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Granger Centre Discussion Paper Series
2011
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 DISCUSSION PAPERS
RECENT PUBLICATIONS |
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11/03 |
Fabrizio Iacone
Stephen J. Leybourne
A. M. Robert Taylor
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On the behaviour of fixed-b trend break tests under fractional integration
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Abstract
Testing for the presence of a broken linear trend when the nature of the persistence in the data is unknown is not a trivial problem, since the test needs to
be both asymptotically correctly sized and consistent, regardless of the order of
integration of the data. In a recent paper, Sayginsoy and Vogelsang (2011) [SV]
show that tests based on fixed-b asymptotics provide a useful solution to this
problem in the case where the shocks may be either weakly dependent or display
strong dependence within the near-unit root class. In this paper we analyse the
performance of these tests when the shocks may be fractionally integrated, an
alternative model paradigm which allows for either weak or strong dependence
in the shocks. We demonstrate that the fixed-b trend break statistics converge to
well-defined limit distributions under both the null and local alternatives in this
case (and retain consistency against fixed alternatives), but that these distribu-
tions depend on the fractional integration parameter d. As a result, it is only
when d is either zero or one that the SV critical values yield correctly sized tests.
Consequently, we propose a procedure which employs d-adaptive critical values to
remove the size distortions in the SV test. In addition, use of d-adaptive critical
values also allows us to consider a simplification of the SV test which is (asymp-
totically) correctly sized across d but can also provide a significant increase in
power over the standard SV test when d = 1. |
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11/02 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Unit root testing under a local break in trend
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Abstract
Recent approaches to testing for a unit root when uncertainty exists over the
presence and timing of a trend break employ break detection methods, so that
a with-break unit root test is used only if a break is detected by some auxiliary
statistic. While these methods achieve near asymptotic efficiency in both
fixed trend break and no trend break environments, in finite samples pronounced
valleys in the power functions of the tests (when mapped as functions of the
break magnitude) are observed, with power initially high for very small breaks,
then decreasing as the break magnitude increases, before increasing again. In
response to this problem we propose two practical solutions, based either on
the use of a with-break unit root test but with adaptive critical values, or on a
union of rejections principle taken across with-break and without break unit root
tests. These new procedures are shown to offer improved reliability in terms of
finite sample power. We also develop local limiting distribution theory for both
the extant and the newly proposed unit root statistics, treating the trend break
magnitude as local-to-zero. We show that this framework allows the asymptotic
analysis to closely approximate the finite sample power valley phenomenon,
thereby providing useful analytical insights. |
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11/01 |
Tomás del Barrio Castro
Denise R. Osborn
A. M. Robert Taylor
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On augmented HEGY tests for seasonal unit roots
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Abstract
The contribution of this paper is twofold. First we extend the large sample results provided for the
augmented Dickey-Fuller test by Said and Dickey (1984) and Chang and Park (2002) to the case of
the augmented seasonal unit root tests of Hylleberg et al. (1990) [HEGY], inter alia. Our analysis
is performed under the same conditions on the innovations as in Chang and Park (2002), thereby
allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale
difference innovations. We show that the limiting null distributions of the t-statistics for unit
roots at the zero and Nyquist frequencies and joint F-type statistics are pivotal, while those of
the t-statistics at the harmonic seasonal frequencies depend on nuisance parameters which derive
from the lag parameters characterising the linear process. Moreover, the rates on the lag truncation
required for these results to hold are shown to coincide with the corresponding rates given in Chang
and Park (2002); in particular, an o(T^1/2) rate is shown to be sufficient. The second contribution
of the paper is to explore the use of data-dependent lag selection methods in the context of the
augmented HEGY tests. Information criteria based methods along with sequential rules, such as
those of Ng and Perron (1995) and Beaulieu and Miron (1993), are compared. |
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2010
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10/05 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Unit root testing under a local break in trend
[Revised to become No. 11/02 above]
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Abstract
It is well known that it is vital to account for trend breaks when testing for a unit
root. In practice, uncertainty exists over whether or not a trend break is present
and, if it is, where it is located. Harris et al. (2009) and Carrion-i-Silvestre et al.
(2009) propose procedures which account for both of these forms of uncertainty.
