mrur-140115.dviDiffusion Models∗
Abstract
This paper analyzes the mean reversion and unit root properties of general
diffusion models and their discrete samples. In particular, we find that the
Dickey-Fuller unit root test applied to discrete samples from a diffusion model
becomes a test of no mean reversion rather than a unit root, or more generally,
nonstationarity in the underlying diffusion. The unit root test has a well defined
limit distribution if and only if the underlying diffusion has no mean reversion,
and diverges to minus infinity in probability if and only if the underlying diffusion
has mean reversion. It is shown, on the other hand, that diffusions are mean-
reverting as long as their drift terms play the dominant role, and therefore,
nonstationary diffusions may well have mean reversion.
This version: January 2014
JEL Classification: C12, C22, C58, G12
Key words and phrases: mean reversion, unit root test, stationarity and nonstationarity,
diffusion model, limit distribution
∗We thank Yacine At-Sahalia, Yoosoon Chang, Oliver Linton, Morten Ørregaard Nielsen, and seminar participants at Indiana, Cambridge, Princeton, 2012 and 2013 Midwest Econometrics Group Meetings, 2013 North American Econometric Society Meeting, and SETA 2013 for many helpful comments.
†Department of Economics, Indiana University. ‡Department of Economics, Indiana University and Sungkyunkwan University.
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1 Introduction
The unit root test was originally developed to test for the unity of the largest root in an
autoregressive time series model. However, the existence of a unit root entails some other
important consequences as well on more general stochastic characteristics of the underlying
time series. In particular, it implies that, every shock having a permanent effect, the effects
of shocks are accumulated and build up a stochastic trend, which eventually makes any time
series with a unit root nonstationary. On the other hand, an autoregressive process becomes
stationary if it has the largest root less than unity in modulus, since every shock has only a
transitory effect and is offset by a future shock with the opposite sign. Accordingly, the unit
root test may also be regarded generally as a test of the nonstationarity or the absence of
mean reversion in the underlying time series. In fact, the unit root test has commonly been
used to test for the nonstationarity or the absence of mean reversion in a much broader
class of time series models than autoregressive models.
Naturally, the unit root test has also been routinely applied to samples that are thought
to be collected discretely at relatively high frequencies from diffusion type continuous time
processes. As is well known, the laws of motions for economic and financial time series
such as stock prices, exchange rates and interest rates, among others, are typically modeled
as diffusions. However, the asymptotics of the unit root test applied to discrete samples
from general diffusion models have long been unknown, except for the simple case that
the underlying diffusion models are driven by Brownian motion and stationary Ornstein-
Uhlenbeck processes respectively for the null and alternative hypotheses, for which the
reader is referred to Shiller and Perron (1985), Perron (1989, 1991) and Chambers (2004,
2008), among others. In particular, it has been completely unknown whether the unit root
test has any discriminatory power in distinguishing general stationary and nonstationary,
or mean-reverting and non-mean-reverting, diffusion models. Moreover, the effects of time
span and sampling frequency on the power of the unit root test have never been theoretically
analyzed for general stationary diffusion models.
In this paper, we investigate the longrun behaviors of general diffusion models including
their mean reversion and unit root properties. Our investigation is extensive and complete.
There were two major difficulties in establishing our asymptotic theory. First, our theo-
retical development needs some basic asymptotics for general stationary and nonstationary
diffusions. In particular, we need to consider the asymptotics for stationary diffusions hav-
ing no proper moments and the nonstationary diffusion asymptotics, and they are only
recently developed by Kim and Park (2013) up to the generality that is required here. Sec-
ond, a diffusion model is completely specified by two functions, drift and diffusion functions,
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which represent respectively the instantaneous rate of change in conditional mean and the
instantaneous conditional volatility. Particularly, they determine all charactersitics of a
diffusion model including its asymptotics or longrun behaviors. However, it is not easy to
see how the asymptotic properties of a diffusion model are related precisely to the drift and
diffusion functions representing its instantaneous characteristics.
A diffusion is either recurrent or transient. A recurrent diffusion visits every point on its
domain in finite time no matter where it starts. If there is a nonempty set of points in its
domain that it cannot reach in finite time with a positive probability, a diffusion is called
transient. In the paper, we mainly consider recurrent diffusions, since they are practically
more relevant and yield well defined asymptotics. Diffusions become transient if they have
trends exploding to their boundaries. Clearly, if transient diffusions have trends in simple
form, we may render them recurrent by detrending. A recurrent diffusion is either positive
recurrent or null recurrent. For a positive recurrent diffusion, the time it takes to hit any
given level has finite expectation. On the other hand, it may take too long for a diffusion
to hit a certain level and the hitting time may have infinite expectation, in which case it
becomes null recurrent. It is well known that positive recurrent diffusions are stationary,
whereas null recurrent diffusions are nonstationary.
In the paper, we show that the Dickey-Fuller test cannot be used to test for nonsta-
tionarity of a diffusion model. The test applied to discrete samples from a diffusion model
may diverge to minus infinity in probability for nonstationary diffusions. Indeed, it tests
for the absence of mean-reverting behavior in the underlying diffusion. The unit root test
has a well defined limit distribution if and only if the underlying diffusion does not have
mean reversion, and it diverges to minus infinity in probability if and only if the under-
lying diffusion has mean reversion. This is true for all recurrent diffusions. On the other
hand, all stationary diffusions have mean reversion regardless of the existence of mean,
and therefore, the test diverges to minus infinity in probability for all stationary diffusions.
However, nonstationary diffusions may also have mean reversion if they have drift terms
dominating their diffusion terms. Consequently, the test does not generally discriminate
stationary and nonstationary diffusions. Rather, it effectively discriminates mean-reverting
and non-mean-reverting diffusions.
Of course, we may use the Dickey-Fuller test to fully discriminate the stationary and
nonstationary diffusions, if we only consider nonstationary and non-mean-reverting diffu-
sions under the null hypothesis. As long as the class of diffusion models are restricted as
such under the null hypothesis, the test becomes consistent against all stationary diffusion
models. Nevertheless, the limit distribution of the test is heavily model-dependent and
depends on the underlying diffusion model in a complicated manner. Therefore, the usual
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Dickey-Fuller critical values cannot be used even in such a restrictive setup. The limit dis-
tribution of the test is generally represented as a functional of a skew Bessel process, and
reduces to the Dickey-Fuller distribution only if the underlying diffusion becomes a Brow-
nian motion in the limit. The test relying on the Dickey-Fuller distribution has a correct
asymptotic size only in this case. Consequently, the Dickey-Fuller test is valid only if we
assume that the underlying diffusion is asymptotically Brownian motion.
The rest of the paper is organized as follows. Section 2 presents the background and
preliminaries that are necessary to understand the subsequent development of asymptotic
theory developed in the paper. The diffusion model and various notions to investigate its
longrun behaviors, and regularly and rapidly varying functions with their useful properties
are introduced. In Section 3, we establish the relationship between the longrun behaviors,
such as recurrence and stationarity, of a diffusion model with its drift and diffusion func-
tions, which are instantaneous parameters representing respectively the rate of change in
conditional mean and volatility. Section 4 explores the mean-reverting behaviors of diffu-
sion models, and establishes the necessary and sufficient condition for the mean reversion of
diffusion model. We consider the Dickey-Fuller unit root test and develop its asymptotics
in Section 5, and show that the unit root test is indeed a test for the absence of mean
reversion. Section 6 concludes the paper, and all mathematical proofs are in Appendix.
Finally, a word on notation. Since our asymptotics are extensive and complicated, we
need to make some notational conventions to facilitate our exposition. The notation “∼”
signifies the asymptotic equivalence. In particular, f(x) ∼ g(x) at x or x implies that
f(x)/g(x) → 1 as x → x or x → x, respectively. On the other hand, “PT ∼p QT ” and
“PT p QT ” just imply that PT /QT →p 1 and PT = op(QT ), respectively, as T → ∞.
Moreover, we follow the notational convention in the markov process literature to use the
same notation for both a measure and its density with respect to Lebesgue measure. As
an example, for a given measure or a density m on D ⊂ R, we write m(D) and m(f)
interchangeably with ∫
D m(x)f(x)dx respectively whenever they are well
defined. Furthermore, as usual, 1{·} is defined as the indicator function. The identity
function on R is denoted by ι, i.e., ι(x) = x, and we write the κ-th order power function as
ικ so that ικ(x) = xκ.
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2.1 Diffusion Model
We consider the diffusion process X given by the time-homogeneous stochastic differential
equation
dXt = µ(Xt)dt+ σ(Xt)dWt, (1)
where µ and σ are, respectively, the drift and diffusion functions, and W is the standard
Brownian motion. We denote by D = (x, x) the domain of the diffusion process X, where
we set x = −∞ or 0 with x = ∞. This causes no loss in generality, since we may simply
consider X − x or −X to allow for a more general case. In what follows, we denote by
xB = x or x the boundary of D. Throughout the paper, we assume
Assumption 2.1. We assume that (a) σ2(x) > 0 for all x ∈ D, and (b) µ(x)/σ2(x) and
1/σ2(x) are locally integrable at every x ∈ D.
Assumption 2.1 provides a simple sufficient set of conditions to ensure that a weak
solution to the stochastic differential equation (1) exists uniquely up to an explosion time.
See, e.g., Theorem 5.5.15 in Karatzas and Shreve (1991).
The scale function of the diffusion process X in (1) is defined as
s(x) =
∫ x
dy, (2)
where the lower limits of the integrals can be arbitrarily chosen to be any point w ∈ D.
