢇ219
0ꢇ106ꢃ
−0ꢇ068
0ꢇ170ꢃ
0ꢇ670
0ꢇ164ꢃ
0ꢇ097
0ꢇ017ꢃ
0ꢇ023
0ꢇ007ꢃ
0ꢇ274
0ꢇ154ꢃ
0ꢇ258
0ꢇ114ꢃ
Jump-ARCH
0ꢇ199
0ꢇ085ꢃ
−0ꢇ161
0ꢇ163ꢃ
0ꢇ590
0ꢇ141ꢃ
Ponderosa and Jeffrey Pines
0ꢇ320
0ꢇ023ꢃ
0ꢇ160
GBM
0ꢇ003
0ꢇ033ꢃ
0ꢇ060
0ꢇ042ꢃ
0ꢇ051
0ꢇ024ꢃ
0ꢇ116
0ꢇ013ꢃ
Jump-GBM
ARCH
0ꢇ514
0ꢇ241ꢃ
−0ꢇ113
0ꢇ088ꢃ
0ꢇ376
0ꢇ063ꢃ
0ꢇ061ꢃ
0ꢇ044
0ꢇ010ꢃ
0ꢇ002
0ꢇ002ꢃ
0ꢇ614
0ꢇ227ꢃ
0ꢇ724
0ꢇ250ꢃ
Jump-ARCH
0ꢇ630
0ꢇ153ꢃ
−0ꢇ151
0ꢇ048ꢃ
0ꢇ208
0ꢇ037ꢃ
Western Hemlock
GBM
−0ꢇ025
0ꢇ053ꢃ
−0ꢇ005
0ꢇ047ꢃ
0ꢇ013
0ꢇ519
0ꢇ038ꢃ
0ꢇ407
0ꢇ052ꢃ
Jump-GBM
ARCH
0ꢇ042
0ꢇ062ꢃ
−0ꢇ485
1ꢇ303ꢃ
1ꢇ472
0ꢇ807ꢃ
0ꢇ096
0ꢇ019ꢃ
0ꢇ028
0ꢇ017ꢃ
0ꢇ714
0ꢇ036ꢃ
0ꢇ031
0ꢇ035ꢃ
0ꢇ214ꢃ
0ꢇ718
Jump-ARCH
0ꢇ380
0ꢇ223ꢃ
−0ꢇ018
0ꢇ118ꢃ
0ꢇ415
0ꢇ113ꢃ
0ꢇ230ꢃ
True Firs
GBM
−0ꢇ025
0ꢇ049ꢃ
0ꢇ106
0ꢇ478
0ꢇ035ꢃ
0ꢇ228
0ꢇ054ꢃ
Jump-GBM
ARCH
0ꢇ503
0ꢇ156ꢃ
−0ꢇ259
0ꢇ136ꢃ
0ꢇ559
0ꢇ082ꢃ
0ꢇ052ꢃ
−0ꢇ040
0ꢇ044ꢃ
0ꢇ105
0ꢇ185
0ꢇ032ꢃ
0ꢇ060
0ꢇ027ꢃ
0ꢇ171
0ꢇ108ꢃ
0ꢇ123
0ꢇ073ꢃ
Jump-ARCH
0ꢇ368
0ꢇ176ꢃ
−0ꢇ419
0ꢇ211ꢃ
0ꢇ484
0ꢇ107ꢃ
0ꢇ061ꢃ
2
2
Note: Let x = Ln S ꢃ − Ln S ꢃ. For the GBMꢉx ∼ N ꢆꢉꢅ ꢃ. For ARCHꢉx ∼ N ꢆꢉh ꢃꢉh = % + %
t+1
x
− ꢆꢃ . Jump process: ꢊ is the arrival rate;
t−1
t
t
t
t
t
t
0
1
2
lnY ∼ N ꢋꢉꢌ ꢃ gives the jump size. The standard error of a parameter is below it in parentheses.
