1. INTRODUCTION
In this paper we consider likelihood based estimation and testing
within a wide class of possibly nonstationary models, including but not
limited to the seasonal fractionally integrated autoregressive moving
average (ARMA) model. In such models, estimators and test statistics
are often found to have nonstandard distributional properties. In
contrast, we show that by adapting time domain procedures and embedding
the models of interest in a general I(d) framework,
instead of the autoregressive alternatives typically considered in the
literature, estimators and test statistics regain the standard
distributions and optimality properties well known from simpler
models.
Several versions of the Wald, likelihood ratio (LR), and score or
Lagrange multiplier (LM) testing procedures have appeared in the
literature on nonstationary models, e.g., when conducting Dickey and
Fuller (1979) type tests for a unit root
(I(1) against I(0)) or testing stationarity
(I(0) against I(1)). For a comprehensive recent
survey, see Phillips and Xiao (1998).
However, these tests have nonstandard limiting distributions that have
to be simulated on a case-by-case basis. Some advances have been made
recently toward achieving efficient tests. Locally optimal and point
optimal tests have been derived for the stationarity hypothesis (e.g.,
Saikkonen and Luukkonen, 1993a, 1993b) and for the unit root hypothesis (e.g.,
Elliott, Rothenberg, and Stock, 1996).
However, these tests still have nonstandard distributions, and no
uniformity results apply.
What is needed is a class of processes that is more general than the
unit root I(1) models and admits the testing of smooth
hypotheses in the sense that the properties of the process do not
differ substantially if the null hypothesis is changed slightly. One
such class is that of fractionally integrated processes. Thus, a
process is I(d) (fractionally integrated of order
d) if its dth difference is I(0); i.e.,
yt ∈ I(d) if
where
denotes the indicator function and et
∈ I(0). A process is I(0) if it is covariance
stationary and its spectrum is bounded and bounded away from zero at
any frequency. Testing H0 : d = 1 in (1)
may be seen as an alternative to unit root testing. We show that in a
fractional integration framework much more desirable properties can be
obtained compared with autoregressive (and possibly seasonal and
fractional) unit root models where test statistics have nonstandard
distributions (see, e.g., Phillips, 1987;
Hylleberg, Engle, Granger, and Yoo, 1990;
Sowell, 1990).
Notable exceptions to the nonstandard tests are Robinson (1994) and Tanaka (1999),
extending earlier work by Robinson (1991) and
Agiakloglou and Newbold (1994), and it is the
Robinson (1994) model that we consider
further in this paper. Robinson (1994)
derives the LM test statistic (of (16) and (23), which follow) in the
frequency domain, claiming that it is more suitable for the analysis,
and shows that the LM test statistic is asymptotically chi-squared
distributed and locally most powerful under Gaussianity. In a
simulation study it is found that when the data generating process
(DGP) is of the fractional type the finite sample performance of the
new test is better than that of existing tests, the opposite being the
case when the DGP is of the autoregressive type. Tanaka (1999) considers the fractional unit root model in
(1) and shows the existence of a local time domain maximum likelihood
estimator (MLE) and derives the LM and Wald tests. Tanaka (1999) shows that the estimator is asymptotically
normal and, under invariance conditions, that the tests are locally
most powerful and indeed asymptotically uniformly most powerful against
one-sided local alternatives. Simulations show that in finite samples
the time domain tests are superior to the frequency domain LM test of
Robinson (1994) with respect to both size and
power. The estimator is also shown to be quite close to its asymptotic
distribution, except in the presence of errors with strong positive
serial correlation.
The main contributions of the present paper when compared with the
previous work of Robinson (1994) are
summarized in the following five points. (i) All the results are
obtained in the time domain, which is most frequently employed by
practitioners, whereas Robinson (1994) favors
a frequency domain approach. The derivation of results and statement of
assumptions in the time domain require different methods than in the
frequency domain. Another reason to consider the time domain is that in
some cases the resulting estimators and tests are more easily applied
than their frequency domain counterparts. (ii) It is of interest to
examine the estimation of the model by maximum likelihood because the
estimator is expected to have good properties. Indeed, it is shown that
standard asymptotics and efficiency apply, which is a great advantage
in applied work. (iii) Whereas Robinson (1994) only considers the LM test, we also consider
the Wald and LR tests and show that standard asymptotics apply to all
the test statistics. (iv) For the submodels with a scalar parameter and
independent and identically distributed (i.i.d.) Gaussian errors, the
LM, LR, and Wald tests are shown to be uniformly most powerful (against
local alternatives) among all invariant and unbiased tests. (v) In a
simulation study based on the well-known fractional unit root model it
is shown that the LR test outperforms the LM and Wald tests with
respect to both size and power.
Contrary to the present paper, Tanaka (1999) considers only a special case of the full
model in Robinson (1994), namely, the
fractional unit root model, and conducts an analysis similar to ours.
We consider the full model.
The paper proceeds as follows. In Section 2 we set up the model and
discuss important special cases. In Section 3 we consider inference
with martingale difference errors and derive the properties of the
estimator and tests, whereas in Section 4 we allow serially correlated
errors. Section 5 presents the results of our Monte Carlo experiments,
and Section 6 concludes. All proofs are collected in the Appendix.
2. THE MODEL
Suppose we observe the real-valued stochastic process
{yt, t = 1,2,…,
n} generated by the linear model
where {xt} is a k × 1
purely deterministic component and {ut} is
an unobserved error component. Two leading cases for the deterministic
terms are xt = 1 and
xt = (1, t)′, which yield
the models yt = β0 +
ut and yt =
β0 + β1 t +
ut, respectively, but other terms such as
seasonal dummies are also allowed for; cf. Assumption 2, which follows.
The unobserved error process {ut} is
assumed to have the generating mechanism
Here, {et} is a stationary and
invertible process with only weakly dependent errors (i.e., no long
memory or nonstationarity) and φ(z, θ) is a function
of the complex variate z and the p × 1
parameter vector
.
The chosen parametrization is such that θ = 0 is the true value,
without loss of generality, and this belongs to the interior of
Θ.
The model is further required to satisfy the following assumptions.
Assumption 1. The function φ(z, θ) is such that (i)
φ(0, θ) = 1 and φ(z, θ) = φ(z) if
and only if θ = 0, where φ(z) = φ(z,
θ)|θ=0. (ii) φ(z, θ) is
twice continuously differentiable in θ in an open convex set
Θ* containing Θ and
where
is the coefficient on zj in the expansion
of ζ(z) = (∂/∂θ)ln φ(z,
θ)|θ=0 in powers of z. (iii) The
function λ(z, θ) = (φ(z,
θ)/φ(z)) × (∂/∂θ)ln
φ(z, θ) is continuous in θ at θ = 0 for
almost all z such that |z| = 1, and,
letting λj(θ) be the coefficient on
zj in the expansion of λ(z,
θ) in powers of z, in a neighborhood N of size
O(n−1/2) of θ = 0,
.
Assumption 2. The k × 1 vector of regressors
xt is nonstochastic and such that
is positive definite for n sufficiently large, where
.
Assumption 3. The innovation sequence in (3),
{et, t =
0,±1,±2,…}, satisfies
for all t ≥ 1, where
is an increasing sequence of σ-algebrae, and
{et2, t =
0,±1,±2,…} is uniformly integrable.
Some comments on the assumptions follow. Assumption 1 is a time
domain equivalent of the assumptions made by Robinson (1994) on the parametric model, where (i) ensures
identifiability of θ and (ii) and (iii) are smoothness conditions
on the parametric model. The unit root process nested in an
autoregressive model is (3) with φ(z, θ) = 1 −
(1 + θ)z, but in this case the right-hand-side inequality
of (4) is not satisfied. Differentiability to any order is easily
verified for all of the examples that follow.
Assumption 2 is a very mild multicollinearity condition on the
regressors. It does not even require the smallest eigenvalue of
Dn to tend to infinity as n
→ ∞, which is usually required in linear regression models
to get consistent estimates of β.
Finally, Assumption 3 ensures that the innovations are such that
are both uniformly integrable martingale difference sequences. This is
more general than i.i.d. and in practice not much more restrictive than
uncorrelatedness. An implication of this assumption is that
in probability (e.g., Hall and Heyde, 1980,
Theorem 2.22), which we will use later. Assumption 3 can be replaced by
any other assumption that gives rise to a weak law of large numbers
(LLN) for {et2} and a central
limit theorem (CLT) in Theorem 3.1, which follows. Thus, we could
presumably relax Assumption 3 to accommodate autoregressive conditional
heteroskedasticity/generalized autoregressive conditional
heteroskedasticity (ARCH/GARCH) type errors (as suggested by an
anonymous referee), which are often found in financial data, where our
methods are especially applicable due to the large amount of data
available (see also Ling and Li, 1997).
A very general model considered by Robinson (1994), and satisfying the preceding assumptions,
is
where for each j, θi(j) =
θl for some l, and for each
l there is at least one j such that
θi(j) = θl;
i.e., there are up to h singularities in the spectral density
of ut and p ≤ h. That
is, we do not require that there is a θj for
each singularity. For example, the quarterly I(1) hypothesis
is given by either one of the functions φ(z, θ) = (1
− z4)1+θ, where we use the
same θ for each of the h = 3 spectral singularities, or
φ(z, θ) = (1 −
z)1+θ1(1 +
z)1+θ2(1 +
z2)1+θ3, where the
integration orders are allowed to be different at different frequencies
under the alternative.
The case considered by Tanaka (1999) is the
fractional unit root model defined by
In this model ζ(z) = ln(1 − z) and
ζj = −j−1 such
that Ψ = π2/6. The weak dependence, unit root,
and I(2) models nested in a fractional integration framework
correspond to (6) with d = 0, d = 1, and d =
2, respectively.
Another important special case of the general model (5) is the
cyclical I(d) or generalized fractional
autoregressive integrated moving average (ARIMA) model of Gray, Zhang,
and Woodward (1989), recently advocated by
Chung (1996), Bierens (2001), and Gil-Alana (2001). This model is generated by the function
where λ is the cyclic frequency of interest. Then d = 1
and θ = 0 corresponds to the cyclic/seasonal unit root at
frequency λ.
Finally, suppose the m-vector
xt is
I(dx) and fractionally
cointegrated and the cointegrating vector is known a priori from
economic theory such that we can treat ut
= α′xt as an observed time
series (when the cointegration vector is unknown and must be estimated,
the results in the present paper do not apply; see Nielsen, 2003). Then the hypothesis
H0 : θ = 0 in (6) with d =
dx corresponds to the null of no
fractional cointegration, and with d = 0 the hypothesis
H0 : θ = 0 corresponds to the null of
fractional cointegration with I(0) equilibrium errors. A
well-known example is the purchasing power parity. Let
xt consist of the time t domestic
log-price, foreign log-price, and the log exchange rate, respectively,
and suppose xt is fractionally integrated
of order d. Then the purchasing power parity predicts that
α = (−1,1,1)′ should be a cointegrating vector and that
the cointegration residuals should be I(0). Imposing α =
(−1,1,1)′ on the data, the last implication can be tested
as in (6) with d = 0.
The preceding examples illustrate the generality of our approach. To
see why standard asymptotics apply, we briefly discuss the data
generating mechanism (see also the discussion by Ling
and Li, 2001, pp. 739–741). When
{ut} is generated by truncation as in (3),
it depends only on the shocks starting at time t = 1 and not
on shocks starting in the infinite past as would otherwise be the case.
