# Change in persistence tests for panels: An update and some new results

**ABSTRACT** In this paper we propose a set of new panel tests to detect changes in persistence. The test statistics are used to test the null hypothesis of stationarity against the alternative of a change in persistence from I(0) to I(1), from I(1) to I(0) and in an unknown direction. The limiting distributions of the panel tests are derived, and small sample properties are investigated by Monte Carlo experiments under the hypothesis that the individual series are independently cross-section distributed. These tests have a good size and power properties. Cross-sectional dependence is also considered. A procedure of de-factorizing, proposed by Stock and Watson (2002), is applied. The defactored panel tests have good size and power. The empirical results obtained from applying these tests to a panel covering 21 OECD countries observed between 1970 and 2007 suggest that inflation rate changes from I(1) to I(0) when cross-correlation is considered.

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**ABSTRACT:**The founding of the Federal Reserve System in 1914 led to a substantial change in the behavior of nominal interest rates. The authors examine the timing of this change and the speed with which it was effected. They then use data on the term structure of interest rates to determine how expectations responded. Their results indicate that the change in policy regime was rapid and that individuals quickly understood the new environment they were facing. Copyright 1987 by American Economic Association.American Economic Review 02/1987; 77(3):358-74. · 2.69 Impact Factor - [Show abstract] [Hide abstract]

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Page 1

Università degli Studi del Molise

Facoltà di Economia

Dipartimento di Scienze Economiche, Gestionali e Sociali

Via De Sanctis, I-86100 Campobasso (Italy)

ECONOMICS & STATISTICS DISCUSSION PAPER

No. 40/07

Change in persistence tests for panels

by

Roy Cerqueti

University of Macerata

Mauro Costantini

ISAE and University of Rome “La Sapienza”

and

Luciano Gutierrez

University of Sassari

The Economics & Statistics Discussion Papers are preliminary materials circulated to stimulate discussion

and critical comment. The views expressed in the papers are solely the responsibility of the authors.

Page 2

Change in persistence tests for panels∗

Roy Cerqueti†, Mauro Costantini‡, Luciano Gutierrez§

Abstract

In this paper we propose a set of new panel tests to detect changes in

persistence. These statistics are used to test the null hypothesis of sta-

tionarity against the alternative of a change in persistence from I(0) to

I(1) or viceversa. Alternative of unknown direction is also considered. The

limiting distributions of the panel tests are derived and small sample prop-

erties are investigated by Monte Carlo experiments under the hypothesis

that the individual series are cross-sectionally independently distributed.

These tests have a good size and power properties. Cross-sectional de-

pendence is also considered. A procedure of de-factorizing proposed by

Stock and Watson (2002) is applied. Monte Carlo analysis is conducted

and the defactored panel tests show to have good size and power. The

empirical results obtained from applying these tests to a panel covering

15 European countries between 1970 and 2006 suggest that inflation rate

changes from I(1) to I(0) when cross-correlation is considered.

Keywords: Persistence, Stationarity, Panel data.

JEL Classification: C12, C23.

∗The authors would like to thank Marco Centoni, Claudio Lupi and Alberto Pozzolo and

other seminar participants at University of Molise, Dept. SEGeS, for valuable comments and

suggestions.

†University of Macerata, Department of Economics and Finance;

roy.cerqueti@unimc.it

‡University of Rome “La Sapienza”, Faculty of Economics, Department of Public

Economics, and Institute for Studies and Economic Analyses (ISAE); E-mail address:

m.costantini@dte.uniroma1.it;

§Corresponding Author: University of Sassari, Department of Agricultural Economics;

E-mail address: lgutierr@uniss.it

E-mail address:

1

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1Introduction

The recent time series literature has shown that the economic and financial

data are characterized by a change in persistence between separate I(1) and

I(0) regime rather than a simply I(1) or I(0) behavior. For example, Cogley and

Sargent (2001) and Emery (1994), using post World War II data, argued that

persistence in U.S. inflation has decreased substantially since the early 1980s.

Strikingly, Emery finds that U.S. inflation in the 1980s can best be described as

white noise. Further evidence of persistence change from I(1) to I(0) behavior

in U.S. inflation is also reported in Kim (2000), Busetti and Taylor (2004) and

Leybourne et al. (2003). Other variables for which changes in persistence have

been observed include real output (e.g Taylor, 2005) and short-term interest

rates (e.g. Mankiw et al., 1987)).