(the mean size of the logarithm of jumps) N = 99 replications. For LM H ꢉH
ꢃ
ꢁ
ꢁꢁ
AJ
appears to be negative, so jumps tend to and LR H ꢉHAJ ꢃ, we report thAeJ PMC
ꢁꢁ
ꢁ
decrease stumpage prices. The precision of (size-exact)ApJ-value while for LR HAJ ꢉHAJ
ꢃ
ꢁ
the parameter estimates is, of course, affected and LR HAJꢁꢉHAJ ꢃ, we give the [LMC,
by the small size of our samples. Even though BMC] p-values which are respectively the
we report them, asymptotic standard errors local bootstrap and the bounds level-exact
(and related t-tests) should not be used to p-values. Looking first at LM H ꢉH ꢃ, we
ꢁꢁ
AJ
ꢁ
assess the precision of estimates because find ARCH(1) effects over the GBAMJ sig-
the theoretical econometric literature cited nificant at the 2% level for Douglas Fir,
above shows that it is impossible to control Western Hemlock, and True Firs, and at 1%
the size of these tests with the models consid- for Ponderosa and Jeffrey Pines. The data do,
ered. This is why we use LR tests and control however, also support the presence of jumps
the level of all tests conducted.
over a simple GBM (LR H ꢉHAJ ꢃ), except
ꢁꢁ ꢁ
Table 3 presents the statistics for the for Ponderosa and Jeffrey PAinJ es: we find sig-
four tests described above calculated with nificant jumps at the 1% level for Douglas

394
May 2002
Amer. J. Agr. Econ.
Table 3. Tests for ARCH and Jumps
LM H ꢉH
ꢃ
LR HAJ ꢉHAJ
ꢃ
LR H ꢉHAJ ꢃ
LR HAJꢁꢉHAJ ꢃ
ꢁꢁ
AJ
ꢁ
ꢁ
ꢁꢁ
AJ
ꢁ
AJ
Douglas Fir
8ꢇ30
!0ꢇ02"
24ꢇ10
!0ꢇ01"
8ꢇ34
!0ꢇ02"
8ꢇ97
!0ꢇ02"
11ꢇ97
!0ꢇ01ꢉ0ꢇ13"
25ꢇ36
!0ꢇ01ꢉ0ꢇ01"
15ꢇ53
!0ꢇ01ꢉ0ꢇ02"
6ꢇ10
!0ꢇ06ꢉ0ꢇ54"
34ꢇ57
!0ꢇ01"
9ꢇ60
!0ꢇ13"
23ꢇ89
!0ꢇ01"
12ꢇ07
!0ꢇ05"
31ꢇ31
!0ꢇ01ꢉ0ꢇ01"
12ꢇ72
!0ꢇ05ꢉ0ꢇ11"
4ꢇ92
!0ꢇ29ꢉ0ꢇ69"
11ꢇ34
!0ꢇ09ꢉ0ꢇ14"
Ponderosa and Jeffrey Pines
Western Hemlock
True Firs
Notes: LM H ꢍH ꢃ is Engle’s Lagrange multiplier test for ARCH(1) effects over a GBM. With reference to (4) and (5), it tests H 4 % = 0 versus
ꢁꢁ
AJ
ꢁ
0
1
AJ
H
4 % > 0, when ꢊ = 0.
A
1
ꢉ
ꢉ
LR H ꢍH ꢃ ≡ 2!L
− L " is the likelihood ratio statistic for testing the null, H , against the alternative hypothesis H . In H , index i refers to
ij
kl
H
H
ij
kl
ij
ꢁ
ij
kl
ꢁ
the absence (when i = Aꢉ% = 0) or presence (when i = A, % > 0) of ARCH effects, and index j refers to the absence (when j = Jꢉꢊ = 0) or presence
1
1
ꢉ
ꢉ
(when j = Jꢉꢊ > 0) of jumps. L
and L
are respectively the maximum of the log-likelihood function under the null and the alternative hypotheses.
H
H
ij
kl
With reference to (4) and (5), LR H ꢉH ꢃ thus tests H 4 % = 0 versus H 4 % > 0 with ꢊ ≥ 0ꢉLR H ꢉH ꢃ tests H 4 ꢊ = 0 versus H 4 ꢊ > 0 when
ꢁ
AJ
0
ꢁꢁ
AJ
ꢁ
AJ
AJ
0
1
A
1
0
A
%
= 0, and LR H ꢉH ꢃ tests H 4 ꢊ = 0 versus H 4 ꢊ > 0 with % ≥ 0.