Each uses what amounts to a pre-test for a trend break, accounting for a trend
break (the associated break fraction estimated from the data) in the unit root
procedure only where the pre-test signals a break. Assuming the break magnitude
is fixed (independent of sample size) these authors show that their methods
achieve near asymptotically ecient unit root inference in both trend break and
no trend break environments. These asymptotic results are, however, somewhat
at odds with the finite sample simulations reported in both papers. These show
the presence of pronounced "valleys" in the finite sample power functions (when
mapped as functions of the break magnitude) of the tests such that power is
initially high for very small breaks, then decreases as the break magnitude increases,
before increasing again. Here we show that treating the break magnitude
as local to zero (in a Pitman drift sense) allows the asymptotic analysis to very
closely approximate this finite sample effect, thereby providing useful analytical
insights into the observed phenomenon. In response to this problem we propose
practical solutions, based either on the use of a with break unit root test but with
adaptive critical values, or on a union of rejections principle taken across with
break and without break unit root tests. The former is shown to eliminate power
valleys but at the expense of power when no break is present, while the latter
considerably mitigates the valleys while not losing all the power gains available
when no break exists. |
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10/04 |
Giuseppe Cavaliere
A. M. Robert Taylor
Carsten Trenkler
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Bootstrap co-integration rank testing: the role of deterministic variables and initial values in the bootstrap recursion
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Abstract
In this paper we investigate the role of deterministic components and initial
values in bootstrap likelihood ratio type tests of co-integration rank. A number
of bootstrap procedures have been proposed in the recent literature some of which
include estimated deterministic components and non-zero initial values in the bootstrap recursion while others do the opposite. To date, however, there has not been
a study into the relative performance of these two alternative approaches. In this
paper we fill this gap in the literature and consider the impact of these choices
on both OLS and GLS de-trended tests, in the case of the latter proposing a new
bootstrap algorithm as part of our analysis. Overall, for OLS de-trended tests our
findings suggest that it is preferable to take the computationally simpler approach
of not including estimated deterministic components in the bootstrap recursion and
setting the initial values of the bootstrap recursion to zero. For GLS de-trended
tests, we find that the approach of Trenkler (2009), who includes a restricted estimate of the deterministic component in the bootstrap recursion, can improve finite
sample behaviour further. |
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10/03 |
Stephan Smeekes
A. M. Robert Taylor
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Bootstrap union tests for unit roots in the presence of nonstationary volatility
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Abstract
Three important issues surround testing for a unit root in practice: uncertainty as
to whether or not a linear deterministic trend is present in the data, uncertainty as to
whether the initial condition of the process is (asymptotically) negligible or not, and uncertainty
over the possible presence, and if so form, of nonstationary volatility in the data.
Assuming homoskedasticity, Harvey, Leybourne and Taylor (2010, Journal of Econometrics,
forthcoming) propose decision rules based on a four-way union of rejections of QD
and OLS detrended tests, both with and without a linear trend, to deal with the first
two problems. In this paper we first discuss, again under homoskedasticity, how these
union tests may be validly bootstrapped using the sieve bootstrap principle combined
with either the i.i.d. or wild bootstrap resampling schemes. This serves to highlight the
complications that arise when attempting to bootstrap the union tests. We then demonstrate
that in the presence of nonstationary volatility the union test statistics have limit
distributions which depend on the form of the volatility process, making tests based on
the standard asymptotic critical values or, indeed, the i.i.d. bootstrap principle invalid.
We show that wild bootstrap union of rejections test are, however, asymptotically valid
in the presence of nonstationary volatility. The wild bootstrap union tests therefore allow
for a joint treatment of all three of the aforementioned problems. |
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10/02 |
Marcus J. Chambers
Joanne S. Ercolani
A. M. Robert Taylor
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Testing for seasonal unit roots by frequency domain regression
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Abstract
This paper develops univariate seasonal unit root tests based on spectral regression estimators.
An advantage of the frequency domain approach is that it enables serial correlation to be treated
non-parametrically. We demonstrate that our proposed statistics have pivotal limiting distributions
under both the null and near seasonally integrated alternatives when we allow for weak dependence
in the driving shocks. This is in contrast to the popular seasonal unit root tests of, among others,
Hylleberg et al. (1990) which treat serial correlation parametrically via lag augmentation of the test
regression. Moreover, our analysis allows for (possibly infinite order) moving average behaviour in
the shocks, while extant large sample results pertaining to the Hylleberg et al. (1990) type tests are
based on the assumption of a finite autoregression. The size and power properties of our proposed
frequency domain regression-based tests are explored and compared for the case of quarterly data
with those of the tests of Hylleberg et al. (1990) in simulation experiments. |
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10/01 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Robust methods for detecting multiple level breaks
in autocorrelated time series
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Abstract
