The role of Skorokhod space in the development of the econometric analysis of time series
This paper discusses the fundamental role played by the Skorokhod space, through its underpinning of functional central limit theory, in the development of the paradigm of unit roots and co-integration. This paradigm has fundamentally affected the way economists approach economic time series as was...
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| Cite this: | The role of Skorokhod space in the development of the econometric analysis of time series / J.R. Mccrorie // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 82–94. — Бібліогр.: 113 назв.— англ. |
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| citation_txt | The role of Skorokhod space in the development of the econometric analysis of time series / J.R. Mccrorie // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 82–94. — Бібліогр.: 113 назв.— англ. |
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| description | This paper discusses the fundamental role played by the Skorokhod space, through its underpinning of functional central limit theory, in the development of the paradigm of unit roots and co-integration. This paradigm has fundamentally affected the way economists approach economic time series as was recognized by the award of the Nobel Memorial Prize in Economic Sciences to Robert F. Engle and Clive W.J. Granger in 2003. Here, we focus on how P.C.B. Phillips and others used the Skorokhod topology to establish a limiting distribution theory that underpinned and facilitated the development of methods of estimation and testing of single equations and systems of equations with possibly integrated regressors. This approach has spawned a large body of work that can be traced back to Skorokhod’s conception of fifty years ago. Much of this work is surprisingly confined to the econometrics literature.
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Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 82–94
UDC 519.21
J. RODERICK MCCRORIE
A ROLE OF THE SKOROKHOD SPACE IN THE DEVELOPMENT
OF THE ECONOMETRIC ANALYSIS OF TIME SERIES
This paper discusses the fundamental role played by the Skorokhod space, through its
underpinning of functional central limit theory, in the development of the paradigm
of unit roots and co-integration. This paradigm has fundamentally affected the way
economists approach economic time series as was recognized by the award of the No-
bel Memorial Prize in Economic Sciences to Robert F. Engle and Clive W.J. Granger
in 2003. Here, we focus on how P.C.B. Phillips and others used the Skorokhod topol-
ogy to establish a limiting distribution theory that underpinned and facilitated the
development of methods of estimation and testing of single equations and systems
of equations with possibly integrated regressors. This approach has spawned a large
body of work that can be traced back to Skorokhod’s conception of fifty years ago.
Much of this work is surprisingly confined to the econometrics literature.
1. Introduction
One important aspect of a time series is that it is but one realization of a multidi-
mensional random variable. From this point of view, an assumption of second-order sta-
tionarity is convenient because it facilitates the inference through laws of large numbers
and central limit theorems in a classical way. An early influence on models of economic
time series was rendered by the book by Grenander and Rosenblatt [35] which was based
on the stationarity about deterministic trend functions, where the inference can be con-
ducted under what today would be called “Grenander conditions” (see, e.g., [43, p. 215]).
These include a requirement that a suitably normalized sample moment matrix of regres-
sors converge to a positive definite limit. Another aspect of the second-order stationarity
assumption is that it permits a Wold decomposition whereby a time series can be repre-
sented as the sum of a linearly regular part involving an infinite-order weighted average
of white noise and a part that is perfectly linearly deterministic [43, p. 137]. This offered
some justification for the then emerging Box–Jenkins methods [9], where autoregressive
integrated moving average (ARIMA) models selected on the basis of the data could be
viewed as approximations to the regular part in the Wold decomposition. These models
produced satisfactory representations of many observed economic time series, at least
for the purpose of prediction. But, as the models were selected purely on the basis of
the data, they lacked a theoretical justification, as if they emerged from a “black box”.
The essential problem faced by the econometrics profession in the late 1960s and 1970s
was that structural econometric models, embodying restrictions from economic theory,
were often outperformed by the black box models. For example, Cooper [14] compared
one-step-ahead forecasts from seven structural models of the U.S. economy against a
näıve forecast from an AR model and found forecasts from the latter were superior in
2000 AMS Mathematics Subject Classification. Primary 60F17, 60G50, 60J15, 60J55, 60J65, 62M10,
62P20.