ꢁ
AJ
1
AJ
A
1
This table reports the values of the test statistics LM H ꢉH ꢃ, LR H ꢉH ꢃ, LR H ꢉH ꢃ, and LR H ꢉH ꢃ as well as MC p-values in
ꢁꢁ
AJ
ꢁ
AJ
ꢁ
AJ
ꢁꢁ
AJ
ꢁ
AJ
ꢁ
AJ
AJ
AJ
brackets. MC p-values are calculated with N = 99 replications. For LM H ꢉH ꢃ and LR H ꢉH ꢃ, we report the PMC (exact-size) p-value. For
ꢁꢁ ꢁꢁ
AJ AJ
ꢁ
ꢁ
AJ
AJ
LR H ꢉH ꢃ and LR H ꢉH ꢃ, we report the [LMC, BMC] p-values which are respectively the local bootstrap and the bound level-exact p-values
ꢁ
AJ
ꢁ
AJ
AJ
AJ
(see section on models and tests).
Fir and Western Hemlock and at 5% for Douglas Fir, strong evidence of ARCH with
True Firs. As mentioned above, both ARCH
effects and jumps could produce the observed
fat tails. Since both tests are exact, rejections
are statistically sound.
a possibility of jumps for Ponderosa and Jef-
frey Pines, strong evidence of ARCH without
jumps for Western Hemlock, and jumps with
a possibility of ARCH for True Firs. Such
differences between the dynamic characteris-
tics of these series may seem surprising. One
possible explanation is that they may have
been affected differently first by speculative
bubbles in stumpage prices that burst during
the 1980s, and later by the Spotted Owl con-
troversy that resulted in harvest restrictions
in some areas.10 Alternatively, our normality
assumption for the logarithm of jumps may
not be adequate and a distribution with fat-
ter tails (such as a t-distribution) or an asym-
metric distribution may be needed to capture
better what happened during this period of
extreme price changes.
We also note that the LM and the LR
tests for ARCH yield different results for
True Firs: the former is significant whereas
the latter fails to reject the no-ARCH null
in the presence of jumps (at 5%). Of course,
the LM test does not take jumps into con-
sideration. Yet there seems to be evidence
in favor of jumps (with or without ARCH).
Alternatively, if ARCH effects (which seem
present) are not accounted for, jumps are
falsely detected in the case of Western Hem-
lock.
To sort out the contributions of ARCH
and jumps, we test for ARCH effects in
the presence of jumps (LR HAJ ꢉHAJ ꢃ) and
ꢁ
for jumps in the presence of ARCH effects
(LR HAJꢁꢉHAJ ꢃ). We find that we cannot
exclude the presence of ARCH over a jump-
GBM model, at the 1% level for Ponderosa
and Jeffrey Pines, and at 2% for Western
Hemlock. If ARCH effects over a jump-
GBM model may possibly be ruled out for
True Firs, our tests do not uphold a defi-
nite (nonspurious) answer to this problem for
Douglas Fir (the bootstrap and MC bounds
p-value are 0.01 and 0.13, respectively).9 On
the other hand, there is strong evidence of
jumps over an ARCH model for Douglas
Fir (at 1%) and to a lesser degree for Pon-
derosa and Jeffrey Pines (between 5% and
11%), and True Firs (between 9% and 14%).
We fail to reject the no-jumps null hypoth-
esis in the presence of ARCH for Western
Hemlock.
To summarize, we find strong evidence of
jumps with a possibility of ARCH effects for
9 Recall that we rely on bounds when testing for ARCH in the
presence of jumps or for jumps in the presence of ARCH. If the
bounds’ p-value ≤ ꢂ, we can conclude that the test is significant,
while it is not significant if the bootstrap p-value > ꢂ. Indeed,
the bootstrap provides an empirical p-value corresponding to a
point estimate for the nuisance parameter. If this estimated p-
value is larger than ꢂ, then a fortiori the largest p-value over the
nuisance parameter space exceeds ꢂ. However, there is no clear
decision when bootstrap p-value < ꢂ < bounds’ p-value.
More generally, our results show that it
may be difficult to distinguish between jumps
10 For more details on these speculative bubbles and on the
management of Federal Forests, see Ando.