In this paper we propose tests for the null hypothesis that a time series process displays a constant level against the alternative that it displays (possibly) multiple changes in level. Our proposed tests are based on functions of appropriately standardized sequences of the differences between sub-sample mean estimates from the series under investigation. The tests we propose differ notably from extant tests for level breaks in the literature in that they are designed to be robust as to whether the process admits an autoregressive unit root (the data are I(1)) or stable autoregressive roots (the data are I(0)). We derive the asymptotic null distributions of our proposed tests, along with representations for their asymptotic local power functions against Pitman drift alternatives under both I(0) and I(1) environments. Associated estimators of the level break fractions are also discussed. We initially outline our procedure through the case of non-trending series, but our analysis is subsequently extended to allow for series which display an underlying linear trend, in addition to possible level breaks. Monte Carlo simulation results are presented which suggest that the proposed tests perform well in small samples, showing good size control under the null, regardless of the order of integration of the data, and displaying very decent power when level breaks occur. |
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2009
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09/05 |
Giuseppe Cavaliere
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Testing for unit roots in the presence of a possible break in trend and non-stationary volatility
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Abstract
In this paper we analyse the impact of non-stationary volatility on the recently developed unit
root tests which allow for a possible break in trend occurring at an unknown point in the sample,
considered in Harris, Harvey, Leybourne and Taylor (2009) [HHLT]. HHLT's analysis hinges on
a new break fraction estimator which, when a break in trend occurs, is consistent for the true
break fraction at rate Op(T^-1). Unlike other available estimators, however, when there is no
trend break HHLT's estimator converges to zero at rate Op(T^-1/2). In their analysis HHLT
assume the shocks to follow a linear process driven by IID innovations. Our first contribution
is to show that HHLT's break fraction estimator retains the same consistency properties as
demonstrated by HHLT for the IID case when the innovations display non-stationary behaviour
of a quite general form, including, for example, the case of a single break in the volatility of the
innovations which may or may not occur at the same time as a break in trend. However, as
we subsequently demonstrate, the limiting null distribution of unit root statistics based around
this estimator are not pivotal in the presence of non-stationary volatility. Associated Monte
Carlo evidence is presented to quantify the impact of a one-time change in volatility on both
the asymptotic and finite sample behaviour of such tests. A solution to the identified inference
problem is then provided by considering wild bootstrap-based implementations of the HHLT
tests, using the trend break estimator from the original sample data. The proposed bootstrap
method does not require the practitioner to specify a parametric model for volatility, and is
shown to perform very well in practice across a range of models. |
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09/04 |
David I. Harvey
Stephen J. Leybourne
Lisa Xiao
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Testing for nonlinear trends when the order of inegration is unknown
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Abstract
We consider testing for the presence of nonlinearities in the mean and/or trend of a time series, approximating the potential nonlinear behaviour using a Fourier function expansion. In contrast to procedures that are currently available, we develop tests that are robust to the order of integration, in the sense that they are asymptotically correctly sized regardless of whether the stochastic component of the series is stationary or contains a unit root. The tests we propose take the form of Wald statistics based on cumulated series, together with a correction factor to line up the asymptotic critical values across the I(0) and I(1) environments. The local asymptotic power and finite sample properties of the tests are evaluated using various different correction factors. We envisage that the testing procedure we recommend should be very useful to applied researchers wishing to draw robust inference regarding the presence of nonlinear trend components in a series. |
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09/03 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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The impact of the initial condition on robust tests for a linear trend
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Abstract
This paper examines the behaviour of some recently proposed robust (to
the order of integration of the data) tests for the presence of a deterministic linear
trend in a univariate times series in situations where the magnitude of the initial
condition of the series is non-negligible. We demonstrate that the asymptotic
size and/or local power properties of these tests are extremely sensitive to the
initial condition. Straightforward modifications to the trend tests are suggested,
based around the use of trimmed data, which are demonstrated to greatly reduce
this sensitivity. |
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09/02 |
Giuseppe Cavaliere
Anders Rahbek
A. M. Robert Taylor
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Co-integration rank tests under conditional heteroskedasticity
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Abstract
In this paper we analyse the properties of the conventional Gaussian-based co-integrating rank tests of Johansen (1996) in the case where the vector of series
under test is driven by possibly non-stationary, conditionally heteroskedastic
(martingale difference) innovations. We first demonstrate that the limiting null
distributions of the rank statistics coincide with those derived by previous authors
who assume either i.i.d. or stationary martingale difference innovations. We
then propose wild bootstrap implementations of the co-integrating rank tests
and demonstrate that the associated bootstrap rank statistics replicate the first-
order asymptotic null distributions of the rank statistics. We show that the
same is also true of the corresponding rank tests based on the i.i.d. bootstrap of
Swensen (2006). The wild bootstrap, however, has the important property that,
unlike the i.i.d. bootstrap, it preserves in the re-sampled data the pattern of
heteroskedasticity present in the original shocks. Consistent with this, numerical
evidence suggests that, relative to tests based on the asymptotic critical values
or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small
samples under a variety of conditionally heteroskedastic innovation processes.