Key words and phrases. Skorokhod space, functional central limit theorems, non-stationary time
series, unit roots and co-integration, Wiener functionals, econometrics.
82
SKOROKHOD SPACE IN TIME SERIES ECONOMETRICS 83
most cases. Naylor et al. [69] even advocated Box–Jenkins models as an alternative
to econometric models. There was also the problem of the proliferation of data-based
models for a given time series that were incompatible with one another from the point
of view of economic theory. In the background, there was a debate about whether or not
the observed aggregate time series, which were manifestly non-stationary, should have
random walk or other trend components removed prior to the estimation given the loss
of a “long run” information that such transformations imply. The introduction of the
concepts of unit roots and co-integration in the context of non-stationary time series
[37,30,38,39,44] helped to resolve some of these issues. The theoretical underpinning of
this work was provided in papers by P.C.B. Phillips [76,75,83] who applied functional
central limit theory based on the Skorokhod space in a way that has firmly established
it as a part of the econometrician’s toolkit. The result has been a distinct literature that
traces back to Skorokhod’s conception of fifty years ago. The purpose of this paper is to
summarize its main contributions for the wider mathematicians’ community.
In Section 2, we briefly discuss the concepts and antecedents of unit roots and co-
integration, and error correction models which Engle and Granger [30] associated with
co-integration. In Section 3, we show how Phillips used the Skorokhod topology to un-
derpin the time series regression with a unit root and how he provided an (asymptotic)
explanation of influential simulation results on spurious regression by Granger and New-
bold [40]. Regression with co-integrated variables is then briefly contrasted with spurious
regression. In Section 4, we discuss the recent work, focusing on functional central limit
theory appropriate for non-linear transformations of non-stationary time series. To our
knowledge, this is the first article to describe this literature as a whole. For reasons of
space, we confine ourselves almost exclusively to a discussion of autoregressive unit roots.
This remains the most important application in econometrics (e.g., [41]), but other work
is relevant, especially under fractional integration [17,19,6,52,61,62,97–99,60,34].
2. Unit Roots and Co-Integration
Intuitively, co-integration tries to bring sense to the notion that two or more time series
when looked at individually appear to be wandering erratically and yet are systematically
wandering together. Casual inspection of many raw economic time series would suggest
they exhibit a trend component, explained possibly by the time series being stationary
about a deterministic trend or possibly because they are non-stationary, say as a result
of their containing of a random walk component. The following idea is difficult to make
precise ([18,57,94,10,32,33,69]), but we could imagine, analogously to the Box–Jenkins
methodology, of “differencing” a time series to induce “stationarity”. More formally, we
could say that if, asymptotically, a time series comprises a sequence of random variables
with the first and second moments which tend to fixed stationary values and whose
covariances tend to stationary values depending only on how far apart the elements are,
then the time series is integrated to order zero, or I(0). A time series could then be said
to be integrated to order d, or I(d), if it must be differenced d times before an I(0) series
results. This would mean, then, that the random walk process
(1) xt = xt−1 + εt, x0 = 0, εt ∼ IID(0, σ2)
is such that it is I(1), because the series of its differences, ∇xt = xt − xt−1, is I(0). See
[45,46] for a non-technical discussion and the comparison of the properties of I(1) and
I(0) series. Much of the early literature was concerned with formulating and interpreting
the tests to discriminate between stationary processes about a deterministic trend and
non-stationary processes containing an autoregressive unit root such as (1) above [28].
This was motivated as a means of trying to avoid applying a wrong filter (either one
of detrending or differencing) to the data which, it was argued, would result in the
84 J. RODERICK MCCRORIE
invalid inference. It is, however, becoming more common today to see both types of the
model as simply alternative representations of the components of an underlying, trending
stochastic process [81,63].
The idea of co-integration, when applied in the context of I(1) and I(0) series, seeks
to relate raw series that are I(1) through a linear combination of them which is I(0).