Saphores, Khalaf, and Pelletier
Jumps and ARCH in Natural Resource Prices
395
and ARCH effects in small data sets. Non- follows a Poisson process with arrival rate ꢊ.
nested tests may be performed, although no We know from option theory that there exists
reliable procedure is available for such prob- an implied value function V Stꢃ that veri-
lems (recall the inference problems docu- fies the following optimality condition when
mented above).
the stand of old-growth forest should be pre-
served:
1
dt
(11) rV Stꢃ = A+ Et!dV Stꢃ"ꢇ
Implications for Harvesting
Old-Growth Timber
Et!·" is the expectation operator at t and
dV Stꢃ is the differential of the unknown
value function. This asset equilibrium con-
dition states that the stand should be pre-
served as long as the flow of amenity and the
expected “capital gains” 1/dtꢃEt!dV Stꢃ"
provide a return equal to the social discount
rate r. Applying a generalization of Itô’s
lemma (Merton), we find
From results reported in the finance litera-
ture (e.g., see Bakshi, Cao, and Chen, and the
references therein), we anticipate that jumps
and ARCH effects in the price of a natural
resource can have important consequences
for its management. In the case of stumpage
prices, they could, for example, impact the
decision to cut a public forest.
Off-the-shelf models for financial assets are
not always adequate for natural resources
problems, however. In this section, we thus
focus on the impact of neglecting jumps,
when they are statistically significant, in the
context of an infinite-horizon, continuous-
time stochastic dynamic model based on the
GBM. We leave the investigation of the
impact of ARCH effects in this framework
for future work because when we move to
continuous time, ARCH effects translate into
complex stochastic volatility models (Duan).
We revisit the classic problem of the opti-
mal timing of cutting a stand of old-growth
forest (e.g., see Reed or Conrad and Lud-
wig) so timber volume is assumed to be con-
stant and equal to unity. This stand generates
a constant amenity A per time period, net of
maintenance costs. Cutting this stand at time
t would provide net revenues St from timber
sales, where St varies stochastically, plus the
present value of the flow of land rents (L per
time period, assumed constant). We consider
a single rotation and denote by r the social
discount rate.
dV Stꢃ
(12) rV Stꢃ = A+ꢄSt
dSt
ꢅ2 d2V Stꢃ
+
St2
2
dSt2
+ꢊꢎY 8V StYtꢃ−V Stꢃ9ꢇ
t
V Stꢃ can be written as the sum of two terms:
V Stꢃ = VP Stꢃ+: Stꢃ. The term VP Stꢃ is a
particular solution of (12). It represents the
present value of the constant flow of amenity
A, and thus
A
(13) VP Stꢃ =
ꢇ
r
The term : Stꢃ represents the value of the
option to cut the stand. We know from option
theory that in the continuation region it ver-
ifies the homogeneous equation associated
with (12):
d: Stꢃ ꢅ2 d2: Stꢃ
(14) r: Stꢃ = ꢄSt
+
St2
dS
2
dSt2
+ꢊꢎY 8: StYtꢃ−: Stꢃ9ꢇ
t
We compare two models, which we apply
to the same data: in the first model, St follows
a GBM with jumps, and in the second one, it
follows a simple GBM. For each model, we
want to find S∗, which separates the values of
St where the stand should be preserved (the
continuation region: St ≤ S∗), from the values
of St where the trees should be cut (the stop-
ping region: St ≥ S∗). We use the theory of
real options to solve these two optimal stop-
ping problems (Dixit and Pindyck).
We also assume that if St ever becomes 0, it
is zero forever so:
(15) : 0ꢃ = 0ꢇ
To find S∗, we need the continuity and the
smooth-pasting conditions, which require V
and V ꢂ to be smooth across the stopping fron-
tier (Dixit and Pindyck):
L
(16)
VP S∗ꢃ+: S∗ꢃ = S∗ +
r
First, let us suppose that net stumpage
price follows a GBM with jumps, as given by
equation (2), and that the arrival of jumps
dVP S∗ꢃ d: S∗ꢃ
+
= 1ꢇ
dS
dS

396
May 2002
Amer. J. Agr. Econ.
The continuity and the smooth-pasting con- of the jumps, so the expected growth rate is
ditions can be rewritten in terms of : to get
(Merton)
ꢃ
ꢄ
A−L
ꢋ
ꢌ
dSt
St
(17) : S∗ꢃ = S∗ −
d:
2
(22)
ꢎ
=ꢄ+ꢊ eꢋ+ꢌ /2 −1
r
(18)
S∗ꢃ = 1ꢇ
where again ꢄ=ꢆ+0ꢇ5ꢅ2. Using parameter
values reported in table 2, we find that the
annual expected stumpage price growth rates
for Douglas Fir and True Firs are 25.5%
and 37.5% respectively for the jump-GBM.