A number of testing procedures have been developed to test against changing

persistence. The most popular of these appear to be the ratio-based persistence

change tests of Kim (2000), Kim et al. (2002), Busetti and Taylor (2004) and

Harvey et al. (2006). These statistics test the null hypothesis that a series

is a constant I(0) process against the alternative that it displays a change in

persistence, either from I(0) to I(1), or viceversa. Kim (2000) and Kim et al.

(2002) proposed residual-based ratio test against changes in persistence in a

time series, focusing on the case of a shift from I(0) to I(1), at some point in the

sample. Kim (2000) also discussed the possibility of I(1) to I(0) shifts but did

not provide tests against such alternative. Busetti and Taylor (2004) proposed

new ratio-based tests and breakpoint estimators which are consistent under I(1)

to I(0) changes, and they demonstrated that the ratio-based tests which are

consistent against changes from I(1) to I(0) are not consistent against changes

from I(0) to I(1), and viceversa, with neither consistent against constant I(1)

processes. Harvey et al. (2006) developed a set of new tests which are based

on modified version of the ratio-base statistics of Kim (2000), Kim et al.(2002)

and Busetti and Taylor (2004). These modifications use the variable addition

approach of Vogelsang (1998), and a recent generalization due to Sayginsoy

(2003), yielding tests which, by design, have the same critical values regardless

of whether the process is I(0) or (near) I(1) throughout. This technique can be

only used with the ratio based test of the null I(0) because other tests of the I(0)

(I(1)) null are based on statistics which are divergent under constant I(1) (I(0))

processes. Hence, the null hypothesis is that of constant persistence (either a

constant I(0) process or a constant I(1) process), and the alternative is that of a

change in persistence. Finally, Costantini and Gutierrez (2007) consider a panel

data companion of the set of recursive ADF unit root tests for single time series

as proposed in Banerjee et al. (1992).

In this paper we propose a set of new panel tests to detect changes in persistence.

These statistics are used to test the null hypothesis of stationarity against the

alternative of a change in persistence from I(0) to I(1) or viceversa. Alternative

of unknown direction is also considered.

The paper is organized as follows. In section 2 we present new panel persis-

tence change tests under the hypothesis of cross-section independence. Section

3 describes the panel tests under cross-section dependence hypothesis. Section 4

2

Page 4

presents Monte Carlo simulations. In section 5 these new panel tests to a panel

of 15 European inflation rate series for the period 1970.1-2006.2 are applied.

Section 6 concludes.

2Persistence tests without cross-section corre-

lation

2.1The model

Consider the following Gaussian unobserved components model for a sample of

N cross-sections observed over T time periods:

yi,t= di,t+ µi,t+ εi,t,i = 1,...,N, t = 1,...,T,

(1)

we allow for the following three cases:

• Case 1: I(0) → I(1)

µi,t= µi,t−1+ 1(t > [Tτ])ηi,t,i = 1,...,N, t = 1,...,T,

(2)

• Case 2: I(1) → I(0)

µi,t= µi,t−1+ 1(t ≤ [τT])ηi,t

i = 1,...,N, t = 1,...,T.

(3)

• Case 3: unknown direction I(0) → I(1) or I(1) → I(0)

where 1(·) is the indicator function, di,tis a deterministic component, εi,tand

ηi,tare mutually independent mean zero iid gaussian process with variance σ2

and σ2

unity vector.

¿From (1)-(2), it can be easily seen that for each cross section i, the data gener-

ating process yields a process which is stationary up to and including time [τT],

with the change-point proportion τ ∈ (0,1), but is I(1) after the break, if and

only if σ2

the data generating process yields a process which is I(1) up to and including

time [τT] but it is stationary after the break, if and only if σ2

Therefore, panel test of stationarity against a shift in persistence from station-

arity to a unit root or viceversa can be framed in testing the null hypothesis:

εi

ηi. For the present, the deterministic components are taken to be the

ηi> 0. From (1)-(3), it can be easily seen that for each cross section i,

ηi> 0.

H0= σ2

ηi= 0, ∀i

(4)

against the alternative hypothesis

H1= σ2

ηi> 0, at least for some i.

(5)

The following assumption plays a key role for the remaining part of the paper.

Assumption 1 The process {µi,t}+∞

1. E[µi] = 0;

i,t=0is such that for each i

3

Page 5

2. E|µi|4< +∞;

3. {µi,t}+∞

i,t=0is φ-mixing with mixing coefficients φmsuch that

∞

?

m=1

φ1−2/γ

m

< +∞;

4. There exists the long-run variance

σ2

µi=

∞

?

j=0

E[µi,j+1µ?

i,1];

5. for each s ∈ (0,1), we have

lim

T→∞V

?1

√T

[sT]

?

t=1

µi,t

?

= sσ2

µi

and

lim

T→∞V

?1

√T

T

?

t=[sT]+1

µi,t

?

= (1 − s)σ2

µi

.

The above conditions have been used by Phillips (1987), Phillips and Perron

(1988) and Phillips and Solo (1992), among others, to prove results on the

asymptotic distribution of a stochastic process. Finally, note that throughout

the next sections we use sequential limits, wherein T → ∞ followed by N → ∞.

2.2Panel ratio-based tests: I(0) → I(1)

In this section we present new panel tests to detect changes in persistence as

in (2) and investigate their asymptotic behavior. We show that panel tests are

standard normal distributed.

Consider the gaussian process (1)-(2). We want to test the null hypothesis H0

in (4) against H1in (5). Let ˜ εi,t, i = 1,...,N and t = 1,...,T, be the residuals

from the regression of yi,ton intercept. If a structural change occurs at time

t = [τT] for τ ∈ (0,1), the following partial sum process can be defined:

Then, we consider the following statistics-test:

S(0)

i,t=?t

j=[Tτ]+1˜ εi,j

j=1˜ εi,j

t = 1,...,[Tτ]; i = 1,...,N,

S(1)

i,t=?t

t = [Tτ] + 1,...,T; i = 1,...,N,

(6)

KT,N(τ) =

√N

σ

·

?(T − [Tτ])−2

[Tτ]−2

·1

N·

N

?

i=1

?T

t=[Tτ]+1S(1)

?[Tτ]

i,t(τ)2

i,t(τ)2

t=1S(0)

i,t(τ)2

− µ

?

,

(7)

where

µ = E

?(T − [Tτ])−2?T

t=[Tτ]+1S(1)

t=1S(0)

[Tτ]−2?[Tτ]

i,t(τ)2

?

4

Page 6

and

σ =

?

?

?

?V

?(T − [Tτ])−2?T

t=[Tτ]+1S(1)

t=1S(0)

i,t(τ)2

[Tτ]−2?[Tτ]

i,t(τ)2

?

.

Fixed i = 1,...,N, t1= 1,...,[Tτ] and t2= [Tτ]+1,...,T, it results S(1)

S(0)

i,t1and

i,t2mutually independent. Therefore

µ =

(T − [Tτ])−2?T

t=[Tτ]+1E[S(1)

t=1E[S(0)

i,t(τ)2]

[Tτ]−2?[Tτ]

i,t(τ)2]

(8)

and

σ =

?

?

?

?(T − [Tτ])−4?T

t=[Tτ]+1V[S(1)

t=1V[S(0)

i,t(τ)2]

[Tτ]−4?[Tτ]

i,t(τ)2]

(9)

Theorem 2 Suppose that Assumption 1 is true under the null hypothesis H0.

Then it results

lim

N→+∞

lim

T→+∞KT,N(τ) = K(τ) ∼ N(0,1).

(10)

Proof. By Theorem 3.1 in Kim (2000), it results

lim

T→+∞

(T − [Tτ])−2

[Tτ]−2

?T

t=[Tτ]+1S(1)

?[Tτ]

i,t(τ)2

t=1S(0)

i,t(τ)2

=(1 − τ)−2?1

τVi(r − τ)2dr

τ−2?τ

0Vi(r)2dr

,

where {Vi}+∞

and identically distributed.

Furthermore, by the hypotheses stated in Assumption 1, we have that

i=1is a sequence of standard brownian bridges that are independent

lim

T→+∞E

?(T − [Tτ])−2

[Tτ]−2

?T

t=[Tτ]+1S(1)

?[Tτ]

?T

i,t(τ)2

t=1S(0)

i,t(τ)2

?

= ¯ µ,

lim

T→+∞V

?(T − [Tτ])−2

?(1 − τ)−2?1

?(1 − τ)−2?1

[Tτ]−2

t=[Tτ]+1S(1)

?[Tτ]

τVi(r − τ)2dr

τ−2?τ

τVi(r − τ)2dr

τ−2?τ

i,t(τ)2

t=1S(0)

i,t(τ)2

?

= ¯ σ2,

with

¯ µ = E

0Vi(r)2dr

?

(11)

and

¯ σ2= V

0Vi(r)2dr

?

(12)

for each i = 1,...,N.

Therefore

lim

N→+∞

lim

T→+∞KT,N(τ) =lim

N→+∞

√N

¯ σ

·1

N·

N

?

i=1

?(1 − τ)−2?1

τVi(r − τ)2dr

τ−2?τ

0Vi(r)2dr

− ¯ µ

?

,

5

Page 7

for i = 1,...,N.

The Central Limit Theorem guarantees that

√N

¯ σ

lim

N→+∞

·1

N·

N

?

i=1

?(1 − τ)−2?1

τVi(r − τ)2dr

0Vi(r)2drNτ−2?τ

− ¯ µ

?

∼ N(0,1),

and the Theorem is completely proved.

The true value of τ is unknown. Under the situation of the true change period

being unknown three transformations of the tests KT,N(τ) defined in (7), for

testing H0against H1with unknown break point [Tτ], can be considered.

• A maximum-Chow-type test as is considered in Davies (1977), Hawkins

(1987), Kim and Siegmund (1989), and Andrews (1993) is

H1(KT,N(τ)) := max

τ∈(0,1)KT,N(τ).

(13)

• The mean score test proposed by Hansen (1991)

H2(KT,N(τ)) :=

?

τ∈(0,1)

KT,N(τ)dτ.

(14)

• The mean-exponential test introduced by Andrews and Ploberger (1994),

that is

H3(KT,N(τ)) := log

??

τ∈(0,1)

exp[KT,N(τ)]dτ

?

.

(15)

The asymptotic distribution of the tests defined in (13), (14) and (15) are given

in the following result.

Theorem 3 The following propositions hold.

(i) It results

lim

T→+∞

lim

N→+∞Hj(KT,N(τ)) = Hj(K(τ)),j = 1,2,3.

(ii) For each j = 1,2,3, we have Hj(K(τ)) ∼ N(0,1).

Proof.

(i) The result follows from the continuous mapping theorem and the continu-

ity of the functionals.

(ii) Since K(τ) ∼ N(0,1), for each τ, then K(τ) is an iid continuous-time

stochastic process. Therefore, we can define the random variable K ∼

N(0,1) such that K(τ) ≡ K, for each τ ∈ (0,1).

Then we have

H1(K(τ)) = max

?

??

τ∈(0,1)(K) = K ∼ N(0,1);

?

exp[K]dτ

H2(K(τ)) =

τ∈(0,1)

Kdτ = K ·

τ∈(0,1)

dτ = K ∼ N(0,1);

?

H3(K(τ)) = log

τ∈(0,1)

?

= log{exp[K]}·

τ∈(0,1)

dτ = K ∼ N(0,1)

The result is completely proved.

6

Page 8

2.3Panel reverse test: I(1) → I(0)

Consider the gaussian process (1)-(3). In this case, the null hypothesis is refereed

to stationary process and the alternative to a shift from I(1) to I(0).

following statistics are proposed:

√N

σ

The

K?

T,N(τ) =

·

?

[Tτ]−2

(T − [Tτ])−2·1

N·

N

?

i=1

?[Tτ]

t=1S(0)

t=[Tτ]+1S(1)

i,t(τ)2

i,t(τ)2− µ

?T

i,t(τ)2

?

,

(16)

where

µ = E

?

[Tτ]−2?[Tτ]

t=1S(0)

t=[Tτ]+1S(1)

(T − [Tτ])−2?T

?

i,t(τ)2

?

and

σ =

?

?

?

?V

[Tτ]−2?[Tτ]

t=1S(0)

t=[Tτ]+1S(1)

i,t(τ)2

(T − [Tτ])−2?T

i,t(τ)2

?

.

The asymptotic distribution of the statistics defined in (16) is the object of the

following result.

Theorem 4 Suppose that Assumption 1 is true under the null hypothesis H0.

Then it results

lim

N→+∞

lim

T→+∞K?

T,N(τ) = K?(τ) ∼ N(0,1).

(17)

Proof. By Theorem 3.1 in Busetti and Taylor (2004), it results

lim

T→+∞

[Tτ]−2?[Tτ]

t=1S(0)

t=[Tτ]+1S(1)

i,t(τ)2

(T − [Tτ])−2?T

V∗∗

i,t(τ)2=

τ−2?τ

0[V∗∗∗

τ[V∗∗

i

(r)]2dr

(1 − τ)−2?1

i(r)]2dr

,

where

i (r) = Vi(r) − Vi(τ) − (r − τ)(1 − τ)−1(Vi(1) − Vi(τ))

V∗∗∗

i

(r) = Vi(r) − rτ−1Vi(τ)

and

Vi(r) = W0(r) + c

??min{r,τ}

0

Wc(s)ds + 1(r > τ)[(r − τ)Wc(τ)]

?