We could imagine series that individually exhibit “persistence” but which together are
attracted to each other towards a statistical equilibrium where linear combinations of
the variables are “stationary”. Co-integration does not explain why the variables are
trending in the first place, but what it does do is to provide a basis of examining the
relationships between I(1) variables or variables integrated to higher order. Often we
can place an economic interpretation on the co-integrating relation. Engle and Granger
[30,38] showed that every co-integrated system can be written as a model that has what
is called an error correction representation, comprising a “balanced” equation in I(0)
variables involving the differences of the variables related to their lagged values but
also to a term that reflects the disequilibrium between the I(1) variables which, under
co-integration, is I(0). Co-integration is a property that can be applied, in principle,
to time series with possibly higher orders of integration, and various representations of
co-integrated systems are useful [86,108,79,42,111].
3. Regression with Non-Stationary Time Series
The essential statistical distributional problem in unit root regression, which Chan
and Wei [11] and Phillips [77] related to the convergence in the Skorokhod topology, can
be exposited in the context of the simple AR(1) model
(2) xt = ρxt−1 + εt (t = 1, . . . , n),
where ρ is the unknown parameter, and εt ∼ NID(0, 1). Consider the joint density
(3) fρ(x) =
(√
2π
)−n
exp
{
−1
2
[
(1 + ρ)2T2 − 2ρT1 + x2
n
]}
,
where
(4) T1 =
n∑
t=1
xtxt−1, T2 =
n∑
t=1
x2
t−1.
The maximum-likelihood estimator (MLE) ρ̂n of ρ is given by
(5) ρ̂n = T1/T2.
If |ρ| < 1, the process is stationary, and [59]
(6)
√
n(ρ̂n − ρ) d−→ N(0, 1− ρ2),
where d−→ denotes the convergence in distribution. When ρ = 1, (6) is not useful as a
basis of testing for a unit root against stationarity; however, it can be shown using a
standard argument (e.g., [42]) that
(7) n−1
n∑
t=1
xt−1εt
d−→ 1
2
(χ2
1 − 1),
which with
(8)
n∑
t=1
x2
t−1 = O(n2)
SKOROKHOD SPACE IN TIME SERIES ECONOMETRICS 85
suggests writing
(9) n(ρ̂n − 1) = n−1
n∑
t=1
xt−1εt
/
n−2
n∑
t=1
x2
t−1,
on substituting (2) into (3). The problem that presents itself is that there is no law of
large numbers with this normalization such that the denominator converges to a con-
stant. What (9) does convey is that if the statistics has a well-defined limit distribution,
which Theorem 1A below confirms, the ML estimator converges, in this case, to this
distribution at a faster rate than it does to a normal distribution in the stationary case.
The asymptotic sampling properties of the MLE in (5) can be described in the following
way, which is more suggestive of a uniform approach.
Theorem 1A. ([59, 112, 3, 24]) Let x0 = 0. Then
(10)
√
In(ρ)(ρ̂n − ρ) d−→
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
N,
ρ
W 2(1)− 1
23/2
∫ 1
0
W 2(r)dr
,
C,
|ρ| < 1,
|ρ| = 1,
|ρ| > 1,
where N is a centered Gaussian random variable with variance 1, C is a standard Cauchy
random variable, W = (W (r), r ≥ 0) is a standard Wiener process, and In(ρ) is the
(expected) Fisher information contained in x1, . . . , xn about the parameter ρ, given by
Eρ(T2). As n→∞,
(11) Eρ(T2) ∼
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
n
1− ρ2
,
n2
2
,
ρ2n
(ρ2 − 1)2
,
|ρ| < 1,
|ρ| = 1,
|ρ| > 1.