Defined as such, the scale function s is uniquely identified up to any increasing affine
transformation, i.e., if s is a scale function, then so is as + b for any constants a > 0 and
−∞ < b < ∞. We also define the speed density
m(x) = 1
(σ2s′)(x) (3)
on D, where s′ is the derivative of s, often called the scale density, which is assumed to
exist. The speed measure is defined to be the measure on D given by the speed density with
respect to the Lebesgue measure. Note, under Assumption 2.1, that both the scale function
and speed density are well defined, and that the scale function is strictly increasing, on D.
Our asymptotic theory depends crucially on the recurrence property of the diffusion
process X. To define the recurrence property, we let τy be the hitting time of a point y in D that is given by τy = inf{t ≥ 0|Xt = y}. We say that the diffusion X is recurrent if P{τy <
∞|X0 = x} = 1 for all x, y ∈ D. The recurrent diffusion X is said to be positive recurrent
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if E[τy < ∞|X0 = x] < ∞ for all x, y ∈ D, and null recurrent if E[τy < ∞|X0 = x] = ∞ for
all x, y ∈ D. Under some regularity conditions on µ and σ, the diffusion X is recurrent if
and only if the scale function s in (2) is unbounded at both boundaries x and x, i.e.,
s(x) = −∞ and s(x) = ∞.
Furthermore, the recurrent diffusion X becomes positive recurrent or null recurrent, de-
pending upon
m(D) < ∞ or m(D) = ∞,
where m is the speed measure defined in (3). A diffusion which is not recurrent is said to
be transient.
Positive recurrent diffusions are stationary. More precisely, they have time invariant
distributions, and if they are started from the time invariant distributions they become
stationary. The time invariant density of the positive recurrent diffusion X is given by
π(x) = m(x)
m(D) .
Null recurrent and transient diffusions are nonstationary. They do not have time invariant
distributions, and their marginal distributions change over time. Out of these two different
types of nonstationary processes, we mainly consider null recurrent diffusions in the paper.
Brownian motion is the prime example of null recurrent diffusions. Typically, transient
processes have upward or downward trends, in which case we may eliminate their trends
using appropriate detrending methods so that they behave like recurrent processes. Like
unit root processes in discrete time, null recurrent processes have stochastic trends and
the standard law of large numbers and central limit theory in continuous time are not
applicable. See, e.g., Jeong and Park (2011) and Kim and Park (2013) for more details on
the statistical properties of null recurrent diffusions.
2.2 Regularly and Rapidly Varying Functions
In the paper, we use extensively the concepts of regularly and rapidly varying functions.
For a function f : (0,∞) → R, we say that it is regularly varying at infinity with index κ,
and write f ∈ RVκ, if for all x > 0 limλ→∞ f(λx)/f(λ) = xκ with some κ ∈ (−∞,∞). In
particular, if κ = 0 and f ∈ RV0, then f is said to be slowly varying at infinity. Moreover,
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we say that f is rapidly varying at infinity with index ∞ or −∞, if
lim inf λ→∞
inf x≥1
λ→∞ sup x≥1
xκf(λ) ≤ 1,
in which case we write f ∈ RV∞ or f ∈ RV−∞, respectively. See Bingham et al. (1993)
for more discussions on regularly and rapidly varying functions, as well as their alternative
concepts and definitions. In our subsequent discussions, we will simply write f ∈ RVκ with
κ ∈ [−∞,∞], if f is regularly or rapidly varying.
It is necessary to define the notions of regular and rapid variations at −∞ and 0, as
well as at ∞, in our paper. To extend the notions of regular and rapid variations, we just
follow the usual convention. For the notions of regular and rapid variations at −∞ for
f on (−∞,∞), we define g from the negative part of f as g(x) = f(−x), and define f
to be regularly and rapidly varying at −∞ with index κ if and only if g is regularly and
rapidly varying at ∞ with index κ. On the other hand, for the notions of regular and rapid
variations at 0 for f on (0,∞), we define g from f as g(x) = f(1/x), and define f to be
regularly and rapidly varying at 0 with index −κ if and only if g is regularly and rapidly
varying at ∞ with index κ.
Throughout the paper, we make the following convention. For function f : D → R
with D = (x, x), we write f ∈ RVκ with two dimensional index κ defined as κ = (κ, κ),
where κ ∈ [−∞,∞] and κ ∈ [−∞,∞] are indexes of regular and rapid variations of f at
x and x. We let xB be either x or x and, once xB is designated as x or x, we use κ to
denote the index of regular or rapid variation at the corresponding boundary, i.e., κ or κ.
For instance, we write κ = 1 at xB = 0, in place of κ = 1 at x = 0, should there be no
confusion. Moreover, equalities and inequalities on κ apply to both κ and κ, and therefore,
κ = 1 implies κ = κ = 1, and κ > 1 implies κ > 1 and κ > 1, respectively.
To effectively deal with the asymptotics of f ∈ RVκ, κ = (κ, κ), on D = (−∞,∞), we
make a notational convention and let
f(λ) = |f(λ)|+ |f(−λ)| (4)
for any numerical sequence λ → ∞, so that we have
f(λx)
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as λ → ∞, where f , called the limit homogeneous function, is given by
f(x) = a|x|κ1{x > 0}+ b|x|κ1{x < 0}
for some constant a, b ∈ R such that |a|+ |b| = 1. The convention (4) greatly simplifies our
exposition and will be made throughout the paper without further reference. Of course,
our convention becomes unnecessary and redundant if we consider nonnegative f defined
on (0,∞).
There are many useful properties of regularly and rapidly varying functions. At any
xB , we have (i) f1 ∈ RVκ1 and f2 ∈ RVκ2
implies f1f2 ∈ RVκ1+κ2 as long as κ1 + κ2 is well
defined, (ii) f ∈ RVκ implies 1/f ∈ RV−κ, if f > 0, and (iii) f ∈ RVκ implies f−1 ∈ RV1/κ,
where f−1 is the generalized inverse of f . Also, it follows from the Karamata’s representation
theorem that f : D → R is RVκ at xB if it may be written in the form
f(x) = a(x) exp
)
for a(x) → c and b(x) → κ as x → xB . Furthermore, any regularly varying function allows
for a simple asymptotic expansion of integrals, which is known as the Karamata’s theorem,
and we have for any f ∈ RVκ
∫ x
f(y)dy ∼
(x), if κ = −1
c, if κ < −1
(x), if κ = −1
c, if κ > −1
at xB = 0, where for some slowly varying function and constant c. The reader is referred
to Bingham et al. (1993) for more details of the regularly and rapidly varying functions.
3 Diffusion Asymptotics
A diffusion is defined by its drift and diffusion functions. Though drift and diffusion func-
tions are infinitesmal parameters that primarily describe instantaneous dynamics, they com-
pletely specify a diffusion, including its longrun behavior. The asymptotics of a diffusion
8
such as recurrence and stationarity, however, are not to be readily obtained from its drift
and diffusion functions, since it is not clearly seen how drift and diffusion functions deter-
mine the longrun behavior of a diffusion. In this section, we develop a new framework we
may conveniently use to find the asymptotics of a diffusion.
To analyze diffusion asymptotics, it is very useful to introduce a function v : D → R
defined as
v(x) = −2xµ(x)
σ2(x) ,
which we call the scale index function in the paper. Note that the scale density s′, which
is well defined by Assumption 2.1, can be written in the form
s′(x) = exp
We assume
Assumption 3.1. (a) v has limit, and (b) σ2 is regularly varying.
Let p = (p, p), p, p ∈ [−∞,∞], be the limit of scale index function ν as x → xB = (x, x),
which plays an important role in diffusion asymptotics. Under Assumption 3.1 (a), it follows
directly from (5) that
s′ ∈ RVp, (6)
due to the Karamata’s representation theorem. Given (6), Assumption 3.1 (b) implies that
m is also regularly or rapidly varying, due to well known properties of regularly and rapidly
varying functions, and we write
m ∈ RVq, (7)
where q = (q, q), q, q ∈ [−∞,∞]. Under Assumption 3.1, the recurrence and stationarity
properties of a diffusion may be readily obtained from two indexes p and q introduced in
(6) and (7).
Assumption 3.2. p 6= −1.
In what follows, we call a diffusion regular if it satisfies Assumption 3.2. The recurrence
of an irregular diffusion is determined by the slowly varying component of s′, and becomes
very complicated to analyze. Moreover, an irregular diffusion has a slowly varying scale
function with a rapidly varying inverse. In the paper, we only analyze diffusion with a scale
function that has a regularly varying inverse. As discussed in Kim and Park (2013), the
asymptotics of a diffusion having a scale function whose inverse is rapidly varying become
nonstandard and generally path-dependent.
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For a regular diffusion, it follows immediately from the Karamata’s theorem that we
have s(±∞) = ±∞ if and only if p > −1, and s(0) = −∞ if and only if p < −1. Therefore,
a regular diffusion on D = (−∞,∞) is recurrent if and only if p > −1. Likewise, a regular
diffusion on D = (0,∞) is recurrent if and only if p < −1 and p > −1. The stationarity of
a recurrent diffusion is given by the integrability of m. Note that m is integrable if q < −1
at boundaries ±∞, and if q > −1 at boundary 0. Consequently, a recurrent diffusion on
D = (−∞,∞) is positive recurrent if q < −1. By the same token, a recurrent diffusion on
D = (0,∞) is positive recurrent if q > −1 and q < −1. If q = −1, however, the stationarity
of a recurrent diffusion is determined by the slowly varying component of m.
It is straightforward to determine the recurrence and stationarity of any given diffusion
using our approach.
