testing in unobserved components models

Journal of Forecastin`

J[ Forecast[ 19\ 0Ð08 "1990#

Copyright Þ 1990 John Wiley + Sons\ Ltd[

Testing in Unobserved ComponentsModels

ANDREW HARVEY�

University of Cambridge, UK

ABSTRACT

This article reviews recent work on testing for the presence of non!stationaryunobserved components and presents it in a uni_ed way[ Tests againstrandom walk components and seasonal components are given and it isshown how the procedures may be extended to multivariate models andmodels with structural breaks[ Many of the test statistics have an asymptoticdistribution belonging to the class of generalized Crame�rÐvon Mises dis!tributions[ A test for the number of common trends\ or equivalently\ co!integrating vectors\ is also described[ Copyright Þ 1990 John Wiley + Sons\Ltd[

KEY WORDS Crame�rÐvon Mises distribution^ co!integration^ Kalman _ltersmoother^ locally best invariant test^ seasonality^ stochastictrend^ structural time series model

INTRODUCTION

There is a vast econometric literature on testing for unit roots[ At the most basic level this entailstesting the null hypothesis that the coe.cient of a _rst!order autoregressive\ AR"0#\ process isone against the alternative that it is less than one[ More generally\ the test is of the null hypothesisthat an AR"p# process contains a unit root against the alternative that it is stationary[ This is theaugmented DickeyÐFuller "ADF# test[ The same test can be used to test the validity of a givenco!integrating relationship\ that is\ one in which the values of the coe.cients are pre!speci_ed[The null hypothesis is that the co!integrating relationship does not hold[ Finally\ tests of thenumber of co!integrating relationships within a set of series may be carried out as in Johansen"0877#[

Unit root testing is normally carried out within a framework of autoregressive models[ Auto!regressions are popular in economics because they are easy to _t\ though they are not withouttheir drawbacks^ see Harvey "0886#[ An alternative approach is to use structural time seriesmodels "STMs#\ as described in Harvey "0878#\ Gersch and Kitagawa "0885#\ and Young "0873#[

� Correspondence to] Andrew Harvey\ Faculty of Economics and Politics\ University of Cambridge\ Austin RobinsonBuilding\ Sidgwick Avenue\ Cambridge CB2 8DD\ UK[

1 A[ Harvey

Copyright Þ 1990 John Wiley + Sons\ Ltd[ J[ Forecast[ 19\ 0Ð08 "1990#

These models are formulated in terms of unobserved components such as trends\ seasonals\ and

cycles[ Multivariate models with common trends may be set up[ Common trends imply co!

integration\ and in some instances a common trends formulation may yield the more natural

economic interpretation[

The most interesting testing issues in structural time series models concern testing the null

hypothesis that a particular component is deterministic against the alternative that it is stochastic

and non!stationary[ The simplest case is the random walk plus noise\ or local level\ model[ The

null hypothesis is that the variance of the disturbance driving the random walk is zero\ so that it

reduces to a constant[ In a sense this can be regarded as the {dual| of the DickeyÐFuller test in

the AR"0# model in that the null is that the process is stationary and the alternative is that it is

non!stationary\ whereas in the DickeyÐFuller test the opposite is true[ "The duality can also be

seen in the context of moving average models since the local level model is equivalent to an

MA"0# in _rst di}erences and the test of a zero variance driving the random walk corresponds

to a test that the MA parameter is strictly non!invertible[# The test for the presence of a random

walk component can be used to assess the validity of a given co!integrating relationship[ The

null hypothesis is that there is co!integration[

The aim of this article is to review recent work on testing for non!stationary unobserved

components and to present it in a uni_ed way[ It should be stressed that while these tests arise

naturally in the context of STMs\ they can be employed quite generally[ One particularly nice

feature of the tests is that many of them have limiting distributions which can be regarded as

belonging to the class of Crame�rÐvon Mises distributions[ Indeed the general form of this class\

given in the paper\ is speci_cally motivated by the tests described here[

The plan of the paper is as follows[ The next section describes the test for a random walk

component[ The main references are MacNeill "0867#\ Nyblom and Ma�kela�inen "0872#\ and

Nyblom "0875\ 0878#[ This test is then extended to deal with situations where the stationary

component is not white noise\ but is serially correlated[ Two approaches are described[ The _rst

is non!parametric\ since it does not involve specifying and estimating a model\ while the second

is parametric[ The non!parametric approach is due to Kwiatkowski et al[ "0881#\ hereafter

referred to as {KPSS|[ Parametric approaches based on AR models and STMs are discussed in

Leybourne and McCabe "0883# and Harvey and Streibel "0886# respectively[ The section con!

cludes by comparing the di}erent approaches and looking at the complications caused by the

presence of other non!stationary components\ such as seasonals[ The third section extends the

random walk tests to multivariate series and sets out the common trends test developed by

Nyblom and Harvey "1999#[ Tests for stochastic seasonality are set out in the fourth section[ A

non!parametric test\ along the lines of KPSS\ was proposed by Canova and Hansen "0883#\ who

contrasted it with the seasonal unit root test of Hylleberg et al[ "0881#[ A test for a stochastic slope\

studied in Nyblom and Harvey "0887#\ is also described[ The section concludes by suggesting how

the various distributions within the Crame�rÐvon Mises class might be classi_ed[ The _fth section

follows Busetti and Harvey "0888# in showing how the tests are a}ected by structural breaks in

a series\ which are modelled by dummy variables[ Although the form of the test statistics is

unchanged\ their asymptotic distributions are altered[ However\ the additive properties of the

Crame�rÐvon Mises distribution suggest a simpli_ed test which is much easier to implement[

Comparison may be made with earlier work by Perron "0878# on the e}ect of breaks on unit

root tests[ The way in which the multivariate tests of the third section are a}ected by breaks is

also described\ following Busetti "0888#[ The _nal section presents conclusions[


Testin` in Unobserved Components Models 2

TESTS FOR THE PRESENCE OF A RANDOM WALK COMPONENT

Univariate random walk plus noise model

Consider a univariate unobserved components model consisting of a random walk plus noise fora set of observations\ yt]

yt �mt ¦ ot\ mt �mt−0 ¦ ht t� 0\ [ [ [ \T "0#

where the h?ts and o?ts are mutually and serially uncorrelated Gaussian disturbances with variancess1

h and s1o respectively[ When s1

h � 9 the random walk becomes a constant level[ Testing thishypothesis is non!standard and this motivated Nyblom and Ma�kela�inen "0872# to consider mostpowerful invariant "MPI# tests[ These tests depend on the signalÐnoise ratio\ q�s1

h:s1o \ but

maximizing power near q�9 leads to the locally best invariant test "LBI# and this does notdepend on q[