If |ρ| ≤ 1, (10) remains valid under the weaker assumptions that x0 is an arbitrary
constant or a random variable with a finite second moment not depending on the sequence
ε = (εt, t ≥ 1), and ε a arbitrary sequence of centered and normalized independent,
identically distributed (i.i.d.) random variables. If |ρ| > 1, the limit distribution depends
on the initial value and, in general, on the particular distribution of each εt, even if ε forms
a sequence of i.i.d. random variables [56]. Different results hold if ε forms a sequence
of i.i.d. random variables with a stable distribution or in the domain of attraction of a
stable law [12,78,67].
If we normalize by the observed rather than the expected Fisher information, the three
limit distributions are reduced to two:
Theorem 1B. ([59, 3, 24]) Let x0 = 0. Then
(12)
√
In(ρ)(ρ̂n − ρ) d−→
⎧⎪⎪⎨⎪⎪⎩
N, |ρ| < 1,
ρ
W 2(1)− 1
2
√∫ 1
0 W
2(r)dr
, |ρ| = 1.
The essential contribution of Chan and Wei [11] and Phillips [77] in the journal paper
immediately following [30] was to set the convergence results in the case |ρ| = 1 to the
86 J. RODERICK MCCRORIE
context of functional central limit theory. Phillips’s approach, which was more obviously
designed as an extension of classical asymptotic theory, has led to the now prevailing two-
stage approach towards deriving the limit distributions pertaining to unit root statistics.
This involves using, in the first stage, the functional central limit theorem (FCLT) to
derive a limit distribution for the (normalized) integrated process itself and then, in the
second stage, to derive the limit distribution of the sample statistics based explicitly on
its construction as a functional of the integrated process. It is in the first stage that the
ideas of Skorokhod are applied because it is here that the classical theory, pertaining
to the convergence in a distribution of random variables, needs to be extended to a
framework involving the weak convergence of random functions. This is needed because
the persistence in random shocks requires us to consider the whole trajectory of the
process and not just its endpoint.
Consider a real-valued stochastic process (xt, t ∈ N ). We should like to consider the
weak convergence in a function space that is complete, to avoid the probability mass
escaping from the space as n → ∞, and separable, because then all the Borel sets of
the space are measurable, and the weak convergence on product spaces is equivalent
to the weak convergence on the component spaces. It is possible to use the space of
continuous functions on the unit interval, C[0, 1], endowed with the uniform metric but,
because most of the functions of interest are not continuous, this involves an awkward
construction that requires extra terms that are defined to make the relevant partial
sum process continuous and are shown to be asymptotically negligible [17,15]. Phillips
[77,76,84] based his work, instead, on the space D[0, 1] of cadlag (continue à droit, limites
à gauche) functions, which contains jumps but not isolated points, but is sufficient for
the problem in hand. This space is not separable under the uniform metric, meaning in
practice there are “too many” sets, on which a probability space can be defined. The
problem is therefore precisely the one Skorokhod [105] resolved fifty years ago: the space
can be rendered separable by what is now popularly called the Skorokhod metric (his J1
metric) defined in such a way that allows functions to be compared “sideways,” as well
as vertically: for x, y ∈ D[0, 1],
(13) dS = inf
λ∈Λ
{
ε > 0 : sup
t
|λ(t)− t| ≤ ε, sup
t
|x(t) − y(λ(t))| ≤ ε
}
,
where Λ denotes the set of functions λ : [0, 1]→ [0, 1]. This is the crucial element in terms
of what is needed to define the Borel σ-algebra; however, the metric space (D[0, 1], ds) is
not complete, and Phillips used a modification of (13) introduced by Billingsley (1968)
that preserves the same topology:
(14) dB = inf
λ∈Λ′
{
ε > 0 : ‖λ‖ ≤ ε, sup
t
|x(t) − y(λ(t))| ≤ ε
}
,
where
(15) ‖λ‖ = sup
t�=s
∣∣∣∣log
λ(t) − λ(s)
t− s
∣∣∣∣ ,
and Λ′ denotes the set of all increasing functions such that ‖λ‖ ≤ ∞.