These high values force us to adopt high dis-
count rates to make cutting worthwhile. They
should be interpreted with caution given that
our parameters are estimated from a small
sample that covers a pretty eventful period in
the Pacific Northwest. We want to take these
events into account, however, if only to cor-
rectly estimate the parameters of the under-
lying GBM (ꢆ and ꢅ) or if we believe that
similar events could occur in the future.
dS
Equation (17) is in fact valid in the entire
stopping region because when St ≥ S∗ cutting
should take place immediately.
Equation (14) with boundary conditions
(15), (17), and (18) cannot be solved ana-
lytically, so we extend Galerkin’s method
(Delves and Mohamed) to solve it numeri-
cally. To our knowledge, this has not been
done before in resource economics. See the
appendix for details.
If, however, St follows a GBM, equation
(14) (where ꢊ has been set to 0) with bound-
ary conditions (15), (17), and (18) is easily
solved. Trying a power function in S, we find
Like Haight and Holmes, we find that the
empirical stumpage price DGP has impor-
tant implications for harvesting timber. From
table 4a, we see that neglecting jumps when
they are indeed present can either lead to
cutting too early (for Douglas Fir) or too
late (for True Firs). To explain this difference,
recall that we have multiplicative, lognor-
mally distributed jumps. Their expected value
is thus exp ꢋ+0ꢇ5ꢌ2ꢃ, which equals 1ꢇ17>1
for Douglas Fir and 0ꢇ90<1 for True Firs.
Jumps thus tend to increase St for the for-
mer and to decrease St for the later. Also
note that old-growth forest would not be cut
with a GBM when the rate of increase of
stumpage value is greater than the discount
rate (the “+ꢃ” for True Firs when the dis-
count rate is 2% above rcritical). A look at
the % change between the stopping values
calculated with the GBM and with the jump-
GBM shows that the difference between the
two can be quite substantial (over 500% for
True Firs when the discount rate is 4% above
rcritical, for example).
To check the robustness of these results,
we conduct a simulation study. For both Dou-
glas Fir and True Firs, we generate 100,000
samples of 97 stumpage prices using the
jump-GBM parameters reported in table 2.
For each sample, we estimate the GBM
parameters ꢆ and ꢅ a∗nd calculate the cor-
responding value of SGBM and the relative
error SJ∗ump-GBM −SG∗ BMꢃ/SJ∗ump-GBM. Table 4b
presents three quartiles of the distribution
of relative errors for different interest rates.
“−ꢃ” means that the stand of old-growth
(19) : Sꢃ = C0S<
where C0 is a constant to be determined
jointly with S∗ and < is given by
ꢀ
− ꢄ−0ꢇ5ꢅ2ꢃ+ ꢄ−0ꢇ5ꢅ2ꢃ2 +2rꢅ2
(20) <=
ꢇ
ꢅ2
The expression of VP in equation (13) is
still valid so, plugging (13) and (19) into (16)
and solving for S∗, we find
<
A−L
(21) S∗ =
ꢇ
<−1
r
To assess the error one could make by fit-
ting a GBM to data generated by a∗jump-GBM
process, we estimate numerically SJump-GBM, the
stopping value for a jump-GBM, and com-
pare it with SG∗ BM, the stopping value for a
GBM calculated from equation (21). We use
our sample data for Douglas Fir and True
Firs and various discount rates. By choosing
appropriate units, we assume that the differ-
ence A−L equals 1.11
A necessary condition to have a finite
stopping value S∗ is that r be larger than
the expected growth rate of stumpage price.
For the GBM, the expected growth rate of
stumpage prices is ꢄ, the infinitesimal drift
in equation (1). For the jump-GBM model,
we also need to account for the contribution
11 If A-L were negative, we see from (21) that it would be opti-
mal to cut immediately.