The LBI test of the null hypothesis that s1h � 9\ against the alternative that s1

h × 9\ can beformulated as

h�T−1 sT

i�0 $si

t�0

et%1

>s1 × c "1#

where

et �yt−y¹\ s1 �T−0 sT

t�0

"yt−y¹#1

and c is a critical value[ In fact\ one initially obtains a form of the statistic with the summationsrunning in reverse\ that is\ from t� i to T\ but it is easily seen that the two statistics are identical[The test can also be interpreted as a one!sided Lagrange multiplier "LM# test[ DickeyÐFuller isa Wald test[ Nyblom and Ma�kela�inen "0872\ p[ 748# give a table of critical values for h[ Bearingin mind that they use a normalizing factor of T−0 instead of T\ the 4) critical value for h is9[351 for sample sizes of both 49 and 099[

The asymptotic distribution of the statistic is found by _rst observing that the partial sum ofdeviations from the mean converges weakly to a standard Brownian bridge\ that is\

s−0T−0:1 sðTrŁ

s�0

escB"r# "2#

where ðTrŁ is the largest integer less than or equal to Tr and B"r# �W"r#−rW"0#\ r $ ð9\0Ł\ withW" = # being a standard Wiener process or Brownian motion[ Hence

hc g0

9

B"r#1 dr "3#

since s1 :p

s1[ This is the Crame�rÐvon Mises distribution\ denoted as CvM[ It is su.cient for theobservations to be independent and identically distributed to yield this asymptotic distribution^see Nabeya and Tanaka "0877#[

The Crame�rÐvon Mises distribution can be represented by a series expansion of independentx1"0# variables\ that is\

3 A[ Harvey


Table I"a#[ Critical values for Crame�rÐvon Mises statistic with k degrees of freedom\ CvM"k#

k0 1 2 3 4 5 6 7 8 09 00

09) 9[236 9[596 9[730 0[952 0[167 0[376 0[582 0[875 1[985 1[184 1[3814) 9[350 9[637 0[999 0[126 0[354 0[575 0[892 1[005 1[215 1[422 1[6280) 9[632 0[963 0[248 0[512 0[763 1[006 1[242 1[473 1[700 2[924 2[145

CvM� g0

9

B"r#1 dr� s�

j�0

"pj#−1x1j "0# "4#

See Nyblom "0878#[ Percentage points for the Crame�rÐvon Mises distribution have been tabu!lated by MacNeill "0867#\ Nyblom and Ma�kela�inen "0872#\ Nyblom "0875#\ and Kwiatkowskiet al[ "0881#[ The 4) critical value is 9[350^ see Table I"a# with k�0[

Time trend

If a linear time trend is included in equation "0# so that

yt �mt ¦bt¦ ot t� 0\ [ [ [ \T "5#

the test statistic\ h1\ is as in equation "1# except that it is formed from the OLS residuals from aregression on a constant and time[ The partial sum of residuals weakly converges to a second!level Brownian bridge\ denoted B1" = #\ where\ as in McNeill "0867#\

B1"r# �W"r#−rW"0# ¦ 5r"0−r# 60

1W"0#−g

0

9

W"s# ds7 "6#

Then

h1c g0

9

B1"r#1 dr "7#

We will refer to this asymptotic distribution as a second!level Crame�rÐvon Mises distribution\and denote it as CvM1[ In the case of any ambiguity the distribution in equation "3# will bereferred to as CvM0[ Again there is a series expansion representation[ The weights are obtainedby changing "pj#−1 to l−1

j \ where l1j−0 � 1jp and l1j is the root of tan"l:1# � l:1 on"1jp\1" j¦ 0#p#\j� 0\1\ [ [ [Critical values are inTable I"b#[

Non!parametric correction for serial correlation

Now suppose that the model is extended so that ot is any indeterministic stationary process[ Inthis case the asymptotic distribution of the NM test statistic remains the same if s1 is replaced bya consistent estimator of the long!run variance

s1L � lim

T:�T−0E $0s

T

t�0

ot11

%� limT:� $ s

T−0

t� −"T−0# 00−=t=T 1 g"t#%� s

�

t� −�

g"t# "8#



where g"t# is the autocovariance of ot at lag t[ Kwiatkowski et al[ "0881# construct such anestimator non!parametrically as

s1L"l# �T−0 s

T

t�0

e1t ¦ 1T−0 s

l

t�0

w"t\l# sT

t�t ¦ 0

etet−t � g¼"9# ¦ 1 sl

t�0

w"t\l#g¼"t# "09#

where w"t\l# is a weighting function\ such as w"t\l# � 0−t:"l¦ 0#\ t� 0\ [ [ [ \ l[ In what followsthis statistic will be referred to as KPSS"l#[

Parametric test

Leybourne and McCabe "0883# attack the problem of serial correlation by introducing laggeddependent variables into the model[ The test statistic obtained after removing the e}ect of thelagged dependent variables is then of the same form as equation "1#[ Leybourne and McCabeshow that their test is consistent to the order T\ the sample size\ compared to the KPSS consistencywhich is of order T:l[ The practical implication\ as demonstrated in their Monte Carlo results"Tables 1 and 2#\ is a gain in power[ However\ more calculation is involved since the coe.cientsof the lagged dependent variables are estimated under the alternative hypothesis and this requiresnumerical optimization[

Since we are testing for the presence of an unobserved component it seems natural to workwith structural time series models[ If the process generating the stationary part of the model wereknown\ the LBI test for the presence of a random walk component could be constructed[ Harveyand Streibel "0886# derive such a test and show how it is formed from a set of {smoothing errors|[A general algorithm for calculating these statistics is the Kalman _lter smoother "KFS# asformulated by de Jong "0877#^ see also Harvey\ Koopman\ and Penzer "0887#[ The smoothingerrors are\ in general\ serially correlated but the form of this serial correlation may be deducedfrom the speci_cation of the model[ Hence a "parametric# estimator of the long!run variancemay be constructed and used to form a statistic which has a Crame�rÐvon Mises distribution\asymptotically\ under the null hypothesis[ An alternative possibility is to use the innovations\that is\ the standardized one!step!ahead prediction errors\ calculated assuming that m9 is _xed[No correction is then needed and\ although the test is not strictly LBI\ its asymptotic distributionis the same and the evidence presented in Harvey and Streibel "0886# suggests that\ in smallsamples\ it is more reliable in terms of size[

As in the LeybourneÐMcCabe test\ the nuisance parameters in the STM need to be estimatedin practice and this is best done under the alternative hypothesis[ This has the compensatingadvantage that since there will often be some doubt about a suitable model speci_cation\ esti!mation of the unrestricted model a}ords the opportunity to check its suitability by the usualdiagnostics and goodness of _t tests[ Once the nuisance parameters have been estimated\ the teststatistic is calculated from the innovations or the smoothing errors with s1

h set to zero[

US GNP and investment

A model consisting of a random walk plus drift trend and a stochastic cycle provides a good _tto the logarithm of quarterly seasonally adjusted US GNP "y# over the period 0840 Q0 to 0874Q3[ The speci_cation of the stochastic cycle is

$ct

c�t%�f $cos lc sin lc

−sin lc cos lc% $ct−0

c�t−0%¦ $kt

k�t% t� 0\ [ [ [ \T "00#

5 A[ Harvey


where lc is frequency in radians and kt and k�t are two mutually uncorrelated white noisedisturbances with zero means and common variance s1

k[ The period corresponding to lc isP� 1p:lc[ The reduced form is a constrained ARMA"1\0# process with the autoregressivepolynomial having complex roots[

The ML estimates of the hyperparameters are

P½ � 10[56\ f½ � 9[83\ s½ k � 9[9943\ s½ h � 9[9967

The test statistic for the null hypothesis that s1h � 9\ taken from Harvey and Streibel "0886#\ is

1[45\ which is greater than the 4) critical value of 9[038[The same model also _ts US Investment "i# well with

P½ � 08[63\ f½ � 9[78\ s½ k � 9[9342\ s½ h � 9[9050

The cycle parameters are similar to those in GNP\ except that the cycle is relatively stronger asindicated by the fact that the ratio sk:sh is 1[70 compared with 9[58 for GNP[ The null hypothesisthat s1

h � 9 is again rejected with the test statistic equal to 9[066[The KPSS"3# and KPSS"01# statistics for GNP are 9[130 and 9[008 respectively[ Thus KPSS"3#

rejects the null hypothesis while KPSS"01# does not[ For Investment KPSS"3# is 9[984 andKPSS"01# is 9[967 so neither rejects[ These results are symptomatic of the low power which canarise with KPSS[ If a plausible model can be found the parametric test is to be preferred[

Testing in the presence of seasonality

An evolving seasonal pattern is often a feature of monthly or quarterly data[ Within a structuraltime series model framework\ seasonality is modelled by a non!stationary component which hasthe property that

S"L#gt ½MA"s−1# "01#

where S"L# � 0¦L¦= = =¦Ls−0 is the seasonal summation operator and s is the number ofseasons[ In an ARIMA framework it is also widely accepted that seasonal e}ects are associatedwith the seasonal summation operator "not with the seasonal di}erence operator#[

The trigonometric form of stochastic seasonality has proved e}ective in modelling a widevariety of seasonal movements^ see Harvey "0878\ ch[ 1#[ In this case

gt � sðs:1Ł

j�0

gj\t "02#

where each gj\t is generated by

$gj\t

g�j\t%� $cos lj sin lj

−sin lj cos lj% $gj\t−0

g�j\t−0%¦ $vj\t

v�j\t%j� 0\ [ [ [ \ ðs:1Ł

t� 0\ [ [ [ \T"03#

where lj � 1pj:s is frequency\ in radians\ and vj\t and v�j\t are two mutually uncorrelated whitenoise disturbances with zero means and common variance s1

v\ which is the same for all j[ For s

even ðs:1Ł �s:1\ while for s odd\ ðs:1Ł � "s−0#:1[ For s even\ the component at j� s:1 collapsesto

gj\t � gj\t−0 cos lj ¦vj\t j� s:1 "04#

The reduced form of gt is as in equation "01#[



The seasonal component may be combined with a trend component\ mt\ and a stationarycomponent or components[ For example\ we may have

yt �mt ¦ gt ¦ct ¦ ot t� 0\ [ [ [ \T "05#

where ct is a cycle as in equation "00#[ If mt is a random walk "plus drift# a test of the nullhypothesis that s1

h is zero may be carried out as before[ The calculations remain valid when s1v is

zero\ meaning that the seasonal component is deterministic[

UK Personal Disposable Income

The random walk plus drift and stochastic cycle model used for US GNP was augmented by atrigonometric seasonal component in order to _t it to UK Personal Disposable Income[ Anirregular component was also included[ The data are for the period 0844 Q0 to 0882 Q1[ TheML estimates of the hyperparameters are

P½ � 25[15\ f½ � 9[83\ s½ k � 9[9977\ s½ h � 9[9958\ s½v � 9[9910\ s½ o � 9[9977

The test statistic for the null hypothesis that s1h � 9 is 9[145\ which is greater than the 4) critical

value of 9[038[The KPSS test can be carried out if the observations are _rst transformed so as to be stationary

under the null hypothesis[ This entails taking seasonal sums so as to give S"L#yt\t� 1\ [ [ [ \T[The LeybourneÐMcCabe test procedure can also be carried out using seasonal sums[ However\approximating the dynamics by lagged dependent variables and a single MA term may proveunsatisfactory because of the presence of a high!order moving average part in S"L#"gt ¦ ot#[ ThisMA will lie close to the invertibility boundary if\ as is often the case\ the seasonal pattern isslowly changing so that s1

v is small relative to s1o [

MULTIVARIATE MODELS

Testing against a multivariate random walk

If yt is a vector containing N time series the Gaussian multivariate local level model is

yt �mt ¦ ot ot ½NID"9\So#"06#

mt �mt−0 ¦ ht ht ½NID"9\Sh# t� 0\ [ [ [ \T

where So is an N×N positive de_nite "p[d[# matrix[ We seek a test of the null hypothesis thatSh � 9^ in other words a test of whether there is any non!stationarity in the system[ Nyblom andHarvey "1999# show that an LBI test can be developed against the homogeneous alternativeSh �qSo^ see Harvey "0878\ section 7[2#[ The test has the rejection region

h"N# � trðS−0CŁ × c "07#

where

C�T−1 sT

i�0 $si

t�0

et% $si

t�0

et%?

and S�T−0 sT

t�0

ete?t "08#

where et � yt−y¹[ Under the null hypothesis\ the limiting distribution of equation "07# is what is

7 A[ Harvey


Table I"b#[ Critical values for Crame�rÐvon Mises statistic withtime trend\ CvM"k#

k0 1 2 3

09) 9[008 9[100 9[185 9[2664) 9[038 9[136 9[221 9[3120) 9[107 9[218 9[317 9[410

sometimes called the Crame�rÐvon Mises distribution with N degrees of freedom\ denotedCvM"N#[ As in the univariate case\ this distribution may be represented by a series expansion[Speci_cally\

h"N# :d

g0

9

B"u#?B"u# du� s�

k�0

"pk#−1x1k"N# "19#

where B is a standard vector Brownian bridge of dimension N[ Selected percentage points forh"N# are given in Table I"a#^ these are published in Nyblom "0878\ table 1#\ with the _gures for4\ 6\ 8 and 00 obtained directly from the author[ Table I"b# gives critical values when a timetrend is present[

Although the test maximizes the power against homogeneous alternatives\ it is consistentagainst all non!null S?hs[ This follows from the result that T−0h"N# has a non!degenerate limitingdistribution[ In fact the limiting distribution depends only on the number of non!zero q?js\ thatis\ the rank of Sh[ The series expansion "without the time trend# is

T−0h"N# :d

tr 0 s�

k�0

"pk#−1vkv?k1−0

0 s�

k�0

"pk#−3vkv?k1 K� 0\ [ [ [ \N "10#

where vk is a K×0 vector such that vk ½NID"9\IK#[The h"N# test can be generalized along the lines of the KPSS test quite straightforwardly by

replacing S in equation "07# by

S"m# � st�l

t� −l

w"t\l#G¼ "t# "11#

where G¼ "t# is the sample autocovariance matrix at lag t\ that is\

G¼ "t# �T−0 sT

t�r ¦ 0

ete?t−t "12#

where w"t\l# is a weighting function as in equation "09#[ Note that G¼ "−t# �G¼ "t#?[ We will referto this test as KPSS"N^l#[ The matrix in equation "11# now estimates the long!run covariancematrix of the noise process[ If l�o"T0:3#\ the limiting distribution of KPSS"N^l# is CvM"N#under the null hypothesis[

Parametric adjustments can also be made by the procedure outlined for univariate models[This requires estimation under the alternative hypothesis\ but is likely to lead to an increase in



power[ In some cases it may be reasonable to estimate key parameters from univariate models\without taking account of the correlations between components in di}erent series[

Common trends and co!integration

If Sh is of full rank\ the multivariate local level model of equation "06# can be written in thefollowing way]

yt � $y0t

y1t%� $IK 9

UUUU Ir% $m$

t

m¹ t%¦ $o0t

o1t% t� 0\ [ [ [ \T "13#

where yt is partitioned into a K×0 vector y0t and an r×0 vector y1t\ with r�N−K\ ot is similarlypartitioned\ UUUU is an r×K matrix of coe.cients and the K×0 vector m$

t follows a multivariaterandom walk

m$t �m$

t−0 ¦ h$t h$

t ½NID"9\S$h# "14#

where h$t is a K×0 vector and S$

h is a K×K positive de_nite matrix[ The r×0 vector m¹ t alsofollows a multivariate random walk with the disturbance vector h¹ t having a p[d[ covariancematrix S¹ h[

If the rank of the covariance matrix Sh is K³N\ an appropriate ordering of the series enablesthe model to be written in the form "13# with m$

t containing K common levels\ or trends\ and S$h

continuing to be positive de_nite while S¹ h � 9 and thus m¹ t 0m¹ 9[ This means that y1t is co!integrated with y0t\ the latter being a set of variables which are not themselves co!integrated[Now consider a set of r linear combinations of the observations\ Ayt\ where A is an r×N matrixpartitioned as A� "A0\ A1# with A1 being r×r[ Then

Ayt �A0y0t ¦A1y1t � "A0 ¦A1UUUU#m$t ¦A1m¹ t ¦A0o0t ¦A1o1t t� 0\ [ [ [ \T "15#

If A satis_es A0 ¦A1UUUU� 9 and S¹ h � 9\ then Ayt is stationary and the rows of A constitute a setof r co!integrating vectors[ The assumption that S$

h is positive de_nite implies that A1 must benon!singular^ see Saikkonen "0882\ p[ 10#[

Suppose that we wish to test the null hypothesis that S¹ h � 9\ i[e[ y1t is co!integrated with y0t\in model "13#[ If we choose A such that A0 ¦A1UUUU� 9\ then the h"N# test applied to Ayt is LBIagainst alternatives in which S¹ h is proportional to UUUUS00UUUU?¦UUUUS01 ¦S10UUUU?¦S11\ where theS?ijs are blocks of So[ The form of A may come directly from economic theory\ or indirectly"though not necessarily uniquely#\ through UUUU\ because of knowledge of common trends[ Forexample\ suppose there is a single common trend in the logarithms of income "y0#\ consumption"y1# and investment "y2#\ so that N�2\ K�0[ With balanced growth UUUU� "0\0#?\ and if the co!integrating relationships are taken to be y1t−y0t and y2t−y0t\ they correspond to the {great ratios|of consumption and investment to income^ see King et al[ "0880#[

The test statistic can be written as

h"r^A# � tr ð"ASA?#−0ACA?Ł "16#

and its limiting distribution under the null hypothesis is\ of course\ CvM"r#[ The test will tend toreject the null hypothesis not only if S¹ h � 9\ but also if the supposed co!integrating vectors areincorrectly speci_ed so that A0 ¦A1UUUU� 9[ Thus the test can be regarded as a test of the validityof a set of pre!speci_ed co!integrating vectors\ in other words it is a test of whether Ayt is

09 A[ Harvey


stationary[ If the test is modi_ed to allow for serial correlation\ it can be used as a general testfor the validity of a speci_ed co!integrating vector[

GNP and Investment

The test used above can also be used to determine whether GNP and Investment exhibit balancedgrowth\ that is\ they are co!integrated with their di}erence\ i−y\ being a stationary process^ seeKing et al[ "0880#[ Fitting a random walk "without drift# plus a stochastic cycle gives theparameter estimates

P½ � 08[25\ f½ � 9[77\ s½ k � 9[9300\ s½ h � 9[9916

A plot of i−y with the smoothed level shows that the change in the level over time appears rathersmall and this is con_rmed by the 9[166 value for the test statistic which is clearly well below the4) critical value of 9[350[

Suppose now that the coe.cients of the variables in the hypothesized co!integrating relation!ships are not pre!speci_ed[ Since A1 must be non!singular\ equation "15# can be multipliedthrough by A−0

1 [ Then\ if there are no overidentifying restrictions in "15#\ it is tempting to testfor the validity of the supposed co!integrating relationships by regressing y1t on y0t[ However\the h"r^A# test described above will not\ in general\ have its usual CvM"r# asymptotic distributionif formed from the OLS residuals[ Ways of constructing valid tests are described by Shin "0883#\Harris and Inder "0882#\ and Choi and Ahn "0884#[

Testing for a speci_ed number of common trends

Suppose we wish to test the null hypothesis that model "06# has a speci_c number of commontrends against the alternative that it has more[ Formally the test is of

H9 ] rank"Sh# �K against H0 ] rank"Sh# ×K K³N

Let l0−= = =−lN be the ordered eigenvalues of S−0C\ given by

=C−ljS= � 9 j� 0\ [ [ [ \N "17#

The h"N# test statistic is the sum of these eigenvalues\ but it can be shown that when the rank ofSh is K$\ the limiting distribution of T−0h"N# is the limiting distribution of T−0 times the sum ofthe K$ largest eigenvalues[ This suggests basing a test of the hypothesis that rank"Sh# �K on thesum of the N−K smallest eigenvalues\ that is\

h"K\N# � lK ¦ 0 ¦= = =¦ lN K� 0\ [ [ [ \N−0 "18#

Then if K$×K the relatively large values taken by the _rst K$−K of these eigenvalues will tendto lead to the null hypothesis being rejected[ This is the common trends test[ Of course\ if we allowK to be zero in equation "18#\ then h"9\N# � h"N#[

The distribution of the common trends test statistic under the null hypothesis is not of theCrame�rÐvon Mises form but it does depend on functions of Brownian motion[ The seriesexpansion is

h"K\N# :d

s�

k�0

"pk#−1u?kuk−tr 0 s�

k�0

"pk#−2ukv?k1 0 s�

k�0

"pk#−3vkv?k1−0

0 s�

k�0

"pk#−2vku?k1where vk and uk are\ respectively\ K×0 and r×0 vectors which are mutually independent



NID"9\ I#[ The signi_cance points for h"K\N# depend on both K and N and are tabulated inNyblom and Harvey "1999#[

Under the alternative hypothesis\ T−0h"K\N# has a limiting distribution which depends onlyon K$−K[ Hence the test based on h"K\N# is consistent[

As with the h"N# test\ the model may be extended to include time trends[ The test can makeno allowance for the fact that there may be common slopes corresponding to the common levels[In the h"r^A# test this possibility can be accommodated quite naturally since it implies thatAb� 9\ and so Ayt includes no time trend[

A non!parametric adjustment for serial correlation may be made in the common trends test inmuch the same way as was suggested for the h"N# test[ All that needs to be done is to replace theeigenvalues in equation "18# by those of S"l#−0C leading to a test statistic which may be denotedas h"K\N^l#[ Parametric corrections can also be made[

The common trends test is testing the null hypothesis that there are K common trends againstthe alternative that there are more[ Equivalently\ it is testing the null hypothesis that there areN−K co!integrating vectors against the alternative that there are less[ The procedures in Johansen"0877# and Stock and Watson "0877# are also testing for the number of co!integrating vec!tors:common trends[ However\ these are VAR!based tests and the common trends test di}ersfrom them just as the KPSS test di}ers from augmented DickeyÐFuller[ Thus the alternative forh"K\N# is in the direction of fewer co!integrating vectors rather than more[ Furthermore in thenon!parametric version of the test no models are being estimated\ either under the null or thealternative and the alternative hypothesis is not that there should speci_cally be one morecommon trend "or one less co!integrating vector#[

Application to stochastic volatility

Harvey\ Ruiz\ and Shephard "0883# consider a multivariate stochastic volatility "SV# model ofthe form

rit �sioit exp"hit:1# i� 0\ [ [ [ \N\ t� 0\ [ [ [ \T

where ot � "o0t\ [ [ [ \ oNt#? is IID"9\So# and ht � "h0t\ [ [ [ \ hNt#? is a multivariate random walk

ht � ht−0 ¦ ht ht ½NID"9\Sh#

Taking the logarithms of the squared observations gives

yt �v¦ ht ¦ o�t t� 0\ [ [ [ \T

where the ith elements in the N×0 vectors yt\ v\ and o�t are log r1it\ log s1

i and log o1it respectively[

The model is now of the form "06# except that it is not Gaussian[ However\ this makes nodi}erence to the asymptotic distribution of the common trends test statistic[

In the application in Harvey et al[ "0883#\ the r?its are the di}erences of 835 logged dailyexchange rates of the dollar against four other currencies starting on 0 November 0870[ Harveyet al[ "0883# proposed a model based on two common trends[ Computing the h"1\3# statistic foryt gives a value of 9[06[ This is less than the 4) critical value of 9[10 and its prob[ value is 09)[Thus the assumption of two common trends is not rejected[ On the other hand\ the hypothesisof one common trend is rejected since h"0\3# � 0[10 and the 4) critical value is 9[43[

01 A[ Harvey


SEASONALITY AND OTHER COMPONENTS

Test against stochastic seasonality

In the second section we examined how to test against a random walk component when non!stationary seasonal e}ects are present[ This leads on to the question of how to test the hypothesisthat a seasonal component is _xed against the alternative that it is stochastic and non!stationary[

Consider a Gaussian model with a trigonometric seasonal component]

yt �m¦ gt ¦ ot t� 0\ [ [ [ \T "29#

where m is a constant[ If ot is white noise\ the LBI test against the presence of a stochastictrigonometric component at any one of the seasonal frequencies\ lj\ apart from the one at p\ is

vj � 1T−1s−1 sT

t�0 $0st

i�0

ei cos lji11

¦ 0st

i�0

ei sin lji11

% j� 0\ [ [ [ \ ð"s−0#1Ł "20#

where s1 is the sample variance of the OLS residuals from a regression on sines and cosines[Canova and Hansen "0884# show that the asymptotic distribution of this statistic is generalizedCrame�rÐvon Mises with two degrees of freedom[ "Actually Canova and Hansen derive the abovestatistic from a slightly di}erent form of the stochastic cycle model in which the coe.cients of asineÐcosine wave are taken to be random walks[ However\ it is not di.cult to show that themodel as de_ned above leads to the same test statistic[# A joint test against the presence ofstochastic trigonometric components at all seasonal frequencies is based on a statistic obtainedby summing the individual test statistics[ This statistic has an asymptotic distribution which isgeneralized Crame�rÐvon Mises with s−0 degrees of freedom[ "The component at p gives rise toa test statistic which has only one degree of freedom[# Critical values are as in Table I[

As shown in Harvey and Streibel "0886\ appendix#\ the inclusion of a time trend does not a}ectthe asymptotic distribution[ However\ the inclusion of seasonal slopes\ as used\ for example\ byProietti "0887#\ will a}ect the distribution in a similar way to a time trend[

Canova and Hansen show how the above tests can be generalized to handle serial correlationby making a correction similar to that in KPSS[ If the model contains a stochastic trend\ thenthe test must be carried out on di}erenced observations[ A parametric test may be carried outby _tting a STM[ If there is a trend it may be a deterministic trend\ a random walk\ with orwithout a drift\ or a trend with a stochastic slope\ as de_ned in the next sub!section[

If desired a joint test against a random walk and stochastic seasonals can be constructed[ Thetest statistic\ obtained by adding the h statistic to the full set of vj statistics\ will have anasymptotic distribution which is Crame�rÐvon Mises with s degrees of freedom[ If the disturbanceis serially correlated\ the test must be modi_ed as discussed above[