An empirical application to the term structure of interest rates is also given. |
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09/01 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Robust methods for detecting multiple level breaks
in autocorrelated time series
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Abstract
In this paper we propose tests for the null hypothesis that a time series process
displays a constant level against the alternative that it displays (possibly) multiple
changes in level. Our proposed tests are based on functions of appropriately standardised sequences of the differences between sub-sample mean estimates from the
series under investigation. The tests we propose differ notably from extant tests for
level breaks in the literature in that they are designed to be robust as to whether the
process admits an autoregressive unit root (the data are I(1)) or stable autoregressive
roots (the data are I(0)). We derive the asymptotic null distributions of our proposed
tests, along with representations for their asymptotic local power functions against
Pitman drift alternatives under both I(0) and I(1) environments. Associated estimators of the level break fractions are also discussed. We initially outline our procedure
through the case of non-trending series, but our analysis is subsequently extended
to allow for series which display an underlying linear trend, in addition to possible
level breaks. Monte Carlo simulation results are presented which suggest that the
proposed tests perform well in small samples, showing good size control under the
null, regardless of the order of integration of the data, and displaying very decent
power when level breaks occur. An empirical application of the methods proposed
in this paper suggests that the majority of the stock price series which comprise the
NASDAQ 100 index display level breaks. |
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[Revised to become No. 10/01 above]
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2008
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08/05 |
Tassos Magdalinos
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Mildly explosive autoregression under weak and strong dependence
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08/04 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Testing for unit roots and the impact of quadratic trends, with an application to relative primary commodity prices
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08/03 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Testing for unit roots in the presence of uncertainty over both the trend and initial condition
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Abstract
We provide a joint treatment of two major problems that surround testing for
a unit root in practice, namely uncertainty as to whether or not a linear deterministic
trend is present in the data, and uncertainty as to whether the initial
condition of the process is (asymptotically) negligible or not. In earlier work
[Harvey, Leybourne and Taylor, 2008] we proposed methods to deal with trend
uncertainty when the initial condition is assumed to be (asymptotically) negligible,
together with methods to deal with uncertainty over the initial condition
when the form of the trend function was taken as known. In each case we recommended
a simple union of rejections-based decision rule. In the first case rejecting
the unit root null whenever either of the quasi-differenced (QD) detrended or QD
demeaned augmented Dickey-Fuller [ADF] unit root tests yields a rejection, and
in the second case if either of the QD and OLS detrended/demeaned ADF tests
rejects. Both approaches were shown to work well. In this paper we extend these
procedures to allow for both trend and initial condition uncertainty, proposing a
four-way union of rejections decision rule based on the QD and OLS demeaned
and the QD and OLS detrended ADF tests. This is shown to work well but
to lack power, relative to the best available test, in some scenarios. A modification
of the basic union, based on auxiliary information including linear trend
pre-test statistics, is proposed and shown to deliver significant improvements. A
by-product of our analysis is that the power functions of the associated trend
function pre-tests are shown to be heavily dependent on the initial condition.
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08/02 |
David Harris
David I. Harvey
Stephen J. Leybourne
Nikolaos D. Sakkas
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Local asymptotic power of the Im-Pesaran-Shin
panel
unit
root test and the impact of initial
observations
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08/01 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Seasonal unit root tests and the role of initial
conditions
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Abstract
In the context of regression-based (quarterly) seasonal unit root tests, we examine the impact of initial conditions (one for each quarter) of the process on
test power. We investigate the behaviour of the OLS detrended HEGY seasonal
unit root tests of Hylleberg et al. (1990) and the corresponding quasi-differenced
(QD) detrended tests of Rodrigues and Taylor (2007), when the initial conditions
are not asymptotically negligible. We show that the asymptotic local power of
a test at a given frequency depends on the value of particular linear (frequency-specific)
combinations of the initial conditions. Consistent with previous findings
in the non-seasonal case (see, inter alia, Harvey et al., 2008, Elliott and Muller,
2006), the QD detrended test at a given spectral frequency dominates on power
for relatively small values of this combination, while the OLS detrended test
dominates for larger values. Since, in practice, the seasonal initial conditions
are not observed, in order to maintain good power across both small and large
initial conditions, we extend the idea of Harvey et al. (2008) to the seasonal case,
forming tests based on a union of rejections decision rule; rejecting the unit root
null at a given frequency (or group of frequencies) if either of the relevant QD
and OLS detrended HEGY tests rejects. This procedure is shown to perform
well in practice, simultaneously exploiting the superior power of the QD (OLS)
detrended HEGY test for small (large) combinations of the initial conditions.