Saphores, Khalaf, and Pelletier
Jumps and ARCH in Natural Resource Prices
397
Table 4. Comparison of Stopping Values Based on (a) Sample Data and (b) A Simulation
Study
Discount Rate in
Excess of rcritical
(a)
SG∗ BM
SJ∗ump-GBM
% Change
Douglas Fir
2%
4%
6%
265
172
127
428
216
144
38%
20%
12%
True Firs
2%
4%
6%
+ꢃ
1601
427
485
246
163
–
−551%
−160%
Quartiles of the Distribution of Relative Errors:
100% SJ∗ump-GBM −SG∗ BMꢃ/SJ∗ump-GBM
Discount Rate in
Excess of rcritical
(b)
25th Percentile
50th Percentile
75th Percentile
Douglas Fir
2%
4%
6%
−ꢃ
−ꢃ
−ꢃ
−ꢃ
−ꢃ
−ꢃ
48%
27%
18%
85%
74%
65%
True Firs
2%
4%
6%
−ꢃ
−161%
−82%
83%
71%
62%
2
ꢋ+0ꢇ5ꢌ
and S
Note: In table 4, r
model. In table 4a, S
=ꢄ+ꢊ e
−1ꢃ is the minimum discount rate that makes cutting the stand of old-growth forest worthwhile with the jump-GBM
critical
∗
∗
are the values of the stand of old-growth forest at which it is optimal to harvest under the GBM and jump-
GBM
Jump-GBM
GBM models respectively. They are calculated using the parameter values shown in table 2, assuming that amenity value is unity. In table 4b, simulated
samples are generated with the jump/GBM parameters from table 2. Results are for 100,000 samples of 97 points each (97 is the size of our stumpage
price samples). “−ꢃ” means that the stand of old-growth forest would not be cut if stumpage prices follow a GBM, because it appreciates too quickly.
forest would not be cut if stumpage prices ments in log prices with more extreme values
than the normal distribution.
follow a GBM, because it appreciates too
quickly; this happens more than 25% of the
time for the parameters considered. Looking
at the 50th percentile, we also see that with
the GBM we would cut too early for Dou-
glas Fir and too late for True Firs, although
the GBM may also lead us to cut too early
on occasion for True Firs (75th percentile).
Finally, we note that the difference between
the stopping values of both models tends to
decrease as the discount rate increases. These
results confirm that omitting jumps when
they are present can lead to large errors.
First, we propose a LR-based methodol-
ogy, based on combining bounds with Monte
Carlo tests, that gives finite-sample, level-
exact p-values when testing for jumps or
ARCH effects. This is particularly useful in
natural resources because sample sizes are
often small. We then analyze four quarterly
time series of stumpage prices from Pacific
Northwest National Forests. Controlling for
seasonal variations and a definition change in
stumpage prices, we find evidence of jumps
and ARCH effects.
Second, we revisit the tree-cutting prob-
lem for old-growth forest when stumpage
prices follow
a GBM with jumps. We
present an algebraic method to solve this
autonomous, infinite-horizon stopping prob-
lem, which turns a complex Bellman equa-
tion into a system of linear equations. We
then show that ignoring jumps, when they
are indeed present, may lead to significantly
suboptimal decisions to harvest old-growth
timber. In empirical work, this illustrates the
importance of investigating the presence of
Conclusions
In this article, we reconsider the represen-
tation of natural resource prices by continu-
ous processes by allowing for the presence of
jumps, which can be due to the arrival of dis-
crete events that cause large price changes.
We also allow for ARCH effects, which like
jumps have been found to generate incre- jumps in natural resource prices.

398
May 2002
Amer. J. Agr. Econ.
ric Applications.” Working paper, CRDE-U.
de Montréal, 1995.
[Received December 2000;
accepted August 2001.]
. “Some Impossibility Theorems in Econo-
metrics with Applications to Structural and
Dynamic Models.” Econometrica, 65(1997):
1365–87.
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ꢁ
ꢁ
f xꢃ= ꢂaj Tj xꢃ
f ꢂ xꢃ= ꢂaꢂj Tj xꢃ
0
0
N
ꢁ
f ꢂꢂ xꢃ= ꢂaꢂjꢂTj xꢃꢇ
0
ꢐꢂ
indicates that the first term of the summation
ꢐꢂꢂ
is halved, and
indicates that both the first and
the last terms are halved. Tj xꢃ≡cos jarccos xꢃꢃ
is the jth Chebychev polynomial. For example, to
find the gi’s 1≤i≤Nꢃ, we calculate
ꢃ
ꢄ
N
ꢁ
2
N
kp
N
ikp
N
gi =
ꢂꢂg cos
cos
ꢇ
k=0
1
Delves and Mohamed show that aj = 2 aꢂj−1 −ajꢂ+1ꢃ,
1
and aꢂj =
a
−aꢂjꢂ+1ꢃꢉj ≥1. In vector form (where
ꢂꢂ
j−1
a 1ꢃ is an2N ×1 vector, A is an N × N +1ꢃ matrix,
and aꢂ is an N +1ꢃ×1 vector)




a1
a2
ꢇ
ꢇ
ꢇ
a0ꢂ
ꢂ








a 1ꢃ =Aaꢂ a 1ꢃ
≡
aꢂ =
a1
ꢇ
ꢇ
ꢇ
1
2i
−1
Zinkhan, F.C. “Option Pricing and Timberland’s
Land-Use Conversion Option.” Land Econ.