Example 3.1. The nonlinear drift diffusion (NLD) on D = (0,∞) is introduced by At-
Sahalia (1996) as a model for spot interest rates. The diffusion is specified as
dXt = (
−1 t
b0 + b1Xt + b2X2 t dWt. (8)
At-Sahalia (1996) derives the conditions for the process to be positive recurrent process, by
explicitly computing boundary values of s and directly examining integrability of m. The
actual derivation is not simple and rather involved. However, his conditions can be readily
obtained if our approach is taken, since it only relies on the indexes (p, q) of s′ ∈ RVp and
m ∈ RVq at both boundaries. Assuming leading coefficients a−1, b0 and a2, b2 respectively
for boundaries xB = 0 and ∞ are nonzero in (8), we may easily find that
p = −2a−1
b0 and p =
∞, if a2/b2 < 0
−∞, if a2/b2 > 0
and q = −p and q = −(p + 2). Recall that, in the absence of slowly varying component,
the necessary and sufficient condition is p < −1 and p > −1 for recurrence, and q > −1
and q < −1 for stationarity. Therefore, we may deduce that the NLD process is positive
recurrent if and only if 2a−1/b0 > 1 and a2/b2 < 0. Note that the stationarity condition is
automatically satisfied if the recurrence condition holds.
Example 3.2. We consider the diffusion model on R given by
dXt = aXt(1 +X2 t )
bdWt (9)
with parameters a, b ∈ R, which we call the generalized Hopfner and Kutoyants model. If
b = 0 or b = 1, it reduces to the model considered by Hopfner and Kutoyants (2003) or Chen
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et al. (2010), respectively. In particular, if a = 0, then the diffusion becomes the driftless
diffusion. Clearly, diffusion (9) satisfies Assumption 2.1, and hence, the weak solution exists
uniquely up to an explosion time. It can be shown that the scale density and speed measure
of the generalized Hopfner and Kutoyants model are simply given by
s′(x) = (1 + x2)−a and m(x) = (1 + x2)a−b.
For the diffusion, we have p = −2a and q = 2a − 2b, and therefore, if we set a < 1/2 it
becomes recurrent on R. Moreover, the diffusion is stationary if and only if q < −1, which
holds if and only if a − b < −1/2. Note that the recurrence and stationarity conditions
become identical if and only if b = 0.
Diffusion asymptotics can be obtained directly from the functional forms of µ and σ2 if
they are regularly varying. To show this, we write
µ(x) ∼ a sgn(x)|x|z1(x) and σ2(x) ∼ b |x|w2(x) (10)
as x → xB for some constants a, b, z and w, where we assume that a and b > 0 are
defined in such a way that slowly varying functions 1 and 2 are both positive near xB and
(x) = 1(x)/2(x) → 1 as x → xB unless it converges to 0 or diverges to ∞. We may easily
deduce from (10) that
b |x|z−w+1(x) (11)
as x → xB . In our subsequent discussions, we assume a 6= 0.
Momentarily, we focus only on the cases of p = 0, p = ∞ or p = −∞. At xB = ±∞,
the recurrence condition is satisfied only when p = 0 or p = ∞, which holds if and only if
VI or DI holds, respectively, where
(VI) z − w + 1 < 0 (or z − w + 1 = 0 and (x) → 0), and
(DI) z − w + 1 > 0 (or z − w + 1 = 0 and (x) → ∞) and a < 0.
On the other hand, at xB = 0, the recurrence condition is satisfied only when p = −∞,
which holds if and only if DO holds, where
(DO) z − w + 1 < 0 (or z − w + 1 = 0 and (x) → 0) and a > 0.
Clearly, the underlying diffusion becomes recurrent only when the recurrence condition
holds at both boundaries, and otherwise it becomes transient.
Once we have any of recurrence conditions VI, DI or DO, the stationarity condition
can also be easily checked. Under VI, the scale density function s′ becomes slowly varying.
Therefore, unless w = 1, the integrability of m = 1/(s′σ2) is completely determined by
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the order of 1/σ2. As a consequence, the stationarity condition holds and fails to hold, if
w > 1 and w < 1, respectively. Note in particular that, under VI, z > 0 implies w > 1,
and therefore, we always have positive recurrence. Clearly, if w = 1, then we have m ∈ RVq
with q = −1, and the slowly varying part of m determines the stationarity of the underlying
diffusion. Under DI and DO, on the other hand, the stationarity condition is always fulfilled
at xB = ±∞ and xB = 0 as long as σ2 is regularly varying, since p = ∞ and p = −∞,
respectively. Consequently, if DI or DO holds, we have positive recurrence, and otherwise,
we have transience. No null recurrence appears. Again, the underlying diffusion becomes
stationary only when the stationarity condition holds at both boundaries, and otherwise it
becomes nonstationary.
Now let
(DV) z − w + 1 = 0 and (x) → 1
as x → xB , so that we have p = −2a/b at xB . Under DV, the necessary and sufficient
condition for recurrence of any regular diffusion becomes 2a/b < 1 at xB = ±∞, and
2a/b > 1 at xB = 0. Moreover, we have m ∈ RVq with
q = 2a
b − w =
b − (z + 1)
at xB. It therefore follows that the stationarity condition holds at xB = ±∞ if 2a/b < z,
and at xB = 0 if 2a/b > z. Consequently, at xB = ±∞ we have
positive recurrence
null recurrence
positive recurrence
null recurrence
2a/b < 1
for z > 1. If z ≥ 1 at xB = ±∞ or z ≤ 1 at xB = 0, any regular diffusion is either positive
recurrent or transient. In this case, no regular diffusion may be null recurrent. In particular,
any regular diffusion with linear drift and regularly varying volatility is positive recurrent
if and only if it is recurrent. In particular, no null recurrent diffusion may have linear drift.
In Figure 1, we characterize the diffusion asymptotics by (a, b, z, w). The top and
bottom panels illustrate the diffusion asymptotics respectively at xB = ±∞ and xB = 0.
12
w
z
b
a
NR
Figure 1: Diffusion asymptotics by (a, b, z, w) at xB = ±∞ (top panels) and at xB = 0 (bottom panels).
In the left top panel, the green area represents the condition DI, whereas the blue and red
areas represent the condition VI. We have stationarity on the green and blue areas, and
nonstationarity on the red area. The right top panel illustrates DV at xB = ±∞, and we
have stationarity and nonstationarity on the blue and red areas, respectively. Similarly, the
green area in the left bottom panel represents DO, and we have stationarity. Moreover,
the right bottom panel illustrates DV at xB = 0, and the blue and red areas represent
stationarity and nonstationarity, respectively.
4 Mean Reversion
In this section, we investigate the mean-reverting behavior of diffusion. The mean-reverting
behavior of a diffusion is closely related to the absence of a unit root in the discrete samples
collected from it. Indeed, we show later in the next section that a diffusion has mean
reversion if and only if it does not have a unit root in its discrete samples. If applied to
discrete samples from a diffusion, the unit root test becomes a test for the absence of mean
reversion rather than a test for the nonstationarity. Below we introduce a formal definition
of mean reversion. Here and elsewhere in the paper, we let X = (X t), X t = t−1 ∫ t 0 Xsds,
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be the sample mean process of X.
Definition 4.1. A diffusion X is said to have mean reversion if
1
CT
∫ T
0 (Xt −XT )dXt →d Z
as T → ∞ for some normalizing sequence (CT ), where Z is a random variable with support
on a subset of (−∞, 0).
For the mean-reverting diffusion X as defined in Definition 4.1, we say that it has strong
mean reversion if Z has a point support, and weak mean reversion otherwise.
The motivation for our definition of mean reversion is clear. We have
MT =
∫ T
)
for 0 = t0 < · · · < tn = T with max1≤r≤n |ti − ti−1| ≈ 0. Therefore, negative MT implies
that (Xti−1 −Xtn) has a negative sample correlation with (Xti −Xti−1
), i.e., the deviation
from sample mean in the current period is negatively correlated with the increment made
in transition to the next period. This occurs if and only if whenever X is observed below its
sample mean it has tendency to increase, and vice versa. In describing the mean-reverting
behavior of diffusion, we may use the recursive mean and define (Xt −X t) as the deviation
from mean, in place of (Xt − XT ) relying on the sample mean XT over the entire time
span. Though we do not show explicitly in the paper, all our subsequent theories are also
applicable for this alternative definition of mean reversion only with some obvious minor
changes.
To analyze the mean reversion property, it is necessary to introduce the asymptotics for
additive functionals of continuous time process ∫ T 0 f(Xt)dt as T → ∞. The asymptotics
are crucially dependent upon the integrability of f in m, or simply, the m-integrability. In
case that f is m-integrable, the required asymptotics are already well known, as we may
find in, e.g., Rogers and Williams (2000) and Hopfner and Locherbach (2003) respectively
for positive recurrent and null recurrent diffusions. Recently, Kim and Park (2013) have
developed in a unified framework the complete asymptotic theory of additive functions of
a diffusion for m-non integrable f , as well as m-integrable f , for both positive and null
recurrent diffusions. Here we use their asymptotics.
With no loss of generality, we may write ∫ T 0 f(Xt)dt =
∫ T 0 fs(X
s t )dt, where fs = f
s−1 and Xs = s(X) is the scale transformation of X, which may be defined as dXs t =
14
ms = 1
Both recurrence and stationarity are preserved under scale transformation. First, X is
recurrent on D if and only if Xs is recurrent on R. Trivially, the scale function of Xs is
identity, since it is already in natural scale, and therefore, Xs is recurrent if and only if its
domain is given by the entire real line R. However, the domain of Xs becomes R if and only
if X is recurrent, i.e., s(x) = −∞ and s(x) = ∞. Second, X is stationary on D if and only
if Xs is stationary on R, since ms(R) = m(D). Indeed, we may easily show by the change
of variables in integrals that ms(fs) = m(f) for any f defined on D.
To effectively introduce our asymptotics, we introduce
Definition 4.2. Let f be a nonintegrable (or m-nonintegrable) regularly varying function
on D. We say that f is strongly nonintegrable (or m-strongly nonintegrable) if f is not
integrable (or not m-integrable) for any slowly varying function . On the other hand,
we say that f is barely nonintegrable (m-barely nonintegrable) if there exists some slowly
varying function such that f is integrable (or m-integrable).