Moreover, our procedure is particularly adept in the seasonal context since, by
design, it exploits the power advantage of the QD (OLS) detrended HEGY tests
at a particular frequency when the relevant initial condition is small (large)
without imposing that same method of detrending on tests at other frequencies. |
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2007
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07/06 |
David I. Harvey
Stephen J. Leybourne
Bin Xiao
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A powerful test for linearity when the order of integration is unknown
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Abstract
In this paper we propose a test of the null hypothesis of time series linearity against a nonlinear alternative, when uncertainty exists as to whether or not the series contains a unit root. We provide a test statistic that has the same limiting null critical values regardless of whether the series under consideration is generated from a linear I(0) or linear I(1) process, and is consistent against nonlinearity of either form, being asymptotically equivalent to the efficient test in each case. Finite sample simulations show that the new procedure has better size control and offers substantial power gains over the recently proposed robust linearity test of Harvey and Leybourne (2007). |
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07/05 |
Richard J. Smith
A. M. Robert Taylor
Tomas del Barrio Castro
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Regression-based seasonal unit root tests
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Abstract
The contribution of this paper is three-fold. Firstly, a characterisation theorem
of the sub-hypotheses comprising the seasonal unit root hypothesis is presented
which provides a precise formulation of the alternative hypotheses against which
regression-based seasonal unit root tests test. Secondly, it proposes regressionbased
tests for the seasonal unit root hypothesis which allow a general seasonal
aspect for the data and are similar both exactly and asymptotically with respect
to initial values and seasonal drift parameters. Thirdly, limiting distribution
theory is given for these statistics where, in contrast to previous papers in the
literature, in doing so it is not assumed that unit roots hold at all of the zero
and seasonal frequencies. This is shown to alter the large sample null distribution
theory for regression t-statistics for unit roots at the complex frequencies,
but interestingly to not affect the limiting null distributions of the regression
t-statistics for unit roots at the zero and Nyquist frequencies and regression Fstatistics
for unit roots at the complex frequencies. Our results therefore have
important implications for how tests of the seasonal unit root hypothesis should
be conducted in practice. Associated simulation evidence on the size and power
properties of the statistics presented in this paper is given which is consonant
with the predictions from the large sample theory. |
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07/04 |
David Harris
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Testing for a unit root in the presence of a possible break in trend
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Abstract
In this paper we consider the issue of testing a time series for a unit root in
the possible presence of a break in a linear deterministic trend at some unknown
point in the series. We propose a break fraction estimator which, in the presence
of a break in trend, is consistent for the true break fraction at rate Op(T^-1) when
there is either a unit root or near-unit root in the stochastic component of the
series. In contrast to other estimators available in the literature, when there is no
break in trend, our proposed break fraction estimator converges to zero at rate
Op(T^-1/2). Used in conjunction with a quasi difference (QD) detrended unit root
test that incorporates a trend break regressor in the deterministic component,
we show that these rates of convergence ensure that known break fraction null
critical values are applicable asymptotically. Unlike available procedures in the
literature this holds even if there is no break in trend (the true break fraction is
zero), in which case the trend break regressor is dropped from the deterministic
component and standard QD detrended unit root test critical values then apply.
We also propose a second testing procedure which makes use of a formal pre-test
for a trend break in the series, including a trend break regressor only where the
pre-test rejects the null of no break. Both procedures ensure that the correctly
sized (near-) efficient unit root test that allows (does not allow) for a break in
trend is applied in the limit when a trend break does (does not) occur. |
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07/03 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Unit root testing in practice: dealing with uncertainty over the trend and initial condition
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Abstract
In this paper we focus on two major issues that surround testing for a unit root
in practice, namely: (i) uncertainty as to whether or not a linear deterministic
trend is present in the data, and (ii) uncertainty as to whether the initial
condition of the process is (asymptotically) negligible or not. In each case simple
testing procedures are proposed with the aim of maintaining good power
properties across such uncertainties. For the first issue, if the initial condition
is negligible, quasi-differenced (QD) detrended (demeaned) Dickey-Fuller-type
unit root tests are near asymptotically efficient when a deterministic trend is
(is not) present in the data generating process. Consequently, we compare a
variety of strategies that aim to select the detrended variant when a trend is
present, and the demeaned variant otherwise. Based on asymptotic and finite
sample evidence, we recommend a simple union of rejections-based decision rule
whereby the unit root null hypothesis is rejected whenever either of the detrended
or demeaned unit root tests yields a rejection. Our results show that this approach
generally outperforms more sophisticated strategies based on auxiliary
methods of trend detection. For the second issue, we again recommend a union
of rejections decision rule, rejecting the unit root null if either of the QD and
OLS detrended/demeaned Dickey-Fuller-type tests rejects. This procedure is also
shown to perform well in practice, simultaneously exploiting the superior power
of the QD (OLS) detrended/demeaned test for small (large) initial conditions. |
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07/02 |
Giuseppe Cavaliere
Anders Rahbek
A. M. Robert Taylor
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Testing for co-integration in vector autoregressions
with non-stationary volatility
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Abstract
Many key macro-economic and financial variables are characterised by permanent changes
in unconditional volatility. In this paper we analyse vector autoregressions with nonstationary
(unconditional) volatility of a very general form, which includes single and
multiple volatility breaks as special cases. We show that the conventional rank statistics
of Johansen (1988,1991) are potentially unreliable. In particular, their large sample distributions
depend on the integrated covariation of the underlying multivariate volatility
process which impacts on both the size and power of the associated co-integration tests,
as we demonstrate numerically. A solution to the identified inference problem is provided
by considering wild bootstrap-based implementations of the rank tests. These do not
require the practitioner to specify a parametric model for volatility, nor to assume that
the pattern of volatility is common to, or independent across, the vector of series under
analysis. The bootstrap is shown to perform remarkably well in practice. |
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07/01 |
David I. Harvey
Stephen J. Leybourne
Bin Xiao
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A powerful test for linearity when the order of integration is unknown
[Revised to become No.
