Temi di discussione

del Servizio Studi

Testing against stochastic trend and seasonality

in the presence of unattended breaks and unit roots

by Fabio Busetti and A. M. Robert Taylor

Number 470 - March 2003

The purpose of the Temi di discussione series is to promote the circulation of working
papers prepared within the Bank of Italy or presented in Bank seminars by outside
economists with the aim of stimulating comments and suggestions.
The views expressed in the articles are those of the authors and do not involve the
responsibility of the Bank.

Editorial Board:
STEFANO SIVIERO, EMILIA BONACCORSI DI PATTI, MATTEO BUGAMELLI, FABIO BUSETTI, FABIO
FORNARI, RAFFAELA GIORDANO, MONICA PAIELLA, FRANCESCO PATERNÒ, ALFONSO ROSOLIA,
RAFFAELA BISCEGLIA (Editorial Assistant).
 7(67,1* $*$,167 672&+$67,& 75(1' $1' 6($621$/,7< ,1 7+(
35(6(1&( 2) 81$77(1'(' %5($.6 $1' 81,7 52276

by Fabio BusettiW and A. M. Robert TaylorWW

$EVWUDFW

This paper considers the problem of testing against stochastic trend and seasonality in the
presence of structural breaks and unit roots at frequencies other than those directly under test,
which we term unattended breaks and unattended unit roots respectively. We show that under
unattended breaks the true size of the Kwiatkowski HW DO (1992) [KPSS] test at frequency zero
and the Canova and Hansen (1995) [CH] test at the seasonal frequencies fall well below the
nominal level under the null with an associated, often very dramatic, loss of power under the
alternative. We demonstrate that a simple modi¿cation of the statistics can recover the usual
limiting distribution appropriate to the case where there are no breaks, provided unit roots
do not exist at any of the unattended frequencies. Where unattended unit roots occur we show
that the above statistics converge in probability to zero under the null. However, computing the
KPSS and CH statistics after pre-¿ltering the data is simultaneously ef¿cacious against both
unattended breaks and unattended unit roots, in the sense that the statistics retain their usual
pivotal limiting null distributions appropriate to the case where neither occurs. The case where
breaks may potentially occur at all frequencies is also discussed. The practical relevance of
the theoretical contribution of the paper is illustrated through a number of empirical examples.
JEL classi¿cation: C12, C22.
Keywords: stationarity tests, structural breaks, pre-¿ ltering, unattended unit roots.
&RQWHQWV
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Score-Based Stationarity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3. Testing with Unattended Structural Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Bias-Corrected Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Tests based on Pre-¿ltered Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4. Unattended Unit Roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6. Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1 Attended and Unattended Structural Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Trending Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7. Empirical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Appendix: proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Tables and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

 Bank of Italy, Economic Research Department.

 University of Birmingham, Department of Economics.
 1
 ,QWURGXFWLRQ

In a recent paper Canova and Hansen (1995) [CH] have developed score-based tests
of the null hypothesis of deterministic seasonality against the alternative of unit roots at
some or all of the seasonal, but not zero, spectral frequencies. Their tests are similar in
spirit to the those against a zero frequency unit root proposed in Kwiatkowski HW DO (1992)
[KPSS]. The statistics upon which these tests are based retain pivotal Cramér-von Mises
limiting null distributions under mixing and seasonally heteroskedastic errors YLD the use of
a non-parametric heteroskedasticity and autocorrelation consistent (HAC) covariance matrix
estimator. More recently, Taylor (2003a) has amalgamated these cases to develop score-based
tests against seasonal and/or zero frequency unit roots.

The above VWDWLRQDULW\ tests can also be derived from the general theory on parameter
stability testing in regression models of Nyblom (1989) and Hansen (1992). We provide a
review of these testing procedures in Section 2. Speci¿cally, in the context of a regression of
the time series variable of interest on a set of zero and seasonal frequency spectral indicator
variables, we consider testing the null hypothesis of ¿xed parameters against the alternative
that (at least one of) the parameters on a given subset, say @, of the spectral indicators evolve
as random walks, such that the process admits unit root behaviour at (at least one of) those
spectral frequencies included in @ .

The testing procedures outlined in Section 2 are conducted under the maintained
hypothesis, say M , that the regressors associated with those spectral frequencies not included
in @ must have ¿xed coef¿cients. In this paper we focus attention on two particular
cases where M is violated, both recognised in the literature to be of considerable practical
relevance. Firstly we consider the case of XQDWWHQGHG VWUXFWXUDO EUHDNV, where some or all
of the parameters on the spectral frequency regressors QRW included in @ display a structural
break at some known or unknown point in the sample. Secondly, we consider the case of
XQDWWHQGHG XQLW URRWV , originally highlighted in Hylleberg (1995) and further developed in
Taylor (2003b), where the process admits unit root behaviour at some or all of the spectral
frequencies not included in @ .

4 We thank Andrew Harvey and Cheng Hsiao for helpful comments on an ealier draft. The views ex-
pressed here are those of the authors and do not necessarily represent those of the Bank of Italy. Email:
busetti.fabio@insedia.interbusiness.it, R.Taylor@bham.ac.uk
 8

Since the work of Perron (1989) on US GNP, it has been known that a process which
is stochastically stationary about a deterministic component subject to structural breaks can
display properties very similar to a unit root process. Indeed, Perron (1989) demonstrates
that the conventional augmented Dickey-Fuller [ADF] tests cannot reject the unit root null
hypothesis, even asymptotically, where a broken trend exists. Although consistent against
stationary processes about a broken level, Perron (1989) shows that the power of the ADF
test is vastly reduced relative to the case where no break occurs. Perron (1989) proposes
modi¿cations of the ADF test constructed so as to be invariant to deterministic breaks at a
known point while Zivot and Andrews (1992), LQWHU DOLD, allow for an unknown break point.
More recently, Busetti and Harvey (2001) have considered the effects of level and trend breaks,
of the type discussed in Perron (1989), on the stationarity tests of KPSS. They show that failing
to account for these breaks renders the KPSS tests severely over-sized, while KPSS-type tests
which explicitly allow for (are invariant to) such breaks will require a different set of critical
values.

Empirical results in Ghysels (1990) suggest that seasonal level shifts are a common
phenomenon in quarterly US macroeconomic time series and argues that such shifts will have
non-trivial consequences on testing for seasonal unit roots. Accordingly, Ghysels (1994),
Smith and Otero (1997) and Franses and Vogelslang (1998), LQWHU DOLD, have extended the
work of Perron (1989) to the case of testing the null hypothesis of seasonal unit roots in series
which undergo seasonal dummy level shifts. Although the standard tests of the seasonal
unit root null hypothesis of Hylleberg HW DO (1990) are shown to be consistent against
stochastically stationary processes which are subject to such shifts, their ¿nite sample power
is vastly reduced. Indeed, in allowing for seasonal dummy level shifts in quarterly real GDP
for fourteen countries, Franses and Vogelslang (1998) ¿nd considerably less evidence for
seasonal unit roots than from the standard tests. Busetti and Harvey (2003a) consider the
effects of structural breaks at the seasonal (but not zero) frequencies on the seasonal frequency
stationarity tests of CH. Paralleling the arguments in Busetti and Harvey (2001), they show
that the standard CH tests are over-sized in such cases and propose modi¿ed statistics which
are invariant to such breaks.

In Section 3 of this paper we derive representations for the limiting null distributions
of the statistics of Section 2 in cases where there are XQDWWHQGHG EUHDNV, but not unattended
 9

unit roots. We demonstrate that in such cases the true asymptotic size of the tests will fall
below the nominal level. This contrasts sharply with cases where some or all of the parameters
on the spectral frequency regressors included in @ display a structural break (we shall term
these DWWHQGHG VWUXFWXUDO EUHDNV in what follows), considered in Busetti and Harvey (2001,
2003a). Monte Carlo simulations reported in Section 5 demonstrate that the asymptotic theory
provides a useful approximation to the behaviour of the statistics in ¿nite-samples and that
the under-sizing phenomenon effects an associated, often very dramatic, loss of ¿nite sample
power under the alternative.

The assumption that only those spectral frequencies included in @ may display unit root
behaviour is clearly untenable in practice, since one cannot know which spectral frequencies
admit a unit root. If one did, one would of course have no need for unit root testing. Busetti
and Taylor (2002) and Taylor (2003b) demonstrate that the statistics against unit roots in @
converge in probability to zero under the null in such cases. Although the tests based on these
statistics are consistent in such cases, their ¿nite sample power is vastly diminished relative
to the case where the maintained hypothesis holds Taylor (2003b) provides considerable
numerical and empirical evidence to illustrate this point. Interestingly, this problem does not
arise with the tests of Hylleberg HW DO (1990) since here one may test for the null hypothesis
of a unit root at a particular spectral frequency (or frequencies) whilst remaining ambivalent
as to the existence or otherwise of unit roots at those frequencies not under test.

From a practical perspective, the impact of unattended breaks and unattended unit roots
on the stationarity tests of Section 2 are just as important as those arising from attended breaks.
As noted above, the latter effect over-sized tests which diverge, even if there are no unit
roots in @. @, this is not an
Since these are tests of the null hypothesis of stationarity in
unreasonable outcome: it draws the practitioner’s attention to non-stationary behaviour in @.
However, care must still be taken since routine differencing in response to rejection of the
null will yield an over-differenced series if the non-stationarity is due to an attended break.
In contrast, unattended breaks and/or unattended unit roots at those frequencies not included
in @2 vastly diminish the likelihood that we can reject the null of stationarity when analysing
series with unit roots in @. The modelling implications of failing to recognise and account
for such behaviour is well-known and further analysis of the data could only be expected to
yield spurious inferences.
 10

In light of these practical problems, we suggest a variety of remedial actions. In the case
where we have unattended structural breaks, but no unattended unit roots, we show in Section
3 that some simple modi¿cations, based on bias-correction, to the existing test statistics can
recover the usual limiting null distributions that pertain where there are no breaks. However,
where there are unattended unit roots we demonstrate in Section 4 that these statistics will
still converge in probability to zero under the null. Pre-¿ltering the data eliminates potential
unattended unit roots and has been shown to be highly effective by Taylor (2003b). We show
that pre-¿ltering simultaneously provides an ef¿cacious remedy for the problem of unattended
structural breaks. We therefore recommend the use of pre-¿ltering as a means of obtaining
robust tests in the presence of unattended breaks and/or unattended unit roots.

In Section 6 we discuss two generalisations. The ¿rst allows for the case where there
are both unattended DQG attended deterministic structural breaks, while the second allows for
deterministic trending variables. The former is of considerable empirical importance because
seasonal level shifts, of the type discussed in the literature on tests of the null of seasonal unit
roots above, may potentially effect both attended and unattended breaks, and hence allow
for cases where structural breaks occur at all frequencies. The practical relevance of our
theoretical results is illustrated in Section 7 where we apply the existing tests against stochastic
trend and seasonality and the modi¿ed versions of these tests suggested in this paper to data
on U.K. marriages and U.K. (log) consumers’ expenditure on tobacco. In the case of UK
marriages we show that tests based on the appropriately modi¿ed statistics yield considerably
more evidence against stationarity than do the existing tests, while the (log) expenditure series
illustrates the case where both attended and unattended breaks occur. Section 8 concludes the
paper, while an Appendix contains proofs of our main results.
 11

 6FRUH%DVHG 6WDWLRQDULW\ 7HVWV

Consider the scalar process +| , | ' c    c A , observed with constant seasonal periodicity
r, generated according to the model

(1) 7   W n | c |  UU(Efc j 2c

+| ' ( | |

where f j2 4, and (7 | is an r-vector of conventional seasonal indicator (dummy)

variables with associated, potentially time-varying, parameter vector  W| .

Rather than work directly with the formulation given in (1), it will prove expedient in
what follows to adopt the following bijective reparameterization of (1) in terms of the spectral
indicator variables of Hannan HW DO (1970) that is,

(2) +| ' ~|  | n | c | ' c    c A

 
In (2), ~| ' 5fc| c 5
c| c c 5 
dr*2oc| is an r-vector of spectral indicator variables that is, 5fc| ' c
5&c| ' EULt 2Z&|*rc t? 2Z&|*r c & ' c c rW, where rW ' r*2   if r is even or dr*2o if


r is odd, while 5r*2c| ' E| if r is even. The ¿rst element of ~| , 5fc| , corresponds to the
zero frequency, bf  f, while the &th pair of spectral indicators, 5 , correspond to the &th &c|

harmonic seasonal frequency, b  2Z&*r, & ' c c rW . Where r is even, the last element of
&

~ , 5 2 , corresponds to the Nyquist frequency, b 2  Z . The exact relationship between 

| r* c| r* |

and  W is given by  ' -3 W, where the full rank Er  r matrix - ' E~ c    c ~  .
| | | r

Consider now the partition of ~| into E~ | c ~2 |  where ~| c ~2| are disjoint sub-vectors
(containing regressors corresponding to disjoint frequencies, but with no particular ordering)
of dimension r and r2 respectively, such that r n r2 ' r Let

 
@  ' & c  ' c 2c c ? c G 5& c| belongs to ~|

be the set of indices of the spectral indicator variables corresponding to the frequencies
included in ~| c  ' c 2 Using this partition we re-write (2), with an obvious notation, as

(3) +| ' ~ |  | n ~2 |  2| n | 

 12

We specify the parameters on ~| to evolve as a (possibly degenerate) vector random walk
that is,

(4)  | '  c|3 n # |c # |  UU(Efc j 2(W 

where (W ' j 2# ( , ( a non-null diagonal positive semi-de¿nite matrix of dimension
Er  r. We assume that the initial value  f is ¿xed, with no loss of generality. We also
assume that # | and | are mutually orthogonal.

In this paper our attention focuses on testing the null hypothesis Mf G j2# ' f against
the alternative M G j2# : f in (3)-(4). As demonstrated in, LQWHU DOLD, CH under M , i+| j
admits unit roots at those spectral frequencies associated with the non-zero diagonal elements
of (. However, the statistical properties of tests of Mf against M will depend crucially on
the speci¿cation of  2|, and it is an exploration of this issue which provides the focus for this
paper. In the remainder of this Section we follow the implicit assumption adopted in CH, KPSS
and Taylor (2003a) that  2| '  2f , a ¿xed r2 -vector of coef¿cients, for all |. This assumption
constitutes the maintained hypothesis, M , discussed in Section 1. Recall that M imposes
the condition that the regressors associated with those spectral frequencies not included in @
must have ¿xed coef¿cients, regardless of the behaviour of the coef¿cient vector  |. If both
Mf and M hold the entire parameter vector  | is ¿xed and we have what will be termed the
QXOO PRGHO

(5) +| ' ~ |  f n ~2 |  2f n | c | ' c c A (

that is, i+| j is a stochastically stationary process around ¿xed (deterministic) seasonal effects.

In Section 3 we consider the effects on the tests of Section 2 when we allow some
or all of the elements of  2| to display a deterministic structural break at some known or
unknown point in the sample. In Section 4 we subsequently consider the case where  2|
follows a random walk. In Section 6.1 we will also discuss the case where deterministic
structural breaks may occur in any of the elements of  | , not just in  2|, thereby allowing for
the interesting case where structural breaks may occur at all frequencies. Indeed, notice that
deterministic structural breaks in the seasonal dummy parameter vector  W| of (1) will effect
structural breaks in either  | or  2| , or both.

Denote by e| , | ' c c Ac the Ordinary Least Squares (OLS) residuals from regressing
SA
j2 ' A 3
+| on ~| and denote by je2 their sample variance, e |'
e2| . As demonstrated in Taylor
 13

(2003a), a locally most powerful invariant (LMPI) test of Mf G j2# ' f against M G j 2# : f in
(3)-(4), with M maintained, rejects for large values of the statistic
3 [A [| [|   4
[ [E A o@Se 5&c e 5&c e
/' /7 &  |'  '  ' Fc
(6) C& 2 D
& MV & MV
A je
2

where & '  if & ' f or & ' r*2, & ' 2 otherwise. Under Mf , / weakly converges to
the right member of (9) below, and is R EA  under M  see Taylor (2003a). Furthermore, it is
not dif¿cult to show that the statistic / of (6) for testing against all of the spectral frequencies
identi¿ed by ~| is precisely the sum of the individual LMPI statistics /
7 & for testing at each
of these frequencies in turn. As we shall see below, this additive property also holds on the
asymptotic null distributions of the tests.

If we weaken the assumptions on i| j to allow for the possibility of weak dependence
and heterogeneity, we will require a non-parametrically modi¿ed version of / in order to
obtain a statistic with a pivotal limiting null distribution. Speci¿c (mixing) conditions on
i j in this case are quite involved
|  full details are given in Taylor (2003a) and are not
repeated here. We denote by l the “long run variance” of the process ~| | c i.e. l
S 

SA
,6A <" A 3. E A
|'
~| | E |'
~| | c  ' c 2c and assume that l is positive de¿nite.
Then from CH and Taylor (2003a), the modi¿ed statistic is given by
% # | $# | $&
[
A
[ [
(7) O ' A 32A o@Se l
3

~ e ~
 e c
|' ' '

where, following CH, we have de¿ned the HAC estimator of l as

3
[
A

(8)  '
l & E*7A K  E c
'3A n

 E  ' A 3 SA
where & E is a kernel function and K ~ |e| e|3 ~ c|3 is the estimator of
|'  n

the autocovariance of ~| | at lag 

For r ' , O of (7) is the # > statistic of KPSS (Equation (13), p.165). Furthermore, for r
even and ~| ' E|, O coincides with the OZ statistic of CH (Equation (17), p.6). Moreover,
for ~| ' 5&c| , & ' c c rW , O is the statistic O Z&*r of CH (Equation (17), p.6), while if ~|
 14

contains all but the ¿rst element of ~| (i.e. ~2| ' ), then O coincides with Os of CH (Equation
(15), p.5).

Suitable choices for the kernel & E may be found in, LQWHU DOLD, Andrews (1991, p.821),
while setting the bandwidth parameter 7A such that 7A $ 4 and 7A *A *2 $ f ensures that
 $R l under both the null and local alternatives see Elliott and Stock (1994). Under the
l
above conditions, Taylor (2003a) establishes the result that under Mf G j 2# ' f
] 
(9) O, r Eo  r Eo_oc
f

where “,” denotes weak convergence and r Eo is a r -dimensional standard Brownian
bridge process. The right member of (9) is a Cramér-von Mises distribution with r degrees
of freedom see Harvey (2001) for further discussion on the Cramér-von Mises family of
distributions. In what follows we will denote this distribution by  Er , upper tail critical
values from which, for   r  2, are provided in Table 1 of CH (page 5).
If we supplant the mixing conditions above by the assumption that i|j is a linear
process, then l is a diagonal matrix containing the spectral generating function of |, denoted
as } Ebc evaluated at each of the frequencies b& c & 5 @. To be precise, we assume that
| '  Eu"| , i"|j a MD sequence satisfying the conditions in Stock (1994, p. 2745), and
S
 Eu   n " '
 u a polynomial in u, the conventional lag operator, u& +|  +|3& ,
S
& ' fc c , satisfying: (i)  Ei Ti 2Z&*rj 9' f, for all & 5 @ , and (ii) "
 '
 m  m 4.2
The ¿rst condition rules out a zero in the spectrum of i| j at any of the frequencies included
in @ , while the second ensures that poles do not exist in the spectrum of i| j. A leading case
which satis¿es the above conditions is the class of ¿nite-order stationary and invertible ARMA
processes. Following Busetti and Harvey (2003a) we also consider the following alternative
non-parametrically modi¿ed statistic
3 [A [| [|   4
[ [E A o@Se 5 e 5&c e F
O ' O&  C&
&c 
|'  '  '
(10) Dc
& MV & MV
A 2e} Eb& 

5 Equivalent conditions must also hold in the mixing case. However, it should be noted that the possibility
of periodic heteroscedasticity, allowed under the mixing conditions, is not permitted.
 15

SA 3
where & is de¿ned below (6), e
} Eb&  ' ' 3A n & E*7A e
 e E  ULt b& c and e e E  '
A 3 SA e|e|3 is the sample autocovariance of the OLS residuals at lag  Notice that
|' n

O and O of (7) coincide if r ' . The statistic O is intuitively appealing since, unlike O, it
retains the additivity property of / . Under the linear process conditions on i j, together with
|

the conditions on & E and 7 stated below (8), O has the usual  Er  limiting distribution
A

of (9) under Mf . Under M , O and O are both  EA*7  see, LQWHU DOLD, CH, KPSS, Taylor
R A 

(2003a) and Busetti and Harvey (2003a). Finally we note that while / is an exact LMPI test of
Mf G j 2# ' f against M G j2# : f in (3)-(4), neither O nor O are locally optimal in any formal
sense, even asymptotically.

 7HVWLQJ ZLWK 8QDWWHQGHG 6WUXFWXUDO %UHDNV

Thus far we have assumed that  2| of (3) is a ¿xed r2 -vector of parameters. Suppose
now that the process i+| j is generated by (3)-(4) but that  2| is generated according to

(11)  2| ' E 2f n w_| Ek c

where _|Ek '  E|  dA koc k 5 Efc  is an indicator variable reÀecting the occurrence of a
deterministic structural break, and w is an r2 -vector of coef¿cients. It is clear that (11) violates
M because, unless w ' f,  2| is not ¿xed for all |. Notice that the end cases of k ' f
and k '  are excluded since these would not yield a within-sample break in the coef¿cient
vector  2| . In Section 6.1 we will subsequently discuss the case where deterministic breaks
may occur in both  2| and  | .

The DGP (3)-(4)-(11) displays potentially non-stationary stochastic seasonality at the

frequencies corresponding to ~| , exactly as in Section 2, but now also displays deterministic
seasonality with a level shift in the seasonal pattern at period dkA o at the frequencies
corresponding to ~2| . Observe that the zero frequency regressor 5fc| '  can be in either
~| or ~2| . Moreover, the components in the vector of level shifts w ' Ewc c wr2  need not be
equal and may contain zeros, so that not all frequencies in ~2| need show a break. Moreover,
if w ' f,  2| '  2f, and so the test based on / of (6) is a LMPI test for Mf G j2# ' f against
M G j 2# : f.

9 f, such that M is violated, and we compute the

Consider now the case where w '
statistic / of (6) without taking into account the occurrence of a level shift that is, there is an
 16

unattended structural break. In this case the test based on / is not LMPI and the asymptotic
distribution of / under Mf is no longer of the form given in (9). Indeed, as we now show in
Proposition 1, it is equal to the right member of (9) multiplied by the factor 6Ekc wc j2 , which
e2 of j 2  Consequently, the pivotal limiting
arises from an asymptotic bias in the estimator j
distribution in (9) can be retained by bias-correcting / as we shall do in Section 3.1, making
use of the work on breakpoint estimation by Bai (1994, 1997).

Proposition 1 /HW +| EH JHQHUDWHG E\  DQG  7KHQ XQGHU Mf G j2# ' fc

] 
(12) / , 6Ekc wc j  2
r Eo r Eo _oc
f

ZKHUH r Eo LV DV GH¿QHG DERYH DQG

(13) 6Ekc wc j 2  ' E n KEkc w*j 23 c

[A

KEkc w  ' *4 A 3 E_| Ek  k E~2 | w 

2 2
(14)
A <"
|'

5HPDUN : e2 , KEkc w, and hence 6Ekc wc j2 ,

Notice from (14) that the asymptotic bias in j
are symmetric in k. Consequently, the representation in (12) for k and that for k
7  E  k
coincide. Where w ' f, the case of no unattended breaks, KEkc f ' f and hence (12) reduces
to the right member of (9), as should be expected.

5HPDUN : As an example of KEkc w , consider the case where either ~2| '  or ~2| ' E| .
Here the bias is given by KEkc w ' kE  kw2 

9 f, and correspondingly,
The asymptotic bias, KEkc w, is strictly positive whenever w '
6Ekc wc j 2 will therefore always be less than unity. This will clearly yield an undersized
test cf. (9). Notice therefore that unattended breaks in the parameters associated with the
spectral regressors in @2 have the opposite effects from unattended breaks in the parameters
associated with @ . In the latter case Busetti and Harvey (2001, 2003a) show that tests against
unit roots in @ are RYHUVL]HG. As a simple illustration of Proposition 1, consider the case
of a model with two seasons (r ' 2), ~| ' Ec E|  , where ~| ' E| , corresponds to


frequency Z, and ~2| '  corresponds to the zero frequency. Suppose that, as a particular
case of (11), a shift in  2| , equal to the standard deviation of the errors, occurs in the middle
of the sample, so that 6EfDc jc j2  ' fH. In this case if / , the test against a unit root at
 17

frequency Z , is run at the nominal (asymptotic) DI level, it would display a true asymptotic
size of hi E j  2DI. In the same example, but with w ' 2j the
feS*fH 3

true asymptotic size would be hi E  feS*fDj  feI Deviations below the
nominal level clearly increase as w is increased. Notice that the same results also apply if we
interchange ~| and ~2| that is, the case where we are testing against a zero frequency unit
root in the presence of an unattended break at frequency Z .

Now consider the case of testing against a unit root at frequency zero in a quarterly model
(r ' e). In this case ~| ' , corresponds to the zero frequency, and ~2| ' E5 c| c E|  ,
where 5c| ' EULt Z|*2c t? Z|*2 , corresponds to the seasonal frequencies. Suppose that
there is a neglected break of size w ' EwW c wW c wW  j at the seasonal frequencies. Table 3.1 gives
the asymptotic biases and true asymptotic size for the test against a zero frequency unit root,
/ , run at the nominal DI level, for various values of the break fraction k and break magnitudes
wW .

As in the preceding example, we see that the true asymptotic size of / deviates further
below the nominal level as the break magnitude wW increases, other things being equal.
Moreover, the tabulated results clearly demonstrate that the degree of size distortion is the
worse the closer is the break date to the middle of the sample. Recall from Remark 1 that
6Ekc wc j 2  is symmetric in k.

6 The asymptotic 8( level upper tail critical value from the FyP +4, distribution is 0.463.
 18

Table 3.1: Asymptotic Bias and Size of /

KEkc w 6Ekc wc j2  |oe r5e

k ' *H 0.219 0.821 f2b
wW '  k ' * e 0.375 0.727 fb
k ' * 2 0.500 0.667 f
k ' *H 0.875 0.533 ffe
wW ' 2 k ' * e 1.500 0.400 ff
k ' * 2 2.000 0.333 fff
k ' *H 3.500 0.222 fff
wW ' e k ' * e 6.000 0.143 fff
k ' * 2 8.000 0.111 fff
If we switch ~| and ~2| in the quarterly example considered above, so that our object
is to test against unit root behaviour at some or all of the seasonal frequencies, and we have
a neglected shift at frequency zero, the true asymptotic size of a test run at the nominal DI
level4 is obtained from the  E distribution as hE E  ff*6Ekc wc j 2c where
6Ekc wc j 2 ' *E n kE  kw2 *j2  For example, if k ' *H and w ' j, the true asymptotic
size for a nominal DI level test is approximately 2DIc while if k ' *2 this drops to HI.
A tabulation of asymptotic sizes and biases for a range of values of k and w in this case is not
provided here, as the patterns are qualitatively similar to those seen in Table 3.1. Full details
can be obtained from the authors, on request.

3.1 %LDV&RUUHFWHG 7HVWV

The bias in the asymptotic null distribution of / detailed in Proposition 1 may be easily
corrected and this can be done in a number of ways which we detail below. Each modi¿cation
is shown to deliver a statistic with the usual pivotal  Er limiting distribution under Mf G
j 2# ' f.

Consider ¿rst the case where there is a structural break at an unknown point in the
sample, kf 5 Efc  An asymptotically unbiased estimator of the error variance is obtained
by minimizing the sum of squared residuals over all the possible break dates. Speci¿cally, let
e| Ek denote the OLS residuals from the ¿tted regression

(15) + | ' ~ |e n ~2 | e

 2 n _| Ek~2 | wc

7 The 8( level upper tail asymptotic critical value from the FyP +6, distribution is 1.010.
 19

SA
e2Ek ' A 3
and let j |'
e| Ek2 be the sample variance of the residuals. Denote by kW the
argument that minimizes this variance that is, kW ' @h} ?u j
e2 Ek.
k

e2EkW is an asymptotically unbiased

In the appendix it is shown that, under Mf G j2# ' fc j
estimator of j 2  Indeed, for the case ~2| ' , Bai (1994, 1997) shows that, under Mf , kW is
a VXSHUFRQVLVWHQW estimator of kf c in the sense that it converges to the true value at rate A
Consequently, and noting the asymptotic invariance of the numerator of / of (6) to the level
shift (see the proof of Proposition 1 in the Appendix), an obvious modi¿ed version of / whose
limiting null distribution is  Er  is given by
[A [| [| 
[ A o@Se 5&c e 5
&c
e
(16) / W ' A 32 & |'  '  '
c
& MV je2 EkW 

e2 by je2 EkW 
which is obtained by replacing in (6) the biased estimator of the error variance j

Notice that in (16) the correction is made only to the denominator of the statistic. One
might then conjecture that a test with superior ¿nite sample properties under level shifts could
be obtained by modifying the numerator of (16) in a similar fashion to the denominator.
Precisely, rather than use the OLS residuals e| , | ' c c A , as in / and / W , one could compute
the partial sums involved in the numerator using the residuals e| EkW , as de¿ned above. This
suggests the statistic
[A [| [| 
[ A o@Se 5&c e EkW   e EkW
5&c 
(17) / WW ' A 32 & |'  '  '

& MV ej 2
EkW 

Consider now the case where the true breakpoint kf is NQRZQ. In this scenario the
best option would be to compute (17) with kW replaced by kf ( we denote this statistic by
/ f  Adapting the results of Busetti and Harvey (2003a), it is easy to show that / f is a LMPI
test statistic for Mf G j2# ' f against M G j2# : f, and that its limiting null distribution is
 Er .

Finally, if no break occurs, both / W and / WW remain asymptotically unbiased and

consistent tests this follows since adding extra regressors asymptotically uncorrelated with
~| has no effect on the limiting behaviour of the statistics cf. Busetti and Harvey (2003a).

The following proposition summarizes the foregoing results.

 20

Proposition 2 8QGHU WKH QXOO K\SRWKHVLV Mf G j 2# ' f UHJDUGOHVV RI WKH H[LVWHQFH RU WLPLQJ RI
D EUHDNSRLQW LH LUUHVSHFWLYH RI WKH YDOXHV RI w DQG k WKH DV\PSWRWLF GLVWULEXWLRQV RI / W c / WW
DUH  Er

5HPDUN : The same result will clearly apply to the modi¿ed statistics / f , provided kf is
indeed the true break date.

5HPDUN : Under the ¿xed alternative, the statistics /c / f c / W and / WW all diverge at rate A
Notice that identi¿cation of the breakpoint in the coef¿cients of ~2| is unimportant in this case,
as the result is driven by the random walk alternative in the coef¿cients of ~| 

Under either the mixing or linear process assumptions on i| j outlined in Section 2,
the non-parametrically modi¿ed statistics O of (7) and O of (10) respectively can be corrected
e  of (7) and
along similar lines as suggested above for / . This simply amounts to replacing l
e  EkW  and e} Eb& ( kW  respectively, constructed using the OLS residuals
}eEb&  of ("") with l
e| EkW , as detailed above. We will denote these statistics with an obvious notation as OW ,
OWW, OW and OWW. The corresponding test statistics derived when the breakpoint is known will
be denoted Of and Of. The limiting properties of these statistics are as given above for the
corresponding bias-corrected variants of / .

3.2 7HVWV EDVHG RQ 3UH¿OWHUHG 'DWD

The bias-corrected tests of Section 3.1 ¿nessed the problem of unattended structural
breaks by explicitly modelling the breaks. In the case of a known break date a LMPI test
can be constructed. Where the break date is unknown, the now standard approach used in
time-series econometrics of estimating across all possible break dates and optimising over the
resulting sequence was proposed. However, rather than attempting to model the breaks, one
might also attack the problem by transforming the data in such a way as to annihilate possible
level shifts. This can be achieved simply by running our stability tests on pre-¿ltered data.

Consider the differencing ¿lter,u2, which reduces, by one, the order of integration at
all of the spectral frequencies identi ed by @2 . As an example, if r ' e with ~ '  and
¿ |

~2 ' EULt Z|*2c t? Z|*2c E  , as in Table 1 above, then u2  E n uE n u2  '
|
|

E n u n u2 n u . We will denote by s the order of this ¿lter in the above example, s ' .
 21

Notice that u2  { *u, where u reduces, by one, the order of integration at each frequency
r

identi¿ ed by @, and {  E  u .

r
r

Now consider SUH¿OWHULQJ the process i+| j generated by (3)-(4)-(11) by u2 , so that

(18) u2 + | ' u2~   n u2~2  2

| | | | n Q
|
c

| '  n sc    c A , where Q
|
 u2 . | u2 to ~2  2 translates the level
The application of | |

shift in the series i+ j into at most s outliers in the series iu2+ j. For example, suppose
| |

~2| ' , so that ~2 |  2| '  2f n w_| Ek, where  2f and w are scalar constants. In such a case
u2 2| 2| 
~   ' E uE n w_ Ek, from which it is clear that ~   ' f for all values of
2f | u2 2| 2|

|, except at | ' dA ko n  where u2 ~2 |  2| ' w. Although the term ~ | | of (4) has been
transformed to u2~  
| | in (18), those regressors in ~| identi¿ed with a particular spectral
frequency span an identical space to the corresponding regressors in u2~| . Consequently, we
may treat u2 ~ | | exactly as if it were ~ |  | in constructing tests against unit root behaviour
at any of the spectral frequencies identi¿ed by ~| .

Unfortunately, we cannot apply the statistic / of (6) to the pre-¿ltered data and
maintain the usual  Er limiting distribution under Mf G j2# ' f. This is because, for
/ to have a  Er  limiting distribution, the error process in the null regression must
be serially uncorrelated under Mf. It is clear from (18) that iQ
|
j is a moving average
process of order s , Es . Notice that iQ
|
j is strictly non-invertible at each of the
spectral frequencies identi¿ed by
S
@2 but not at those identi¿ed by

@  that is, lQ
 
SA
,6A <" A 3. E A
|'s
~| Q
|
E |' s
~| Q
|
 is positive de¿nite. Consequently, we may
consider the non-parametrically modi¿ed statistic
% # | $# $&
[
A
[ [
|

(19) OQ ' A 32A o@Se e

l Q 3 ~ eQ

~  eQ

c
|'s ' s 's

where eQ
|
, | ' sc    c A are the OLS residuals from the regression of u2 +| on ~| , and leQ
 is

as de¿ned in (8), replacing e| by eQ in the expression for K E . Similarly, O of (10) may be
|
Q
modi¿ed in an obvious way to produce the corresponding statistic O 

As discussed above, the application of u2 to ~2  2 | | transforms the level shift into at

most s outliers. Asymptotically, these are singletons. Consequently, they have no effect on
the limiting distribution of OQ of (19). It is then straightforward to show that asymptotically,
 22

under Mf , OQ and OQ behave exactly as O and O of Section 2, under the mixing and linear
process assumptions, respectively, placed on i j, and hence iQ j.
| |

In Section 5 we give Monte Carlo evidence into the ¿nite sample properties of the
modi¿ed statistics developed in this Section. Our results suggest that, for the sample sizes
typically seen in practical applications of these statistics, the proposed corrections perform
well in practice. In cases where there are no unattended unit roots, the bias-corrected statistic
/ WW seems, overall, to present the best option in cases where the true break date is unknown,
although the non-parametrically modi¿ed tests based on the pre-¿ltered data perform almost
as well and have the added advantage that they are also robust to serially correlated errors and
unattended unit roots, the latter issue discussed further in the next section.

 8QDWWHQGHG 8QLW 5RRWV

In this section we consider the case where the process i+| j is generated by (3)-(4) but
where  2| now follows the vector random walk process,

(20)  2| '  2c|3 n V| c | ' c c Ac

 2f ¿xed, with V|   UU(Efc j2 (2W, independent of | and #| , and where (2W ' j2V (2 , (2
a non-null diagonal positive semi-de¿nite matrix of dimension Er2  r2 .

As with the case of the unattended structural break model of Section 3, it is clear that,
unless j2V ' f, (20) violates the maintained hypothesis, M . Speci¿cally, the DGP (3)-(4)-
(20) can display unit root behaviour at DQ\ of the spectral frequencies, regardless of whether
the associated spectral indicator variable belongs to @ or @2. This is an important empirical
issue since in practice one cannot know which of the spectral frequencies admit a unit root.
In Busetti and Taylor (2002) and Taylor (2003b) it is demonstrated that if +| is generated by
(3)-(4)-(20) with j 2V : f then under Mf G j 2# ' f, / of (6) is R EA 3  while O of (7) and O of
(10) are R EEA 7A 3. Consequently, each of these statistics converges in probability to zero
under Mf when there are unattended unit roots. In the following proposition we show that the
same result is true for the bias-corrected statistics of Section 3.1.

Proposition 3 /HW +| EH JHQHUDWHG E\  ZLWK j2V : f 7KHQ XQGHU Mf G j 2# ' fc / W 

W
/ WW DQG / f DUH DOO RI R EA 3  ZKLOH OW  OWW  Of O  OWW DQG Of DUH RI  EEA 7
R A 3
 23

5HPDUN : Although the bias-corrected statistics of Section 3.1 were shown to be robust
with respect to unattended structural breaks, the results of Proposition 3 show that in the
presence of unattended unit roots at any spectral frequency associated with ~2| that is, where
j2V : f, they will converge in probability to zero, under Mf G j 2# ' f. This occurs because the
variance estimators used in constructing these statistics, as do those used in the test statistics
of section 2, diverge, albeit at differing rates when j2V : f. Indeed, as is demonstrated in the
Appendix, exactly the same result holds if  2| contains a mixture of random walk elements and
determinstically breaking elements, provided at least one of its elements evolves as a random
walk.

However, the pre-¿ltered statistic OQ of (19) (all of what follows also applies to OQ)
discussed in Section 3.2 to combat the problem of unattended level shifts is also simultaneously
a curative against unattended unit roots indeed it has been originally suggested in Taylor
(2003b) for this purpose. Recall that the pre-¿lter u2 is such that it reduces, by one, the
order of integration at all of the spectral frequencies identi ed by @2 . It is therefore clear
¿

that the ltered series iu2 + j will not admit unit root behaviour at any of the frequencies
¿ |

identi ed with @2 . When i+ j is generated according to (3)-(4)-(20) the error process in (18)
¿ |

is u2 ~2  2 n u2  , which we denote by QW, which is clearly an  Es  process. The process
| | | |

iQWj is strictly non-invertible at all of the spectral frequencies identi ed by @2 if j2 ' f,

|
¿ V

while, if j2 : f, iQW j will be strictly non-invertible at those frequencies associated with zero
V |

diagonal elements in (2  cf. Harvey (1989,pp.54-70) for rigorous discussion on this point. As
in Section 3.2, the serial correlation in the error process precludes the use of / of (6). However,
we may directly apply the pre-¿ltered statistic OQ of (19) since the matrix lQW
 , the long run

variance of ~|QW
|
, is positive de¿nite.

The results from Sections 3.2 and above demonstrate that the pre-¿ltered statistic OQ of
(19) retains the usual  Er limiting distribution of (9) under Mf, even if some subset of
the parameters on the spectral frequency regressors in @2 display HLWKHU level shifts or evolve
as random walks. This in contrast to the bias-corrected statistics / W , / WW , / f , OW, OWW, Of
OW, OWW and Of of Section 3.1, which all converge in probability to zero in the presence of
unattended unit roots. The pre-¿ltered statistics therefore appear particularly appealing from a
practical point of view.
 24

 1XPHULFDO 5HVXOWV

In this section we use Monte Carlo simulations methods to investigate the ¿nite sample
properties of the foregoing tests against unit roots in @ in cases where the parameters
associated with those spectral regressors not included in @ display deterministic structural
breaks. All experiments were programmed using the random number generator of the matrix
programming language Ox 2.20 of Doornik (1998), over Dfc fff Monte Carlo replications. All
tests were run at the nominal DI asymptotic level, although other choices of the nominal level
did not alter the results qualitatively.

Speci¿cally, we now investigate the ¿nite sample properties of the tests based on the
non-bias-corrected statistic / , the bias-corrected statistics / W c / WW and / f , and the statistic
based on pre-¿ltered data, OQ , against data generated by (3)-(4)-(11). In the context of this
DGP we considered a range of possible breakpoint magnitudes wc breakpoint locations k and
signal-to-noise ratios j # . We considered samples of size A ' ff and A ' 2ff observations.

In the context of quarterly time series data, we have focused attention on two cases
which we consider to be of most relevance for applied workers, each of which subsequently
arises in the empirical examples of Section 6: (i) testing against a unit root at frequency zero
when there is a break equal to w ' EwW c w Wc wW  j at the seasonal frequencies (ii) testing jointly
against unit root behaviour at the seasonal frequencies when there is a break equal to w Wj in
the underlying level of the process. Other scenarios are clearly possible, for example the case
where breaks may occur at all frequencies. We will discuss such possibilities further in Section
6.1. Simulation results for the bias-corrected tests are summarised in Tables 5.1a and 5.1b for
case (i), and Tables 5.2a and 5.2b for case (ii). Corresponding results for the tests based on
pre-¿ltered data are reported in Table 5.3a for case (i) and Table 5.3b for case (ii). For the
non-parametric test on pre-¿ltered data, OQ , we have followed the recommendation given on
page 12 of CH and used a Newey and West (1987) HAC estimator employing a Bartlett kernel
with lag truncation 7A ' c ec Dc S for the case of the zero frequency test and 7A ' fc c 2c 
for the test against stochastic seasonality. Results for the tests based on pre-¿ltered data are
reported only for A ' ff. Results for A ' 2ff are omitted as they parallel the progression
from A ' ff to A ' 2ff seen in the bias-corrected tests that is, power under the alternative
is improved and rejection frequencies under the null are much closer to nominal levels. These
results are available from the authors on request.
 25

Consider ¿rst Table 5.1a, which reports results for case (i) above with A ' ff. The
columns labelled j # ' f contain the empirical rejection frequencies (empirical size) of the
tests, using the asymptotic DI critical values, under the null hypothesis of a constant level
Mf G j 2# ' f. In the ¿rst row where wW ' f, the maintained hypothesis M holds and
there is no break in the seasonal, while for the remaining rows M is violated by a variety of
seasonal level shifts. In the absence of a break, wW ' f, the results in Table 5.1a demonstrate
that the ¿nite sample rejection frequency of / under Mf is slightly above that predicted by
the asymptotic theory. As shown in Proposition 1, the presence of a level shift biases the
asymptotic null distribution of / below the nominal level. In ¿nite samples the bias in /
appears to be slightly smaller than that predicted by Proposition 1, although this is somewhat
artifactual given the slight over-sizing seen for wW ' f. As an example, for wW '  and
k ' *2 the empirical rejection frequency under Mf for a sample of 100 observations is seen
from Table 5.1a to be 1.62%, while the asymptotic size in this case is approximately 1.3%,
the latter obtained from the tables of the  E distribution, and reported in Table 3.1. The
bias-corrected statistics / Wc / WW and / f all seem to work quite well in practice. Where the true
breakpoint is known, / f generally outperforms the other bias-corrected tests and is unaffected
by the value of wW , as should be expected given that it is a LMPI test in this scenario. Where the
true breakpoint is unknown, / WW seems to be preferable to / W, at least in terms of their relative
properties under Mf . Interestingly, in all but the case of a massive shift in the seasonal pattern
(e.g. 8 times the standard deviation of the error, unlikely to remain unnoticed in the data),
knowledge of the existence and the timing of the break does not seem to provide any great
advantage in terms of the properties of the tests under Mf . Indeed, where wW ' fH both / W
and / WW are seen to be rather over-sized. This appears to be a small sample effect, attributable
to the estimation of wW in (15), since these distortions are vastly diminished for A ' 2ff cf.
Table 5.1b.

All three bias-corrected tests appear to display quite reasonable power5 properties under
M G j 2# : f. In cases where the wc k combination does not yield an over-sized bias-corrected
test, power is largely comparable, for a given value of j # , with that of the LMPI statistic /
when wW ' f. Where the bias-corrected tests were over-sized, one sees correspondingly higher
power and that should of course be borne in mind when assessing the results in the Tables.

8 Strictly speaking, what is reported is not ¿nite sample power because the tests are constructed using the
8(
asymptotic, rather than exact, critical values.
 26

Notice, however, that, as predicted by Remark 4, this effect is wiped out as j # increases, and
has all but vanished for j # ' fD. The three bias-corrected tests display very similar power
properties under M , again remaining close to that observed for these tests in the absence of a
seasonal shift. In contrast, the power of the biased test / depends very heavily on the values
of wW and k, even for large values of j # . For example, where j # ' f and no break occurs,
/ has power of 59.44%, but this falls below 0.00 % for wW ' fH, k ' fD. Notice that, as
predicted by Proposition 1, the loss of power in / becomes more pronounced as the breakpoint
approaches the middle of the sample that is, as k approaches fD.

The results for case (i) and A ' 2ff are displayed in Table 5.1b. The power of all
of the bias-corrected tests is improved, relative to A ' ff, as would be expected from the
asymptotic theory in Section 3. Moreover, the properties of the bias-corrected tests under
the null are much closer to the nominal level than for A ' ff. In turn the effect of wc k
on the power of the bias-corrected tests is further diminished, with little or no impact seen
for j #  ffD. When wW  e it appears that all of the bias-corrected tests display power
comparable with that appropriate for the LMPI test in the absence of shifts. The ¿nite sample
size of the biased test / also gives a closer match with the asymptotic size obtained from the
quantiles of the  E distribution, reported in Table 3.1. The effects of wc k on the power
of / are also diminished relative to A ' ff, but remain considerable.

Tables 5.2a and 5.2b contain the results for the case of testing against unit roots at
the seasonal frequencies when there is a break in the underlying level of the process. The
properties of the tests are broadly in line with those seen for case (i) in Tables 5.1a and 5.1b.
For A ' ff, however, the bias-corrected tests display less reliable size properties for wW  e.
Interestingly, / W outperforms / WW for k ' *H and k ' *2c with this pattern being reversed
when k ' *e. As with case (i), these ¿nite-sample effects are much diminished when A is
increased to 2ff. Notice that, for a given value of j # , the empirical powers observed in Tables
5.2a-5.2b are generally higher than the corresponding entries in Tables 5.1a-5.1b. This is due to
the fact that under the alternative hypothesis there are two non-stationary cycles at the seasonal
frequencies, as opposed to the previous case where only one non-stationary component was
present under the alternative.

Consider now the properties of the pre-¿ltered test OQ for case (i), or pre-¿ltered KPSS
test, reported in Table 5.3a for A ' ff. The Monte Carlo results here are directly comparable
 27

with those in Table 5.1a, as the same set of random numbers (and, hence, the same arti¿cially
generated observations) were used. It clearly emerges that the outcome of the pre-¿ltered test
is largely unaffected by the existence, location or magnitude of a structural break. The size of
the pre-¿ltered test clearly depends on the lag truncation parameter, 7A . For smaller values
of 7A the test is rather over-sized this is expected given that the error term is now an E
process see Section 3.2. However, for 7A  e the pre- ¿ ltered test displays size properties
which are largely comparable with those of the bias-corrected tests / W and / WW, yet avoids the
rather bad over-sizing problem noted above for those tests when wW ' H. For 7A ' S, the
size properties of the pre-¿ltered test are largely comparable with those of the test based on
knowledge of the true breakpoint, / f, regardless of wW . Other things being equal, test power
declines as 7A is increased, again as would be expected. In practice all of the tests will need
to correct for serially correlated innovations and so this observation will not be con¿ned to the
pre-¿ltered tests. However, as with / f , the power of the pre-¿ltered test for a given value of
7A is little affected by the values of wW and k.

The results for case (ii), the CH test applied to data pre-¿ltered by u2  E  u, are
reported in Table 5.3b for A ' ff. In this case, pre-¿ltering the data appears to have very
little effect on the properties of the test. The size of the pre-¿ltered test in this case is very
close to the nominal level, even for 7A ' , with power only slightly below that of the LMPI
test / f  For example, when j # ' fc w W ' 2c k ' *2 and 7A ' c the power of OQ is
77.23% against 80.41% for / f . Increasing 7A tends, other things being equal, to reduce the
empirical rejection frequency of the pre-¿ltered test as would be expected.

Although not reported here we also investigated the effects of unattended unit roots on
the ¿nite sample properties of the bias-corrected tests, / W , / WW and / f of Section 3.1. Under
Mf G j 2# ' f all of the tests were severely under-sized with an associated dramatic loss of
power under M G j 2# : f, relative to the results reported in Tables 5.1a-5.2b. Similar patterns
were observed for the OW, OWW, Of OW, OWW and Of tests, whose behaviour closely paralleled
those reported for the O test under unattended unit roots in Taylor (2003b). In contrast, the
Q
pre- ltered tests, OQ and O , behaved very similarly to the results reported in Tables 5.3a and
¿

5.3b.

To summarize, the reported simulation evidence has shown that in the case of unattended
structural breaks, by using the bias-corrected tests / W c / WW even in ¿nite samples one can
 28

achieve almost the same size and power properties as the LMPI test / fc based on the
knowledge of the breakpoint, provided the magnitude of the break is not too large (wW  e),
and providing there are no unattended unit roots. Similar ¿ndings apply to the test OQ , based
on pre-¿ltered data, which has the added advantage of simultaneously proving an effective
remedy for the problem of unattended unit roots. Pre-¿ltering was found to be particularly
effective when testing against non-stationarity at the seasonal frequencies, where the pre-¿lter
is the simple ¿rst difference operator. We therefore recommend the use of pre-¿ltering as a
means of obtaining robust tests in the presence of unattended breaks and/or unattended unit
roots.

 *HQHUDOLVDWLRQV

6.1 $WWHQGHG DQG 8QDWWHQGHG 6WUXFWXUDO %UHDNV

As explained in Busetti and Harvey (2001, 2003a), the KPSS and CH stationarity tests
of Section 2 are likely to be severely oversized if there are (attended) structural breaks in the
parameters associated with @ . Indeed, it is known from the work of Nyblom (1989) that these
statistics will diverge in such cases. Where the breakpoint is known and all attended breaks
occur at the same point in the sample, Busetti and Harvey (2001, 2003a) demonstrate that
constructing the KPSS and CH statistics as in Section 2 but replacing e| , | ' c c A , by the
OLS residuals from the regression of +| on ~| and _|Ek~|, where _| Ek is as de¿ned below
(11), renders these statistics exact invariant to the attended breaks, with a further modi¿cation
employed to remove the dependence of the limiting null distribution of the statistics on the
break location. However, these modi¿cations change the limiting null distribution of the
statistics from  Er  to  E2r  see Busetti and Harvey (2001, 2003a) for full details.
In what follows, we will refer generically to the tests based on the modi¿ed statistics of Busetti
and Harvey (2001, 2003a) as BH tests. When the breakpoint is not known Busetti and Harvey
(2003b) suggest estimating the break by minimizing, over all possible break-dates, the error
variance from the regression of +| on ~| and _| Ek~| , and then use the resulting estimate,
k say, in constructing the BH statistic. Notice that this parallels our construction of bias-
corrected tests in Section 3.1.

It is straightforward to show that the BH tests outlined above will suffer from the
problems of unattended breaks and unattended unit roots in exactly the same way as did
the tests of Section 2 that is, under Mf G j2# ' f they will be under-sized in the presence
 29

of unattended breaks and will converge in probability to zero under unattended unit roots.
However, running the BH tests on appropriately pre-¿ltered data, in exactly the same way as
was proposed in Sections 3.2 and 4 in the context of the tests of Section 2, will simultaneously
treat the problems of both attended and unattended breaks (including the possibility of breaks
at all frequencies) and will also be robust against unattended unit roots. If there are no
unattended unit roots and both the attended and unattended breaks occur at the same point
in the sample, an alternative strategy is to compute the BH statistic using the residuals from
the regression of +| on ~| and _| Ek~| , where k may be known or estimated. The  E2r 
limiting null distribution remains appropriate under each of these approaches. However, the
BH tests based on pre-¿ltering have a further advantage in that they will also be robust to cases
where the attended and unattended breaks are located in different points in the sample.

6.2 7UHQGLQJ 9DULDEOHV

The inclusion of a full set of spectral time trends ~| | at the frequencies b 5 @ has the
effect of changing the asymptotic null distributions of the statistics presented in the previous
sections from standard Cramér-von Mises distributions into second level Cramér-von Mises
distributions with r degrees of freedom again see Harvey (2001) for representations of these
distributions. Appropriate critical values are tabulated in Taylor (2003a) and Nyblom and
Harvey (2000). In particular, an unattended structural break will imply the same asymptotic
bias factor 6Ekc wc j2  as in Proposition 1 but applied to this new distribution again the bias
will disappear if the bias-corrected statistics of Section 3.1 or the pre-¿ltered statistic of Section
3.2 are adopted.

Busetti and Harvey (2003a) and Taylor (2003a) have shown that a modelled time trend at
frequency zero does not affect the asymptotic null distribution of the CH statistic for stochastic
seasonality. By extending their argument to our framework, it follows that modelled time
trends at the frequencies b 5 @2 will have no effect under the maintained hypothesis M  .
However, the presence of a neglected level shift in this case will change the bias term derived
in Proposition 1, but again the same asymptotic critical values will apply for the appropriate
bias-corrected and pre-¿ltered tests. Note that not only will the pre-¿ltered test statistics of
Section 3.2 be asymptotically unaffected by the presence of spectral time trends and level
shifts at the frequencies b 5 @2, but as pre- ¿ ltering reduces spectral time trends to spectral
 30

indicators, the spectral trends need not be included in the regression model for the pre-¿ltered
data. Finally, a neglected slope change at the frequencies b 5 @2 can be dealt with either by
computing bias-corrected statistics in the same spirit of Section 3.1 or by double pre-¿ltering,
i.e. by applying the operator u2 of Section 3.2 twice.

 (PSLULFDO ([DPSOHV

As a ¿rst example we consider testing at frequency zero for a well-known series

characterized by a structural break in the seasonal patterns. Figure 7.1 shows the quarterly,
seasonally unadjusted, series of marriages registered in the UK from 1958q1 to 1982q4 the
source of the data is the U.K. Monthly Digest of Statistics. As emphasised in Busetti and
Harvey (2003a), there is a structural break in the seasonal pattern in this series, occurring in
the ¿rst quarter of 1969. This is seen very clearly in Figure 7.1 and was due to a change in
U.K. taxation laws, which effected a switch from winter marriages to marriages in the spring
quarter.

Busetti and Harvey (2003a) considered testing against stochastic seasonality for this
series, taking the known seasonal break into account. They found some evidence for non-
stationary stochastic seasonality. In the context of this paper we are interested in testing against
stochastic non-stationarity at frequency zero, given that there may be a seasonal structural
break and/or non-stationary seasonal cycles present in the data. Regressing the ¿rst difference
of the series on the complete set of spectral indicators and the break dummies at the seasonal
e ' H2c a long run variance estimate,
frequencies yielded a regression standard error j
eu- EH ' .bS and a break magnitude
computed using the ¿rst 8 residual autocovariances, j
ew ' Eb2Dc fec 2Sb c with associated |-ratios ESbSc eec S . The structural
break then appears suf¿ciently big to affect the behaviour of the test at frequency zero if
computed without either bias-correcting or pre-¿ltering the data by u2  E n u n u2 n u .
 31

Table 7.1: Tests at frequency zero for UK marriages

SA =0 SA =1 SA =2 S A =3 SA =4 SA =5 SA =8 10% 5% 1%

O 0.534 0.523 0.577 0.569 0.428 0.364 0.279 0.347 0.463 0.739
Of 1.608 1.037 0.758 0.598 0.488 0.413 0.294 0.347 0.463 0.739

OW 1.535 0.989 0.731 0.577 0.472 0.399 0.285 0.347 0.463 0.739
OWW 1.600 1.031 0.762 0.601 0.492 0.416 0.297 0.347 0.463 0.739
OQ 2.349 1.212 0.823 0.628 0.511 0.434 0.280 0.347 0.463 0.739

Setting @ ' ifj, Table 7.1 reports the values taken by the KPSS-type statistic O and
the modi¿cations thereof proposed in Section 3: Ofc OWc OWW and OQ. These statistics were
constructed using the OLS residuals from the regression of the data on ~|. As in Section
5.2, we have used a Newey and West (1987) HAC estimator with a Bartlett kernel with lag
truncation parameter 7A . The 10%, 5%, 1% asymptotic critical values appropriate for each
reported test are provided in the last three columns of Table 7.1.

Given the empirical results in Busetti and Harvey (2003a) and the simulation results
of Section 5 one would anticipate that the KPSS-type statistic O will provide less signi¿cant
outcomes than bias-corrected and pre-¿ltered versions of O. From the ¿rst row of Table 7.1
we observe that the null hypothesis of stationarity around ¿xed seasonal effects is not rejected
using the O statistic at the 1% signi¿cance level for any value of 7A and not rejected at the 5%
level for 7A  e. On the other hand, the non-parametric bias-corrected statistic Of, which uses
knowledge of the breakpoint in the seasonal, rejects the null at the 5 % level for all 7  e A

and at the 1 % level for 7  2. The same inferences are drawn using the OWW statistic which
A

does not assume knowledge or existence of the breakpoint. The OW statistic behaves similarly
but rejects the null only at the 5 % level for 7 ' 2. The pre- ltered statistic OQ also provides
A ¿

stronger evidence than O against the null hypothesis here a value of 7   needs to be
 A

chosen as pre-¿ltering, in this case, would turn an otherwise white noise error into an MA(3)
process. Overall, the bias-corrected and pre-¿ltered statistics yield more signi¿cant outcomes
than the KPSS-type statistic, O.

As a second example, which entails testing at the seasonal frequencies, consider Figure
7.2 which graphs the logarithm of real quarterly seasonally unadjusted U.K. consumers’
expenditure on tobacco goods for the period 1975q1 1996q1, the data obtained from the U.K.
O.N.S. macroeconomic database. Fitting the Basic Structural Model (BSM) of Harvey (1989,
 32

p.47) to these data yielded the following values of the hyperparameters6 (standard deviations
of the level, slope, seasonal and irregular components respectively): eHc ff2c fSc fc each
multiplied by f32. Notice, in particular that the hyperparameter associated with the level is
approximately four times as large as that associated with the seasonal component. The Box-
Ljung statistic with 9 lags and 6 degrees of freedom, 'Ebc Sc was 7.39, with R-value 0.29. The
extracted seasonal component is depicted in Figure 7.3. It is quite clear from both Figures 7.3
and 7.2 that the seasonal pattern in the data is not ¿xed through time. In particular, there is a
clear structural break in the seasonal pattern around 1980-1981.

Table 7.2 reports the values taken by the pre-¿ltered CH-type seasonal stability statistics,
OQ and OQ, together with the corresponding BH-type statistics, denoted by EK and EK,
discussed in Section 6.1. The former allow for unattended unit roots and unattended breaks,
while the latter also control for the possibility of attended breaks. It is clear from Figures
7.2 and 7.3 that attended breaks will be an important issue in this application. By setting
@ i j
' c 2 , we have considered joint tests against non-stationary stochastic seasonality at
either the harmonic seasonal frequency, b  Z*2, or the Nyquist frequency, b2  Z, or both.
We also report the corresponding individual frequency tests against unit roots at the harmonic
and Nyquist frequencies which obtain on setting @ ' ij and @ ' i2j, repectively.

9 Imposing the restriction of a ¿xed slope did not alter the estimates of the remaining hyperparameters.
 33

Table 7.2: Tests at the seasonal frequencies for tobacco expenditure

V 7A 'f 7A ' 7A '2 7A ' 7A 'D 7A 'H fI DI I

tc 2 nQ
nQ
1.941 1.551 1.411 1.377 0.973 0.783 0.846 1.010 1.350

GQ
3.672 3.194 3.091 2.940 2.020 1.553 0.846 1.010 1.350

GQ
2.015 2.038 2.031 2.001 1.857 1.870 1.490 1.680 2.120

nQ
2.011 1.972 1.946 1.928 1.887 1.904 1.490 1.680 2.120
t
nQ
3.270 2.145 1.547 1.295 0.949 0.722 0.610 0.749 1.070

GQ
3.324 3.324 2.343 2.041 1.485 1.093 0.610 0.749 1.070

GQ
0.898 0.904 0.841 0.795 0.747 0.779 1.070 1.240 1.600

nQ
0.951 0.951 0.887 0.858 0.814 0.845 1.070 1.240 1.600
t2
GQ
3.279 1.841 1.263 0.981 0.697 0.509 0.347 0.463 0.739
1.198 1.211 1.008 0.919 0.799 0.717 0.610 0.749 1.070

The above statistics were constructed using the OLS residuals from the regression of the
appropriately pre-¿ltered data on ~| .7 In the case of the joint seasonal statistic the appropriate
pre-¿lter is u2 
 u, since @2
E ' ifj. For @ ' ij, u2  E  u2 while
u2  E  uE n u2 for @ ' i2j. Graphs of the three pre-¿ltered series are provided
in Figures 7.4-7.6. Again we have used a Newey and West (1987) HAC estimator with a
Bartlett kernel and lag truncation parameter 7A . The 10%, 5%, 1% asymptotic critical values
appropriate for each reported test are provided in the last three columns of Table 7.2. Due to
the sizeable random walk component estimated in the BSM, we will only consider pre-¿ltered
tests against seasonal unit roots in this application. The bias-corrected tests of Section 3.1 are
clearly inappropriate.

The pre-¿ltered CH-type statistics reported in Table 7.2, which do not allow for attended
breaks at the seasonal frequencies, all provide a consistent body of evidence against the null:
there is only one case reported where the null cannot be rejected at the 10 % level, and in most
cases the null can be rejected at the 1 % level. In the case of the joint seasonal tests and the
tests against unit root behaviour at the harmonic seasonal frequency, the evidence from the
OQ statistics are stronger that from the OQ statistic. Overall, application of the pre-¿ltered
CH tests strongly indicates the presence of non-stationary behaviour at both the harmonic and
Nyquist seasonal frequencies.

: The reported statistics were constructed from a regression without a linear time trend since, as was noted
u
in Section 6.2, the ¿lter 5 will remove a linear time trend from the data. Although not reported here, we also
computed the pre-¿ltered tests with an unnecessary time trend ¿tted. This yielded much less evidence against the
null, reÀecting the usual argument on the reduction of the power of tests in the presence of additional nuisance
parameters.
 34

The BH-type statistics, EK and EK, reported in Table 7.2 allow us to investigate whether
Q
the evidence found against seasonal stability provided by the OQ and O statistics in the series
of (log) tobacco expenditure is attributable to seasonal unit root behaviour or purely to the
deterministic break in the seasonal pattern. In computing these series we took the breakpoint
as unknown. In each case the breakpoint was estimated as discussed in Section 6.1, and was
identi¿ed at 1980q4 for the joint test and at 1981q1 and 1980q3 for the tests at frequencies
Z*2 and Z, respectively.

The joint BH-type test against unit roots at frequencies Z and Z*2 again provides
consistent evidence against the null of stochastic seasonal stationarity, allowing for seasonal
level shifts: both variants of the test reject at the 5 % level for all values of 7A reported. Neither
variant of the BH-type test against unit roots at the harmonic seasonal frequency Z*2 is able
to reject the null, even at the 10% level, regardless of the value of 7A . In contrast, the BH-
type test against unit roots at frequency Z points strongly towards seasonal unit root behaviour
at frequency Z , even on allowing for seasonal level shifts. Consequently, the tests which
explicitly allow for unattended unit roots and both attended and unattended breaks provide a
strong indication of unit root behaviour at the Nyquist frequency but no indication of unit root
behaviour at the harmonic seasonal frequency. In contrast, the standard tests would have led
the practitioner to conclude that unit roots were present at all of the seasonal frequencies.

 &RQFOXVLRQV

In this paper we have investigated the effects of structural breaks on tests against
stochastic trend and seasonality. In contrast to the existing literature we have considered the
effects of structural breaks at frequencies RWKHU than those being subject to stability testing. We
have shown that such breaks alter both the large sample and ¿nite sample distributions of the
stationarity statistics, effecting a severe under-sizing problem in the tests with an associated,
often very dramatic, loss of power under the alternative. This effect coincides with what is seen
when there are unattended unit roots, but is in contrast to the case where there is a structural
break at the frequencies under test, where an over-sizing problem occurs.

We have suggested two methods of modifying the stationarity tests to remove these
problems. The ¿rst, appropriate, for the case of structural breaks, involves bias-correcting
the original stability test. The second, effective against both structural breaks and unattended
 35

unit roots, is achieved by running the stability tests on pre-¿ltered data. Both are shown to
recover the usual limiting null distribution, appropriate to the case where there are no breaks
or unattended unit roots. We have reported simulation evidence suggestive that the corrected
statistics perform well in practice, adequately correcting test size under the null, without
sacri¿cing test power under the alternative. Applications of the original and modi¿ed tests to
data on U.K. marriages and U.K. consumers’ expenditure on tobacco were considered. These
demonstrated the practical relevance of the problems discussed in this paper and the usefulness
of the remedial suggestions which we have made.
 $SSHQGL[ SURRIV

3URRI RI 3URSRVLWLRQ : Let e c e

 2 be the OLS estimators of the regression coef¿cients for
~| c ~2| from (5). Using standard arguments it is not dif¿cult to show that, under Mf G j2# ' f,
    
e   f f j2 73 f
A

2
e 2   2f  kw , f f j2 723
c

SA
where 7 ' *4 A 3 |'
~|~| c  ' c 2 Note that 7 c 72 are diagonal matrices with either
A <"
 or 
2 on the main diagonal precisely, they contain 3
&

for the element(s) corresponding to
the & -th spectral regressor.
[|
Then we can show that the partial sum process ~ e converges weakly to a
 '

Brownian bridge, on scaling. Write the OLS residuals as

(21) e| ' | n ~2 | w_| Ek  ~ | Ee

    f   ~2 | Ee
 2   2f  

Since e2   2f $ R
kw and from the orthogonality relations between each pair of spectral
indicators we have from (21) that

3 [
dA oo
3 [
dA oo
3 [
dA oo
j 3 7 2 A 3 2 j 3 7 2 A 3 2 ~| |  j37 2 A 3     f  n JR E
~| ~ | A 2 Ee
  
 ~| e| '  
|' |' |'

(22) , `r Eo  o`r E  r Eoc

using the Functional Central Limit Theorem of Chan and Wei (1988). It then follows directly
from (22) and an application of the Continuous Mapping Theorem (CMT) that
# | $ ]
[ [
A
[ [
| 
(23) j 32 A 32 & A o@Se 5&c e 5 e
&c 
, r Eo  r Eo_oc
& MV |'  '  '
f

LH the numerator of the statistic / is asymptotically unaffected by the level shift.

The factor 6Ekc wc j 2 in the asymptotic null distribution of / is due to a bias in

estimating j 2  After taking the square of (21), it is easy to show that

je2 $ j 2 n KEkc wc

R
 37

where
[
A

KEkc w ' *4 A 3

A <"
E_| Ek  k2 E~2 w2  |
|'

The stated result then follows directly, using an application of the CMT.

3URRI RI 3URSRVLWLRQ : Let kf 5 Efc  be the true breakpoint location parameter in the model
(3)-(11). As we want to compute regressions for the whole range of breakpoints, it is useful to
express the null model in terms of a breakpoint parameter k not necessarily equal to the true
value:

(24) +| ' ~ |  f n ~2 | 2f n _| Ek~2 | w n W| c

(25) W| ' E| n E_| Ekf   _| Ek ~2 | w 

e2 Ek '
Let e|Ek be the OLS residuals from regression (24) and denote by j
SA
A 3 |'
e| Ek2 their sample variance. Note that (24) can be viewed as a regression with
e2Ek
omitted variables. Then, it is a standard result that j  je2Ekfc and thus, under Mf G
j2# ' fc je EkW ?u je Ek $ j2 
2 2 R

Then we have that under Mf G j 2# ' f and for every k 5 Efc 

[
dA oo
~| e| Ek , r Eo (
3
j3 7 2 A 3 2

(26) 
|'

this follows from extending the result of Busetti and Harvey (2003a) on the null distribution
of / in the presence of modelled breaks in the trend (i.e. at frequency zero) together with the
argument of Proposition 1 on the presence of neglected level shifts, which in this case would
correspond to the term E_| Ekf   _| Ek ~2 |w

It therefore follows directly from (26) and an application of the CMT that
# $ ]
[ [
A
[
|
[
| 
j32A 32 & A o@Se 5&c e Ek 5 e Ek
&c 
, r Eo r Eo_o   Er c
& MV |'  '  '
f
(27)
 38

e2EkW  and by applying the CMT, we obtain

independently of k Using the previous results for j
that the right member of (27) is the asymptotic null distribution for both / W of (16) and / WW of
(17).

e2 EkW  is still asymptotically unbiased and

If no breakpoint actually occurs, under Mf , j
thus, by the result of Busetti and Harvey (2003a), the limiting null distributions of / W and / WW
are  Er  as before.

3URRI RI 3URSRVLWLRQ : Under the conditions of Proposition 3, (24)-(25) is appropriate on

substituting W| in (24) by E| n ~2 | E 2|   2f n E_ Ekf  _ Ek ~2 w.
| | |
Again denote the
OLS residuals from (24) by e| Ek, which therefore satisfy
 
e| Ek ' W|  ~ | Ee   f  ~2 | Ee2   2f  _|Ek~2 | w  w
(28) ' | n _| Ekf ~2 | w  _ Ek~2 w  ~ Ee    f  ~2 Ee 2   2  c
| | | | |

  c e 2 and w denote the OLS estimators of the regression coef¿cients for ~| c ~2| and
where e
_| Ek~2| respectively from (24). Routine algebra establishes the result that

3 [
dA oo
j3 7 2 A 3 2 ~| e| Ek , r Eo  n R Ec

(29) 
|'

SdA oo SdA oo
since, independently of k, A 3*2 |'
 2   2|  and A 3*2
~|~2 | Ee |'
~|E_| Ek~2 | w are
SdA oo
both R E while A 3*2 |'
~| E_| Ekf ~2 | w ' JRE. Notice that it was not necessary to
impose the condition that w ' f in deriving (29) cf. Remark 5. Consequently,
# | $ ]
[ [
A
[ [
| 
j32 A
(30) 32 & A o@Se 5&c e Ek 5 e Ek
&c 
, r Eo  r Eo_o n R Ec
& MV |'  '  ' f

i.e. the numerator of (6) remains R EA 2 , but when appropriately scaled differs from the
 Er distribution by the presence of an additional R E random variable. Using the
same development as in KPSS (p.168), it is straightforward to demonstrate that the long-run
e Ek and e} Eb& ( k both diverge at rate R EA 7A  for all k, and hence for
variance estimators l
k ' kW , when j2V : f, regardless of w. The OLS variance estimator je2Ek is a special case
of the long-run estimator e
} Ef( k for 7A ' f and, hence, is R EA  for all k. Consequently,
W WW
/ W of (16) and / WW of (17) are both R EA 3 while OW , OWW , O and O are all R EEA 7A 3 ,
 39

their respective orders in probability differing through the respective divergence rates of the
variance estimators used in their construction.
Table 5.1a: Empirical rejection probabilities for the bias-corrected tests of stationarity at frequency zero in the presence of an unattended break at the seasonal
frequencies.* The sample size is 100.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

∗∗ ∗∗ ∗∗ ∗∗ ∗∗
ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0

θ∗=0 5.69 7.24 6.80 N.A 7.00 8.60 8.01 N.A 31.48 34.46 33.45 N.A 59.44 61.98 61.01 N.A 96.66 97.14 96.85 N.A
α=1/8 3.20 6.47 6.07 6.14 4.06 7.86 7.29 7.51 25.83 33.20 32.00 32.37 54.40 60.99 59.88 60.31 95.93 96.91 96.61 96.73
∗
θ =1 α=1/4 2.09 6.64 6.12 6.15 2.82 7.81 7.34 7.62 22.34 33.18 31.99 32.46 50.98 60.87 59.81 60.16 95.34 96.86 96.59 96.73
α=1/2 1.62 6.68 6.11 6.16 2.03 7.95 7.40 7.58 19.69 33.21 32.30 32.37 48.48 60.91 59.95 60.34 94.95 96.89 96.54 96.79
α=1/8 0.69 6.45 6.04 6.14 0.92 7.73 7.23 7.51 14.51 33.09 31.90 32.37 42.45 60.89 59.75 60.31 93.91 96.84 96.59 96.73
∗
θ =2 α=1/4 0.19 6.68 6.24 6.15 0.23 8.00 7.42 7.62 8.52 33.42 32.28 32.46 34.11 60.95 59.93 60.16 91.90 96.93 96.67 96.73
α=1/2 0.04 6.89 6.33 6.16 0.08 8.11 7.56 7.58 5.43 33.58 32.51 32.37 29.06 61.22 60.06 60.34 90.22 96.90 96.62 96.79
α=1/8 0.00 6.94 6.57 6.14 0.00 8.48 7.82 7.51 1.84 34.03 33.07 32.37 19.06 61.64 60.53 60.31 86.57 96.97 96.73 96.73
∗
θ =4 α=1/4 0.00 7.74 7.16 6.15 0.00 9.15 8.54 7.62 0.23 35.13 34.08 32.46 9.80 62.34 61.22 60.16 80.23 97.11 96.82 96.73
α=1/2 0.00 8.36 7.85 6.16 0.00 9.85 8.99 7.58 0.07 36.28 35.21 32.37 5.37 62.99 62.08 60.34 75.16 97.04 96.78 96.79
α=1/8 0.00 10.05 9.32 6.14 0.00 11.36 10.62 7.51 0.00 38.44 37.58 32.37 1.09 65.25 64.14 60.31 65.60 97.35 97.13 96.73
∗
θ =8 α=1/4 0.00 14.30 13.92 6.15 0.00 16.13 15.28 7.62 0.00 43.94 42.77 32.46 0.11 68.92 67.88 60.16 53.48 97.55 97.34 96.73
α=1/2 0.00 20.74 20.03 6.16 0.00 22.58 21.48 7.58 0.00 49.88 48.66 32.37 0.00 72.56 72.02 60.34 45.83 97.61 97.35 96.79

* The DGP corresponds to equations (2.1)-(2.2)-(3.1) for quarterly data, &1={0}.

Table 5.1b: Empirical rejection probabilities for the bias-corrected tests of stationarity at frequency zero in the presence of an unattended break at the seasonal
frequencies.* The sample size is 200.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

∗∗ ∗∗ ∗∗ ∗∗ ∗∗
ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0

θ∗=0 5.55 6.07 5.93 N.A 10.06 10.99 10.65 N.A 60.47 62.01 61.41 N.A 85.33 86.01 85.65 N.A 99.74 99.80 99.78 N.A
α=1/8 2.97 5.77 5.67 5.70 6.53 10.55 10.17 10.33 55.09 61.18 60.64 60.91 82.14 85.65 85.24 85.52 99.64 99.76 99.76 99.76
∗
θ =1 α=1/4 1.78 5.80 5.64 5.71 4.75 10.53 10.17 10.39 51.58 61.20 60.53 60.93 80.00 85.68 85.15 85.58 99.58 99.76 99.75 99.76
α=1/2 1.27 5.82 5.66 5.72 3.70 10.65 10.19 10.41 48.82 61.37 60.65 60.97 78.33 85.70 85.19 85.51 99.54 99.75 99.74 99.75
α=1/8 0.44 5.74 5.65 5.70 1.77 10.52 10.09 10.33 42.61 61.19 60.54 60.91 74.09 85.60 85.18 85.52 99.42 99.76 99.75 99.76
∗
θ =2 α=1/4 0.10 5.86 5.66 5.71 0.64 10.62 10.20 10.39 34.69 61.30 60.63 60.93 67.68 85.67 85.20 85.58 99.12 99.74 99.74 99.76
α=1/2 0.01 5.89 5.70 5.72 0.24 10.73 10.38 10.41 29.57 61.53 60.76 60.97 62.63 85.70 85.22 85.51 98.82 99.76 99.73 99.75
α=1/8 0.01 5.93 5.77 5.70 0.02 10.84 10.39 10.33 19.28 61.70 60.90 60.91 53.16 85.79 85.42 85.52 97.90 99.76 99.75 99.76
∗
θ =4 α=1/4 0.00 6.18 6.02 5.71 0.00 11.13 10.86 10.39 9.61 62.11 61.47 60.93 41.03 86.05 85.62 85.58 95.90 99.75 99.75 99.76
α=1/2 0.00 6.63 6.38 5.72 0.00 11.69 11.18 10.41 5.54 62.54 61.98 60.97 33.60 86.23 85.72 85.51 94.12 99.75 99.75 99.75
α=1/8 0.00 7.04 6.70 5.70 0.00 12.33 11.78 10.33 1.49 63.33 62.71 60.91 21.44 86.67 86.43 85.52 89.85 99.77 99.76 99.76
∗
θ =8 α=1/4 0.00 8.21 7.92 5.71 0.00 13.95 13.40 10.39 0.14 65.13 64.60 60.93 10.39 87.35 87.08 85.58 83.35 99.80 99.78 99.76
α=1/2 0.00 9.66 9.36 5.72 0.00 16.09 15.71 10.41 0.03 66.81 66.35 60.97 5.74 87.97 87.64 85.51 78.14 99.80 99.80 99.75

* The DGP corresponds to equations (2.1)-(2.2)-(3.1) for quarterly data, &1={0}.

Table 5.2a: Empirical rejection probabilities for the bias-corrected tests of stationarity at the seasonal frequencies in the presence of an unattended break at
frequency zero.* The sample size is 100.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

∗∗ ∗∗ ∗∗ ∗∗ ∗∗
ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0

θ∗=0 5.23 6.61 6.03 N.A 6.27 7.84 7.18 N.A 39.59 42.93 40.84 N.A 80.14 81.70 80.48 N.A 99.81 99.84 99.77 N.A
α=1/8 3.29 6.09 6.17 5.52 4.05 7.30 7.13 6.58 33.90 41.96 40.30 40.29 76.76 81.39 80.17 80.40 99.78 99.82 99.78 99.82
∗
θ =1 α=1/4 2.43 6.46 5.82 5.50 2.93 7.78 6.80 6.60 29.96 42.36 39.93 40.15 73.99 81.41 79.90 80.42 99.67 99.83 99.75 99.82
α=1/2 1.73 6.14 5.68 5.57 2.11 7.44 6.87 6.62 27.00 41.88 39.77 40.20 71.82 81.29 80.09 80.41 99.67 99.83 99.75 99.82
α=1/8 0.87 5.97 7.26 5.52 1.21 7.17 8.46 6.58 20.18 41.76 41.42 40.29 66.76 81.19 80.21 80.40 99.59 99.82 99.77 99.82
∗
θ =2 α=1/4 0.22 7.00 5.89 5.50 0.28 8.33 6.97 6.60 12.07 43.26 40.33 40.15 56.76 81.41 80.01 80.42 99.46 99.80 99.75 99.82
α=1/2 0.03 6.28 6.14 5.57 0.05 7.43 7.22 6.62 7.69 42.25 40.53 40.20 49.48 81.24 80.14 80.41 99.35 99.84 99.78 99.82
α=1/8 0.00 6.71 14.37 5.52 0.00 8.01 15.92 6.58 2.53 43.25 48.11 40.29 33.95 81.76 82.19 80.40 98.86 99.84 99.79 99.82
∗
θ =4 α=1/4 0.00 10.73 7.43 5.50 0.00 12.29 8.63 6.60 0.32 47.35 43.66 40.15 16.26 82.88 81.40 80.42 97.80 99.78 99.78 99.82
α=1/2 0.00 7.81 9.22 5.57 0.00 9.36 10.50 6.62 0.05 46.13 45.08 40.20 9.00 82.74 82.07 80.41 96.84 99.84 99.76 99.82
α=1/8 0.00 12.67 54.50 5.52 0.00 14.49 55.37 6.58 0.00 53.45 71.71 40.29 1.88 85.82 88.82 80.40 93.59 99.87 99.80 99.82
∗
θ =8 α=1/4 0.00 35.70 21.50 5.50 0.00 37.41 23.86 6.60 0.00 66.21 60.89 40.15 0.12 89.28 88.62 80.42 86.01 99.76 99.83 99.82
α=1/2 0.00 23.06 32.40 5.57 0.00 25.42 34.70 6.62 0.00 64.17 65.71 40.20 0.00 89.95 89.53 80.41 79.71 99.86 99.80 99.82

* The DGP corresponds to equations (2.1)-(2.2)-(3.1) for quarterly data, &1={1,2}.

Table 5.2b: Empirical rejection probabilities for the bias-corrected tests of stationarity at the seasonal frequencies in the presence of an unattended break at
frequency zero.* The sample size is 200.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

∗∗ ∗∗ ∗∗ ∗∗ ∗∗
ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0 ω ω∗ ω ω0

θ∗=0 5.30 5.87 5.57 N.A 10.61 11.84 11.50 N.A 81.37 82.30 81.69 N.A 97.96 98.06 98.00 N.A 100.00 100.00 100.00 N.A
α=1/8 3.55 5.70 5.40 5.38 7.64 11.41 11.05 10.87 77.91 81.83 81.40 81.57 97.36 98.06 97.94 97.98 100.00 100.00 100.00 100.00
∗
θ =1 α=1/4 2.44 5.69 5.54 5.43 5.88 11.32 11.23 10.92 75.39 81.83 81.50 81.58 96.90 98.08 97.89 97.97 100.00 100.00 100.00 100.00
α=1/2 1.77 5.63 5.52 5.44 4.82 11.33 11.18 10.94 73.23 82.07 81.55 81.52 96.58 98.03 97.88 97.96 100.00 100.00 100.00 100.00
α=1/8 0.92 6.20 5.35 5.38 2.76 11.89 11.10 10.87 67.19 81.92 81.26 81.57 95.20 98.10 97.93 97.98 100.00 100.00 100.00 100.00
∗
θ =2 α=1/4 0.27 6.15 6.00 5.43 0.99 11.75 11.51 10.92 57.68 81.98 81.65 81.58 92.78 98.08 97.86 97.97 100.00 100.00 100.00 100.00
α=1/2 0.05 5.87 5.98 5.44 0.42 11.53 11.47 10.94 50.77 82.21 81.80 81.52 90.61 98.03 97.85 97.96 100.00 100.00 100.00 100.00
α=1/8 0.02 8.56 5.69 5.38 0.09 14.90 11.56 10.87 35.43 82.56 81.65 81.57 84.28 98.15 97.94 97.98 100.00 100.00 100.00 100.00
∗
θ =4 α=1/4 0.00 8.10 7.73 5.43 0.00 13.85 13.82 10.92 19.31 82.92 82.51 81.58 72.81 98.12 98.01 97.97 99.99 100.00 100.00 100.00
α=1/2 0.00 7.04 7.58 5.44 0.00 13.20 13.78 10.94 10.92 83.23 82.79 81.52 64.17 98.09 97.89 97.96 99.96 100.00 100.00 100.00
α=1/8 0.00 21.43 7.80 5.38 0.00 28.53 14.08 10.87 2.89 85.84 83.88 81.57 42.59 98.55 98.26 97.98 99.83 100.00 100.00 100.00
∗
θ =8 α=1/4 0.00 18.11 17.63 5.43 0.00 25.45 25.39 10.92 0.23 86.23 85.83 81.58 21.28 98.52 98.35 97.97 99.27 100.00 100.00 100.00
α=1/2 0.00 14.81 16.93 5.44 0.00 22.82 25.06 10.94 0.02 88.10 87.45 81.52 11.99 98.69 98.56 97.96 98.50 100.00 100.00 100.00

* The DGP corresponds to equations (2.1)-(2.2)-(3.1) for quarterly data, &1={1,2}.

Table 5.3a: Empirical rejection probabilities for the pre-filtered KPSS test in the presence of an unattended break at the seasonal frequencies.* The sample size
is 100.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

6 7 =3 6 7 =4 6 7 =5 6 7 =6 6 7 =3 6 7 =4 6 7 =5 6 7 =6 6 7 =3 6 7 =4 6 7 =5 6 7 =6 6 7 =3 6 7 =4 6 7 =5 6 7 =6 6 7 =3 6 7 =4 6 7 =5 6 7 =6

θ∗=0 10.62 8.44 7.06 6.16 11.66 9.41 8.09 7.06 35.96 32.27 29.40 27.33 58.44 54.47 51.18 48.39 84.61 78.74 74.38 70.79
α=1/8 10.67 8.26 6.98 6.14 11.80 9.65 8.12 7.11 35.91 32.29 29.43 27.30 58.59 54.38 51.05 48.35 84.64 78.72 74.38 70.71
∗
θ =1 α=1/4 10.56 8.34 7.00 6.20 11.77 9.50 8.08 7.01 35.85 32.23 29.55 27.26 58.53 54.41 51.01 48.22 84.71 78.74 74.41 70.73
α=1/2 10.55 8.40 7.15 6.21 11.56 9.42 8.00 7.09 36.02 32.06 29.42 27.26 58.50 54.33 51.15 48.41 84.63 78.71 74.40 70.83
α=1/8 10.60 8.24 6.93 6.11 11.81 9.51 8.16 7.19 35.67 32.23 29.60 27.48 58.34 54.20 50.92 48.13 84.49 78.81 74.44 70.71
∗
θ =2 α=1/4 10.37 8.22 6.85 6.15 11.77 9.35 8.01 7.02 35.55 31.96 29.48 27.13 58.29 54.16 50.72 48.04 84.64 78.73 74.39 70.70
α=1/2 10.35 8.20 7.00 6.04 11.33 9.22 7.92 6.94 35.75 31.87 29.20 27.03 58.38 54.05 51.06 48.23 84.60 78.76 74.40 70.84
α=1/8 10.46 8.37 6.99 6.09 11.67 9.43 8.13 7.14 35.19 31.61 29.33 26.93 57.62 53.55 50.35 47.72 84.55 78.72 74.49 70.65
∗
θ =4 α=1/4 9.91 7.98 6.83 6.00 11.30 9.14 7.93 6.89 34.82 31.19 28.53 26.46 57.43 53.42 49.97 47.47 84.45 78.60 74.46 70.63
α=1/2 9.51 7.49 6.38 5.48 10.57 8.47 7.29 6.40 34.44 30.95 28.28 26.07 57.55 53.49 50.39 47.63 84.38 78.71 74.29 70.70
α=1/8 10.25 8.67 7.68 6.74 11.67 9.88 8.42 7.48 32.72 29.75 27.37 25.69 54.73 51.29 48.32 46.08 83.79 78.46 74.17 70.47
∗
θ =8 α=1/4 8.86 7.21 6.16 5.42 9.86 8.41 7.17 6.33 31.42 28.09 25.68 24.09 54.10 50.60 47.54 45.06 84.04 78.45 73.96 70.18
α=1/2 6.58 5.25 4.53 3.98 7.54 6.13 5.23 4.46 30.32 26.99 24.64 22.89 54.32 50.60 47.72 45.07 83.96 78.47 74.09 70.46

* The DGP corresponds to equations (2.1)-(2.2)-(4.1) for quarterly data, &1={0}.

Table 5.3b: Empirical rejection probabilities for the pre-filtered (joint) CH test in the presence of an unattended break at frequency zero.* The sample size is
100.

ση=0 ση=0.01 ση=0.05 ση=0.1 ση=0.5

6 7 =0 6 7 =1 6 7 =2 6 7 =3 6 7 =0 6 7 =1 6 7 =2 6 7 =3 6 7 =0 6 7 =1 6 7 =2 6 7 =3 6 7 =0 6 7 =1 6 7 =2 6 7 =3 6 7 =0 6 7 =1 6 7 =2 6 7 =3

θ∗=0 13.53 6.08 4.26 3.29 15.21 7.36 5.38 4.06 53.93 39.51 34.70 30.99 85.38 77.55 74.40 71.94 99.70 98.91 98.84 98.69
α=1/8 13.57 6.02 4.32 3.26 15.26 7.43 5.28 4.06 53.68 39.04 34.34 30.81 85.35 77.42 74.41 71.97 99.71 98.92 98.85 98.72
∗
θ =1 α=1/4 13.39 6.19 4.35 3.26 15.30 7.43 5.45 4.13 53.57 39.52 34.61 30.99 85.39 77.46 74.38 71.85 99.71 98.91 98.82 98.69
α=1/2 13.43 6.15 4.23 3.25 15.13 7.31 5.30 4.10 53.60 39.38 34.42 30.74 85.26 77.43 74.33 71.82 99.71 98.90 98.81 98.68
α=1/8 13.51 6.17 4.47 3.53 15.33 7.37 5.34 4.26 53.18 38.83 34.34 30.77 84.92 77.27 74.04 71.67 99.72 98.91 98.84 98.68
∗
θ =2 α=1/4 13.18 6.08 4.24 3.29 15.15 7.41 5.44 4.06 53.18 39.08 34.39 30.55 85.04 77.24 74.11 71.62 99.72 98.92 98.76 98.70
α=1/2 13.14 5.91 4.16 3.14 14.72 7.08 5.04 3.98 53.07 38.73 33.84 30.31 84.98 77.23 73.99 71.48 99.71 98.88 98.77 98.70
α=1/8 13.53 6.44 4.80 3.82 15.59 7.58 5.64 4.63 52.10 38.43 33.53 30.05 84.13 76.15 73.06 70.66 99.70 98.93 98.79 98.66
∗
θ =4 α=1/4 12.52 5.79 4.11 3.10 14.43 7.12 5.16 3.91 51.34 37.88 33.10 29.50 83.87 76.35 73.14 70.26 99.69 98.87 98.71 98.57
α=1/2 11.72 5.19 3.74 2.87 13.15 6.26 4.47 3.51 50.69 36.96 32.02 28.80 84.01 76.07 72.73 70.10 99.67 98.86 98.75 98.65
α=1/8 13.49 6.67 4.86 3.86 15.48 7.75 5.85 4.69 49.95 37.09 32.40 28.74 81.68 73.66 70.66 67.89 99.68 98.87 98.69 98.56
∗
θ =8 α=1/4 10.89 5.22 3.61 2.74 12.74 6.24 4.54 3.35 47.09 34.46 29.68 26.10 80.84 72.76 69.34 66.66 99.58 98.78 98.51 98.38
α=1/2 9.12 3.91 2.92 2.14 10.34 4.87 3.42 2.67 45.21 32.31 28.00 24.90 80.36 72.64 69.40 66.37 99.62 98.76 98.60 98.42

* The DGP corresponds to equations (2.1)-(2.2)-(4.1) for quarterly data, &1={1,2}.

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and the time pattern of income inequality, Giornale degli economisti e Annali di economia, Vol. 58
(2), pp. 183-239, TD No. 350 (April 1999).
L. GUISO, A. K. KASHYAP, F. PANETTA and D. TERLIZZESE, Will a common European monetary policy
have asymmetric effects?, Economic Perspectives, Federal Reserve Bank of Chicago, Vol. 23 (4),
pp. 56-75, TD No. 384 (October 2000).

2000
P. ANGELINI, Are Banks Risk-Averse? Timing of the Operations in the Interbank Market, Journal of Money,
Credit and Banking, Vol. 32 (1), pp. 54-73, TD No. 266 (April 1996)
F. DRUDI and R: GIORDANO, Default Risk and optimal debt management, Journal of Banking and Finance,
Vol. 24 (6), pp. 861-892, TD No. 278 (September 1996).
F. DRUDI and R: GIORDANO, Wage indexation, employment and inflation, Scandinavian Journal of
Economics, Vol. 102 (4), pp. 645-668, TD No. 292 (December 1996).
F. DRUDI and A. PRATI, Signaling fiscal regime sustainability, European Economic Review, Vol. 44 (10),
pp. 1897-1930, TD No. 335 (September 1998).
F. FORNARI and R. VIOLI, The probability density function of interest rates implied in the price of options,
in: R. Violi, (ed.) , Mercati dei derivati, controllo monetario e stabilità finanziaria, Il Mulino,
Bologna, TD No. 339 (October 1998).
D. J. MARCHETTI and G. PARIGI, Energy consumption, survey data and the prediction of industrial
production in Italy, Journal of Forecasting, Vol. 19 (5), pp. 419-440, TD No. 342 (December
1998).
A. BAFFIGI, M. PAGNINI and F. QUINTILIANI, Localismo bancario e distretti industriali: assetto dei mercati
del credito e finanziamento degli investimenti, in: L.F. Signorini (ed.), Lo sviluppo locale:
un'indagine della Banca d'Italia sui distretti industriali, Donzelli, TD No. 347 (March 1999).
A. SCALIA and V. VACCA, Does market transparency matter? A case study, in: Market Liquidity: Research
Findings and Selected Policy Implications, Basel, Bank for International Settlements, TD No. 359
(October 1999).
F. SCHIVARDI, Rigidità nel mercato del lavoro, disoccupazione e crescita, Giornale degli economisti e
Annali di economia, Vol. 59 (1), pp. 117-143, TD No. 364 (December 1999).
G. BODO, R. GOLINELLI and G. PARIGI, Forecasting industrial production in the euro area, Empirical
Economics, Vol. 25 (4), pp. 541-561, TD No. 370 (March 2000).
F. ALTISSIMO, D. J. MARCHETTI and G. P. ONETO, The Italian business cycle: Coincident and leading
indicators and some stylized facts, Giornale degli economisti e Annali di economia, Vol. 60 (2), pp.
147-220, TD No. 377 (October 2000).
C. MICHELACCI and P. ZAFFARONI, (Fractional) Beta convergence, Journal of Monetary Economics, Vol.
45, pp. 129-153, TD No. 383 (October 2000).
R. DE BONIS and A. FERRANDO, The Italian banking structure in the nineties: testing the multimarket
contact hypothesis, Economic Notes, Vol. 29 (2), pp. 215-241, TD No. 387 (October 2000).
2001
M. CARUSO, Stock prices and money velocity: A multi-country analysis, Empirical Economics, Vol. 26 (4),
pp. 651-72, TD No. 264 (February 1996).
P. CIPOLLONE and D. J. MARCHETTI, Bottlenecks and limits to growth: A multisectoral analysis of Italian
industry, Journal of Policy Modeling, Vol. 23 (6), pp. 601-620, TD No. 314 (August 1997).
P. CASELLI, Fiscal consolidations under fixed exchange rates, European Economic Review, Vol. 45 (3),
pp. 425-450, TD No. 336 (October 1998).
F. ALTISSIMO and G. L. VIOLANTE, Nonlinear VAR: Some theory and an application to US GNP and
unemployment, Journal of Applied Econometrics, Vol. 16 (4), pp. 461-486, TD No. 338 (October
1998).
F. NUCCI and A. F. POZZOLO, Investment and the exchange rate, European Economic Review, Vol. 45 (2),
pp. 259-283, TD No. 344 (December 1998).
L. GAMBACORTA, On the institutional design of the European monetary union: Conservatism, stability
pact and economic shocks, Economic Notes, Vol. 30 (1), pp. 109-143, TD No. 356 (June 1999).
P. FINALDI RUSSO and P. ROSSI, Credit costraints in italian industrial districts, Applied Economics, Vol. 33
(11), pp. 1469-1477, TD No. 360 (December 1999).
A. CUKIERMAN and F. LIPPI, Labor markets and monetary union: A strategic analysis, Economic Journal,
Vol. 111 (473), pp. 541-565, TD No. 365 (February 2000).
G. PARIGI and S. SIVIERO, An investment-function-based measure of capacity utilisation, potential output
and utilised capacity in the Bank of Italy’s quarterly model, Economic Modelling, Vol. 18 (4), pp.
525-550, TD No. 367 (February 2000).

F. BALASSONE and D. MONACELLI, Emu fiscal rules: Is there a gap?, in: M. Bordignon and D. Da Empoli
(eds.), Politica fiscale, flessibilità dei mercati e crescita, Milano, Franco Angeli, TD No. 375 (July
2000).

A. B. ATKINSON and A. BRANDOLINI, Promise and pitfalls in the use of “secondary" data-sets: Income
inequality in OECD countries, Journal of Economic Literature, Vol. 39 (3), pp. 771-799, TD No.
379 (October 2000).
D. FOCARELLI and A. F. POZZOLO, The determinants of cross-border bank shareholdings: An analysis with
bank-level data from OECD countries, Journal of Banking and Finance, Vol. 25 (12), pp. 2305-
2337, TD No. 381 (October 2000).
M. SBRACIA and A. ZAGHINI, Expectations and information in second generation currency crises models,
Economic Modelling, Vol. 18 (2), pp. 203-222, TD No. 391 (December 2000).
F. FORNARI and A. MELE, Recovering the probability density function of asset prices using GARCH as
diffusion approximations, Journal of Empirical Finance, Vol. 8 (1), pp. 83-110, TD No. 396
(February 2001).
P. CIPOLLONE, La convergenza dei salari manifatturieri in Europa, Politica economica, Vol. 17 (1), pp. 97-
125, TD No. 398 (February 2001).
E. BONACCORSI DI PATTI and G. GOBBI, The changing structure of local credit markets: Are small
businesses special?, Journal of Banking and Finance, Vol. 25 (12), pp. 2209-2237, TD No. 404
(June 2001).
G. MESSINA, Decentramento fiscale e perequazione regionale. Efficienza e redistribuzione nel nuovo
sistema di finanziamento delle regioni a statuto ordinario, Studi economici, Vol. 56 (73), pp. 131-
148, TD No. 416 (August 2001).

2002
R. CESARI and F. PANETTA, Style, fees and performance of Italian equity funds, Journal of Banking and
Finance, Vol. 26 (1), TD No. 325 (January 1998).
C. GIANNINI, “Enemy of none but a common friend of all”? An international perspective on the lender-of-
last-resort function, Essay in International Finance, Vol. 214, Princeton, N. J., Princeton University
Press, TD No. 341 (December 1998).
A. ZAGHINI, Fiscal adjustments and economic performing: A comparative study, Applied Economics,
Vol. 33 (5), pp. 613-624, TD No. 355 (June 1999).
F. ALTISSIMO, S. SIVIERO and D. TERLIZZESE, How deep are the deep parameters?, Annales d’Economie et
de Statistique,.(67/68), pp. 207-226, TD No. 354 (June 1999).
F. FORNARI, C. MONTICELLI, M. PERICOLI and M. TIVEGNA, The impact of news on the exchange rate of
the lira and long-term interest rates, Economic Modelling, Vol. 19 (4), pp. 611-639, TD No. 358
(October 1999).
D. FOCARELLI, F. PANETTA and C. SALLEO, Why do banks merge?, Journal of Money, Credit and Banking,
Vol. 34 (4), pp. 1047-1066, TD No. 361 (December 1999).
D. J. MARCHETTI, Markup and the business cycle: Evidence from Italian manufacturing branches, Open
Economies Review, Vol. 13 (1), pp. 87-103, TD No. 362 (December 1999).
F. BUSETTI, Testing for stochastic trends in series with structural breaks, Journal of Forecasting, Vol. 21
(2), pp. 81-105, TD No. 385 (October 2000).

F. LIPPI, Revisiting the Case for a Populist Central Banker, European Economic Review, Vol. 46 (3), pp.
601-612, TD No. 386 (October 2000).
F. PANETTA, The stability of the relation between the stock market and macroeconomic forces, Economic
Notes, Vol. 31 (3), TD No. 393 (February 2001).
G. GRANDE and L. VENTURA, Labor income and risky assets under market incompleteness: Evidence from
Italian data, Journal of Banking and Finance, Vol. 26 (2-3), pp. 597-620, TD No. 399 (March
2001).
A. BRANDOLINI, P. CIPOLLONE and P. SESTITO, Earnings dispersion, low pay and household poverty in
Italy, 1977-1998, in D. Cohen, T. Piketty and G. Saint-Paul (eds.), The Economics of Rising
Inequalities, pp. 225-264, Oxford, Oxford University Press, TD No. 427 (November 2001).

2003

P. CASELLI, P. PAGANO and F. SCHIVARDI, Uncertainty and slowdown of capital accumulation in Europe,
Applied Economics, Vol. 35 (1), pp. 79-89, TD No. 372 (March 2000).

E. GAIOTTI and A. GENERALE, Does monetary policy have asymmetric effects? A look at the investment
decisions of Italian firms, Giornale degli Economisti e Annali di Economia, Vol. 61 (1), pp. 29-59,
TD No. 429 (December 2001).

FORTHCOMING

L. GAMBACORTA, Asymmetric bank lending channels and ECB monetary policy, Economic Modelling, TD
No. 340 (October 1998).

F. SCHIVARDI, Reallocation and learning over the business cycle, European Economic Review, TD No.
345 (December 1998).
F. LIPPI, Strategic monetary policy with non-atomistic wage-setters, Review of Economic Studies, TD No.
374 (June 2000).
P. ANGELINI and N. CETORELLI, Bank competition and regulatory reform: The case of the Italian banking
industry, Journal of Money, Credit and Banking, TD No. 380 (October 2000).

P. CHIADES and L. GAMBACORTA, The Bernanke and Blinder model in an open economy: The Italian case,
German Economic Review, TD No. 388 (December 2000).

P. PAGANO and F. SCHIVARDI, Firm size distribution and growth, Scandinavian Journal of Economics, TD.
No. 394 (February 2001).
 M. PERICOLI and M. SBRACIA, A Primer on Financial Contagion, Journal of Economic Surveys, TD No.
407 (June 2001).
M. SBRACIA and A. ZAGHINI, The role of the banking system in the international transmission of shocks,
World Economy, TD No. 409 (June 2001).

L. GAMBACORTA, The Italian banking system and monetary policy transmission: Evidence from bank level
data, in: I. Angeloni, A. Kashyap and B. Mojon (eds.), Monetary Policy Transmission in the Euro
Area, Cambridge, Cambridge University Press, TD No. 430 (December 2001).

M. EHRMANN, L. GAMBACORTA, J. MARTÍNEZ PAGÉS, P. SEVESTRE and A. WORMS, Financial systems and the
role of banks in monetary policy transmission in the euro area, in: I. Angeloni, A. Kashyap and B.
Mojon (eds.), Monetary Policy Transmission in the Euro Area, Cambridge, Cambridge University
Press. TD No. 432 (December 2001).

D. FOCARELLI, Bootstrap bias-correction procedure in estimating long-run relationships from dynamic

panels, with an application to money demand in the euro area, Economic Modelling, TD No. 440
(March 2002).

D. FOCARELLI and F. PANETTA, Are mergers beneficial to consumers? Evidence from the market for bank
deposits, American Economic Review, TD No. 448 (July 2002).
F. BUSETTI and A. M. ROBERT TAYLOR, Testing against stochastic trend and seasonality in the presence of
unattended breaks and unit roots, Journal of Econometrics, TD No. 470 (February 2003).