Following Definition 4.2, we say that a null recurrent diffusion X is strongly nonstation-
ary if its speed density m is strongly nonintegrable, and barely nonstationary if its speed
density m is barely nonintegrable. We assume that ms and s−1 have +∞ as their domi-
nating boundary. This assumption is not restrictive and made just for the convenience of
exposition.
Now we provide the fundamental lemma for the mean reversion of diffusion. There we
consider two conditions.
(DD) σ2 is either m-integrable or m-barely nonintegrable.
Clearly, ST is a condition related to the stationarity of X, and it holds if and only if X is
stationary or barely nonstationary. On the other hand, DD provides a condition that we
have a dominating drift in X, as will be shown more precisely below. It is easy to see that
ST holds if and only if q ≤ −1 at xB = ±∞ and q ≥ −1 at xB = 0, and DD holds if and
only if p ≥ 1 at xB = ±∞. Recall that we assume p 6= −1 throughout the paper. Note that
recurrence requires p < −1 at xB = 0, and therefore, σ2 is always m-integrable at xB = 0.
We have
Lemma 4.1. ST or DD holds if and only if
∫ T
1
2
∫ T
for all large T .
Lemma 4.1 alludes that X has mean reversion if and only if one of the two conditions ST and
DD holds. This is indeed true, as will be shown below. Note that we have (Xt−XT )dXt =
(1/2) (
)
, due to Ito’s formula, and d[X]t = σ2(Xt)dt. Therefore, the
only if part of Lemma 4.1 follows straightforwardly, if ST holds with stationary X having
all required moments, since in this case XT ,X0 and XT are all of Op(1), and (XT −XT ) 2
and (X0 −XT ) 2 become negligible in the limit compared to [X]T =
∫ T 0 σ2(Xt)dt. Lemma
4.1 shows that the asymptotic negligibility of (XT − XT ) 2 and (X0 − XT )
2 is indeed a
necessary and sufficient condition for ST or DD.
To better understand our condition DD, we write (Xt−XT )dXt = (Xt−XT )µ(Xt)dt+
(Xt −XT )σ(Xt)dWt. It can be deduced that
Lemma 4.2. DD holds if and only if
∫ T
∫ T
0 (Xt −XT )µ(Xt)dt
for all large T .
Lemma 4.2 shows that DD holds if and only if drift plays a dominating role in mean reversion
of a diffusion. Together with Lemma 4.1, Lemma 4.2 implies that DD holds if and only if
∫ T
∫ T
1
2
∫ T
for all large T .
More explicit asymptotics on mean reversion are given in the following theorem. For
locally integrable f on R, we define f [λ] as
f [λ] =
Note in particular that, for f on D, we have
(msfs)[λ] =
16
whenever it is well defined. If f is m-integrable, we have (msfs)[λ] → m(f) as λ → ∞. We
let (λT ) be the normalizing sequence satisfying
T = λTms[λT ] or λ2 Tms(λT ) (12)
depending upon whether or not ST holds, and subsequently define (CT ) from (λT ) as
CT =
λ2 T (msσ
neither ST nor DD holds
(13)
Furthermore, we let the stopping time τ be defined as
τ = inf {
(14)
depending upon whether or not ST holds, where L is the local time of Brownian motion
B. Finally, following our convention, we denote by f the limit homogeneous function of
regularly varying f on R.
Theorem 4.3. If DD holds, we have
1
CT
∫ T
1
2 L(τ, 0).
as T → ∞. On the other hand, if ST holds and DD does not hold, we have
1
CT
∫ T
2
∫ τ
as T → ∞.
Theorem 4.3 provides the details of mean-reverting behavior of diffusion. Both limit distri-
butions in Theorem 4.3 have support on (−∞, 0), since L(t, 0) > 0 a.s. for all t > 0, and ∫
|x|<msσ2 s(x)dx > 0 for any > 0 as well as τ > 0 a.s. If ST holds, we have L(τ, 0) = 1 a.s.
In fact, if both ST and DD hold, we may easily deduce that X has strong mean reversion.
Furthermore, if in particular m is integrable and σ2 is m-integrable so that X is stationary
and Eσ2(Xt) < ∞, we have
1
17
as T → ∞. Note that (ms)[λ] → m(D) and (msσ 2 s)[λ] → m(σ2) as λ → ∞, and therefore,
CT = π(σ2)T , as well as L(τ, 0) = 1 a.s.
If neither ST nor DD holds, we would have a quite different asymptotics. Let Y = s(X)
be the scale transformation of X and define Y T by Y T t = λ−1
T YTt with the normalizing
sequence (λT ) in (12), we have Y T →d Y as T → ∞ in the space C[0, 1] of continuous
functions with uniform topology, where using Brownian motion B and its local time L we
may represent the limit process Y as
Y t = B At with At = inf
{
}
.
To obtain the asymptotics for a general diffusionX, we write it asX = s−1(Y ) and note that
s−1 ∈ RVκ with κ ∈ (0, 2) in this case. Therefore, if we define XT as XT t = XTt/s
−1(λT ) =
XT = s−1(λTY
Theorem 4.4. If neither ST nor DD holds, we have
1
CT
∫ T
∫ 1
t
as T → ∞, where X is the limit process of X and X 1 =
∫ 1 0 X
t dt.
The limit distribution in Theorem 4.4 does not have support inside (−∞, 0), and therefore,
we do not have mean reversion if neither ST nor DD holds.
Now it is clear from Theorems 4.3 and 4.4 that diffusion has mean reversion if and only
if either ST or DD holds. Moreover, we have strong mean reversion if and only if both ST
and DD hold, and weak mean reversion if and only if only one of ST and DD holds.
It is worth noting that the normalizing sequences (λT ) and (CT ) introduced respectively
in (12) and (13) can be defined as
T = λTms[λT ] and CT = λT (msσ 2 s)[λT ] (15)
in all cases, up to some adjusting constants. Originally, we use them respectively only
for the case of ST and DD. For the definition of (λT ) in case that ST does not hold and
18
m ∈ RVq with q > −1, we may easily deduce from the Kamarata’s theorem that
λTms[λT ] = λT
1 + κ λ2 Tms(λT )
for ms ∈ RVκ with κ > −1 where κ = (q − p)/(1 + p) since ms = (m/s′) s−1. We may
obtain similar results for the definition of (CT ). In case ST holds and DD does not hold,
we have
∫
2 s)(λT )
for msσ 2 s ∈ RVκ with κ > −1 where κ = −2p/(1 + p) since msσ
2 s = (1/s′)2 s−1. Finally,
we have (
T (msσ 2 s)(λT ) ∼ (1− p2)λT (msσ
2 s)[λT ]
s′ s−1(λ) )
by Karamata’s theorem with (
1/s′ s−1(λ) )2
= (msσ 2 s)(λ).
Example 4.1. For an illustration, we consider the generalized Hopfner and Kutoyants
model (9). As discussed in Example 3.2, we have p = −2a and q = 2a− 2b, and therefore,
if we set a < 1/2, it becomes recurrent on R. Therefore, DD and ST holds if and only if
p ≥ 1 and q ≤ −1, which holds if and only if a ≤ −1/2 and a− b ≤ −1/2 for the diffusion,
respectively. Consequently, the diffusion has mean reversion if and only if either a ≤ −1/2
or a − b ≤ −1/2, and if both conditions hold, it has strong mean reversion. However, if
a > −1/2 and a− b > −1/2, the diffusion has no mean reversion.
In Figure 2, we provide the simulated sample paths of four different diffusions with =
1/250 and T = 40, which correspond to daily observations for 40 years. The sample path
of stationary Ornstein-Uhlenbeck process is in Figure 2 (a). For the Ornstein-Uhlenbeck
process, we have p = ∞ and q = −∞. Thus it satisfies both ST and DD, and hence,
it has strong mean reversion. Figure 2 (b) and (c) present the sample paths of driftless
diffusions with q = −1.6 and q = −0.8, respectively. Recall that all driftless diffusions have
p = 0. The driftless diffusion in Figure 2 (b) satisfies ST but not DD, and it has weak mean
reversion. On the other hand, the driftless diffusion in Figure 2 (c) satisfies neither ST nor
DD, and therefore, it is nonstationary and has no mean reversion. Finally, Figure 2 (d)
presents the sample path of the generalized Hopfner and Kutoyants model with parameter
values a = −1.4 and b = −1, which yields p = 2.8 and q = −0.8. The diffusion satisfies DD
but not ST, and therefore, it has mean reversion, though it is nonstationary.
Remark 4.1. Clearly, we may extend the concept of mean reversion in Definition 4.1 to
discrete time series. As an example, we consider the fractionally integrated process (xi)
19
t ) 2/5dWt
t ) −2dt+(1+X2
Figure 2: Sample Paths of Various Diffusions
with order −1/2 < d < 3/2, i.e., dxi = εi, where (εi) is an i.i.d. random sequence with
mean zero. It is well known that (xi) is stationary if and only if −1/2 < d < 1/2, whereas it
is nonstationary if d ≥ 1/2, see, e.g., Baillie (1996). Under appropriate moment conditions
on (εi), it follows from Sowell (1990) and Marmol (1998) that, as N → ∞,
1 N
∑N i=1(xi−1 − xN )xi →p (2δ − 1/1− δ)Eu2i < 0,
1 N
∑N i=1(xi−1 − xN )xi →p −Eu2i /2 < 0,
1 σ2
∫ 1 0 (Wt −W 1)dWt,
1 σ2
1 /2−W δ 1W
δ 1 > 0,
(
motion defined as W δ t = (1/Γ(1+ δ))
∫ t 0 (t− s)δdWs. Consequently, (xi) has mean reversion
if and only if −1/2 < d < 1. For d ≥ 1, (xi) has no mean reversion, and if d > 1, in
particular, (xi) has anti-mean reversion.