07/06 above]
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Abstract
In this paper we propose a test of the null hypothesis of time series linearity against a nonlinear alternative, when uncertainty exists as to whether or not the series contains a unit root. We provide a test statistic that has the same limiting null critical values regardless of whether the series under consideration is generated from a linear I(0) or linear I(1) process, and is consistent against nonlinearity of either form, being asymptotically equivalent to the efficient test in each case. Finite sample simulations show that the new procedure has good size control and offers substantial power gains over the recently proposed robust linearity test of Harvey and Leybourne (2007). |
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2006
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06/06 |
Tae-Hwan Kim
Paul Mizen
Alan Thanaset
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Forecasting changes in UK interest rates
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Abstract
Making accurate forecasts of the future direction of interest rates is a vital element when
making economic decisions. The focus on central banks as they make decisions about the future
direction of interest rates requires the forecaster to assess the likely outcome of committee
decisions based on new information since the previous meeting. We characterize this process
as a dynamic ordered probit process that uses information to decide between three possible
outcomes for interest rates: an increase, decrease or no-change. When we analyze the predictive
ability of two information sets, we find that the approach has predictive ability both in-sample
and out-of-sample that helps forecast the direction of future rates. |
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06/05 |
Tassos Magdalinos
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On the inconsistency of the unrestricted estimator
of the information matrix near a unit root
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Abstract
The unrestricted estimator of the information matrix is shown to be inconsistent for an autoregressive process with a root lying in a neighbourhood of unity with radial length proportional or smaller than 1/n, i.e. a root that takes the form rho=1+c/n^alpha, alpha>=1. In this case the information evaluated at rho-hat_n converges to a non-degenerate random variable and contributes to the asymptotic distribution of a Wald test for the null hypothesis of a random walk versus a stable AR(1) alternative. With this newly derived asymptotic distribution the above Wald test is found to improve its performance. A non local criterion of asymptotic relative efficiency based on Bahadur slopes has been employed for the first time to the problem of unit root testing. The Wald test derived in the paper is found to be as efficient as the Dickey Fuller t ratio test and to outperform the non studentised Dickey Fuller test and a Lagrange Multiplier test. |
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06/04 |
Giuseppe Cavaliere
A. M. Robert Taylor
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Testing for a change in persistence in the presence of non-stationary volatility
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Abstract
In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0) to I(1) behaviour or vice versa, of, inter alia, Kim (2000). We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical application to a variety of measures of U.S. price inflation data is provided. |
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06/03 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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Testing for a unit root when uncertain about the
trend
[Revised to become No. 07/03 above]
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Abstract
In this paper we consider the issue of testing for a unit root when it is uncertain as to whether or not a linear deterministic trend is present in the data. The Dickey-Fuller-type tests of Elliott, Rothenberg and Stock (1996), based on (local) GLS detrended (demeaned) data, are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. We consider a variety of strategies which aim to select the demeaned variant when a trend is not present and the detrended variant otherwise. Asymptotic and finite sample evidence demonstrates that some sophisticated strategies which involve auxiliary methods of trend detection are generally outperformed by a simple decision rule of rejecting the unit root null whenever either the GLS demeaned or GLS detrended Dickey-Fuller-type tests reject. We show that this simple strategy is asymptotically identical to a sequential testing strategy proposed by Ayat and Burridge (2000). Moreover, our results make it clear that any other unit root testing strategy, however elaborate, can at best only offer a rather modest improvement over the simple one. |
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06/02 |
David I. Harvey
Stephen J. Leybourne
Nikolaos D. Sakkas
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Panel unit root tests and the impact of initial observations
[Revised to become No. 08/02 above]