67(1991):317–25.
Aiꢉi
=
Aiꢉi+2
=
2i
Aiꢉj =0 otherwise.
Equation (A.2) links a0 to the aꢂ , and a0ꢂ to the
aꢂꢂ. For example, if c11 +c12 =0, wejhave
j
Appendix
ꢑ
ꢈ
2
Galerkin’s Method
a0
=
e1 − d11 d12ꢃT
+ c11 c12ꢃT 1ꢃA aꢂ
c11 +c12
ꢊ ꢒ
With Galerkin’s method (Delves and Mohamed),
we solve numerically for the unknown func-
tion f xꢃ in the integro-differential equation with
boundary conditions:
aꢂ0 =2 f21e1 +f22e2ꢃ
ꢂ
ꢋ
ꢓ
ꢌ ꢋ
ꢌ
+ ht −2 f21 f22 CT 2ꢃA11 +DT 1ꢃ
A
(A.1) P xꢃf ꢂꢂ xꢃ+Q xꢃf ꢂ xꢃ+R xꢃf xꢃ
ꢃ
ꢄ
ꢔꢅ
ꢍ
1
0ꢇ5
0ꢇ5
+D
ht aꢂꢂꢇ
+
k xꢉuꢃf uꢃdu=g xꢃ
−1≤x≤1
−1
ꢎ
ꢏꢎ
ꢏ
c11 c12
f −1ꢃ
ꢎ
ꢏ
(A.2)
c21 c22
f 1ꢃ
0ꢇ5T0 −1ꢃ T1 −1ꢃ T2 −1ꢃ···
•
T≡
ꢎ
ꢏꢎ
ꢏ
ꢎ
ꢏ
d11 d12
d21 d22
f ꢂ −1ꢃ
f ꢂ 1ꢃ
e1
e2
0ꢇ5T0 1ꢃ
T1 1ꢃ
T2 1ꢃ···
+
=
ꢇ
is a 2× N +1ꢃ matrix, T kꢃ is T without its first
k columns;
• A11 is A without its first line and column;
P xꢃ, Q xꢃ, R xꢃ, k xꢉuꢃ, and g xꢃ are “well
behaved” functions.
We first replace P xꢃ, Q xꢃ, R xꢃ, g xꢃ, f xꢃ,
f ꢂ xꢃ, and f ꢂꢂ xꢃ by their truncated Chebychev
decomposition:
1
4
1
• ht is the 1× N +1ꢃ vector: 0 0 − 4 0 ··· 0ꢃ;
ꢎ
ꢏ
f11 f12
f21 f22
•
is the inverse of
N
N
ꢎ
ꢏꢎ
ꢏ
ꢎ
ꢏꢎ
ꢏ
ꢁ
ꢁ
P xꢃ= ꢂpj Tj xꢃ
Q xꢃ= ꢂqj Tj xꢃ
c11 c12
0ꢇ5 −1
0ꢇ5 1
d11 d12
d21 d22
0 1
0 1
+
0
0
c21 c22
N
N
ꢁ
ꢁ
R xꢃ= ꢂrj Tj xꢃ
g xꢃ= ꢂgj Tj xꢃ
which is required to be nonsingular to have a
well posed problem; and
0
0

400
May 2002
Amer. J. Agr. Econ.
• aꢉaꢂꢉ and aꢂꢂ are the vectors of Chebychev coeffi-
cients for f xꢃꢉf ꢂ xꢃ, and f ꢂꢂ xꢃ respectively.
Here, we want to solve for : •ꢃ and for S∗ in
(14) to (18). We first change variables: s ≡Ln Sꢃ
(and so s∗ =Ln S∗ꢃꢃ, D Ln Sꢃꢃ≡: Sꢃ, Ln Y ꢃ≡Z.