20
5 Unit Root Test
In the section, we consider testing for a unit root in discrete samples from diffusion models.
More precisely, we let the discrete sample
(X, ...,Xi, ...,XN)
of size N be collected from the diffusion process X defined in (1) at times , . . . , N over
interval [0, T ] with T = N, and test where there is a unit root in the sample. For the
brevity of notation, we will simply write
xi = Xi
for all i = 1, . . . , n with x0 = X0.
To test for a unit root in (xi), we may use the first-order autoregression without any
augmented lags, since X is a markov process. In particular, we consider the regression
xi = α+ βxi−1 + εi, (16)
where is the usual difference operator, and test the null hypothesis β = 0 against the
alternative hypothesis β < 0 using the least squares regression. The least square estimator
and the t-statistic for β in (16), denoted respectively as β and t(β), are given by
β =
i=1(xi−1 − xN )2
and
)−1/2 ,
where xN is the sample mean of (xi) and σ2 is the usual estimator for the variance of
regression errors (εi).
In the usual discrete time setup, the Dickey-Fuller test based on the t-statistic t(β) from
regression (16) is widely used to test for the null hypothesis of a unit root, i.e., β = 0, against
the alternative hypothesis of stationarity, i.e., β < 0. Under the null hypothesis of a unit
root, it has nonstandard, yet well-defined, limit distribution, as long as some mild regularity
conditions are satisfied for the innovations (εi). The limit distribution, which we call the
Dickey-Fuller distribution, is usually represented as a functional of Brownian motion. On
the other hand, under the alternative hypothesis of stationarity, it diverges to negative
21
infinity in probability. Consequently, it provides a test for unit root nonstationarity that is
consistent against a wide class of stationary time series. Therefore, we may say, loosely yet
generally, that a unit root time series is nonstationary, and that a stationary time series
does not have a unit root.
The Dickey-Fuller test has also routinely been applied to samples that are thought to
be collected discretely from diffusion type continuous time processes. However, little is
known about its asymptotic behavior for discrete samples obtained from general continuous
time processes. In particular, except for simple models such as Brownian motion and
Ornstein-Uhlenbeck process, it has been completely unknown whether the test can be used
to effectively discriminate nonstationary diffusions from stationary diffusions. To facilitate
our discussions on the asymptotic behavior of the Dickey-Fuller test, we will simply say in
what follows that a diffusion has a unit root if and only if t(β) is stochastically bounded
and the test is expected to suggest the non-rejection of the unit root hypothesis at least
with some positive probability.
5.1 Continuous Time Approximation of Unit Root Test
In this section, we present the primary asymptotics for the unit root test. We start with
some technical assumptions. We let ι denote the identity function, i.e., ι(x) = x.
Assumption 5.1. ι, µ, σ are all majorized by ω : D → R.
In Assumption 5.1, we require the existence of an envelop function for µ, σ and ι so that we
may effectively control them near the boundaries. Indeed, the envelop function ω always
exists since we may simply set ω(x) = max {µ(x), σ(x), x}. In our asymptotics, we require that the sampling interval be sufficiently small relative
to the extremal bounds of various functional transforms of X over time interval [0, T ].
Following At-Sahalia and Park (2011), we define
T (f) = max 0≤t≤T
|f(Xt)|
for some function f : D → R. For the identity function, we have T (ι) = max0≤t≤T |Xt| and T (ι) becomes the asymptotic order of extremal process of X. It is obvious that we
have [T (ι)]k = T (ιk) for any nonnegative k. More generally, for any regularly varying f ,
we may obtain the exact rate of T (f) from the asymptotic behavior of extremal process.
For instance, the extremal processes of Ornstein-Uhlenbeck process and Feller’s square root
process are respectively of orders Op( √ log T ) and Op(log T ), and the extremal process of the
general driftless diffusion process is of order Op(T ). Thus if f is regularly varying and cT is
22
the order of extremal process, then we have T (f) = f(cT ). The order of extremal process is
known for a wide class of diffusions. For instance, under some regularity conditions on µ and
σ2, it is well known that the extremal processes of positive recurrent diffusions are of order
Op(s −1(T )), to which the reader is referred to, e.g., Davis (1982). Moreover, asymptotic
orders for the extremal processes of general null recurrent diffusions are obtained by Stone
(1963), Jeong and Park (2011) and Kim and Park (2013).
Assumption 5.2. 1/2T (ω2)T √
log 1/[T (ω2)] → 0.
Assumption 5.2 makes it necessary to have → 0. If we fix T , → 0 is indeed the
necessary and sufficient condition for Assumption 5.2. Clearly, we may also allow T → ∞ as long as → 0 sufficiently fast. In this case, our asymptotic results will be more relevant
for the case where is sufficiently small relative to T . Our asymptotics in the paper are
derived under the condition → 0 and T → ∞ jointly. For Assumption 5.2 to hold, it
suffices to have = O(T−2−) for any > 0, if X is bounded so that T (ω2) is a constant.
The condition appears to be mild enough to yield asymptotics generally relevant for a very
wide range of empirical analysis relying on samples collected from diffusion models. For
daily observations over ten years, as an example, we have ≈ 1/250 and T−2 = 1/100.
Our subsequent asymptotics hold jointly in and T as long as they satisfy Assumption
5.2 as → 0 and T → ∞. In particular, we do not use sequential asymptotics, requiring
→ 0 and T → ∞ sequentially.
The primary asymptotics for β and t(β) are given by the following lemma.
Lemma 5.1. Let Assumption 5.1 and 5.2 hold. We have
,
[X] 1/2 T
)1/2
for all sufficiently small relative to T .
The limit theory in Lemma 5.1 holds for all small enough relative to T as long as and
T satisfy Assumption 5.2. Therefore, we expect that they provide good approximations
for finite sample distributions, whenever is relatively small compared with T . Note that
we do not assume T = ∞ to obtain the asymptotics in Lemma 5.1. The asymptotics we
have in Lemma 5.1 will be referred in the paper to as the primary asymptotics. The joint
asymptotics for → 0 and T → ∞, which are presented below, may be obtained simply
by taking T -limits to our primary asymptotics.
23
Our primary asymptotics in Lemma 5.1 make it clear that we have β →p 0 whenever
→ 0 fast enough relative to T . If, in particular, T is fixed, we have β →p 0 for any
diffusion X as long as → 0. For discrete samples from any diffusion, we will therefore
always observe a root getting close to unity as we collect samples frequently enough and
becomes sufficiently small. However, even in this case, the unit root test will not necessarily
support the presence of a unit root. If we let → 0 with fixed T , t(β) remains to be random
and the unit root test will yield completely random conclusion, regardless of the asymptotic
properties of the underlying diffusion. The unit root test will be totally uninformative in
this case. Indeed, it is very natural and not surprising at all that the unit root test does not
yield any information on the asymptotics, such as mean reversion and unit root properties,
of the underlying diffusion unless T becomes large. In discussions to follow, we let T → ∞ and consider large T asymptotics.
5.2 Asymptotics of Unit Root Test
Now we establish the asymptotics for the unit root test.
Theorem 5.2. Let Assumption 5.1 and 5.2 hold. If neither ST nor DD holds, we have
T−1β →d
∫ 1 0 (X
2dt )1/2
as → 0 and T → ∞. On the other hand, if either ST or DD holds, we have
T−1β →p −∞ and t(β) →p −∞
as → 0 and T → ∞.
Theorem 5.2 shows that Nβ and t(β) have nondegenerate limit distributions if and only
if neither ST nor DD holds. If either ST or DD holds, Nβ →p −∞ and t(β) →p −∞.
Note in particular that T−1 = N . In other words, the normalized estimator of β and its
t-ratio, the two most commonly used statistics for the unit root, may not reject the unit
root hypothesis even in the limit if and only if neither ST nor DD holds. If either ST or
DD holds, both statistics are expected to reject the unit root hypothesis with probability
one in the limit. Clearly, the necessary and sufficient condition we require here to have the
evidence of a unit root from the unit root test is precisely the same as the necessary and
24
Rejection Probabilities 10.1% 14.2% 20.3% 29.9%
Table 1: Rejection probabilities of Dickey-Fuller test for nonstationary diffusion having dominating drift
sufficient condition for the absence of mean reversion. That is, we have a unit root and the
unit root test supports the presence of a unit root, if and only if the underlying diffusion
does not have mean reversion. In particular, the test of a unit root in discrete samples from
diffusion does not provide any clue for, or against, the nonstationarity of the underlying
diffusion.
It is clear from Theorem 5.2 that the unit root test cannot be used to test for nonsta-
tonarity. Null recurrent diffusion will not have a unit root if DD holds and it has mean
reversion due to the dominant role of drift in its longrun behavior. As an illustrative ex-
ample, we consider the generalized Hopfner and Kytoyants model in (9) with a = −7/5
and b = −1. With the given parameter values, X has a dominating drift though it is non-
stationary, i.e., DD holds though ST is not satisfied. Therefore, it is mean-reverting and
the unit root test is expected to reject the unit root hypothesis. To demonstrate this, we
compute the actual rejection probabilities of the unit root test and report them in Table 1.
The reported rejection probabilities are obtained by simulating 10,000 realizations of the
diffusion with = 1/250 and T = 10, 20, 40, 80, which correspond to daily observations
for the periods of 10, 20, 40, 80 years, respectively. We use the 5% critical value from
the Dickey-Fuller distribution. As expected, the actual rejection probabilities exceed the
nominal level for all T , and they increase with T .