The two-stage approach to establishing the limiting distributions of sample statis-
tics considers, in the first stage, a real-valued stochastic process (xt, t ∈ N ) such that
n−1/2x[rn] ⇒ σW (r), σ > 0, where [rn], r ∈ [0, 1], denotes the integer part of rn, “⇒”
denotes the weak convergence in D[0, 1], as described above, and W represents the stan-
dard Brownian motion on [0, 1]. The approach is set up such that one of the variety of
such FCLT’s could be employed [26,31,65,66,48–50,113]. With econometric time series
in mind, Phillips chose to use the conditions by Herrndorff [49] which allow for certain
SKOROKHOD SPACE IN TIME SERIES ECONOMETRICS 87
types of weakly dependent and heterogeneously distributed disturbances. Later, Phillips
and Solo [91] advocated an approach based on ε following a linear process. Today, we
might instead use the conditions based on results by Doukhan and Louhichi [21,4], Beare
[5], de Jong and Davidson [22], or Davidson [16].
In the second stage, an argument based on the continuous mapping theorem [95] is
used to show that, for continuous real-valued functions T on R,
(16) n−1
n∑
t=1
T (n−1/2xt)
d−→
∫ 1
0
T (σW (r))dr.
For the denominator in (8),
(17)
n−2
n∑
t=1
x2
t−1 = n−1
n∑
t=1
(
n−1/2
t−1∑
s=1
εs
)2
= n−1
n∑
t=1
⎛⎝n−1/2
[rn]∑
s=1
εs
⎞⎠2 (
t− 1
n
≤ r ≤ t
n
)
d−→
∫ 1
0
W (r)2dr,
using (16). For the numerator,
(18)
n−1
n∑
t=1
xt−1εt =
n∑
t=1
(
n−1/2
t−1∑
s=1
εs
)
n−1/2εt
d−→ σ2
∫ 1
0
W (r)dW (r).
The integral in (18) is an Itô integral equal to 1
2 (W 2(1)− 1), which agrees with (7). The
joint convergence of (17) and (18) gives (10) up to a normalization.
Expressions involving functionals of Brownian motion such as (10) and (12) can be
derived using the above two-stage approach for many relevant time series models, the
most important being the unit root model that includes a constant, and a constant and
a time trend (see [15, 42 (Ch. 17)]). With the densities of (10) and (12), we could use,
in principle, a transformation theorem to generate the densities pertaining to the other
models [1]. Deriving and computing these densities, however, is not a tractable problem
either analytically or numerically. Indeed, except in the simple model above [2,64], we
do not have analytic expressions for the asymptotic moments, densities, or distributions.
This means, in practice, critical values for hypothesis tests are constructed by simulating
the densities from the Wiener functionals, and, for general ARMA models, additional
methods need to be employed, often using a decomposition by Beveridge and Nelson
[7] under linear process assumptions [91]. This gives what are called Phillips–Perron
tests [90] and augmented Dickey–Fuller tests (see [42, Ch. 17)], [107]). More recently,
in an attempt to improve the asymptotic nature of expressions such as (10) and (12),
computationally intensive methods based on the bootstrap [73,72] have been designed.