,

where W is a standard Wiener process. Hypotheses stated in Assumption 1

imply

?

?

with

¯ µ = E

(1 − τ)−2?1

7

lim

T→+∞E

[Tτ]−2?[Tτ]

[Tτ]−2?[Tτ]

t=1S(0)

t=[Tτ]+1S(1)

i,t(τ)2

(T − [Tτ])−2?T

(T − [Tτ])−2?T

?

i,t(τ)2

?

?

= ¯ µ,

lim

T→+∞V

t=1S(0)

t=[Tτ]+1S(1)

i,t(τ)2

i,t(τ)2

= ¯ σ2,

τ−2?τ

0[V∗∗∗

τ[V∗∗

i

(r)]2dr

i(r)]2dr

?

(18)

Page 9

and

¯ σ2= V

?

τ−2?τ

0[V∗∗∗

τ[V∗∗

i

(r)]2dr

(1 − τ)−2?1

i(r)]2dr

?

(19)

for each i = 1,...,N.

Hence

lim

N→+∞

for i = 1,...,N.

The Central Limit Theorem guarantees that

√N

¯ σ

lim

T→+∞K?

T,N(τ) =lim

N→+∞

√N

¯ σ

·1

N·

N

?

i=1

?

τ−2?τ

0[V∗∗∗

τ[V∗∗

i

(r)]2dr

(1 − τ)−2?1

i(r)]2dr

− ¯ µ

?

,

lim

N→+∞

·1

N·

N

?

i=1

?

τ−2?τ

0[V∗∗∗

τ[V∗∗

i

(r)]2dr

(1 − τ)−2?1

i(r)]2dr

− ¯ µ

?

∼ N(0,1),

and the Theorem is completely proved.

In the next result, the asymptotic distributions of the transformations H1, H2

and H3of the test K?are given.

Theorem 5 The following propositions hold.

(i) It results

lim

T→+∞

lim

N→+∞Hj(K?

T,N(ˆ τ)) = Hj(K?(τ)),j = 1,2,3.

(ii) For each j = 1,2,3, we have Hj(K?(τ)) ∼ N(0,1).

Proof. Analogous to the proof of Theorem 3.

2.4Panel tests with unknown direction

We now discuss the case of unknown direction of changes in persistence. Three

panel tests are developed and their asymptotic distributions are derived. The

tests are:

√N

σ∗

j

i=1

where

˜KT,i=(T − [Tτ])−2

[Tτ]−2

˜K?

µ∗

Hj(˜KT,i),Hj(˜K∗

?

The asymptotic distributions of these tests are now derived.

Mj,∗

T,N=

·1

N·

N

?

[max{Hj(˜KT,i),Hj(˜K∗

T,i)} − µ∗

j],j = 1,2,3; (20)

?T

t=[Tτ]+1S(1)

?[Tτ]

i,t(τ)2

t=1S(0)

i,t(τ)2

,

T,i= (˜KT,i)−1,

T,i)}

?

j= E

??

j = 1,2,3; i = 1,...,N,

σ∗

j=

V

?

Hj(˜KT,i),Hj(˜K∗

T,i)}

j = 1,2,3; i = 1,...,N.

Theorem 6 It results

lim

N→+∞

lim

T→+∞Mj,∗

T,N∼ N(0,1).

Proof. It is a direct consequence of the Central Limit Theorem.

8

Page 10

2.5Modified panel tests

In this section we propose panel tests that are based on the modified version of

the statistics developed in subsections 2.2-2.4. These tests have the same critical

value in the limit as the corresponding unmodified tests under the null hypoth-

esis H0, and the same limiting critical value is also appropriate under the the

alternative hypothesis H1. The modification proposed has no asymptotic effect

under the null H0, so that the limiting distribution of the modified tests is the

same of the corresponding unmodified tests. Under the alternative hypothesis,

the asymptotic distribution of the tests is affected by this modification, but the

last is chosen such that the limiting critical value is precisely the same as under

the null. The panel tests developed are:

MHd

j(KT,N(τ)) := exp(−bJ1,N,T) · Hj(KT,N(τ)),

MHd

j = 1,2,3;(21)

j(K?

T,N(τ)) := exp(−bJ1,N,T) · Hj(K?

MMj∗

where b is a finite constant and J1,N,T is the arithmetic mean on N of the

truncated sequences of T−1times the Wald statistic J(i)

hypothesis ςi,k+1= ··· = ςi,9= 0 in panel regression

?