We should also note that the unit root test does not have any nontrivial power in
discriminating diffusions with and without drift. As we have seen earlier, recurrent diffusion
with linear drift is positive recurrent. Therefore, as long as it is recurrent, any diffusion is
stationary if it has a linear drift. This, in turn, implies that the unit root test rejects the
unit root hypothesis for any recurrent diffusion with linear drift. However, in general, the
unit root test does not have any discriminatory power for or against the presence of drift
in diffusion. It is clear that for a driftless diffusion may also be mean-reverting, and has no
unit root, as long as it is stationary or barely nonstationary.
The limit distribution for the unit root test t(β) is model-dependent. In particular, its
limit distribution is generally different from the Dickey-Fuller distribution and the standard
critical values for the unit root test cannot be used. For an illustration of model dependency
of the limit distribution of the unit root test, we consider the generalized Hopfner and
25
−0.5
0
0.5
1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −4.2
−4
−3.8
−3.6
−3.4
−3.2
−3
−2.8
q
DF
Figure 3: Illustration of the model dependency in limiting distributions under nonstation- arity. The data were generated according to generalized Hopfner and Kutoyants model (9). The plot on the upper-left presents 5% quantiles of t-statistics for (p, q) ∈ [−1, 1]× [−1, 1], and its corresponding Contour plot is on the upper-right panel. In the plot on the lower panel, we compare the critical value from the Dickey-Fuller distribution with 5% quantiles of t-statistics for p = −0.9, 0, 0.9 and q ∈ [−1, 1].
Kutoyants model in (9). For the generalized Hopfner and Kutoyants model, we have p =
−2a and q = 2a− 2b. As shown in Theorem 5.2 (a), different choice of (p, q) gives different
limiting distributions. We conduct simulations for the process (9) with various combinations
of (p, q) ∈ [−1, 1]× [−1, 1] to examine the model dependency in null distributions. For each
experiment, we simulate 10,000 realizations with = 1/250 and T = 40 which correspond
to daily data for 40 years. Figure 3 shows the 5% quantiles of t-statistics. There appears to
be considerable amount of model dependency in the finite sample distribution. As we can
see in Figure 3, the critical values from the Dickey-Fuller distributions may give misleading
conclusions. In particular, if both p and q are close to, but not less than, -1, then 5%
quantiles of t-statistics from (9) are significantly smaller than the critical value from the
Dickey-Fuller distributions. For example, if p = −0.9 and q = −0.9, then the 5% quantile
of t-statistics from (9) is about −3.7, whereas corresponding critical value from the Dickey-
Fuller distributions is about −2.9.
26
Of course, our limit distribution of the unit root test reduces to the Dickey-Fuller dis-
tribution in case the limit process X of the underlying diffusion X becomes Brownian
motion. Clearly, X becomes Brownian motion if and only if σ2 and s′ are constant func-
tions. Therefore, we need to have p = 0 and q = 0 at both boundaries. Moreover, it is
required that σ2(−λ)/σ2(λ) → 1, and
s′(−λ)
|x|≤λ
dXt = Xt
becomes Brownian motion in the limit.
If used to test for nonstationarity of the underlying diffusion, the unit root test is
consistent against all mean-reverting diffusions. However, the test becomes consistent and
has discriminating power only when the time span T goes to infinity. In particular, it is well
expected from our theory that the test will not be consistent if we just increase the sample
size N = T/ by decreasing the sampling interval (or equivalently, increasing sampling
frequency 1/) with fixed T . This was first observed in Shiller and Perron (1985), and
further analyzed later by Perron (1989, 1991), for the test of a unit root in samples from
Brownian motion against those from Ornstein-Uhlenbeck process.
Remark 5.1. In the paper, we mainly consider stock variables, for which we assume that
the discrete samples (xi) are obtained from the underlying diffusion at equally spaced time
intervals and let xi = Xi. However, we may also analyze the discrete samples for flow
variables, which are given as
xi =
∫ i
(i−1) Xtdt
from the underlying diffusion X. The asymptotics of the unit root test for the discrete
samples from continuous time flow variables is investigated by Chambers (2004, 2008),
under the assumption that they are generated by either Brownian motion or Ornstein-
Uhlenbeck process. For some related simulation studies of the unit root test for the samples
from continuous time processes, the reader is referred to Choi (1992), Choi and Chung
(1995) and Boswijk and Klaassen (2012).
27
Remark 5.2. Let (xi) be a fractionally integrated process with order d introduced in
Remark 4.1. Under appropriate moment conditions on (εi), Sowell (1990) and Marmol
(1998) show that t(β) →p −∞ if d < 1, and t(β) →p ∞ if d > 1, as N → ∞. As is well
known, if d = 1, the t-ratio converges in distribution to the Dickey-Fuller distribution as
N → ∞. Therefore, we may use the unit root test to investigate the presence of mean
reversion in fractionally integrated processes.
6 Conclusion
In this paper, we study the longrun behaviors of general diffusion models and find many
interesting results. In particular, we show that the Dickey-Fuller statistic cannot be used to
test for nonstationarity of the underlying diffusion. Rather, it becomes a test for no mean
reversion of the underlying diffusion, if applied to discrete samples from a diffusion. A dif-
fusion has a unit root in its discrete samples if and only if it does not have mean reversion,
and a nonstationary diffusion may also be mean-reverting if it has a drift dominating its
diffusion. The test would at least have the perfect discriminatory power against all station-
ary diffusions, if we only consider nonstationary and non-mean-reverting diffusions under
the null hypothesis. However, even in this case, the limit distribution of the test becomes
model-dependent and the usual Dickey-Fuller critical values are not applicable. For the unit
root test applied to discrete samples from a diffusion model to be strictly valid as a test for
nonstationarity of the underlying diffusion, therefore, we should further restrict the class of
diffusion models considered under the null hypothesis to be nonstationary diffusions that
become Brownian motions in the limit.
28
Appendix A Useful Lemmas
In the following, we assume that X is recurrent diffusion satisfying Assumption 2.1, 3.1
and 3.2. Thus s′ ∈ RVp with p 6= −1, and m ∈ RVq. Moreover, we let Xs be the scale
transformed process of X, i.e., Xs t = s(Xt) for all t ≥ 0, and denote ms by the speed
measure of Xs, which is given by ms = 1/(s′σ)2 s−1.
To facilitate our exposition, we make some notational conventions. For locally integrable
f on R, we define f [λ] = ∫
f(x)1{|x| < λ}dy. Note that if f is either integrable or
barely nonintegrable, then f [λ] is slowly varying. Moreover, we let (λT ) and (dT ) be the
normalizing sequence satisfying dT = s−1(λT ) and
T =
λ2 Tms(λT ), otherwise.
Recall that λT ∼ Tm(D) if X is stationary, whereas λT = o(T ) if X is nonstationary. Using
λT , we define Tf by
Tf =
λT (msfs)[λT ], if f is either m-I or m-BN
λ2 T (msfs)(λT ), if f is m-SN.
where we mean I, BN and SN by integrable, barely nonintegrable and strongly nonintegrable,
respectively. Sometimes, we write f [a, b] = ∫ b a f(x)dx for locally integrable function f .
Lemma A1. (a) Let f and g2 be m-I or m-SN. Then we have
1
Tf
∫ T
{
L(τ, 0), if f is m-I ∫ τ 0 msfs(Bt)dt, if f is m-SN
and
1 √
Tg2
∫ T
1/2 s gs(Bt)dBt, if g2 is m-SN
where B is Brownian motion with local time L, N is standard normal variate independent
of B, and τ is time change defined as
τ =
inf {
s| ∫
1
0 f(Xt)dt →d KL(τ, 0)
where K = limλ→∞(msfs)[λ]/|msfs|[λ], and L and τ are defined in part (a). In particular,
if λ(msfs)(λ) (msfs)[λ], then
1
Tf
∫ T
where L and τ are defined in part (a).
(c) Let f be m-I with m(f) = 0. If (mf)[x, x] is square integrable, then
1√ λT
)1/2
N
where N is standard normal variate independent of Brownian local time L.
(d) If ST holds, then L(τ, 0) = 1 with probability one.
Proof for Lemma A1. The statements in the lemma follows from Kim and Park (2013), and
we omit the proof here.
Lemma A2. (a) If ST holds, then XT = op(dT ) and XT = op(dT ) for large T .
(b) If DD holds, then XT = op( √
λT (msσ2 s)[λT ]) for large T .
Proof for Lemma A2 (a). For the first statement, we note that XT = Op(T|ι|/T ) due to
Lemma A1, where T|ι| is determined by the m-integrability of ι. Thus it suffice to show
that T|ι|/T = o(dT ).
Trivially, if ι is m-I, then T|ι|/T = o(dT ) since T|ι| ∼ λTm(|ι|) = O(T ) and dT → ∞. On
the other hand, if s′ is rapidly varying, then m is rapidly decreasing, and hence, ι is always
m-I. In other word, if ι is m-non integrable, then s′ is regularly varying, which implies that
s−1 is regularly varying but not slowly varying. Thus if ι is m-BN, then s−1(λT )/λ T → ∞
for some > 0, and therefore, we have T|ι|/T = o(dT ) because
T|ι|
s−1(λT ) → 0
for some slowly varying , since both ms[λ] and |msιs|[λ] are slowly varying.
Now let ι be m-SN. Note that if m is I on D, then ms[λT ] → m(D) and λTms(λT ) → 0
since ms is I on R. Moreover, if m is BN, then λTms(λT )/ms[λT ] → 0 by Karamata’s
30
theorem for barely nonintegrable function (see, e.g., Proposition 1.5.9 in Bingham et al.
(1993)). In both cases, we have
T|ι|
ms[λT ] → 0 (A.1)
since s−1 = ιs, which complete the proof for XT = op(dT ).
The second statement XT = op(dT ) follows immediately from Kim and Park (2013),
and the proof is therefore omitted here.