We replace −ꢃ by sinf. A second change of vari-
ables
We find:
(A.3) a=Aꢂaꢂ +ꢀꢉ
aꢂ =Aꢂꢂaꢂꢂ +ꢁ
2
s∗ +sinf
w≡
s−
s∗ −sinf
s∗ −sinf
f wꢃ≡D sꢃ
g wꢃ≡G sꢃ
with
ꢃ
2e1
2 f21e1 +f22e2ꢃ 0···0
ꢄ
leads to
ꢀt =
ꢁt =
0···0
and
c11 +c12
(A.6) P wꢃf ꢂꢂ wꢃ+Q wꢃf ꢂ wꢃ+R wꢃf wꢃ
ꢋ
ꢌ
ꢇ
ꢍ
1
+
k wꢉuꢃf uꢃdu=g wꢃ
−1≤w≤1
−1
For 0 ≤i≤N, we then multiply the resulting equa-
√
tion by Ti xꢃ/ 1−x2ꢃ and integrate between −1
and 1. We obtain
f −1ꢃ=0
f ꢂ 1ꢃ=
s∗ −sinf
∗
es
(A.4) Paꢂꢂ +Qaꢂ + R+ꢊBꢃa = gꢇ
2
A−L
∗
f 1ꢃ=es
−
For 0 ≤i≤Nꢉ 0<j ≤N, the coefficients of PꢉQꢉR,
and B are
r
with
pi
qi
ri
ꢃ
ꢄ
2
ꢅ2
2
Pi0 =
Bi0 =
Qi0 =
Ri0 =
2
2
2
P wꢃ=
2
s∗ −sinf
ꢃ
ꢄ
N−1
ꢁ
B
N2
sB
ꢃ
ꢄ
ꢅ2
2
sin
N
Q wꢃ= ꢄ−
R wꢃ=− ꢊ+rꢃ
s=1
2
s∗ −sinf
ꢃ
riB
ꢃ
ꢄ
ꢃ
ꢄꢄ
N
ꢃ
ꢄ
ꢁ
s∗ −sinf
s∗ −sinf
ꢋ
ꢌ
rB
sB
×
ꢂꢂk cos
ꢉcos
k wꢉuꢃ=ꢊ
'
u−wꢃ−
N
N
r=0
2ꢌ
2ꢌ
ꢃ
ꢄ
ꢃꢓ
s∗ −sinf
s∗ +sinf
×cos
pi+j +p
g wꢃ=ꢊexp
w+
N
2
2
ꢔꢄ
ri+j +r
ꢌ2
ꢀi−jꢀ
ꢀi−jꢀ
Pij =
Rij =
+
+ꢋ
2
2
2
ꢂ
ꢃ
ꢄ
ꢅ
qi+j +q
ꢀi−jꢀ
s∗ −sinf
2ꢌ
ꢋ
Qij =
× ꢏ
1−wꢃ− −ꢌ −1
2
ꢌ
ꢃ
ꢄ
ꢃ
ꢄ
2B N−1
N2
sjB
N
sB
N
ꢁ
A−L
Bij =
cos
sin
+ꢊ
s=1
r
ꢂ
ꢃ
ꢄꢅ
ꢃ
riB
ꢃ
ꢄ
ꢃ
ꢄꢄ
s∗ −sinf
ꢋ
ꢌ
N
ꢁ
rB
sB
×
ꢂꢂk cos
ꢉcos
× 1−ꢏ
1−wꢃ−
ꢇ
2ꢌ
N
N
r=0
ꢃ
ꢄ
' zꢃ and ꢏ zꢃ are respectively the density and
the cumulative distribution of the standard nor-
mal. Compared to (A.1), we have one extra
boundary conditions because s∗ is unknown. To
solve, we extend Galerkin’s method: we pick a
value of s∗, find f from the first three equations
of (A.6), check if the fourth equation of (A.6)
is satisfied, and iterate with other values of s∗
until it is. A GAUSS program, available upon
request, was written to implement this procedure.
With sinf =−4ꢇ0 and N =60 (number of terms in
the Chebychev expansions), we obtain satisfactory
numerical convergence.
×cos
ꢇ
N
Substituting (A.3) into (A.4) gives the linear sys-
tem in aꢂꢂ:
(A.5) !P+ Q+ R+ꢊBꢃAꢂꢃAꢂꢂ"aꢂꢂ
= g− Q+ R+ꢊBꢃAꢂꢃꢁ− R+ꢊBꢃꢀꢇ
Once we know aꢂꢂ, we calculate a from a=
Aꢂ Aꢂꢂaꢂꢂ +ꢁꢃ+ꢀ.