Under (3), there are two fundamentally different approaches to allow
for nonstationarity that lead to different asymptotic results. Ling and
Li (2001) consider the fractional unit root
model (6) assuming that d ∈ (−½,½),
the stationary region, and allowing unit roots in the autoregressive
polynomial a(z). Standard asymptotics is obtained for
the fractional difference parameter, but the estimates of the unit
roots have nonstandard Dickey–Fuller type distributions. On the
other hand, Robinson (1994) and Tanaka (1999) capture the unit root through the fractional
difference parameter d and assume that a(z)
is stationary. We follow this practice in the present paper. Because no
unit root must be estimated in a(z) we avoid the
nonsmooth behavior of the model near the unit roots, and this admits
standard asymptotics in our setting.
In the subsequent analysis we first consider the case where
{et} is a martingale difference sequence,
and then we treat the full model in which
{et} is allowed to follow an ARMA
process.
3. INFERENCE WITH MARTINGALE DIFFERENCE
ERRORS
The Gaussian log-likelihood function of (2) and (3) is
apart from constant terms. The asymptotic results derived in this
section impose only Assumption 3 on the error process. Gaussianity is
not necessary for most of our results and is used only to choose a
likelihood function and to show efficiency.
Because only θ is of interest we concentrate out the nuisance
parameter (β′, σ2). This does not influence
the results, and in fact
is asymptotically uncorrelated with
(see the formula for the information matrix (21), which follows). The
concentrated likelihood is
where
and capital letters denote the appropriate matrices of observations;
e.g., X is the n × k matrix with
xt′ as the tth row. Here,
β(θ) and σ2(θ) are functions of θ.
They define the estimator
of (β′, σ2). We shall also need
which is the estimator of (β′, σ2) under the
true value of θ. Note that β(θ) is just the ordinary
least squares estimator in a regression of φ(L,
θ)yt on φ(L,
θ)xt and σ2(θ) is
the usual maximum likelihood variance estimator for the residual
process φ(L, θ)(yt
− β(θ)′xt).
Note that the estimate of β need not be consistent in our model.
One such case occurs when xt = 1 in the
fractional unit root model (6) with d = 1. Then
is inconsistent, but this has no influence on inference based on
(see Robinson, 1994; also see the Appendix).
In fact, what we need in the proofs is the relation
which follows under Assumption 2 by definition of
.
3.1. Estimation
In this section we show the existence of a local MLE and derive the
limiting distribution theory following the approach of Sargan and
Bhargava (1983) and Tanaka (1999). In particular, we consider the conditional
sum of squared residuals objective function (9).
In the following we find it convenient to consider maximizing
where
.
Assume first that we are in a neighborhood of the true value, i.e., that
there exists a δ such that
(the existence of δ will be proved shortly). Then we can show the
following results.
THEOREM 3.1. Let Assumptions 1–3 be satisfied and let
g(θ) be given by (13). Then, under
,
where Z is a p-dimensional standard normal random
vector.
Next, we prove the existence of a local MLE
such that
following Sargan and Bhargava (1983) and
Tanaka (1999). Let ι be a p
× 1 direction vector, i.e., satisfying ∥ι∥ = 1, where
∥·∥ is the Euclidean norm, and let δ =
∥δ∥ι. Generalizing the scalar approach by Sargan and
Bhargava (1983) and Tanaka (1999), it suffices to show that
for any direction vector ι, ε > 0, and n ≥
n0 (n0 fixed) and for some
∥δ∥ > 0. Note that
is the directional derivative of g at
,
i.e., the rate of change of g at
in the direction ι.
Thus, for all direction vectors ι, moving some distance
∥δ∥ in the direction ι from the true value, the
directional derivative of g in the same direction ι should
be negative for sufficiently large n. In the one-dimensional
case ι = ±1 and (14) reduces to the corresponding conditions
of Sargan and Bhargava (1983) and Tanaka
(1999). It follows from Theorem 3.1 that
which can be made arbitrarily small by selecting ∥δ∥
large. Thus, (14) holds by appropriate choices of ∥δ∥ and
n0, and the existence of the local MLE
is ensured.
THEOREM 3.2. Under Assumptions 1–3, there exists a local
maximizer
of the concentrated likelihood (9) that satisfies, as n →
∞,
and under the additional assumption of Gaussianity of
{et},
is asymptotically efficient in the sense that its asymptotic variance
attains the Cramér–Rao lower bound.
This asymptotic normality result stands in sharp contrast, e.g., to
the nonstandard Dickey–Fuller distribution. In that case,
,
and thus
,
where W(t) is a standard Brownian motion and ⇒ is
weak convergence (see, e.g., Phillips, 1987;
Phillips and Xiao, 1998). Furthermore, if a
constant term is included in the Dickey–Fuller model the
distribution changes. This is not the case in our model, where the
limiting distribution is independent of the nuisance parameter
(β′, σ2).
The additional assumption of Gaussianity allows a strengthening of
the results. Thus,
is asymptotically the best estimator in the class of all
-consistent
and asymptotically normal estimators. This result also is in contrast
with those usually found in the theory of nonstationary time series.
The simple asymptotic distribution in Theorem 3.2 makes it easy to
construct p-dimensional confidence ellipsoids for θ or
conduct Wald-type tests of hypotheses on θ. This is examined in
detail in the next section.
3.2. Hypothesis Testing
Suppose we wish to test the hypothesis
where θ0 is set to zero because otherwise we would
get trivial asymptotic distributions under the null. Robinson (1994) considers the LM test in a frequency domain
framework. We now consider all the classical likelihood-based (Wald,
LR, LM) tests (see Engle, 1984) in the time
domain.
From Theorem 3.2, the Wald test statistic is
We denote by a tilde an estimator under the null hypothesis. The
(quasi) LR test statistic is given by
(see equation (9)). Finally, to derive the LM test statistic
where η = (β′, σ2,
θ′)′, we note that
whereas the other two partial derivatives vanish. Here,
is the jth sample autocorrelation of
.
The diagonal block of the Fisher information matrix corresponding to
θ is
so the Fisher information matrix in (19) evaluated at
is n times
which is invertible for n sufficiently large by (4) and
Assumption 2. The diagonal blocks corresponding to β and
σ2 follow using that
are martingale differences, respectively. In Tanaka (1999), ζj =
j−1 and Ψ = π2/6. We
allow for more general weights to the autocorrelations in
,
corresponding to the more flexible model represented by the function
φ(z, θ). The expression
is a truncated version of Ψ, which is asymptotically equivalent to
Ψ. Thus, the LM test statistic is
In the fractional unit root model (6) where
ζj = j−1 we have
Ψ100 = 1.5831, Ψ500 = 1.6294, and
Ψ = Ψ∞ = π2/6 =
1.6449.
We derive the distribution of the test statistics under the more
general assumption of local (Pitman) alternatives given by the
sequence
with δ a fixed p × 1 vector.
THEOREM 3.3. Let Assumptions 1–3 be satisfied and let
T denote the W, LR, or LM test statistics
given by (17), (18), and (22). Then, under (23), it holds that
as n → ∞. The three tests are consistent and
asymptotically equivalent; i.e., if T1 and
T2 are any two of the statistics then
T1 − T2 → 0 in
probability. Under the additional assumption of Gaussianity they are
locally most powerful.
Usually in nonstandard tests such as the Dickey–Fuller test,
the three test statistics are not equivalent. From the proof we note
that the equivalence of the tests depends crucially on the information
matrix equality, which holds asymptotically in our model but does not
hold when the unit root is nested in an autoregressive alternative.
Thus, we find that unusually simple asymptotic tests can be performed
in this model using the chi-squared distribution. Also, we can easily
calculate the asymptotic local power of the three test statistics,
which we state as a corollary.
COROLLARY 3.1. Under the conditions of Theorem 3.3 it holds that,
under
,
as n → ∞, where
χp,1−α2 is the 100(1
− α)% point of the χp2
distribution and Fδ′Ψδ is the
distribution function of the
χp2(δ′Ψδ)
distribution.
Using Corollary 3.1 we can compare the finite sample performance of
the tests with the approximation offered by asymptotic theory, and we
shall discuss this in Section 5.
Next, we show that even stronger results can be obtained in a
subclass of models.
3.3. Uniformly Most Powerful Tests
While the general theory discussed previously applies for
multidimensional θ, even stronger results are obtained in the
special case of scalar θ, e.g., (6) or (7), which we now consider
briefly. Following the reasoning in Elliott et al. (1996) and Tanaka (1999),
we derive the power envelope for the two-sided testing problems under
invariance and unbiasedness conditions and show that this two-sided
power envelope is equal to (24), i.e., that this power is achieved by
our tests. The unbiasedness condition is new because Elliott et al.
(1996) and Tanaka (1999) only consider one-sided tests and thus do not
need unbiasedness.
In particular, we assume that the errors are Gaussian and that the
model in (3) is characterized by a scalar parameter θ. This rules
out the general model in (5) but still applies to most of the models in
Section 2. The testing problem is invariant to any transformation of
the type y → ay + Xb
,
or in the parameter space,
Thus, we shall restrict attention to the family of tests that are
invariant to the group of transformations in (25) (see Lehmann, 1986, Chap. 6).
Assume that the DGP is given by (2) and (3), with true parameter
value θ0n =
for some fixed c. Now consider testing the hypothesis
H0 : θ = 0 against the sequence of local alternatives
for some fixed δ. This is a test of a simple null vs. a simple
alternative with nuisance parameter (β′, σ2).
Then we can apply invariance arguments to (β′,
σ2) and the Neyman–Pearson lemma tells us (e.g.,
Lehmann, 1986, p. 338) that the test that
rejects the null when
becomes large is most powerful invariant (MPI) with respect to the
group of transformations (25). As in the previous section,
are residuals under H0 and H1,
respectively. The next theorem derives the limiting distribution of
Mn under local alternatives.
THEOREM 3.4. Let Mn denote the MPI test
statistic (26), with
(c a fixed scalar) instead of θ0 = 0. Let
Assumptions 1 and 2 be satisfied and suppose the error process is
i.i.d. Gaussian. Then, under the sequence of local alternatives
(δ a fixed scalar), it holds that
as n → ∞, where Z is a standard normal
variable.
Thus, invariance arguments have reduced the testing problem to the
consideration of the statistic Mn, and the
power envelope of all invariant tests is the power of M(δ,
δ). Obviously, the results in Tanaka (1999) apply with little change to the corresponding
one-sided testing problem in our setup and this power envelope is
achieved by one-sided versions of our tests. However, because we
consider mainly the two-sided testing problem, we cannot hope to
achieve the same power envelope, and thus the following results differ
from those in Tanaka (1999), where only
one-sided hypotheses are considered.
To find a test statistic that applies against two-sided alternatives
we invoke the principle of unbiasedness (see Lehmann, 1986, Ch. 4) to construct an MPI unbiased test.
Unbiasedness requires that the power of the test never falls below the
nominal significance level for any point in the alternative. Because
for varying c the family of distributions
M(c, δ) is normal, it satisfies the requirement
that it be strictly totally positive of order three (STP3;
see Lehmann, 1986, p. 119), and hence the
power envelope of all invariant and unbiased tests of
H0 : θ = 0 against
is given by Π(δ) = 1 − P(C1,
α(δ) < M(δ, δ) < C2,
α(δ)) (Lehmann, 1986, p. 303),
where the constants are determined by
A test whose asymptotic power attains the power envelope for all
points δ is asymptotically uniformly most powerful invariant
unbiased. The following theorem shows that the power envelope of all
invariant and unbiased tests is given by (24), i.e., that this power is
achieved by our tests.
THEOREM 3.5. Let Assumptions 1 and 2 be satisfied and suppose the
error process is i.i.d. Gaussian. Then the asymptotic Gaussian power
envelope of all invariant (with respect to (25)) and unbiased tests of
H0 : θ = 0 against
(δ a fixed scalar) is given by (24). Thus, the W,
LR, and LM tests are uniformly most powerful (against local
alternatives) among all invariant and unbiased tests.
This result is in stark contrast to the results of Saikkonen and
Luukkonen (1993a, 1993b) and Elliott et al. (1996), among others, whose tests are only point
optimal invariant, i.e., tests that have maximal power against a single
prespecified (local) point in the alternative. Our criterion is against
all possible (local) alternatives.
Furthermore, Theorem 3.5 also applies to the test statistic in
Robinson (1994) and thus generalizes his
result, too, because he only shows that his test is locally most
powerful.
4. INFERENCE WITH SERIALLY CORRELATED
ERRORS
Now we extend the basic model to allow for weakly dependent (ARMA)
errors. In particular, we work with the following assumption.
Assumption 4. {et} is generated by an ARMA
model of the form
where {εt} satisfies Assumption 3. Here
a(z) and b(z) are finite
polynomials without common roots and all roots strictly outside the
unit circle. The coefficients in the autoregressive and moving average
polynomials are collected in the q × 1 parameter vector
ψ.
This assumption follows Tanaka (1999);
Tanaka in addition assumes that {εt} is
i.i.d. Thus, we offer more generality in this respect too, because of
our martingale difference assumption on
{εt}.
Collect the parameters of the dynamic part of the model in the vector
γ = (θ′, ψ′)′ with true value
γ0 = (0′, ψ0′)′ and
let c(z, ψ) =
a(z)b−1(z).
Analogously to ζ(z, θ), define ξ(z,
γ) = ∂ ln(φ(z, θ)c(z,
ψ))/∂γ and
.
Note that ξj =
(ζj′,
cj′)′ with
ζj defined as before and
cj defined as the coefficient on
zj in the expansion of ∂ ln
c(z,
ψ)∂ψ|ψ=ψ0 in
powers of z. As in Assumption 1(ii) we define
with
.
It is easily shown that Φ is the Fisher information for ψ
under Assumption 4; e.g., if {et} is an
AR(1) process with coefficient a then
cj =
−aj−1 and Φ = (1 −
a2)−1. Finally, corresponding to
(4), we assume that
which in particular implies that Ξ is nonsingular.
The log-likelihood function in the case of serially correlated errors
is, except for constants,
to be compared with (8). The concentrated likelihood function for
γ = (θ′, ψ′)′ becomes
except for constants, where
and
are now defined in terms of the functions (34) and (35). Corresponding
to (13) we consider the function
4.1. Estimation
The analysis of the model with serially correlated errors proceeds in
the same way as with martingale difference errors as discussed
previously. Thus, we are able to show the existence of a local MLE
satisfying
and to prove joint asymptotic normality of
.
Under Gaussianity we achieve efficiency as before.
THEOREM 4.1. Under Assumptions 1, 2, and 4 and (31) there exists
a local maximizer
of the concentrated likelihood (33) that satisfies, as n →
∞,
Under the additional assumptions of Gaussianity of
{εt} and correct (minimal) specification (all
elements of ψ0 are nonzero),
is asymptotically efficient in the sense that its asymptotic variance
attains the Cramér–Rao lower bound.
Based on this theorem it is possible to create joint (p +
q)-dimensional confidence ellipsoids for θ and ψ
that take into account the asymptotic correlation between the estimates
represented by the matrix κ. This is important for inference, not
only on θ but also on ψ. Usually, in applied work one would
determine the appropriate filtration of data (i.e., the function
φ(z, θ)) by Dickey–Fuller tests or similar
methods and then treat the filtered data as if it were observed, i.e.,
as if the correct filter were known a priori. The resulting inference
on ψ is incorrect, because the correlation between θ and
ψ is ignored. When applying Theorem 4.1, this pretesting problem
is avoided because θ and ψ are estimated jointly.
When inference on θ is of interest, the asymptotic marginal
distribution of
can be immediately derived from the theorem.
COROLLARY 4.1. Under the conditions of Theorem 4.1,
as n → ∞.
In parallel with Corollary 4.1,
(by the partitioned matrix inverse formula), and in the special case
where φ is not present this reduces to Φ−1,
which is the Fisher information on ψ. Thus, the well-known
asymptotic efficiency of the MLE in pure ARMA models comes out as a
special case of our results. More important, Theorem 4.1 with φ
present demonstrates the joint efficiency in the generalized model.
To illustrate the loss of efficiency in estimation of θ stemming
from serially correlated errors, consider again the fractional unit
root model. Suppose we know that the errors are not serially correlated
but simply are martingale differences. Then the asymptotic variance of
is 6/π2 by Theorem 3.2. If instead it is known that
the errors exhibit serial correlation of the AR(1) or MA(1) type with
coefficient a, then the asymptotic variance of
is the inverse of (π2/6) − ((1 −
a2)/a2)(ln(1 −
a))2 by Corollary 4.1.
Figure 1 shows the relative efficiency of
these two estimates as a function of the serial correlation parameter,
a. This is calculated as
Relative efficiency of
in the presence of first-order autoregressive or moving average errors.
which has a minimum at a = 0.684. This suggests that
moderate levels of a best replicate the behavior of the
(weighted) autocorrelations of a fractionally integrated process. The
point a = 0 shows that the relative efficiency allowing for
serial correlation when it is not present is 0.392, as noted by Tanaka
(1999).
4.2. Hypothesis Testing
We now consider the testing problems (16) and (23) in the presence of
serially correlated errors, where again only θ is of interest. The
Wald, LR, and LM test statistics are
where
are evaluated at
are evaluated at
,
the estimate of ψ under the null, and
is defined in terms of the jth sample autocorrelation of
.
It is obvious from the expressions for the test statistics and
that the LM test is not necessarily the simplest to apply in practice.