Perhaps the most celebrated application of functional limit theory based on the weak
convergence in the Skorokhod topology to econometrics was the explanation offered by
Phillips [76] of simulation results by Granger and Newbold [40]. Specifically, suppose we
estimate the parameters α and β by OLS in the model
(19) yt = α+ βxt + εt (t = 1, . . . , n)
88 J. RODERICK MCCRORIE
where both xt and yt are generated by independent random walks. Granger and Newbold
found in experiments on a sample sizes of 50 and 100 (constrained by the then limits
of computer technology) that the coefficient of determination, R2, was very high (in the
standard setting, indicating that most of the variation in y is explained by the variation
in x), the Durbin–Watson (DW) statistics was extremely low (in the standard setting,
suggesting among other possibilities the residuals were serially correlated), and tests of
the significance of β were seriously biased towards the rejection of no relationship between
y and x. Phillips showed under the FCLT that
(20) β̂
d−→
(
σv
σw
)
SXY
SXX
, n−1/2tβ
d−→ SXY√
SXXSY Y − S2
XY
,
(21) R2 d−→ S2
XY
SXXSY Y
, DW
d−→ 0,
where
SXY =
∫ 1
0
V (t)W (t)dt−
∫ 1
0
V (t)dt
∫ 1
0
W (t)dt,(22)
SXX =
∫ 1
0
W 2(t)dt−
(∫ 1
0
W (t)dt
)2
,(23)
SY Y =
∫ 1
0
V 2(t)dt−
(∫ 1
0
V (t)dt
)2
,(24)
σw, σv > 0, and the independent Wiener processes W (t) and V (t) arise from the com-
ponent-wise application of the FCLT. The usual t-statistics does not therefore possess a
limit distribution but diverges as the sample size increases; the bias in this test towards
the rejection of no relationship based on a given nominal critical value will increase with
n; the DW d-statistics converges in probability to zero; and R2 has a non-degenerated
limit distribution as n→∞.
A regression model like (19) that appears to find relations that do not really exist is
called a spurious regression. The problem here has two elements which co-integration
seeks to resolve. Firstly, under the null hypothesis, the model says that yt is equal to a
constant plus a disturbance term, and so the null is false, because yt is truly generated
as a random walk. It is not unusual for tests to reject false null hypotheses even when
the alternative is false [20]. Secondly, the standard asymptotic results do not hold when
at least one of the regressors is I(1) [76,28]. Our focus is principally on the development
of the FCLT for integrated processes, and so we will mention just two other papers here.
Sims, Stock and Watson [106] showed in a vector autoregression containing some unit
roots and possibly time trends that it is not the case the t-statistics on every parameter
involving an I(1) variable follows a non-standard distribution asymptotically: parameters
that can be written as coefficients on mean-zero, non-integrated regressors are root n
consistent and asymptotically normal. Elliot, Rothenberg and Stock [29] proposed a
family of point-optimal tests that dominates the basic unit root tests when a time series
has an unknown mean or a linear trend, although Robinson [98] showed if the unit root
is nested in a model against a different class of alternatives that include fractionally
differenced processes, a standard efficiency theory based on the chi-square distribution
is possible (see [107] for a fuller discussion of hypothesis testing.)
We would expect that residuals from a spurious regression such as (19) would be I(1).
But if y and x were instead co-integrated variables, we would expect the residuals from
SKOROKHOD SPACE IN TIME SERIES ECONOMETRICS 89
(19) to be I(0). This means that the residuals can be used as the basis of testing for co-
integration: we subject the residuals to a unit root test under the null hypothesis that the
variables are not co-integrated. Limiting distribution theory based on Wiener functionals
can be constructed using the FCLT’s above for the time series regression with mixtures
of integrated processes with an unknown degree of co-integration (see [54],[80],[13],[42]).
One problem with the asymptotic theory of non-linear ML estimation in integrated and
co-integrated systems is that the classical proofs of consistency based on the uniform
convergence of the objective function over the parameter space generally are not applied
owing to the objective function diverging at different rates in the parameter space. Even
if the consistency can be established, it can be difficult to deduce the limit distribution
of the ML estimator from a conventional Taylor series expansion of the score vector.
Saikkonen [101] resolved these difficulties by constructing a statistical theory, where,
in addition to the consistency, a result on the order of consistency of the estimator
of the long-run parameter of the model (pertaining to the co-integrating relation) is
available, and the standardized sample information matrix satisfies a suitable stochastic
equicontinuity condition. This program of work was completed with two recent papers
[102,103] allowing the consistency of the reduced form parameters to be established
without assuming the identifiability of the structural parameters and establishing a limit
distribution theory without assuming the identifiability of the parameters in the short-
run dynamics (i.e., differenced terms and lags of differenced terms not involving the
co-integrating relation). These conditions are especially suitable from the point of view
of econometric time series.
4. Other Recent Work
The application of the FCLT to (16) is related to the context of linear regression involv-
ing integrated time series. Recently, the focus has switched to non-linear transformations
of integrated time series, and an asymptotic theory of inference that is applicable to the
non-linear regression has been developed by Park and Phillips [74,75], de Jong [22], de
Jong and Wang [23], Pötscher [95], and Berkes and Horváth [6]. Alongside this, related
analytic tools on the local time density and hazard functions of the limiting Brown-
ian motion of a standardized integrated process have been developed by Phillips [82,83].
Park and Phillips [74] showed that (16) holds for a class of functions they called “regular”
that is wider than the class of continuous functions, although their class does not contain
every bounded and measurable function and does not include locally bounded functions
such as T (x) = log |x| or T (x) = |x|α(−1 < α < 0). de Jong [22] established (16) under
local integrability for a different class, covering the two functions above and allowing T
to have poles but not encompassing all the regular functions in [74]. The problem was
elegantly resolved by Pötscher [95] who established that (16) holds under the condition
that the process satisfies the FCLT (as in the “first stage” described above) and then
only under the additional condition that T is locally integrable. Since the integral in
(16) exists almost surely (a.s) and is finite a.s. if and only if T is locally integrable [55,
pp. 216–217], the latter condition is minimal. The integral in (16) can be equivalently
expressed in terms of the local time:
(25)
∫ 1
0
T (σW (r))dr =
∫ ∞
−∞
T (σs)L(1, s)ds, a.s.,
where L(t, s) is the Brownian local time [96, Ch. 6] which, intuitively, is a spatial density
that records the relative sojourn time of the process W (t) at the spatial point s over the
time interval [0, t]. Equation (25) is known as the occupation times formula [96, p. 224],
and it opens up the possibility of using the sojourn times and spatial densities as the
basis of a theory of non-parametric co-integrating regression [110].
90 J. RODERICK MCCRORIE
Suppose (16) holds for a function H , and the function T satisfies T (λx) = ν(λ)H(x)
for all λ > 0 and all x ∈ R. Then (16) applied to H can be rewritten as
(26) (nν(n−1/2))−1
n∑
t=1
T (xt)
d−→
∫ 1
0
H(σW (r))dr.
Expression (26) can still be established if the function T satisfies the equation approx-
imately, in the sense that ν(λ)−1T (λx) L1−→ H(x) as λ → ∞, thereby providing a basis
for convergence results for non-linear functions of the unnormalized integrated processes.
This approach is described by Park and Phillips [74] for their “regular” functions and by
de Jong and Whang [23] under the conditions in [22], but the result holds more gener-
ally under just Pötscher’s basic condition that H is locally integrable. This can then be
applied to the context of the non-linear regression with integrated time series [75].
Another topic of recent interest, in the light of the recent observed behaviour in com-
modity prices and “financial exuberance” in stock markets, is works on mildly explosive
processes [87–89,58,36]. This has involved the “moderate deviations from unity” model
which is specified as
xt = ρnxt−1 + εt, (t = 1, . . . , n)(27)
ρn = 1 +
c
nα
, α ∈ (0, 1), c ∈ R,(28)
initialized at some x0 = op(nα/2) independent of σ(ε1, . . . , εn), where εt is either a se-
quence of i.i.d. (0, σ2) random variables with finite νth moment (ν > 2α−1) [87] or a
sequence of weakly dependent random variables [88]. Strictly speaking, since the au-
toregressive parameter ρn is a sequence of the sample size n, (28) is a triangular array
(xnt, 1 ≤ t ≤ n, n ∈ N ), and we adopt the notation xt and the notation x0, for con-
venience. The idea of model (27) and (28) is “to smooth the passage through unity”
relative to (2) such that the roots belong to larger (more moderate) deviations from
unity compared with conventional local-to-unity roots [77,11]. The boundary value as
α → 1 includes the conventional local-to-unity case, and the boundary value as α → 0
includes the stationary or explosive AR(1) process depending on the value of c. Phillips
and Magdalinos [87,88] combined a functional law to a diffusion on D[0,∞) with a cen-
tral limit law to a Gaussian random variable to establish the limit distribution of the
centered and normalized serial correlation coefficient
(29) ρ̂n − ρ =
(
n∑
t=1
xt−1
)2 n∑
t=1
xt−1εt.
For c < 0, they established an n(1+α)/2 rate of convergence to the asymptotic normality,
bridging the
√
n and n convergence rates for the stationary and conventional local-to-
unity cases. For c > 0, the limit distribution is a Cauchy one and is invariant to both
the distribution and the dependence structure of innovations. The rate of convergence is
nαρn
n, bridging the n and (1 + c)n convergence rates for the conventional local-to-unity
and explosive cases. Paper [88] contains possibly the first general invariance principle in
statistics that is applicable to explosive processes.
This current program of work seems to offer the immediate challenge of either design-
ing a model offering a uniform limit theory for autoregression, or at least constructing
an optimality theory that is applied uniformly across parameter values. Shiryaev and
Spokoiny [104] have already shown that the two distributions in (12) can be reduced to
one—a centered Gaussian variable with variance one—by considering a sequential max-
imum likelihood estimator. Phillips and Han [85], in an approach based on taking the
SKOROKHOD SPACE IN TIME SERIES ECONOMETRICS 91
first differences, offer the estimates that have virtually no finite sample bias and are not
sensitive to initial conditions, and the Gaussian central limit theory is applied as the
autoregressive coefficient passes through unity with uniform root n convergence (see also
[36]). There are still, of course, difficulties that need to be overcome [51,92–94]. It should
also be borne in mind that, while we have focused on the econometric literature that has
been spawned by the application of the FCLTs based on the Skorokhod topology, this
involves an asymptotic approach, and the true objective remains characterizing the finite
sample properties of an appropriate statistics.
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E-mail : mccrorie@st-andrews.ac.uk
|
| id | nasplib_isofts_kiev_ua-123456789-4539 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-26T02:44:54Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mccrorie, J.R. 2009-11-25T11:05:39Z 2009-11-25T11:05:39Z 2008 The role of Skorokhod space in the development of the econometric analysis of time series / J.R. Mccrorie // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 82–94. — Бібліогр.: 113 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4539 519.21 This paper discusses the fundamental role played by the Skorokhod space, through its underpinning of functional central limit theory, in the development of the paradigm of unit roots and co-integration. This paradigm has fundamentally affected the way economists approach economic time series as was recognized by the award of the Nobel Memorial Prize in Economic Sciences to Robert F. Engle and Clive W.J. Granger in 2003. Here, we focus on how P.C.B. Phillips and others used the Skorokhod topology to establish a limiting distribution theory that underpinned and facilitated the development of methods of estimation and testing of single equations and systems of equations with possibly integrated regressors. This approach has spawned a large body of work that can be traced back to Skorokhod’s conception of fifty years ago. Much of this work is surprisingly confined to the econometrics literature. en Інститут математики НАН України The role of Skorokhod space in the development of the econometric analysis of time series Article published earlier |
| spellingShingle | The role of Skorokhod space in the development of the econometric analysis of time series Mccrorie, J.R. |
| title | The role of Skorokhod space in the development of the econometric analysis of time series |
| title_full | The role of Skorokhod space in the development of the econometric analysis of time series |
| title_fullStr | The role of Skorokhod space in the development of the econometric analysis of time series |
| title_full_unstemmed | The role of Skorokhod space in the development of the econometric analysis of time series |
| title_short | The role of Skorokhod space in the development of the econometric analysis of time series |
| title_sort | role of skorokhod space in the development of the econometric analysis of time series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4539 |
| work_keys_str_mv | AT mccroriejr theroleofskorokhodspaceinthedevelopmentoftheeconometricanalysisoftimeseries AT mccroriejr roleofskorokhodspaceinthedevelopmentoftheeconometricanalysisoftimeseries |