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Abstract
In this paper we show that panel unit root tests based on OLS detrending have inferior power relative to tests based on GLS detrending when the deviations of the initial observations from the deterministic components of the series are small. This ranking, however, is reversed for larger deviations. We propose a hybrid panel unit root test that captures the desirable power features of both approaches across the range of initial conditions. |
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06/01 |
David I. Harvey
Stephen J. Leybourne
A. M. Robert Taylor
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A simple, robust and powerful test of the trend hypothesis
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Abstract
In this paper we develop a simple test procedure for a linear trend which does not require knowledge of the form of serial correlation in the data, is robust to strong serial correlation, and has a standard normal limiting null distribution under either I(0) or I(1) shocks. In contrast to other available robust linear trend tests, our proposed test achieves the Gaussian asymptotic local power envelope in both the I(0) and I(1) cases. For near-
I(1) errors our proposed procedure is conservative and a modification for this situation is suggested. An estimator of the trend parameter, together with an associated confidence interval, which is asymptotically efficient, again regardless of whether the shocks are I(0) or I(1), is also provided. |
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Recent Publications by Granger Centre Internal Fellows
Forthcoming
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DISCUSSION PAPERS
RECENT PUBLICATIONS |
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 Cavaliere, G., Rahbek, A. and Taylor, A. M. R., "Bootstrap determination of the co-integration rank in VAR models",
Econometrica, forthcoming
Cavaliere, G., Georgiev, I. and Taylor, A. M. R., "Wild bootstrap of the mean in the infinite variance case", Econometric
Reviews, forthcoming
Cavaliere, G., Taylor, A. M. R. and Trenkler, C., "Bootstrap co-integration rank testing: the role of deterministic
variables and
initial values in the bootstrap recursion", Econometric Reviews, forthcoming
Chambers, M., Ercolani, J. and Taylor, A. M. R., "Testing for seasonal unit roots by frequency domain regression",
Journal of Econometrics, forthcoming
del Barrio Castro, T, Osborn, D. and Taylor, A. M. R., "On augmented HEGY tests for seasonal unit roots",
Econometric Theory, forthcoming
Eberhardt, M., Helmers, C. and Strauss, H., "Do spillovers matter when estimating private returns to R&D?", Review of
Economics and Statistics, forthcoming
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Testing for unit roots in the presence of uncertainty over both the
trend and initial condition", Journal of Econometrics, forthcoming
Lee, K.C. and Shields, K., " Decision-making in hard times: what is a recession, why do we care and when do we
know
we are in one?", North American Journal of Economics and Finance, forthcoming
Rodrigues, P. M. M. and Taylor, A. M. R., "The flexible Fourier form and local GLS de-trended unit root tests", Oxford
Bulletin of Economics and Statistics, forthcoming
Smeekes, S. and Taylor, A. M. R., "Bootstrap union tests for unit roots in the presence of nonstationary volatility",
Econometric Theory, forthcoming
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2012
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Unit root testing under a local break in trend", Journal of
Econometrics, 167, 140-167
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2011
Ahmad, A. H., Harvey, D. I. and Pentecost, E. J., "Exchange rate regime verification: an alternative method of testing
for regime changes", Economics Letters, 113, 96-98
Cavaliere, G., Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Testing for unit roots in the presence of a possible
break in trend and non-stationary volatility", Econometric Theory, 27, 957-991
Eberhardt, M. and Teal, F., "Econometrics for grumblers: a new look at the literature on cross-country growth empirics",
Journal of Economic Surveys, 25, 109-155
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Testing for unit roots and the impact of quadratic trends, with an
application to relative primary commodity prices", Econometric Reviews, 30, 514-547
Marsh, P. W., "Saddlepoint and estimated saddlepoint approximations for optimal unit root tests", Econometric Theory,
27, 1026-1047
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2010
Cavaliere, G., Rahbek, A. and Taylor, A. M. R., "Bootstrap sequential determination of the number of common
stochastic trends under conditional heteroskedasticity", Estudios de Economia Aplicada, 28, 519-552
Cavaliere, G., Rahbek, A. and Taylor, A. M. R., "Co-integration rank testing under conditional heteroskedasticity",
Econometric Theory, 26, 1719-1760
Cavaliere, G., Rahbek, A. and Taylor, A. M. R., "Testing for co-integration in vector autoregressions with non-stationary
volatility", Journal of Econometrics, 158, 7-24
Clements, M. P. and Harvey, D. I., "Combining probability forecasts", International Journal of Forecasting, 27, 208-223
Clements, M. P. and Harvey, D. I., "Forecast encompassing tests and probability forecasts", Journal of Applied
Econometrics, 25, 1028-1062
Garratt, A. and Lee, K. C., "Investing under model uncertainty: decision based evaluation of exchange rate forecasts in
the US, UK and Japan", Journal of International Money and Finance, 29, 403-422
Harris, D., Harvey, D. I., Leybourne, S. J. and Sakkas, N. D., "Local asymptotic power of the Im-Pesaran-Shin panel
unit root test and the impact of initial observations", Econometric Theory, 26, 311-324
Harvey, D. I., Kellard, N. M., Madsen, J. B. and Wohar, M. E., "The Prebisch-Singer hypothesis: four centuries of
evidence", Review of Economics and Statistics, 92, 367-377
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Robust methods for detecting multiple level breaks in
autocorrelated time series", Journal of Econometrics, 157, 342-358
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "The impact of the initial condition on robust tests for a linear
trend", Journal of Time Series Analysis, 31, 292-302
Harvey, D. I., Leybourne, S. J. and Xiao, L., "Testing for nonlinear deterministic components when the order of
integration is unknown",Journal of Time Series Analysis, 31, 379-391
Marsh, P. W., "A two-sample nonparametric likelihood ratio test", Journal of Nonparametric Statistics, 22, 1053-1065
Marsh, P. W., "Some geometry for the maximal invariant in time series regressions", Advances and Applications in
Statistical Science, 1, 105-124
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2009
Cavaliere, G. and Taylor, A. M. R., "A note on testing covariance stationarity", Econometric Reviews, 28,
364-371
Cavaliere, G. and Taylor, A. M. R., "Bootstrap M unit root tests", Econometric Reviews,
28, 393-421
Cavaliere, G. and Taylor, A. M. R., "Heteroskedastic time series with a unit root", Econometric Theory, 25, 1228-1227
Clements, M. P. and Harvey, D. I., "Forecast combination and encompassing", in Palgrave Handbook of
Econometrics, Volume 2: Applied Econometrics, eds. Mills, T. C. & Patterson, K., Palgrave Macmillan, pp. 169-198
Garratt, A., Lee, K. C., Mise, E. and Shields, K., "Real time representations of the UK output gap in the presence of
model uncertainty", International Journal of Forecasting, 25, 81-102
Harris, D., Harvey, D. I., Leybourne, S. J. and Sakkas, N. D., "Local asymptotic power of the Im-Pesaran-Shin panel
unit root test and the impact of initial observations", Econometric Theory, 26, 311-324
Harris, D., Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Testing for a unit root in the presence of a possible
break in trend", Econometric Theory, 25, 1545-1588
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Simple, robust and powerful tests of the breaking trend
hypothesis", Econometric Theory, 25, 995-1029
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Unit root testing in practice: dealing with uncertainty over the
trend and initial condition" (with commentaries and rejoinder), Econometric Theory, 25, 587-667
Lloyd, T. A, McCorriston, S., Morgan, C. W., Rayner, A. J. and Weldegebriel, H. T., "Buyer power in UK food retailing:
a ‘first-pass’ test", Journal of Agricultural and Food Industrial Organisation, 7, Issue 1, Article 5
Marsh, P. W., "Comment on 'Unit root testing in practice: dealing with uncertainty over the trend and initial
condition' [Econometric Theory, 25, 587-636]", Econometric Theory, 25, 637-643
Marsh, P. W., "The properties of Kullback-Leibler divergence for the unit root hypothesis", Econometric Theory, 25,
1662-1681
Smith, R. J., Taylor, A. M. R. and del Barrio Castro, T., "Regression-based seasonal unit root tests", Econometric
Theory, 25, 527-560
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2008
Cavaliere, G. and Taylor, A. M. R., "Bootstrap unit root tests for time series with non-stationary volatility",
Econometric Theory, 24, 43-71
Cavaliere, G. and Taylor, A. M. R., "Testing for a change in persistence in the presence of non-stationary volatility",
Journal of Econometrics, 147, 84-98
Cavaliere, G. and Taylor, A. M. R., "Time-transformed unit root tests for models with non-stationary volatility", Journal
of Time Series Analysis,
29, 300-330
Garratt, A., Lee, K. C., Mise, E. and Shields, K., "Real time representations of the output gap", Review of Economics
and Statistics, 90, 792-804
Garratt, A., Lee, K. C. and Vahey, S., "Real time probability forecasts of UK macro events", National Institute
Economic Review, 203, 78-90
Harris, D., McCabe, B. P. M. and Leybourne, S. J., "Testing for long memory", Econometric Theory, 24, 143-175
Harvey, D. I., Leybourne, S. J. and Taylor, A. M. R., "Seasonal unit root tests and the role of initial conditions",
Econometrics Journal, 11, 409-442
Harvey, D. I., Leybourne, S. J. and Xiao, B., "A powerful test for linearity when the order of integration is
unknown", Studies in Nonlinear Dynamics and Econometrics, 12, Issue 3, Article 2
Lloyd, T. A. and Morgan, C. W., "Market power in UK food retailing", EuroChoices, 6, 20-28
Marsh, P. W., "Conditional information in projections of Gaussian vectors", Communications in Statistics, 38, 332-339
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