This paper proposes simple Hausman-type tests to check for bias in the log-periodogram regression of a time series believed to be long memory. The statistics are asymptotically standard normal on the null hypothesis that no bias is present, and the tests are consistent.Long memory, log periodogram regression, Hausman test.

James Davidson

Philipp Sibbertsen

English

Research Papers in Economics

Economics Department 
Discussion Papers Series 
ISSN 1473 – 3307 
Tests of Bias in Log-Periodogram 
Regression 
 
 
 
James Davidson and Philipp Sibbertsen 
 
 
 
Paper number 08/05 
URL: http://business-school.exeter.ac.uk/economics/papers/ Tests of Bias in Log-Periodogram Regression
James Davidson,
Department of Economics, University of Exeter, Exter EX4 4PU, UK
Philipp Sibbertsen,
Department of Economics, University of Hannover, D-30167 Hannover, Germany
Abstract
This paper proposes simple Hausman-type tests to check for bias in the log-periodogram
regression of a time series believed to be long memory. The statistics are asymptotically
standard normal on the null hypothesis that no bias is present, and the tests are consistent.
JEL Classi￿cation: C12, C22
Keywords: Long memory, log periodogram regression, Hausman test.
1 Introduction
Long memory models, and speci￿cally fractionally integrated models, are a popular method of
representing the persistence characteristics of time series. Inference about long memory is often
based on the results of the narrow band log-periodogram regression, ￿rst introduced by Geweke
and Porter-Hudak (1983) (henceforth, GPH). This method has applications both to testing for
the presence of long memory in individual time series and to testing for fractional cointegration
(Hassler et al., 2006). However, as shown by Agiakloglou et al. (1993) inter alia, the presence
of short-run dynamic components can still severely bias the GPH estimator in ￿nite samples,
falsely indicating the existence of long memory. A stable autoregressive root is nominally a ￿ short
run￿component, yet if close to unity it approaches observational equivalence with long memory.
There is, moreover, a trade-o⁄ that investigators must make in the matter of bandwidth choice,
between excessive bias and lost e¢ ciency.
In this context, it appears desirable to have a means of testing whether a signi￿cant bias is
present. In this paper we propose simple tests of the Hausman (1978) type. Numerous variants of
the basic log-periodogram estimator have been proposed, notably by Phillips and Shimotsu (2002)
who advocate a frequency-grouping approach with a ￿xed-e⁄ects treatment of the intercept,
Andrews and Guggenberger (2003) who include polynomial terms in the frequencies in the narrow-
band regression, and Moulines and Soulier (1999) (henceforth MS) who advocate a broad-band
regression with dummy variables alone controlling for the short-run e⁄ects. In this note we
con￿ne attention to two cases, although our idea could in principle be generalized. Section 2
derives the test in the context of GPH, showing consistency and deriving the null distribution.
Some simulations are reported in Section 3. Section 4 extends to idea to the MS estimator, and
Section 5 contains concluding remarks.
12 The Test
Consider the class of covariance stationary Gaussian processes fYtg whose spectrum takes the
form
f(￿) = j1 ￿ exp(￿i￿)j￿2df￿(￿); ￿￿ ￿ ￿ ￿ ￿ (2.1)
where jdj < 0:5 is the fractional integration parameter, and f￿(￿) represents the short-term
correlation structure of the model. We assume the regularity conditions of Hurvich et al. (1998)
(henceforth HDB). Given a sample Y1;:::;YT, GPH suggested estimating d in (2.1) from the
regression
logI(￿k) = c + dXk + "k; k = 1;:::;m (2.2)
for some m ￿ [T=2] where Xk = ￿2log(sin￿k), ￿k = 2￿k=T denotes the kth Fourier frequency,