Testing against a stochastic slope

A stochastic trend component is de_ned as

mt �mt−0 ¦bt−0 ¦ ht ht ½NID"9\s1h#

"21#bt �bt−0 ¦ zt zt ½NID"9\s1

z#

where the level and slope disturbances\ ht and zt\ respectively\ are mutually uncorrelated[ We nowconsider how to test H9 ]s1

z � 9 against H0 ]s1z × 9[



Table II[ Asymptotic critical values "×099# of inte!grated random walk test\ z"k#

k 09) 4) 0)

0 9[471 9[797 0[2551 0[990 0[167 0[8102 0[258 0[570 1[2763 0[602 1[944 1[701

If s1h is assumed to be zero the trend\ mt\ is an integrated random walk\ or {smooth trend|[ If ot

is white noise\ the test statistic\ derived in Nyblom and Harvey "0887#\ is

z�0

T3s1sT

t�0 $st

s�0

ss

r�0

er%1

"22#

The derivation of the asymptotic distribution of the integrated random walk test statistic followsalong similar lines to that of h leading to

z:d

g0

9 0gu

9

B1"r# dr11

du "23#

The second!level Brownian bridge comes about because the model under the null hypothesiscontains a time trend as in equation "5#[

As with the random walk test statistics\ there is a series expansion for the right!hand side ofequation "23#[ Table II shows the asymptotic critical values[

A comparison of the power of the test with the test against the random walk plus drift can befound in Nyblom and Harvey "0887#[ Somewhat surprisingly there appears to be little gain inpower from using z[

If the white noise process\ ot\ is serially correlated\ a KPSS!type correction can be made[ Aparametric test can be carried out by _tting a fully speci_ed model[

If s1h × 9\ then\ when s1

z � 9 the trend reduces to a random walk plus drift[ Di}erencing yields

Dyt �bt−0 ¦ ht ¦Dot t� 1\ [ [ [ \T "24#

with ht ¦Dot being invertible[ Thus the KPSS test of the second section of this paper can be usedto test whether bt−0 is a random walk[ If the parametric model is _tted the smoothing errors orinnovations can be used to construct a statistic of the form given in equation "22#\ but\ of course\the distribution will be the same as that of the h statistic "without drift#\ that is CvM[

A family of Crame�rÐvon Mises distributions

The asymptotic distributions described earlier suggest a family of Crame�rÐvon Mises distri!butions\ denoted CvMp ¦ 0"k#\ dependent on degrees of freedom\ k\ and whether or not a constant"p�9# or a time trend "p�0# is _tted[ The multivariate statistic "equation "08##\ has anasymptotic CvM0"N# distribution\ while the CanovaÐHansen test for seasonality is CvM0"s−0#["Recall that when p�9\ the p¦0 subscript is often dropped[# MacNeill "0867\ p[ 320# considers_tting polynomials of degree p¦0 and tabulates CvMp ¦ 0"0# for p� 9\0\ [ [ [ \ 4[ He also gives

03 A[ Harvey


the corresponding critical values for Brownian motion "which could be classed as p�−0\ i[e[no deterministic component is _tted#^ these are 0[085\ 0[545 and 1[677 for 09)\ 4) and 0)respectively[

The test for a stochastic slope\ introduced above\ suggests a further generalization of the familyof Crame�rÐvon Mises distributions\ by bringing in the number of iterated partial sums of theresiduals\ m[ We might therefore extend the terminology to CvMp ¦ 0"k^m#[ Then the standardCrame�rÐvon Mises distribution is CvM0"0^ 0#\ while equation "23# is CvM1"0^1#[

The Crame�rÐvon Mises distribution with p�9\ m�0 and k degrees of freedom may beexpanded as

CvM0"k^ 0# �CvM"k# � s�

j�0

"pj#−1x1j "k# "25#

with the interesting corollary that\ because of the additive property of chi!square distributions\the sum of two independent random variables with distributions is CvM"k0# and CvM"k1# isCvM"k0 ¦k1#[ The same additivity property holds for other members of the family[

It follows from the series expansion in equation "25# that EðCvM"N#Ł �N:5 andVarðCvM"n#Ł �N:34[ As N:�\ each chi!square distribution may be approximated by a normaland so CvM"N# may also be approximated by a normal[ Hence the 4) critical value for large N

may be approximated by N:5¦ 0[534zN:34[ For N�3\ this yields 0[048 as opposed to thevalue of 0[126 given in Table I"a#[ For N�00\ the approximate value is 1[535 as against 1[628[

TESTS WHEN BREAKS ARE PRESENT

The addition of explanatory variables to equation "0# may not change the asymptotic distributionof the h statistic if they satisfy certain conditions\ for example weak dependence conditions ifthey are stochastic[ However\ the dummy variables needed to account for structural breaks dochange the distribution[ Note that omitting such dummies can cause the test to reject the nullhypothesis even though no random walk is present^ see Nyblom "0878#[

Assume there is a structural break in the trend at time t¦0\ and let l� t:T denote the fractionof the sample before the break occurs[ Consider the following models]

ð0Ł yt �mt ¦ dwt ¦ ot "26#

ð1Ł yt �mt ¦bt¦ dmwt ¦ db"wtt# ¦ ot "27#

ð1aŁ yt �mt ¦bt¦ dmwt ¦ ot "28#

ð1bŁ yt �mt ¦bt¦ dbzt ¦ ot "39#

where mt is a random walk\ ot is white noise and

wt � 69 for t¾ t

0 for t× tand zt � 6

9 for t¾ t

t−t for t× t

There is no slope in model ð0Ł and so the only break is in the level[ The other models all containa time trend[ In model ð1Ł there is a structural change in both the level and the slope[ Model ð1aŁcontains a break in the level only while ð1bŁ corresponds to a piecewise linear trend[



LBI test against a random walk

Under Gaussianity the LBI "and one!sided LM# test statistics for H9 ]s1h � 9 against

HA ]s1h × 9 in the above models are of the same form as equation "1#[ They will be denoted as

hi"l#\ i�0\ 1\ 1a\ 1b\ where the subscript i indicates that the residuals depend on the model\ 0\1\ 1a or 1b\ and l denotes that the statistic has been constructed for a speci_c value of thebreakpoint location parameter and that its asymptotic distribution depends on it[

As in the previous section\ the limiting distribution can be derived by looking at the asymptoticproperties of the process followed by the partial sum of residuals[ This will converge to a limitingprocess*de_ned on an underlying Wiener process*that will depend on l and collapse to a"second!level# Brownian bridge when l�9 or l�0[ The asymptotic distribution of hi"l# is thenthe integral on the unit interval of the square of this process[ The form of these distributions isgiven in Busetti and Harvey "1999#[

The upper tail percentage points of the above asymptotic distributions are reported in Busettiand Harvey "1999# for di}erent values of l[ The _gures for l: 9 or l: 0 correspond to thecritical values for the Crame�rÐvon Mises distributions given in Table I[ As expected\ the per!centage points*as functions of l*are symmetric around l�0:1\ which is also the minimumfor models ð0Ł\ ð1Ł and ð1bŁ[