Proof for Lemma A2 (b). In the proof, we only consider the case where ι is m-SN because
the results for m-I or m-BN ι can be shown in a similar way used in the proof of part (a).
Now let ι be m-SN. If ST holds, then XT = op(dT ) due to Lemma A2 (a). On the other
hand, if ST does not hold, then we can deduce from Lemma A1 that XT = Op(dT ) because
we have dT = Tι/T since Tι = λ2 T (msιs)(λT ), dT = s−1(λT ) = ιs(λT ) and T = λ2
Tms(λT ).
In all cases, we have XT = Op(dT ). Thus it suffice to show that dT / √
λT (msσ2 s)[λT ] → 0
when DD holds.
s′(x) dx = (1/s′)[dT ]
which is slowly varying satisfying λ/ (s′(λ)(1/s′)[dT ]) → 0 due to Karamata’s theorem for
integrable or barely nonintegrable function. Thus
dT √
)1/2
→ 0 (A.2)
since s(λ) ∼ λs′(λ)/(1 + p) by Karamata’s theorem for strongly nonintegrable function,
which complete the proof.
XT
∫ T
XT
∫ T
Proof for Lemma A3 (a). Since m(σ2) = ∫
D 1/s′(x)dx, σ2 is m-I or m-BN if and only if
1/s′ is I or BN, respectively. Moreover, DD holds if and only if p ≥ 1 at xB = ±∞. Here,
we don’t need to consider the m-integrability of σ2 around the zero boundary because σ2
is always integrable around zero boundary because (mσ2)[x, x] = ∫ x 0 1/s′(y)dy < ∞ for any
x ∈ D since p < −1 at xB = 0. In the following, we will consider each of m-I and m-BN
cases separately.
(m-I σ2) Note that if σ2 is m-I, then x/s′(x) → 0 as x → x and x since m(σ2) = ∫
D 1/s′(x)dx < ∞ and s′ is regularly varying due to Assumption 3.1. Then ιµ is also m-I
and satisfies m(σ2) = −2m(ιµ) because
m(σ2) =
= −2m(ιµ)
due to the integration by parts with s′′(x) = −2µ(x)s′(x)/σ2(x). Thus it follows from
Lemma A1 that
2 m(σ2)L(τ, 0) (A.3)
where L and τ are defined in Lemma A1.
Moreover, σ2 is m-I implies that µ is m-I and m(µ) = 0 since
m(µ) =
= 0.
It is easy to see that s′′ is regularly varying function by Assumption 3.1 with the represen-
tation theorem for regularly varying function. Then we have
(mµ)[x, x] =
2s′(x)
due to Karamata’s theorem for integrable function (see, e.g., Proposition 1.5.10 in Bingham
et al. (1993)). Therefore, (mµ)[x, x] is square integrable since s′ ∈ RVp with p ≥ 1, and
hence, we have
λT ). (A.4)
by Lemma A1 (c). The stated result for m-I σ2 follows immediately from (A.3) and (A.4)
with Lemma A2 (b).
32
(m-BN σ2)We note that ιµ ism-BN and (msσ 2 s)[λ] ∼ −2(msιsµs)[λ] as λ → ∞ because
−2(msιsµs)[λ] = −2
s′ s−1(−λ)
lim λ→∞
−2(msιsµs)[λ]
s′ s−1(−λ)
= 1 (A.5)
by Karamata’s theorem for barely nonintegrable function. Moreover, we have λT (msιsµs)(λT ) (msιsµs)[λT ] because dT /s
′(dT ) −2(msιsµs)[λT ] due to (A.5), and
λT (msιsµs)(λT ) = λT (mιµ)
(s′)2(dT ) ∼ −2pdT
(1 + p)s′(dT )
since mιµ = −2ιs′′/(s′)2 ∼ −2p/s′ and s ∼ ιs′/(1 + p) by Karamata’s theorem (see, e.g.,
Proposition 1.5.8 in Bingham et al. (1993)). It then follows from Lemma A1 (b) that
1
0 Xtµ(Xt)dt →d −1
2 L(τ, 0). (A.6)
On the other hand, we can show that m(µ) = 0 and (mµ)[x, x] is square integrable as
in the proof for m-I σ2. Thus we have (A.4), and therefore, the stated result for m-BN σ2
follows immediately from (A.6) and Lemma A2 (b).
Proof for Lemma A3 (b). We can show that ιµ is m-SN if σ2 is m-SN in similar ways in
the proof for Lemma A3 (a). Then, by Lemma A1 (a), we have
1
0 msιsµs(Bt)dt.
On the other hand, the m-integrabilities of ι and µ can be arbitrary. We will only prove
both ι and µ are m-SN since other cases can be driven with a similar way. Let ι and µ
be m-SN. Then the stochastic order of XT
∫ T 0 µ(Xt)dt is given by λ4
T (m 2 sιsµs)/T due to
33
Lemma A1 (a). However, we can deduce from (A.1) that
λ4 T (m
{
λTms(λT )/ms[λT ] → 0, if m is either I or BN
1, if m is SN
since T = λ2 Tms(λT ) for SN m, and this provide the stated result.
Lemma A4. If DD holds, then
∫ T
∫ T
for large T .
Proof for Lemma A4. Note that if DD holds, then ι2σ2 can be m-I, m-BN or m-SN. For
each case, we can deduce from Lemma A1 that
∫ T
Op( √
λT (msι2sσ 2 s)[λT ]), if ι2σ2 is m-I or m-BI
Op(λT (m 1/2 s ιsσs)(λT )), if ι2σ2 is m-SN
(A.7)
where (msι 2 sσ
2 s)[λT ] is slowly varying when ι2σ2 is m-I or m-BN. If ι2σ2 is either m-I or
m-BN, then we should have m-I σ2. Then it follows from Lemma A3, (A.3) and (A.7) that √
λT (msι2sσ 2 s)[λT ]/λT → 0, which provide the desired results for m-I or m-BN ι2σ2.
Now let ι2σ2 be m-SN. Then it follows from (A.2) that
λT (m 1/2 s ιsσs)(λT )
λT (msσ2 s)[λT ]
since m 1/2 s ιsσs = (ι/s′) s−1 and msσ
2 s [λT ] = (1/s′)[dT ]. The stated result for m-SN ι2σ2
follows from Lemma A3, (A.6), (A.7) and (A.8), which completes the proof.
Lemma A5. If DD does not hold, then
∫ T
∫ T
for large T .
Proof for Lemma A5. If DD does not hold, then ι2σ2 is m-SN, and hence, the stochastic
order of ∫ T 0 (Xt−XT )σ(Xt)dWt is given by λT (m
1/2 s ιsσs)(λT ) as shown in (A.7). Moreover,
34
ιµ is also m-strongly nonintegrable, and the stochastic order of ∫ T 0 (Xt − XT )µ(Xt)dt is
λ2 T (msιsµs)(λT ) as shown in the proof of Lemma A3 (b). Then we have
λ2 T (msιsµs)(λT )
= λT (m
2(1 + p)
since −2dTµ(dT )/σ 2(dT ) → p by Assumption 3.1, and s(dT )/[dT s
′(dT )] → 1/(1 + p) by
Karamata’s theorem, from which we have the stated result since p 6= −1.
Lemma A6. If ST holds, then
∫ T
1
2
∫ T
∫ T
0 σ2(Xt)dt
. (A.9)
By applying Lemma A1 to the last term in (A.9), we have
1
Tσ2
∫ T
s(Bt)dt, otherwise. (A.10)
(X2 T −X2
0 )− 2XT (XT −X0) = op(d 2 T ) (A.11)
due to Lemma A2 (a). It then follows from (A.2) that d2T = o (
λT (msσ 2 s)[λT ]
)
when DD
holds. On the other hand, if DD does not hold, then we have d2T = O(λ2 T (msσ
2 s)(λT ))
→ (1 + p)2 (A.12)
since s(λ) ∼ λs′(λ)/(1 + p) and p 6= −1. In all cases, we have d2T = O(Tσ2), and hence, the
stated result follows from (A.9), (A.10) and (A.11).
35
∫ T
1
2
∫ T
for large T .
Proof for Lemma A7. Due to Lemma A6, we only need to prove the statement for SN m.
If m is SN, then
XT = Op(λ 2 T (msιs)(λT )/T ) = Op(dT )
by Lemma A1, and hence, we have
(X2 T −X2
0 )− 2XT (XT −X0) = Op(d 2 T ) (A.13)
since Xt = Op(dT ) for all t ∈ [0, T ] by Kim and Park (2013). However, if DD holds, we
have (A.2), and hence, the stated result follows from (A.9), (A.10) and (A.13).
Lemma A8. If f is a twice differentiable function, then we have
N ∑
∫ T
0 f(Xt)dt+Op(T (f ′µ)T ) +Op(T (f ′′σ2)T ) +Op(T (f ′σ)T 1/2))
as T → ∞ and → 0.
N ∑
=
by applying Ito’s lemma to f(Xt)− f(X(i−1)), where
Ra T =
|Ra T | =
by changing the order of the integrals.
As for Rb T , we have
Rb T =
(i−1) (i− s)f ′σ(Xs)dWs
by changing the order of the integrals. Define a continuous martingale D by
Dt =
∫ t
(j−1) (i− s)f ′σ(Xs)dWs
for t ∈ [(j − 1), j], i = 1, 2, ..., so that DT = Rb T . Then the quadratic variation [D] of D
follows that
[D]T =
The stated result follows from (A.14) with (A.15) and (A.16).
Lemma A9. Let D be a continuous martingale defined as Dt = ∫ t 0 f(Xs)dWs. If T (f2) →
0, then we have
Proof for Lemma A9. The quadratic variation of D satisfies
sup |t−s|≤
∫ t
s f2(Xr)dr = Op(T (f2)) (A.17)
uniformly in 0 ≤ s ≤ t ≤ T . Then, by the DDS (or Dambis, Dubins-Schwarz) theorem (see
Revuz and Yor (1999), page 496), we may represent D as Dt = B [D]t where B is the
DDS Brownian motion of D. Thus we have
sup |t−s|≤
|B [D]t −B [D]s|
= sup |t−s|≤
(2 ([D]t − [D]s) log 1/([D]t − [D]s)) 1/2 (A.18)
due to the modulus of continuity of Brownian motion (see Borodin and Salminen (1996),
page 50). The stated result follows immediately from (A.17) and (A.18).
Lemma A10. Let f be a twice differentiable function. Then we have the following:
(a) We have
0 fg(Xt)dt+Op(T (gf ′µ)T ) +Op(T (gf ′′σ2)T )
+Op(T (gf ′σ)T 1/2))
as T → ∞ and → 0.
as T → ∞ and → 0.















= Op(T (f ′µ)T ) +Op(T (f ′′σ2)T ) +Op(T (f ′σ)T 1/2))
which, in view of (A.19), completes the proof for the first part.
For the second statement, we have
∫ T
by applying Ito’s lemma to f(Xt)− f(X(i−1)), where
Ra T =
Da t =
j−1 ∑
i=1
(j−1) f ′σ(Xs)dWsdWr
for t ∈ [(j − 1), j], i = 1, 2, ..., so that Da T = Ra
T and Db T = Rb
T . Then, for the quadratic
39
[Da]T = N ∑
log 1/[T ((f ′σ)2)] )
due to Lemma A9. Therefore, the state result for the second part follows immediately from
(A.20), (A.21) and (A.22) with the constructions of Da and Db.
Lemma A11. We have
N ∑
as T → ∞ and → 0.





from which, we have the first statement.
For the second part, we define a continuous martingale D by
Dt =
for t ∈ [(j − 1), j], i = 1, 2, ..., so that
DT =
N ∑
[D]T =
Lemma A12. If T (f2) → 0, then we have
N ∑
as T → ∞ and → 0.
Proof for Lemma A12. The first part follows immediately from Lemma A9. For the second
part, we define a continuous martingale D by
Dt =
for t ∈ [(j − 1), j], i = 1, 2, ..., so that
DT =
N ∑
[D]T = N ∑
T ((fg)2)T log 1/[T (f2)] )
due to Lemma A9, which completes the proof of the second part.
41
i=1
x2i +Op(T (µ2)T ) +Op(T (σ2)) +Op(T (µσ)T 1/2)
as T → ∞ and → 0, where xN = N−1 ∑N
i=1xi. Moreover, if T (σ2) → 0, then
N ∑
+Op
N ∑
However, we have
]
,
and hence,
N(xN )2 = Op(T (µ2)T ) +Op(T (σ2)) +Op(T (µσ)T 1/2) (A.24)
which provides the first statement.
Now, for the first term in (A.23), we have
N ∑
N ∑
)
)
Ra T =
Rb T =
Ra T =
log 1/[T (σ2)] )
due to Lemma A11 and Lemma A12, which completes the proof of the second part since
[X]T = ∫ T 0 σ2(Xt)dt.
Appendix B Proofs for Theorems
Proof for Lemma 4.1. Due to Lemma A6 and A7, it suffice to prove that
∫ T
1
2
∫ T
0 σ2(Xt)dt (B.1)
when neither ST nor DD hold. To prove (B.1), we let X be strongly nonstationary, and
define XT by XT t = XTt/dT . Then, it follows from Kim and Park (2013) that XT →d X
in C[0, 1], where X = s−1(B A) with Brownian motion B having local time L, and time
change A is defined as
At =
XT
dT =
1
TdT
∫ T
and hence
0 )− 2X 1(X
1
d2T
∫ T
1
d2T
∫ T
0 )− 2X 1(X
from which we have (B.1), which complete the proof.
Proof for Lemma 4.2. The statement in the lemma follows immediately from Lemma A4
and A5.
Proof for Theorem 4.3. The stated results follows immediately from Lemma A1 (a), (b)
and Lemma 4.1.
Proof for Theorem 4.4. Note that if DD does not hold, then the limiting process X of XT
is semi-martingale with quadratic variation [X] given by
[X]t =
(s−1)′−
)2 (Bs)ds,
where (s−1)′− is the left-hand derivative of s−1. The reader is referred to Kim and Park
(2013) for more detailed discussions on the limiting process X. If we show
∫ At
0
0 msσ2
s(Bt)dt (B.6)
for all t ∈ [0, 1], then the stated result follows from (B.5) since
∫ 1
0 )− 2X 1(X
To show (B.6), we note that msσ 2 s =
(
since msσ 2 s = (1/s′ s−1)2 and (s−1)′ =
1/s′ s−1. Moreover, if p 6= 0, then (s−1)′ is ultimately monotone, and hence, (s−1)′(x) ∼
44
s−1(x)/[x(1 + p)] due to monotone density theorem (see, e.g., Theorem 1.7.2 in Bingham
et al. (1993)). On the other hand, if p = 0, then (s−1)′ is slowly varying. In both cases, we
have s−1′(x) = (1 + p)(s−1)′−(x) for all x ∈ R\{0} since
(s−1)′(x) = a|x|−p/(1+p)1{x > 0}+ b|x|−p/(1+p)1{x ≤ 0},
(s−1)′−(x) = 1
(
a|x|−p/(1+p)1{x > 0}+ b|x|−p/(1+p)1{x ≤ 0} )
for some constant a, b ∈ R such that |a| + |b| = 1. Therefore, we have (B.6), and this
complete the proof.
Proof for Lemma 5.1. First, we consider β. For the proof, we note that
N ∑
0 Xtdt+Op(T (µ)T ) +Op(T (σ)T 1/2), (B.7)
N ∑
∫ T
0 Xtdt+Op(T (ιµ)T ) +Op(T (σ2)T ) +Op(T (ισ)T 1/2) (B.8)
due to Lemma A8. Moreover, we can deduce from Lemma A10 that
N ∑
Rσ T = Op(T (µσ)T 1/2) +Op
(
.
Then the stated result for β follows immediately from (B.7), (B.8) and (B.9) with
Assumption 5.1 and 5.2.
N ∑
= N ∑
]2
, (B.10)
i=1 xi−1.
For the second term in (B.10), we can deduce from (B.7), (B.8) and (B.9) with Assump-
tion 5.1 and 5.2 that
[
]2
=
[
]2
(1 + op(1)). (B.11)
Moreover, we have
∫ T
∫ T
1/2T (ισ)T 1/2) = op(1) (B.12)
due, in particular, to Assumption 5.2. It follows from (B.10), (B.11) and (B.12) with Lemma
A13 that
= 1
T
∫ T
0 σ2(Xt)dt(1 + op(1)) (B.13)
due to Assumption 5.1 and 5.2. The stated result for t(β) follows immediately from (B.13)
with the proof of the first part of this lemma.
Proof for Theorem 5.2. For a strongly nonstationary X, we have
1
Td2T
∫ T
2dt (B.14)
in a similar way of (B.2). Note in particular that Td2T = λ2 T (msι
2 s)(λT ). On the other hand,
if X is stationary or barely nonstationary, then we have
1
Tι2
∫ T
→p 1, if ι2 is m-BI
→d
(B.15)
46
by Lemma A1. In the following, we first prove the case where (a) neither ST nor DD holds,
and then we consider the case where (b) either ST or DD holds.
Part (a) We let nether ST nor DD hold. Then it follows immediately from Lemma 5.1,
Theorem 4.4 and (B.14) that
T−1β ∼p
1 Td2T
→d
1)
2dt
as T → ∞ and → 0. Moreover, it follows from Lemma 5.1, Theorem 4.4, (B.14) and
(B.4) with (B.6) that
(
)1/2 →d
2dt )1/2
as T → ∞ and → 0, which completes the proof for the first case where nether ST nor
DT holds.
Part (b) Now let either ST or DD holds. Then we may write
T−1β ∼p TTσ2
1 Tι2
(B.16)
and
(
)1/2 . (B.17)
If we show TTσ2/Tι2 → ∞, then the stated result follows immediately from Lemma 5.1,
Theorem 4.3, (B.14), (B.15) and (A.10) with (B.16) and (B.17).
To complete the proof, we note that
TTσ2
Tι2 =
T (msσ 2 s)[λT ]/(msι
2 s)[λT ], if σ2 is m-I or m-BN and ι2 is m-I or m-BN,
TλT (msσ 2 s)(λT )/(msι
2 s)[λT ], if σ2 is m-SN and ι2 is m-I or m-BN,
T (msσ 2 s)[λT ]/λT (msι
2 s)(λT ), if σ2 is m-I or m-BN and ι2 is m-SN,
T (msσ 2 s)(λT )/(msι
2 s)(λT ), if both σ2 and ι2 are m-SN.
Recall that (msfs)[λ] is slowly varying if f is either m-I or m-BN. Thus if ι2 is either m-I
or m-BN, then TTσ2/Tι2 → ∞. Thus we only need to examine the cases where ι2 is m-SN.
47
T (msσ 2 s)[λT ]
λT (msι2s)(λT ) =
dT
≡ aT bT
since msι 2 s = (ι2m/s′) s−1. Moreover, we have bT → ∞ due to (A.2) and
aT =
λTms(λT )/dTm(dT ) → 1, otherwise
by Karamata’s theorem.
Now let σ2 be m-SN. Then ST should be satisfied, and we have
T (msσ 2 s)(λT )
d2T =
d2T
(1 + p)dT =
m[dT ]
(1 + p)dTm(dT ) → ∞
by Karamata’s theorem. Therefore, we have TTσ2/Tι2 → ∞ in all cases, from which we
have the stated result.
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