and
I(￿) :=
1
2￿T
￿ ￿
￿ ￿
￿
T X
t=1
Yt exp(￿it￿)
￿ ￿
￿ ￿
￿
2
:
To minimize bias due to the omission of the unknown function f￿, GPH suggested setting m =
O(T1=2), in the hope that in this narrow band of low frequencies, the variations in f￿ are small. Let
^ d denote this estimator. De￿ning aj = Xj ￿ ￿ X where ￿ X = m￿1 Pm
j=1 Xj, and SXX =
Pm
k=1 a2
j,
HDB derive the bias expression
E(^ d ￿ d) =
1
SXX
m X
k=1
ak logf￿
k +
1
SXX
m X
k=1
akE("k)
= ￿
2￿2
9
f￿00
(0)
f￿(0)
m2
T2 + o
￿
m2
T2
￿
+ O
￿
log3 m
m
￿
: (2.3)
They further show that for m = o(T4=5),
p
m(^ d ￿ d)
d ! N
￿
0;
￿2
24
￿
:
There is evidently a trade-o⁄ between e¢ ciency and bias in the choice of m. HDB show
that the mean squared error is minimized by setting m = CT4=5 where, the constant C depends
inversely on f￿00
(0)=f￿(0). Hurvich and Deo (1999) propose a plug-in estimator of C, but their
Monte Carlo evidence shows that the asymptotic optimality criteria are successful only when
there is a limited amount of short-run dependence. When the bias component is large enough to
dominate the MSE, the GPH narrow-bandwidth proposal proves more e⁄ective.
Knowledge of the magnitude of the bias term would obviously be desirable in guiding the
choice of bandwidth. Our suggestion is, in essence, to compare the results of setting two di⁄erent
bandwidths. Let the null hypothesis have the form
H0 : f￿(￿) = constant; 0 ￿ ￿ ￿ ￿ ￿ ￿ (2.4)
where ￿ is a bound to be speci￿ed. The test statistics we consider are of the form
TS =
^ d1 ￿ ^ d2
SE(^ d1 ￿ ^ d2)
(2.5)
where ^ d1 and ^ d2 are estimators of d constructed with m1 and m2 < m1 periodogram ordinates,
respectively, and SE(￿) denotes a suitable estimator of the standard error. Letting Si =
Pmi
j=1 a2
ij
2for i = 1 and 2 respectively, we can adapt HDB￿ s formula (5) to give
Var(^ d1 ￿ ^ d2) = E
0
@ 1
S1
m1 X
j=1
a1j"j ￿
1
S2
m2 X
j=1
a2j"j
1
A
2
= E
0
@
m2 X
j=1
￿
a1j
S1
￿
a2j
S2
￿
"j +
1
S1
m1 X
j=m2+1
a1j"j
1
A
2
=
￿2
6
0
@
m2 X
j=1
￿
a1j
S1
￿
a2j
S2
￿2
+
1
S2
1
m1 X
j=m2+1
a2
1j
1
A + o
￿
1
m2
￿
: (2.6)
The last member of (2.6), less the small-order term, is used to provide the standard error in (2.5).
For a test of hypothesis (2.4) we need to set m1 such that
￿m1 = 2￿m1=T ￿ ￿:
and then to choose m2 so as to optimize the power if the test. Applying expression (2.3) yields
E(^ d1 ￿ ^ d2) =
2￿2
9
f￿00
(0)
f￿(0)
m2
1
T2
￿
1 ￿
m2
2
m2
1
￿
+ o
￿
m2
1
T2
￿
+ O
￿
log3 m2
m2
￿
: (2.7)
Suppose we set m1 = CT￿ in this expression for some ￿ ￿ 1 and also set m2 = Km1, where
K 2 (0;1) is another constant to be chosen. Substituting these choices into (2.7) yields
E(^ d1 ￿ ^ d2) =
2￿2
9
f￿00
(0)
f￿(0)
T2￿￿2 ￿
1 ￿ K2￿
+ o(T2￿￿2) + O
￿
log3 T￿
T￿
￿
: (2.8)
Also observe that (2.6) implies
Var(^ d1 ￿ ^ d2) =
￿2
6
"￿
1
m1
￿
1
m2
￿2
m2 +
m1 ￿ m2
m2
1
#
+ o
￿
1
m2
￿
=
￿2
6T￿
￿
1
K
￿ 1
￿
+ o(T￿￿): (2.9)
Compare the leading terms in these expressions. For a consistent test, it is necessary that
jE(^ d1￿ ^ d2)j=
q
Var(^ d1 ￿ ^ d2) ! 1 when f￿00
(0) 6= 0, and according to (2.8) and (2.9), this requires
2￿ ￿ 2 > ￿￿=2 or, in other words, ￿ > 4=5. Subject to this condition, we should then choose K
to maximize the expression (1 ￿ K2)=
￿
K￿1 ￿ 1
￿1=2, so as to make jE(^ d1 ￿ ^ d2)j=
q
Var(^ d1 ￿ ^ d2)
as large as possible with any given T. The solution to this problem on [0;1] can be veri￿ed
numerically to be about K = 0:64. The following theorem establishes su¢ cient conditions for
the statistic to have a normal limiting distribution under the null hypothesis. (See the Appendix
for proof.)
Theorem 2.1 If Yt is a Gaussian process satisfying (2.1), hypothesis (2.4) holds, and m1 ￿
[T￿=2￿], then
TS
d ! N(0;1):
3Note that the Gaussianity is required since the CLT is established by the method of moments,