For models ð0Ł and ð1Ł the asymptotic distributions can be characterized in terms of twoindependent Crame�rÐvon Mises distribution[ To see that this is the case\ _rst notice that we canrewrite the statistic as

hi"l# � l1

st

t�0 0st

s�0

e111

t1s1¦ "0−l#1

sT

t�t ¦ 0 0 st

s�t ¦ 0

es11

"T−t#1s1i� 0\1 "30#

since from the OLS orthogonality conditions

sT

s�0

es � sT

s�t ¦ 0

es � 9\

and this implies

st

s�0

es � 9[

Then it is easy to see that for model ð0Ł

es � 6ys−y¹0 for s¾ t

ys−y¹1 for s× t

where

y¹0 � t−0 st

t�0

yt

and

05 A[ Harvey


y¹1 � "T−t#−0 sT

s�t ¦ 0

yt

are the averages in the _rst and second subsamples respectively[ A similar result holds for modelð1Ł where the two sets of residuals are obtained from regressing on a constant and a time trend[Thus the residuals are independent across subsamples and

hi"l#c l1 g0

9

ðBi"r#Ł1 dr¦ "0−l#1 g

0

9

ðB?i"r#Ł1 dr i� 0\1 "31#

where B0" = # and B?0" = # are independent Brownian bridges and B1" = # and B?1" = # are independentsecond!level Brownian bridges[ Hence the statistics are weighted averages of two independentCrame�rÐvon Mises distributions[

This is a very simple way to characterize the asymptotic distribution and it is trivially gen!eralizable to the case of more than one break[ Note that if the breaks are equispaced thedistribution of the statistic "when multiplied by four# converges to the sum of two randomvariables with independent Crame�rÐvon Mises distributions[ Of course\ assuming equispacedbreaks is not appropriate in general[ However\ the same additivity property can be obtained aftera slight modi_cation of the test statistic\ as suggested in the following section[ By doing this thedependence on the parameter l in the asymptotic distribution can be eliminated[

A simpli_ed test

Bearing in mind the additivity property of the Crame�rÐvon Mises distribution noted above\Harvey and Busetti "1999# propose the following test statistics for models ð0Ł and ð1Ł]

h�i �

st

t�0 0st

s�0

es11

t1s1¦

sT

t�t ¦ 0 0 st

s�t ¦ 0

es11

"T−t#1s1i� 0\1 "32#

Thus the weights in equation "31# are eliminated[ The statistics still depend on the location ofthe breakpoint\ but their asymptotic distributions do not since

h�i c 6CvM"1# for i� 0

CvM1"1# for i� 1"33#

Not having to consult a table giving the distribution of the test statistic for all the possiblevalues of l is a big advantage^ compare the unit root tests in Perron "0878#[ Furthermore the testimmediately generalizes to cases where there are several structural breaks[ If there are k breaksat times t0 � l0T³= = =³tk � lkT the test statistic is

h�i "k# � sk ¦ 0

j�0

s

tj

t�tj−0 ¦ 0 0 st

s�tj−0 ¦ 0

es11

"tj−tj−0#1s1

i� 0\1 "34#

where t9 �9 and tk ¦ 0 �T[ The distribution of this statistic converges to a "second!level# gen!eralized Crame�rÐvon Mises distribution with k¦0 degrees of freedom\ that is\ CvMi"k¦ 0#\



i�0\ 1^ see Table I[ The advantage is now even greater since constructing tables for all patternsof k breakpoints would be extremely cumbersome[

How good is this simpli_ed test< Busetti and Harvey "1999# compare the LBI test\ based onhi"l#\ and the simpli_ed test "equation "32##\ in terms of size and power by a Monte Carloexperiment[ For l�9[4 the two tests are the same while for other values of l the size and powerare comparable\ with the LBI being clearly superior only in the region close to the null hypothesisand for the breakpoint near the beginning or end of the sample[ The experiment was repeatedfor a data!generating process with two structural breaks\ and no slope\ with the breaks locatedin a variety of positions[ The conclusions are similar to those reached for the case of a singlebreak\ with the simpli_ed test having a size close to the nominal and power comparable with theLBI test[

Example*~ow of the Nile

Annual data on the volume of the ~ow of the Nile "in cubic metres ×097# is shown in Harvey et

al[ "0887#[ Fitting a mean and computing the test statistic "equation "1## gives a value of h�1[416\indicating a clear rejection of the null hypothesis that there is no random walk component^ theasymptotic 4) critical value is 9[350[ The KPSS test gives the same result with the statistics forl�2 and 6 being 0[099 and 9[624\ respectively[ However\ it is known that the _rst Aswan damwas constructed in 0788 and if a level intervention is included\ neither the LBI nor the simpli_edtest rejects the null hypothesis\ since h0"l# � 9[977 and h�0 � 9[290[ In fact a simple constant levelplus noise model with a break at 0788 provides a good _t to the data[

Multivariate models

Busetti "0888# extends the tests of the third section of this paper to deal with situations wherethere are breaks in some or all of a set of N time series[ He shows that a simpli_ed version of thetest against a multivariate random walk can be constructed by allowing for a break in all theseries at the same point in time^ this statistic has the CvM"1N# asymptotic distribution[

Seasonality

The presence of seasonal dummies will not a}ect the asymptotic distributions of the test statisticsdescribed in this section[ If the seasonal pattern evolves according to a non!stationary process\it can be modelled explicitly as suggested above or rendered stationary by an appropriatetransformation[

The presence of breaks in the level and:or slope will not a}ect the asymptotic distribution ofthe CanovaÐHansen test[ However\ a break in the seasonal pattern will[ The asymptotic dis!tribution of the LBI test statistic could be derived[ This would be tedious to do\ but the simpli_edtest statistic\ based on the residuals obtained before and after the break\ has a CvM"1s−1#asymptotic distribution[ With k breaks in the seasonal pattern the degrees of freedom would bek"s−0#[

CONCLUSIONS

This article has presented a uni_ed treatment of tests against non!stationary unobserved com!ponents[ Many of the statistics have asymptotic distributions belonging to a general class ofCrame�rÐvon Mises distributions\ characterized\ in particular\ according to degrees of freedom[

07 A[ Harvey


Parametric versions of the tests based on _tting structural time series models have particularlyattractive properties\ but\ by making corrections for serial correlation\ either non!parametricallyor by _tting lagged variables\ the tests can be used even if a full unobserved componentsframework is not adopted[

The non!stationarity in question appears under the alternative rather than under the nullhypothesis[ In the unit root tests\ such as augmented DickeyÐFuller\ the situation is reversed\but for unobserved components models this is not the natural way to proceed[ Furthermore\ asthe results in Laybourne and McCabe "0883\ p[ 051# illustrate\ the size and power properties ofunit root tests can be very poor for such models[

ACKNOWLEDGEMENTS

I am grateful to Jukka Nyblom for supplying the critical values in Table I"a# which are notpublished in his paper[ I would also like to thank Fabio Busetti\ David Harris\ Paul Newboldand Steven Leybourne for helpful comments[ The paper was originally written for presentationat the Second International Colloquium on Economic Dynamics and Economic Policy in Brasiliain May 0888[ I|m grateful to the organizers\ Joanilio Teixeira and Francisco Carneiro\ forencouraging me to write the paper and for commenting on it[

REFERENCES

Busetti F[ 0888[ Testing for "common# nonstationary components in multivariate time series with breakingtrends[ Mimeo\ LSE[

Busetti F\ Harvey AC[ 1999[ Testing for the presence of a random walk in series with structural breaks[Journal of Time Series Analysis "to appear#[

Canova F\ Hansen BE[ 0884[ Are seasonal components constant over time< A test for seasonal stability[Journal of Business and Economic Statistics 02] 126Ð141[

Choi I\ Ahn BC[ 0884[ Testing for cointegration in systems of equations[ Econometric Theory 00] 841Ð872[de Jong P[ 0877[ A cross!validation _lter for time series models[ Biometrika 64] 483Ð599[Gersch W\ Kitagawa G[ 0872[ The prediction of time series with trends and seasonalities[ Journal of Business

and Economic Statistics 0] 142Ð153[Harris D\ Inder B[ 0883[ A test of the null hypothesis of co!integration[ In Non!stationary Time Series

Analysis and Co!inte`ration\ Hargreaves C "ed[#[ Oxford University Press] Oxford[Harvey AC[ 0878[ Forecastin`\ Structural Time Series Models and the Kalman Filter[ Cambridge University

Press] Cambridge[Harvey AC[ 0886[ Trends\ cycles and autoregressions[ Economic Journal 096] 081Ð190[Harvey AC\ Ruiz E\ Shephard N[ 0883[ Multivariate stochastic variance models[ Review of Economic

Studies 50] 136Ð153[Harvey AC\ Koopman SJ\ Penzer J[ 0887[ Messy time series[ Advances in Econometrics 02] 092Ð032[Harvey AC\ Streibel M[ 0886[ Testing for nonstationary unobserved components[ Mimeo[Hylleberg S\ Engle R\ Granger CWJ\ Yoo BS[ 0889[ Seasonal integration and co!integration[ Journal of

Econometrics 33] 104Ð127[Johansen S[ 0877[ Statistical analysis of cointegration vectors[ Journal of Economic Dynamics and Control

01] 02Ð43[King ML\ Hillier GH[ 0874[ Locally best invariant tests of the error covariance matrix of the linear

regression model[ Journal of the Royal Statistical Society\ Series B 36] 87Ð091[King RG\ Plosser CI\ Stock JH\ Watson MW[ 0880[ Stochastic trends and economic ~uctuations[ American

Economic Review 70] 708Ð739[



Kitagawa G\ Gersch W[ 0885[ Smoothness Priors Analysis of Time Series[ Springer!Verlag] Berlin[Koopman SJ\ Harvey AC\ Doornik JA\ Shephard N[ 0884[ STAMP 4[9 Structural Time Series Analyser\

Modeller and Predictor[ Chapman and Hall] London[Kwiatkowski D\ Phillips PCB\ Schmidt P\ Shin Y[ 0881[ Testing the null hypothesis of stationarity against

the alternative of a unit root] How sure are we that economic time series have a unit root< Journal ofEconometrics 33] 048Ð067[

Leybourne SJ\ McCabe BPM[ 0883[ A consistent test for a unit root[ Journal of Business and EconomicStatistics 01] 046Ð055[

MacNeill IB[ 0867[ Properties of sequences of partial sums of polynomial regression residuals with appli!cations to tests for change of regression at unknown times[ The Annals of Statistics 5] 311Ð322[

Nabeya S\ Tanaka K[ 0877[ Asymptotic theory of a test for the constancy of regression coe.cients againstthe random walk alternative[ Annals of Statistics 05] 107Ð124[

Nyblom J[ 0875[ Testing for deterministic linear trend in time series[ Journal of the American StatisticalAssociation 70] 434Ð438[

Nyblom J[ 0878[ Testing for the constancy of parameters over time[ Journal of the American StatisticalAssociation 73] 112Ð129[

Nyblom J\ Harvey AC[ 0887[ Testing against smooth stochastic trends[ Mimeo[Nyblom J\ Harvey AC[ 1999[ Tests of common stochastic trends[ Econometric Theory 05 "to appear#[Nyblom J\ Ma�kela�inen T[ 0872[ Comparison of tests for the presence of random walk coe.cients in a

simple linear model[ Journal of the American Statistical Association 67] 745Ð753[Perron P[ 0878[ The Great Crash\ the oil price shock\ and the unit root hypothesis[ Econometrica 46] 0250Ð

0390[Phillips PCB\ Hansen BE[ 0889[ Statistical inference in instrumental variables regression with I"0# processes[

Review of Economic Studies 46] 88Ð014[Proietti T[ 0887[ Seasonal heteroscedasticity and trends[ Journal of Forecastin` 06] 0Ð06[Saikkonen P[ 0882[ Estimation of cointegration vectors with linear restrictions[ Econometric Theory 8] 08Ð

24[Saikkonen P\ Luukkonen R[ 0882[ Testing for a moving average unit root in autoregressive integrated

moving average models[ Journal of the American Statistical Association 77] 485Ð590[Shin Y[ 0883[ A residual based test of the null of cointegration against the alternative of no cointegration[

Econometric Theory 09] 80Ð004[Stock JH\ Watson MW[ Testing for common trends[ Journal of the American Statistical Association 72]

0986Ð0096[Young P[ 0873[ Recursive Estimation and Time Series Analysis[ Springer!Verlag] Berlin[

Author|s bio`raphy]Andrew Harvey is Professor of Econometrics at Cambridge University\ having previously been Professor ofEconometrics at the London School of Economics[ He has held visiting positions at the University of BritishColumbia and the University of California at Berkeley[ He has published widely in statistics\ econometrics\_nance and operations research and has been on the editorial board of a number of journals including theJournal of Econometrics\ Journal of the American Statistical Association\ Journal of Time Series Analysisand Journal of Business and Economic Statistics[ He is the author of well!known textbooks in econometricsand time series analysis and has written a monograph Forecastin`\ Structural Time Series Models and theKalman Filter[

Author|s address]Andrew Harvey\ Faculty of Economics and Politics\ University of Cambridge\ Austin Robinson Building\Sidgwick Avenue\ Cambridge CB2 8DD\ UK[

testing in unobserved components models

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