Under the null hypothesis, Harvey et al. (2006) show that

T,N(τ)),j = 1,2,3;(22)

T,N:= exp(−bJ1,N,T) · Mj∗

T,N

j = 1,2,3;(23)

1,Tfor testing the joint

yi,t= εi,t+

9

j=k+1

ςi,jtj+ error,t = [τT] + 1,...,T; i = 1,...,N.

(24)

lim

T→+∞J(i)

1,T= 1,

∀i = 1,...,N.

Therefore, since we assumed independence and identical distribution with re-

spect to the cross-sectional dimension i, we have

lim

T,N→+∞J1,N,T= lim

T,N→+∞

1

N

N

?

i=1

J(i)

1,T= 1.

Under the alternative hypothesis, following Harvey et al. (2006), we modify the

tests (21), (22) and (23) by introducing

JN,min:= min

τ∈(0,1)J1,N,[τT].

We define

MHj,min(KT,N(τ)) := exp(−bJN,min) · Hj(KT,N(τ)),

MHj,min(K?

MMj∗

and

Jmin:=

j = 1,2,3;(25)

T,N(τ)) := exp(−bJN,min) · Hj(K?

min,T,N:= exp(−bJN,min) · Mj∗

T,N(τ)),j = 1,2,3;(26)

T,N

j = 1,2,3; (27)

lim

N→+∞JN,min.

9

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By rewriting the asymptotic analysis under the alternative hypothesis of Harvey

et al. (2006), by using the fact that the asymptotic distributions of the tests

Hj(KT,N(τ)), Hj(K?

Theorem 3, Theorem 5 and Theorem 6), for each j = 1,2,3, we have

T,N(τ)), and Mj∗

T,N, j = 1,2,3 are standard gaussian (see

lim

N→+∞T−2(MHj,min(KT,N(τ)) − MHd

j(KT,N(τ))) =

= T−2

lim

N→+∞(MHj,min(KT,N(τ)) − MHd

= T−2

lim

j(KT,N(τ))) =

N→+∞{1 − exp[−bJN,min]}MHd

= T−2{1 − exp[−bJmin]}

Analogously, it results

j(KT,N(τ)) =

lim

N→+∞MHd

j(KT,N(τ)) = op(1)Op(1) = op(1).

lim

N→+∞T−2(MHj,min(K?

T,N(τ)) − MHd

j(K?

T,N(τ))) = op(1)

and

lim

N→+∞T−2(MMj∗

min,T,N− MMj∗

T,N) = op(1).

The tests is then consistent under the alternative hypothesis.

2.6Estimation of the break

In this subsection we present a procedure to estimate the unknown change point.

Consider the following estimator:

ΛN,T(τ) =

1

N

N

?

i=1

?T

t=[Tτ]+1˜ µ2

?[Tτ]

i,t/(T − [Tτ])2

t=1˜ µ2

i,t/[τT]

.

(28)

In order to explore the asymptotic behavior of the estimated unknown change

point, the following assumption is required.

Assumption 7 Let ˜ µi,s+1,˜ µi,s+2,....., ˜ µi,s+m, for s ∈ 0,...,T − 1 and m ≤ T −

s be a sequence of stationary variables. Assume that m−1?s+m

Now, let ˆ τ be such that:

?

The following theorem shows asymptotic properties of ˆ τ:

t=s+1˜ µ2

i,t→ E[µ2

i]

for E[µ2

i] < ∞, ∀i = 1,...,N.

ˆ τ =argmaxτ∈(0,1)ΛN,T(τ)

?

.

(29)

Theorem 8 Suppose that Assumption 7 holds. Under the alternative hypothe-

sis, it results

(ˆ τ − τ) = op(1),

T(ˆ τ − τ) = Op(1),

(30)

(31)

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Proof. Since

M(i)

T(τ) :=

?T

t=[Tτ]+1˜ µ2

?[Tτ]

ˆ τ = argmaxτ∈(0,1)M(i)

?1

i,t/(T − [Tτ])2

t=1˜ µ2

i,t/[τT]

≥ 0,

∀i = 1,...,N,

then

T(τ) ⇒

?

⇒ ˆ τ = argmaxτ∈(0,1)

N·

N

?

i=1

M(i)

T(τ)= argmaxτ∈(0,1)ΛN,T(τ).

Therefore, Theorem 3.5 in Kim (2000) guarantees the thesis.

3Persistence test with cross-section correlation

Previous derivations are valid under the assumption that the units are cross-

section independent. However, this requirement is rarely likely to be satisfied

in empirical economic applications where the countries or regions depend each

other. In order to generalize the framework of the paper we have extended our

approach to account for the presence of common factors as in Stock and Watson

(2002), Bai (2003) and Bai and Ng (2004).

Let as before yi,tbe the observation on the i-th cross section unit at time t for

i = 1,...,N, t = 1,...,T and suppose that it is generated according to the

following linear heterogeneous panel data model:

yi,t= di,t+ µi,t+ εi,t

µi,t= µi,t−1+ 1(t > [τT])ηi,t, if τ ∈ (0,1)

εi,t= Ftλi+ ui,t

(32)

(33)

(34)

where Ftdenotes a stationary (1×m)-vector of unobserved common factors, λi

indicates the vector of loadings and ui,tis a stationary process. For the present,

the deterministic component di,tis taken to be the unity vector. The following

assumptions are required.

Assumption 9 (i) for non-random λi,?λi? ≤ M; for random λi, E?λi?4≤

M,

(ii)

N

1

?N

i=0λiλ

?

i⇒?

Π, a (m × m) positive matrix.

Assumption 10 The error ui,t, the factor Ftand the loadings λiare mutually

independent.

Assumption 9 ensures that the factor loadings are identifiable. The estimation

of the common factors are done as in Stock and Watson (2002), i.e. using prin-

cipal components. Specifically, the principal component of F = (F1,F2,...,FT),

denoted as˜F, is

largest eigenvalues of the (T×T) matrix of demeaned and stardardized ˜ yi˜ yi?. Un-

der the normalization˜F˜F?/T = Ir, the estimated loading matrix is˜?=˜F?˜ yi/T.

˜ zi,t= ˜ yi,t−˜Ft˜λi

√T times the first r eigenvectors corresponding to the first r

Therefore, the estimated residuals are defined as

(35)

11

Page 13

From the data generating process (32)-(34) and (35), it can be easily seen that

for each cross section i, the process ˜ zi,tis stationary up to and including time

[τT] but is I(1) after the break, if and only if σ2

Thus our strategy is to apply the panel test statistics presented in section 2 to

the de-factored data ˜ zi,t.

ηi> 0.

4Monte Carlo simulation results

In this section we use Monte Carlo experiments to examine finite sample prop-

erties of the panel persistence tests. We consider two sets of Monte Carlo ex-

periments. The first set focuses on the model (1)-(2), i.e where we assume

cross-section independence, while the second set of experiments is based on the

model (32)-(35) where we allow for the presence of dependence across the differ-

ent units in the panel. We start the analysis considering the empirical rejection

frequencies of the tests when the data are generated according to the I(0)-I(1)

switch data generating process embraced in (1)-(2) under the hypothesis of

cross-sectional independency. As in Busetti and Taylor (2004) we investigate

the impact of varying the signal-noise-ratio among σηi= 0,0.5,0.10,0.25,0.50

and σεi ∼ U [0.5,1.5] and the breakpoint among τ = 0.3,0.5,0.7. The sim-

ulation results were performed 1000 Monte Carlo replications and the RNDN

function of Gauss 6.0. As is often used in the literature, for all the tests we

fix?= [0.2,0.8] and T = 50,100 and N = 1,10,30,50. In Table 1 we present

have been used to standardize the panel tests. Their values have been computed

using 50000 replications.

the moments of Kim’s (2000, 2002) and Busetti and Taylor’s (2004) tests which

Table 1 about here

The size results for the benchmark model (1)-(2) are reported in table 2.

Table 2 about here

All the panel test statistics seem to have good size for both small and large T,N.

Looking at power of the tests, see table 3, table 4 and table 5 many interesting

things emerge.

Table 3 about here

Table 4 about here

Table 5 about here

First of all, panel tests have better properties than single time series tests.

Comparing the power of panel tests derived along the line of Kim (2000), Busetti

and Taylor (2004) and Harvey et al. (2006) we don’t find significant differences.

As expected, the power of tests grows, with the exception of reversed panel

tests, as the signal to noise ratio rise and the smaller is τ. This occurs because

the higher is σηthe stronger is the random walk component. We have that the

smaller is τ, the greater is the proportion of the sample containing a random

walk component. Finally, the previous finding are reversed for the panel reversed

tests. This depends on because we are testing a change from I(0) to I(1). We do

12

Page 14

not report for brevity the results for the cases of changes from I(1) to I(0), they

are available upon request. Here as expected, the power of tests grows largely

for the reversed tests and it is striking that the results mimic those in Tables

3-5. Thus H1(K∗), H2(K∗) and H3(K∗) show better properties than H1(K),

H2(K) H3(K) and M1∗, M2∗and M3∗.