using the approach pioneered by Robinson (1995).
Provided 4=5 < ￿ ￿ 1, the test is consistent for the null hypothesis in (2.4) for any ￿ > 0.
Setting ￿ = 1 corresponds to testing the case ￿ = ￿. In other words ^ d1 is obtained from the
broad band regression including all the periodogram points. In this form, the null hypothesis
takes the strict form that the spectrum depends on the single dynamic parameter d, such that
the fractional di⁄erences are white noise. On the other hand, by choosing ￿ < 1 we obtain a
test of the hypothesis simply that f￿00
(0) = 0, and hence that f￿ is constant in an arbitrary
neighbourhood of the origin. To see how this works in practice, note that the width of the
interval of [0;￿] from which periodogram points for the estimators ^ d1 and ^ d2 are drawn is of
order m1=T = O(T￿￿1), and hence shrinks to zero as T ! 1. Heuristically, we can say that in a
sample of size T, f￿ needs to be constant in a region of this width to approximate the limiting null
distribution of the statistic. The smaller ￿ is set, the more rapid is this convergence and hence
the greater potential generality of the null, although the power of the test grows correspondingly
more slowly. Note the interesting fact that the test depends on comparing two estimators which
are both consistent, under both null and alternative. Their t ratios nonetheless diverge under the
alternative (even when d = 0, note) with ￿ in the (4=5;1) range. By taking the di⁄erence of two
such estimators we remove the dependence on d, and so obtain the null distribution. However, the
bias term still converges to 0 more slowly than its standard deviation, so that statistic TS yields
a consistent test. We note that in the class of ARFIMA(p;d;q) models only the case p = q = 0
yields a restriction corresponding to a case of H0. The possibility of short-run autocorrelations
existing, compatibly with the null distribution, can be realized only by invoking a wider class
of DGPs. Nonetheless, we note that it is straightforward to construct such processes in the
frequency domain, and to simulate them, by applying an inverse Fourier transform to a sample
of suitably heteroscedastic normal innovations. The generality in the choice of nulls a⁄orded by
(2.4) is therefore not inconsiderable.
3 Monte Carlo Evidence
Extensive simulation results (available on request from the authors) suggest that our test has
ample power to detect the existence of bias, at least in the context of linear Gaussian models.
However, we note the unusual feature of our test, that in many applications of interest the null
hypothesis is known to be technically false, and the test will therefore always reject when the
sample is large enough. We therefore need to emphasize that the question of interest is not
￿Is there bias?￿but rather ￿Is there important bias?￿ , where ￿ important￿is to be interpreted,
typically, in the context of bandwidth choice in relation to sample size. Ideally, failure to reject
a false null should imply that the bias is small enough to be innocuous from the viewpoint of
standard inference on d, while a rejection should indicate a need for corrective action. The choice
of signi￿cance level needs to be geared to these implicit trade-o⁄s.
With these issues in mind, we present evidence of the performance of the test in the context of
implementing a pretest estimator. Figures 1-3 show the results in 30,000 replications of GPH esti-
mations in samples of 500 observations. The data generation processes have the ARFIMA(1;d;0)
form
(1 ￿ ￿L)(1 ￿ L)dYt = ut s NI(0;1) (3.1)
with the AR coe¢ cient ￿ set to values 0, 0.25, 0.5 and 0.75, and either d = 0 or d = 0:25. Note
that in this case f￿(￿) = (1 ￿ 2￿cos￿ + ￿2)￿1 and
f￿00
(0)
f￿(0)
=
￿2￿
(1 ￿ ￿)2 (3.2)
4so that the bias is related directly to the magnitude of ￿:
In addition to GPH estimators with bandwidths m = 22 ￿ [T0:5] and m = 78 ￿ [T0:7],
we construct a composite estimator de￿ned to equal the second case if the bias test rejects at
the nominal 20% level (asymptotic criterion) and otherwise the ￿rst case. This conservative
signi￿cance level was chosen on the basis of some preliminary experiments. Figure 1 shows the
RMSE of the three estimators for the four cases of ￿ and d = 0. The same simulations, performed
with d = 0:25, yielded e⁄ectively identical results, and are not shown. Figure 2 shows the rejection
rates using the standard asymptotic criterion of the conventional Wald signi￿cance tests on d when
the null hypothesis is true, with d = 0. Figure 3 shows a case of the alternative, the proportion
of rejections in the corresponding cases with d = 0:25.
4 Broadband Estimator
We have adopted the GPH estimator for our exposition as the best-known case of log-periodogram
regression, but analogous tests could be constructed for other variants. The MS estimator uses
the whole range of frequencies. MS expand logf￿ in Fourier series to pth order, yielding terms
of the form hj; j = 1;:::;p where hj(￿k) = cos(j￿k)=
p
￿. Assuming the expansion
logf￿(￿) =
1 X
j=1
￿jhj(￿) (4.1)
they show that if the coe¢ cients decline exponentially, such that j￿jj = O(￿j) for 0 < ￿ < 1,
then if p = pT = O(logT), the mean squared error of the estimator of d is of O(logT=T). MS
show that
p
T=pT(^ dMS ￿ d)
d ! N
￿
0;
￿2
6
￿
:
A Hausman-type test can be constructed for this estimator similarly, although only to test the
strict version of the null hypothesis. The cosine expansion of f￿ corresponding to (2.4) with
￿ < ￿ cannot be constructed with only a ￿nite number of terms. To obtain a statistic having
the form (2.5) for this case, choose ^ d1 to be the estimator computed in the regression of the
log periodogram points logI(￿k) onto (Xk;1), and ^ d2 to be the estimator computed from the
regression of I(￿k) onto (Xk;1;h1(￿k);:::;hpT(￿k)). The asymptotic variance of ^ d1 ￿ ^ d2 under
H0 is straightforwardly shown to be the di⁄erence of their asymptotic variances, by the usual
Hausman (1978) variance formula. The counterpart of Theorem 2.1 for this case follows directly
from MS Theorem 1.
5 Concluding Remarks
This paper proposes simple diagnostic procedures to detect bias in log periodogram regressions.
The tests are very easy to implement as routine outputs of a log-periodogram regression package.1
In particular, being nonparametric, they are easier to apply than goodness-of-￿t tests based on
a speci￿ed spectral functional form (e.g. Chen and Deo 2004, Paparoditis 2000). We have
focused in our simulation experiments on linear Gaussian (ARFIMA) alternatives, in which an
autoregressive root may ￿ mimic￿long memory, but recent research (Davidson and Sibbertsen 2005,
Hsieh et al. 2007) has considered nonlinear processes giving rise to long memory. The properties
of long-periodogram regression in this context are still poorly understood, but the availability of
diagnostic techniques assume special importance in the context of model uncertainty.
1They are implemented in the package Time Series Modelling 4.26 (Davidson 2008)
5Appendix: Proof of Theorem 2.1
We adapt the proof of MS Theorem 1 to show that both
p
m1(^ d1 ￿ d) and
p
m2(^ d2 ￿ d) are
asymptotically normal under H0. Applying the continuous mapping theorem, these results extend
directly to yield the stated limiting distribution of the TS statistic.
The MS theorem is given for the case of a broadband log-periodogram regression containing
pT dummy variables representing the leading terms of the expansion in (4.1). The MS estimator
allows local smoothing of the periodogram by using averages of m ￿ 1 consecutive periodogram