The implementation of the Wald and LR test statistics is
straightforward when we can estimate the model under both the null and
alternative and should not be a problem, given the methods available in
the previous sections. In particular, the LR test is attractive because
there is no need to calculate Ψ, κ, and Φ.
Similar to the calculation of the infinite-order moving average
coefficients in standard ARMA models, the calculation of κ and
Φ can be quite cumbersome when the model in Assumption 4 is more
complex than just an AR(1) or MA(1) model (see also the discussion in
Tanaka, 1999). To overcome this issue, one
could employ the numerical approximations
which of course have the same asymptotic properties as W and
LM. However, because Ψ, κ, and Φ, and thus
W and LM, can be calculated for any given parameter
value (say, γ) by numerically expanding ∂ ln
φ(z, θ)c(z,
ψ)/∂γ at γ = γ in powers of
z using a computer, we do not consider
further.
The asymptotic distribution of the tests under local alternatives and
with serial correlation is given by the following theorem.
THEOREM 4.2. Let Assumptions 1, 2, and 4 and (31) be satisfied
and let T denote the W, LR, or LM test
statistics (39), (40), and (41). Then, under (23), it holds that
as n → ∞. The three tests are consistent and
asymptotically equivalent, and under the additional assumption of
Gaussianity they are locally most powerful.
This theorem shows that the tests are still locally most powerful,
even in the presence of serially correlated errors. Setting κ =
Φ = 0, i.e., when no serial correlation is present and ψ is
not estimated, generates Theorem 3.3 as a special case. As with
Corollary 3.1 in the case without serial correlation, we can easily
calculate the asymptotic local power, giving us a benchmark against
which to compare the power of the tests in finite samples.
COROLLARY 4.2. Under the conditions of Theorem 4.2 it holds that,
under
,
as n → ∞, where
χp,1−α2 is the 100(1
− α)% point of the χp2
distribution and
Fδ′(Ψ−κ′Φ−1κ)δ
is the distribution function of the
χp2(δ′(Ψ −
κ′Φ−1κ)δ) distribution.
Using Corollary 4.2, Figure 2 shows the
local power functions against positive alternatives for the fractional
unit root model with different specifications of AR(1) errors. Because
δ only enters (42) through δ2, the power functions
are symmetric. The starred line is the local power function when the
errors are a martingale difference sequence and this is known (i.e.,
using Corollary 3.1). The dotted, dashed, and solid lines correspond to
AR(1) specifications of the errors with coefficient a =
−0.5, a = 0, and a = 0.5, respectively. In the
case a = 0, the errors are a martingale difference sequence,
but an AR(1) error process is estimated.
Asymptotic local power functions with martingale difference and
first-order autoregressive or moving average errors.
The local power of the tests in the model with a = 0.5 is
much lower than for the other specifications. On the other hand, the
power loss in the model with a = −0.5 is small. This is
in accordance with the results in Section 4.1; cf. (38) and Figure 1.
5. FINITE SAMPLE PERFORMANCE
In this section, we compare the asymptotic local power functions
derived in the previous sections to the finite sample rejection
frequencies by means of Monte Carlo experiments.
The model we use for the simulation study is the well-known
fractional unit root model with an AR(1) error:
where {εt} is i.i.d. standard normal. This
model is also studied in simulations by Robinson (1994) and Tanaka (1999).
In addition to this fractional DGP, Robinson (1994) also considers an autoregressive DGP and
finds that his test is dominated by Dickey–Fuller type tests in
the latter case.
We concentrate on comparing the finite sample performance of the
three test statistics (Wald, LR, and LM). Tanaka (1999) documents that the time domain LM test
outperforms Robinson's (1994) frequency
domain LM test, so we do not consider the frequency domain test here.
The properties of the estimator
in this model are examined by Tanaka (1999),
who finds that in the case without serial correlation the behavior of
the local MLE is very close to the asymptotic distribution. However,
with serially correlated errors the performance of the local MLE
degrades, and especially in the case of strong positive serial
correlation the performance is poor. This is expected based on (38) and
Figure 1.
Throughout, we fix the nominal level (type I error) at 0.05 and the
number of replications at 5,000. We consider the sample sizes
n = 100 and n = 500. The former is typical for
macroeconomic time series and the latter (or even larger) for financial
time series. For each experiment, 5,000 samples of size n =
500 were generated using the rann, diffpow, and armagen routines in Ox
version 3.00 including the Arfima package version 1.01 (see Doornik, 2001; Doornik and Ooms,
2001). For the smaller sample size, n = 100, we used
the first 100 out of the 500 observations from each sample.
Figures 3, 4,
5, 6 present the
simulated finite sample power functions of the test statistics for
different specifications of the error term in (44) (the tables
containing the numerical values used to construct the figures can be
obtained from the author upon request). For each value of θ, the
asymptotic local power has been calculated by setting
in Corollaries 3.1 and 4.2 and is reported under the heading Limit. In
all the figures, the left-hand-side figures (a) and (c) present the
simulated power functions of the tests calculated as in Sections 3.2
and 4.2, whereas the simulated power functions in the right-hand-side
figures (b) and (d) are calculated using size corrected critical
values.
First, consider the case of martingale difference errors shown in
Figure 3, i.e.,
{et} = {εt}. In
this case, all the finite sample rejection frequencies are very close
to the asymptotic local power, except the LM test in the small sample
n = 100, which has lower power than the LR and Wald tests.
Finite sample power functions with martingale difference
errors.
When the errors are serially correlated the differences between the
test statistics are more apparent. With negative serial correlation
a = −0.5 (Figure 4), and with
a = 0 (Figure 5), i.e., when there
is no serial correlation in the DGP but an AR(1) is estimated, the LM
test loses power compared to the LR and Wald tests, and the Wald test
tends to be oversized in the small sample, which is also reflected by
its very low size corrected power for n = 100 in Figure 5(b).
Finite sample power functions with AR(1) errors with coefficient
a = −0.5.
Finite sample power functions with AR(1) errors with coefficient
a = 0.
In Figure 6 the errors are positively
serially correlated with a = 0.5. From the previous sections
we know that the asymptotic local power is much lower in this case than
with negative or no serial correlation. As Figure
6 shows, this is also the case for the finite sample rejection
frequencies (note that the scaling along the vertical axis is different
in Figures 6(a) and (b), compared with the other
plots). In the small sample, n = 100, there are severe distortions,
especially to the LM and Wald tests. The LM test completely loses power
against negative alternatives, with rejection frequencies even lower
than the nominal size, and the Wald test is severely oversized. When
n = 500 the situation improves, but the LM test still has the
lowest power and the Wald test is still severely oversized.
Finite sample power functions with AR(1) errors with coefficient
a = 0.5.
Unreported simulations (which can be obtained from the author upon
request) show that, not surprisingly, the performance of the LR test
(with n = 100) is very bad when relevant deterministic terms
are left out and that the inclusion of irrelevant mean and/or trend
terms decreases power against positive values of θ. This is well
known from AR-based unit root tests such as the Dickey–Fuller
test, where a mean (and trend) must be included if any power against
nonzero mean (and trend) is desired. However, it is worth noting that,
unlike in our model, the distribution of Dickey–Fuller type test
statistics changes when deterministic terms are included.
Overall, the simulations show that the improvement with respect to
both size and power when considering n = 500 instead of
n = 100 is substantial. Thus, one would expect very good
performance of the tests in financial applications, where samples are
often many times larger. In such cases, the power loss resulting from
the estimation of serially correlated errors would also be of less
importance. It was also found that generally the LM test has lower
power than the Wald and LR tests and that the Wald test is often
severely oversized. We have stressed the possibility of conducting
simple asymptotic inference in our model, using the chi-squared tables,
and because this property is lost if size corrected critical values
must be employed, this weighs heavily against the Wald test.
Even though we concentrated on the simple and well-known fractional
unit root model in the present simulation study, similar relative
performance is to be expected in more complicated models such as the
general model in (5). Thus, the LR test is expected to outperform the
Wald and LM tests with respect to both size and power also in more
complicated models.
6. CONCLUSION
We have considered likelihood inference in a wide class of
potentially nonstationary univariate time series models. In such cases,
inference is usually drawn in an autoregressive framework and
nonstandard asymptotics apply.
In this paper we have shown that, when the estimation and testing
problems are embedded in a fractional integration framework, standard
asymptotics apply and desirable statistical properties of likelihood
inference reemerge. In particular, there exists a local MLE that is
asymptotically normal, and the classical likelihood-based tests (Wald,
LR, and LM) are consistent and asymptotically chi-squared distributed
under local alternatives. Under the additional assumption of
Gaussianity, the local MLE is asymptotically efficient, and the tests
are locally most powerful. Furthermore, in the scalar parameter case
with i.i.d. Gaussian errors, our tests achieve the asymptotic Gaussian
power envelope of all invariant and unbiased tests; i.e., they are
asymptotically uniformly most powerful (against local alternatives)
among all invariant and unbiased tests.
The Monte Carlo study shows that with sample sizes typical for
macroeconomic time series the tests perform reasonably well, and with
larger sample sizes such as those usually found in finance applications
the performance of the tests is very good and their rejection
frequencies very close to the asymptotic local power. In our Monte
Carlo study the LR test dominates with respect to both size and power
in finite samples. The LR test also has attractive computational
features when serially correlated errors are allowed for, because it
avoids a quite cumbersome calculation of covariance matrices.
The results derived in this paper could also be applied to the
problem of testing for fractional cointegration when the cointegrating
vector is known a priori, e.g., from economic theory. When the
cointegrating vector must be estimated the results in this paper no
longer apply. This presents an interesting avenue for further research
which is currently under investigation by the author (see, e.g., Nielsen, 2003).
APPENDIX: PROOFS
Proof of Theorem 3.1. First, by noting that
it is immediate that the denominator in g(θ) is
by Assumption 3. The last two terms are asymptotically negligible
because
by Assumption 2 and (12) and
using also uncorrelatedness of {et}.
The numerator in g(θ) can be written as
By the mean value theorem we have, for some θ* =
θ*(t, n) such that 0 ≤
∥θ*∥ ≤ ∥θ∥,
where the last term has mean zero and variance
by Assumption 1(iii) and dominated convergence. As in Robinson (1994, p. 1435), it follows that
uniformly in t. Using (A.5) we get that (A.3) is
For a fixed m > 0, consider the p-vector
and the p × p matrix
.
By Assumption 3,
and applying an LLN,
in probability. The vector sequence {vt} is
a martingale difference sequence with respect to the filtration
because vt is
measurable and integrable and
.
for all t. Using Assumption 3,
,
and by application of a martingale difference CLT (e.g., Brown, 1971; Hall and Heyde, 1980, Chap. 3.2), we establish that
Because
can be made arbitrarily small by choosing m large by (4), we
can apply Bernstein's lemma (e.g., Hall and Heyde, 1980, pp. 191–192) to conclude that (A.3)
converges in distribution to
δ′(2Ψ1/2Z −
Ψδ)σ2. Because
we have proven the first statement of the theorem if we show that
(A.2) is asymptotically negligible.
Thus, (A.2) can be written as
by (A.5), where
.
Now, (A.9) is
where
uniformly in t by the same analysis as for
ut, and
using (A.11) and Assumption 2. Now the second term of (A.10) is
by uncorrelatedness of {et}, Assumption 2,
(4), (12), and (A.12). The same arguments apply to the first term of
(A.10) and to the terms in (A.8).
Next, we examine
The expression in the first set of parentheses is
using (A.5),
as in (A.1) by Assumption 2, (12), (A.11), and (A.12).
Defining the function ζ(z, θ) =
(∂/∂θ)ln φ(z, θ), the second sum
in (A.13) is
where (A.18) converges in distribution to
Ψ1/2Zσ2 as in (A.6).
Applying (A.5) to (A.17) we see that it equals
,
which converges in probability to Ψδσ2 as in
(A.6).
Thus, we need to show that (A.15) and (A.16) are asymptotically
negligible. First, write (A.15) as
using (A.5). The first and third terms are
Op(n−1/2) by
(4) and the arguments applied to (A.14), and the second and fourth
terms are
Op(n−1/2) by
combining the arguments applied to the first term and those applied to
(A.4). Rewriting (A.16) as
using (A.5), we note that it is asymptotically negligible by the same
arguments as applied to (A.4). This establishes the second statement of
the theorem.
The second derivative is
which is equal to
by (A.14). Combining the preceding arguments it can be shown that the
last two terms are both op(1) whereas the
first term converges in probability to −Ψ. This completes
the proof. █
Proof of Theorem 3.2. By Theorem 3.1(iii) and Assumption 1,
g(θ) is asymptotically a concave function of
,
the sphere in p-dimensional Euclidean space centered at the
origin with radius
.
Hence, by Theorem 3.1 and the subsequent analysis,
is asymptotically the unique maximizer of W(δ) in
,
and its asymptotic distribution is given by (15) by the usual expansion.
Under Gaussianity of {et}, (8) is the true
likelihood. The limiting Fisher information is then given by
which is the inverse of the asymptotic variance as required. █
Proof of Theorem 3.3. Though the equivalence of the test
statistics is well known in standard testing problems, we have stressed the
nonstandard nature of our model, and thus we start by showing equivalence.
By the mean value theorem
where θ* is an intermediate value. This implies that
W − LM → 0 in probability by Theorem 3.1.
Similarly, by a Taylor expansion of the likelihood
and thus LR − W → 0 in probability by
Theorem 3.1(iii).
The asymptotic distribution of the test statistics follows directly
from the previous theorems. Under the local alternatives (23) we set
by Theorem 3.2. Then the Wald test is
by Theorem 3.2. Similarly,
by Theorem 3.1(ii) and (21), and
by Theorems 3.1(i) and 3.2.
Under the additional assumption of Gaussianity the tests are locally
most powerful because the noncentrality parameter is maximal by Theorem
3.2 and the formula for the information matrix (21). █
Proof of Corollary 3.1. This is immediate from Theorem 3.3.
█
Proof of Theorem 3.4. Following the arguments of the previous
sections and those in Tanaka (1999) and using (A.5)
we find that
uniformly in t. Thus, the denominator of (26) normalized by
n−1 converges to σ2 in
probability as n → ∞, and the numerator
by the same arguments as those in the proof of Theorem 3.1. As
before, it can be shown that this is unaffected by the presence of the
regressors and the result follows. █
Proof of Theorem 3.5. Consider first (28), which implies that (in
this context φ is the density function of the standard normal
distribution)
with the nontrivial solution C1, α(δ) =
−C2, α(δ) −
2δ2Ψ. Now determine the constants by (27):
where Z is a standard normal random variable. Thus,
C2, α(δ) is the solution to
i.e.,
,
where Z1−α/2 is the 100(1 −
α/2)% point of the standard normal distribution.
The power envelope is given by
where the last line follows by squaring both sides of the inequality,
χ1,1−α2 is the 100(1 −
α)% point of the χ12 distribution, and
Fδ2Ψ is the distribution
function of the χ12(δ2Ψ)
distribution. █
Proof of Theorem 4.1. The proof proceeds along the same lines as
those of Theorems 3.1 and 3.2. By the same arguments it can be shown
that the results are unaffected by the presence of the regressors, so
we assume here that {ut} is observed.
Under
,
we first show that
where Z is a (p + q)-dimensional standard
normal random vector.
It is immediate that the denominator in g(γ) converges
in probability to σ2 by Assumption 4. By the mean value
theorem we have, for some γ* =
γ*(t, n) partitioned as
γ* = (θ*′, ψ*′)′ and such that
∥γ0∥ ≤ ∥γ*∥ ≤
∥γ∥,
where λν(z, ψ) = (∂ ln
c(z, ψ)/∂ψ)(c(z,
ψ)/c(z, ψ0)) and
λ(z, θ) is defined in Assumption 1(iii). Denoting by
λν, j(ψ) the coefficient on
zj in an expansion of
λν(z, ψ) in powers of z and
by N a neighborhood of size
O(n−1/2) around
ψ0,
because a(z, ψ) and b(z,
ψ) have roots that are outside the unit circle. Thus, as in (A.5)
it follows that
uniformly in t.
Hence, the numerator in g(γ) is
Define for a fixed m > 0 the (p +
q)-vector
and the (p + q) × (p + q) matrix
.
As in the proof of Theorem 3.1,
in probability, and
by application of a martingale difference CLT. Part (i) now follows
by Bernstein's lemma.
To prove (ii) we notice that the first term in
is (σ2 +
op(1))−1 by (A.19) and
write the second term in (A.21) as
The last term converges in distribution to
Ξ1/2Zσ2 as in (A.20), and by
application of (A.19) the difference of the first two terms is
,
which converges in probability to Ξμσ2 as in
(A.20).
The result (iii) follows exactly as in the proof of Theorem 3.1.
Next, it follows as in Section 3.1 that (14) holds with δ
replaced by μ and g replaced by the function in Section 4.
Thus, the existence and uniqueness in
of a local MLE
satisfying
are ensured, and its distribution is given by (36) from the usual
expansion.
Efficiency follows directly from (iii), which is the Fisher
information under Gaussianity of {εt}.
█
Proof of Corollary 4.1. Apply the partitioned matrix inverse
formula to Ξ. █
Proof of Theorem 4.2. This follows straightforwardly by applying
the arguments in the proof of Theorem 3.3 to the results in Theorem 4.1 and
its proof. █
Proof of Corollary 4.2. This is immediate from Theorem 4.2.