We now present the empirical size of tests when cross-section dependence is

included in the model as in equations (32)-(34). We consider two levels of cross

section dependence where we generate λi∼ iidU [0,0.20] as an example of ”low

cross section dependence”, and λi∼ iidU [−1,3] to represent the case of ”high

cross section dependence”. The results are reported respectively in Table 6 and

7. As to be expected the extent of over-rejection of the tests very much depend

on the degree of cross section dependence. Both for low as well as for strong

cross section dependence the panel tests are distorted with over-rejection which

grows as the degree of cross section dependence rise. Thus panel tests that do

not allow for cross section dependence can be seriously biased if the degree of

cross section dependence is large.

Table 6 about here

Table 7 about here

As previously reported to take into account of cross section dependence we use

the method propose in Stock and Watson (2002). The method basically consists

in filtering out the individual-specific cross-sections yitby the factor component

computed using the principal component method. The number of factors are

computed using a methodology proposed in Bai and Ng (2002). Specifically

throughout the Monte Carlo simulations analysis the number of factors are

computed using the IC(3) criterion proposed in Bai and Ng (2002) with a

maximum number of five factors.

In Tables 8-11 we present respectively the size and power of defactored panel

tests using the Stock and Watson (2002) methodology. Looking at the results

we note first that the tests have now generally good size, although some sign of

oversize are noted especially for Hj(K∗) tests. As expected, the power of tests

grows for larger values of T and N.

5 Empirical applications

We apply the panel tests described in this paper to a panel of 15 European quar-

terly inflation rate series observed for the period 1970.1-2006.21. The series are

calculate as first difference of the logarithm of the (seasonally adjusted) con-

sumer price index. The data are taken from OECD Main Economic Indicators.

In table 12 panel tests results are reported. Looking at the results we note first

that for first set of test statistics, i.e. test statistics which are computed not

taking into account possible cross-sectional dependence, reverse tests strongly

suggest a change of persistence from I(1) to I(0) process. Change in persistence

are also evidenced by Mjtests as well as their modified version. Mixed response

1The countries included in the panel are Austria, Belgium, Denmark, Finland, France,

Germany, Greece, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden, United

Kingdom.

13

Page 15

are obtained from H1(K) and H2(K) test statistics. While the former tests reject

the null hypothesis for a change from I(0) to I(1) the latter tests do not reject

the null hypothesis for a process which I(0) or I(1) throughout. In order to take

into account for possible cross-dependence across the countries, we first compute

the number of factors. The IC(3) criterion suggests three factors. Thus we use

the estimated factors and factor loadings to compute˜ˆ zi,tas in equation (35).

Looking at the results of panel test statistics we note that the previous results

are now partially reversed. Here both H1(K) and H2(K) test statistics do not

reject the null hypothesis allowing for a process which is I(0) or I(1) throughout

the sample of analysis. Reversed test statistics indicate a change from I(1) to

I(0) and Mjstrongly reject the null hypothesis of constant persistence (either

constant I(0) or constant I(1)). Given the previous results we conclude that the

inflation process is characterized by a change in persistence from I(1) to I(0).

Using the cross-sectional dependence adjusted series, the change of persistence,

computed using (29), was around the 1984.4.

6Conclusions

In this paper we present new panel persistence change tests which are based on

modified time series version of the ratio-based statistics by Busetti and Taylor

[2004. Test of stationarity against a change in persistence. Journal of Econo-

metrics 123, 33-66]. These statistics are used to test the null hypothesis of

stationarity against the alternative of a change in persistence from I(0) to I(1)

or viceversa. Alternative of unknown direction is also considered. Asymptotic

distributions of the new panel tests under the hypothesis of cross-section inde-

pendence are derived and Monte Carlo analysis suggest that these tests perform

very well. Cross-section dependence is also considered.

We show first that when testing for a change in persistence from I(0) (I(1))

to I(1) (I(0)) panel tests have better properties than the single time series tests.

Secondly, we report the importance of taking into account for possible cross-

sectional dependence when computing the panel test statistics, especially for

highly dependent panels. Finally, we apply the panel tests to a panel of EU 15

inflation rates observed during the period 1970.1 - 2006.2. The outcomes were

consistent with a change of persistence from I(1) to I(0) around the 1984.4.

14

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