points (their notation) to construct the regressand, and includes the maximum number of KT =
[T=2m] periodogram points in the regression.2 For simplicity, MS￿ s smoothing factor m can
validly be set to 1. We then modify the conditions by setting, respectively, KT = m1 and
KT = m2. We also set pT = 0, although note that in reading the MS proof, pT (or pn in MS￿ s
notation) should be replaced by 1 whenever it appears in orders-of-magnitude expressions.
Since, by hypothesis H0, the f￿(￿j) are constant for ￿j < ￿ = 2￿m1=T, the bias terms
containing l￿
p in MS equation (C.2) can be replaced by 0, and the tail sums in MS equations
(C.6) treated similarly. With these modi￿cations, MS Theorem 1 holds for
p
m1(^ d1 ￿ d) and
p
m2(^ d2 ￿ d) under the stated assumptions, noting that MS Assumption 2 is equivalent to (2.1)
under hypothesis H0, and likewise MS Assumption 3 (summability of the cosine weights) is
subsumed under our assumptions.
References
Agiakloglou, C., P. Newbold and M. Wohar (1993) Bias in an estimator of the fractional di⁄erence
parameter, Journal of Time Series Analysis 14, 235￿ 246
Andrews D. W. K. and P. Guggenberger, A Bias-reduced log-periodogram regression estimator
for the long-memory parameter, Econometrica 71, 675-712.
Chen, W. S. and R. S. Deo (2004) A Generalized Portmanteau Goodness-of-Fit Test for Time
Series Models. Econometric Theory 20, 382-416.
Davidson, J. (2007) Time Series Modelling 4.26 at http://www.timeseriesmodelling.com
Davidson, J. and P. Sibbertsen (2005) Generating schemes for long memory processes: regimes,
aggregation and linearity" Journal of Econometrics 128 (2) 253-282
Geweke, J. and S. Porter-Hudak (1983) The estimation and application of long-memory time
series models. Journal of Time Series Analysis 4, 221-237
Hassler, U., Marmol, F. and C. Velasco (2006) Residual log-periodogram inference for long-run
relationships. Journal of Econometrics 130, 165-207
Hausman, J. A. (1978) Speci￿cation Tests in Econometrics, Econometrica 46, 1251-1272.
Hsieh, M.-C., Hurvich, C. M. and P. Soulier (2007) Asymptotics for duration-driven long range
dependent processes. Journal of Econometrics141, 913-949
Hurvich, C. M. and R. Deo (1999) Plug-in selection of the number of frequencies in regression
estimates of the memory parameter of a long memory time series, Journal of Time Series Analysis
20, 331￿ 341
Hurvich, C. M., R. Deo and J. Brodsky (1998) The mean squared error of Geweke and Porter-
Hudak￿ s estimator of a long memory time series, Journal of Time Series Analysis 19, 19-46
2For clarity, we substitute here our notation T for sample size, in place of MS￿ s n:
6Moulines, E. and P. Soulier (1999) Broad ban log-periodogram estimation of time series with
long-range dependence. Annals of Statistics 27, 1415-1439
Paparoditis, E. (2000) Spectral Density Based Goodness-of-Fit Tests for Time Series Models.
Scandinavian Journal of Statistics 27(1), 143-176
Phillips, P. C. B. and K. Shimotsu (2002) Journal of Time Series Analysis 23, 57-93.
Robinson, P. (1995) Log-periodogram regression of time series with long-range dependence, An-
nals of Statistics 23, 1048-1072.
7Figure 1. Root Mean Squared Error of GPH Estimates, d = 0, T = 500 
Figure 2. Rejection Frequencies of 5% Significance Test, d = 0, T = 500 
Figure 3. Rejection Frequencies of 5% Significance Test, d = 0.25, T = 500 
8
0.0
0.2
0.4
0.6
0.8
1.0
0 0.25 0.5 0.75
M = 22
M = 78
Composite
φ
0.0
0.2
0.4
0.6
0.8
1.0
0 0.25 0.5 0.75
M = 22
M = 78
Composite
φ
0.00
0.10
0.20
0.30
0.40
0.50
0 0.25 0.5 0.75
M = 22
M = 78
Composite
φ

Tests of Bias in Log-Periodogram Regression

http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0805.pdf

Tests of Bias in Log-Periodogram Regression

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics