Recursive and sequential test for unit roots and

BANCO DE ESPAÑA
RECURSIVE AND SEQUENTIAL TEST FOR UNIT ROOTS AND
STRUCTURAL BREAKS IN LONG ANNUAL GNP SERIES
Anindya Banerjee, Juan J. Dolado and John W. Galbraith
SERVICIO DE ESTUDIOS
Documento de Trabajo nº 9010
RECURSIVE AND SEQUENTIAL TESTS FOR UNIT ROOTS AND
STRUCTURAL BREAKS IN LONG ANNUAL GNP SERIES (*)
Anindya Banerjee (**)
Juan J. Dolado (***)
John W. Galbraith (****)
(*) James Stock very kindly provided the data series used. We are grateful to Sieve Ambler, David Hendry and
Mark Rush for especially helpful comments. Galbraith thanks FeAR for research support under grant
number NC-0Q47.
(•• ) Department of Economics College of Bus. Admin. University of Florida Gainesville. FL 32611 USA.
("U)
Servicio de Estudios, Banco de Espafia. Alcala, 50. 28014 Madrid. SPAIN.
(* .. * ) Department of Economics McGill University. 855 Shcrbrooke St. W. Montreal, QC H3A 217 CANADA.
Banco de Espafia. Servicio de Estudios
Oocumenlo de Trabajo n." 9010
In publishing this series, the Banco de Espana seeks to
diffuse worthwhile studies that help acquaint readers better
with the Spanish economy.
The analyses. opinions and findings of these studies re·
present the views of their authors; they do not necessarily
coincide with those of the Banco de Espana.
ISBN: 84-779 3"()S7·8
Deposito legal: M. 43464·1990
Imprenta del Banco de Espana
Abstract
Recur s i ve and sequentia.l estimation procedures are applied
to each of two long annual data ser'ies on the logarithm of real per'
capi tal U. S, GNP, each of which includes the G,'eat Depre s s ion, The
recursively--calculated Augmented Dickey-Fuller statistic for' a unit
root,
and
several
sequentially-calculated
F-type s tatistics
for
str'uctural break s. are compared wi th cri tical values appropriate to
such sets of statistics, The results for the long data series are
consistent with those found on shorter (post"",ar) data sets; we do
not find statis tically significant evidence of a trend break ,
-
5
-
A number of recent p&pers,
in particular Stock and Watson
(1986) ,,,,d Perr'on ..nd Phillips (1987),
logarithm
of real
per'
capital
test the hypothesis that the
gross national
product
(GNP) in
the
United StaLes follows a stochastic tn�nd by testing for a unit root in
:.:Ieviations
fr'om
non-par'ametric
Phillips
deterministic
variants,
developed
(1988) and Perron
b�st for a unit rout,
(1988,
by
tr'end,
Both
Phillips
1989),
papers
(1987),
employ
PerTon
and
of the Dickey--Fuller (1979)
Stock and Watson (1986) extract a i, 5% annual tr'end gr'owth
to
induce cln
test
that
(1987)
taking
does not allow for
u':$e a
non-Zero
approximately driftless series,
more
dr'ift.
account
gener'dl
of
Lhe
a fitted tr'end,
Perr'on and
tests have bNm
proposed
version of
Non-pararnetr'ic
dnd use a for'm of
presence
the
of
the
Phillips
lest which accomodates a
as way
autocorrelations
first-differenced representation of the process, However,
in
of
the>
recent work
by SChWH('t (1989) suggests that such non·-parametric corrections do not
perform ...ell even in large samples, The simulations in Schwert (1989)
show that for some (especially MA) err'or processes,
the non"·parametric
variants of the Dickey-Fuller tests are characterised by low power and
incor'r'ect sizes, in samples as large as
1000,
and that thQ lJse of the
Augmented Dickey-·Fuller (ADF) test is to be preferred to the use of
non-parametrica lly corrected Dickey -Fuller test s tati sties.
Stock and Watson
(1986) consider three separate real per
capital eNP series and test for the existence of a unit r'oot in
1
each , The tests are conducted in different sub-·�samples and the
results an� var'ied.
They conclude that there is little or no evidence
against a unit root in the post-1919 sub-samples,
- 6-
Pen'on "nd Phill ips (198 7 ) , using the lIIore gene,'al version
of
the
test ,
conclude
that
using
the
N-P
series
does
not
allow
rejection of the null hypothesis 'of a unit r'oot ei ther before or after
World
n,
War
pre,,·World war
The
II
F-·S
series
perlll i ts
rejection
data but not on the post--World War
i� possible for the full sample ) ,
The
evidence
purpose
of
this
paper
is
of
II
the
null
on
data (rejection
to re--examine the previous
in the detail allowl�d by recurs ive estimation procedures. A
line of argument i n i t iated by Perron (198 9 ) , states that tests for a
unit n;,lot \lJhich use lhe full satnple are biased in favour of accepting
the un i t root hypothesis i f the series has a structural break at some
intermed iate date. Such struc tural bn�aks may take the fonn of changos
in the mean level of
while
Cl.
t rend,
ser i�s Iilay
( that
is,
Cl.
series or changes in tr'end gr'owth rates , Thus
be s tationcH'Y around
Cl.
broken Il\{�an or a broken
the mean or slope coeffic ient takes on two or more
diffQt'�rlt values in d i ffen�nt Piu'ts of the sample, and thQ deviation
of the series from thi s changing tr'end
sub,·-samples) ,
standar'd
unit
r'I,)ot
i s stationary in each of the
te'St which do (It..lt takQ account of
these breaks occuring i n sub--samples of the series will be dogged by
low
power.
By
standard
uni t
r'oot
tes ts we
mean
thosa
proposed
by
D i ckey and Fuller (1979) and Fuller (1976 ) ,
Pen'on ( 1989 ) proposed modifying the standard D ickey-Fuller
test by
i.ncluding d ummy variables
in the Dick ey�-·Fuller regression, i n
order to allow for a break i n the trend and mean. He computed cr i t ical
values
appr'opriate
cr' i t ical values ,
to
found
this mod i fied
regr'ession
and,
using the new
in favour of a structural br'cak in a majori ty
of the t ime series inves t i gated earlier by Nelson and Plosser (1974 ) ,
An
impor'l;ant criticism of the Per-ron approach,
identi fied
ori ginally by Christiano (198 8 ) was that the break date in the series
was assum(�d by Per-ron to be known so Lhat thre is
Cl
the testing pr'ocedure which should be accounted for.
pre·-test bias in
In
general , i t i s
-7-
more r'easonable not to assume a priori knowledge of the br'eak date but
r'ather to allow its estimation to be part of the empirical exer'cise.
This is the view taken by Chdstiano (1988) and is also the pr'emise of
the
present
seems
paper,
natur'al,
min imum
To
"ndogeneise
following
values
the
Christiano,
choice
to
pay
of
the brnak date it
attention
to maximum
0
in a sequence of test statistics constructed through
the two procedur'es descr'ibed below,
Recursive
procedur'es
for
testing
for'
a
unit
root
or
structural break have their most natural uses in this setting,
tests date,
in their moder'n fon",
to Brown,
Cl
Such
Durbin and Evans (1975)
and their use of the cumulative sums of squares statistic, and Quandt
(1960),
Recent
Ploberger,
paper's
by
Kramer,
Ploberger
and
Alt
(1988)
and
Kr'amer and Kontrus (1989) have extended the Brown, Durbin
and Evans analysis to dynamic models,
Chdstiano (1988) and Banerjee,
Lumsdaine and Stock (1989) (BLS henceforth) deal with the case where,
under the null hypothesis,
the data generation process is assumed to
have a unit root.
Two different clases of statistics are computed to test for
the
presence
class,
of
of
a
unit
root
in
N-P
and F-S series,
The first
commonly termed IIrecursiue statistics", consists of a sequence
statistics
which
is
constructed
starting from some minimum size,
Lhe
the
recursive
regression
procedure .
using
each
The
of
by
the
sample,
by one data point at each stage of
recursive
these
incrementing
nested
algorithm estimates the same
samples,
given
the
minimum
.ample size required to estimate the first regression i n the sequence,
The
critical
values
of
the
test
statistics
used
for this class of
tests are given in BLS (1989),
The
sequence
stage .
of
second
class,
statistics
termed
computed
by
" sequential
using
the
statistics" ,
full
sample
uses
at
Cl
each
The regY'essions in this second sequence differ from each other
in postulating varying dates
for the break,
The first regr'ession in
-8-
break date of k ' where 1 < k < T,
o
O
with sucessive regressions in the sequence postulating k ' k + 1
O
O
T
k
as dales for the break, SOllle of the critical values used for
O
these tests appear in Christiano (1988) and in BLS (1989), We provide
the
sequence
uses
a
postulated
•
.
.
.
.
•
-
additional cr'itical values in section 3 of this paper.
Section 2 describes
some
Section
detail.
the
recursive and
describes
3
2
provides appropiate critical values
.this
sequential tests in
asymptotic
pr'oper'ties
Section 4 presents
of the tests when applied to the N-P and F-S series,
concludes.
In viow of the critici�lIls in Schwer't
and
the n:!sults
and section 5
we
(1989)
limit the
discussion to extended ve r s ions of t he Augmented Dickey-..Fuller· tests
'
and d o not compute dony non-parametr'ic vers ions.
·A
denotes
"::;"
word
on
the
notation
is
The
symbol
"_)11
weak conven3ence of l:he associated probability measur'es, and
signifies equality in distribution.
.8rownian motion with unit
indexed
necessary.
by
wr ' i tt en
Stochastic processes such as
variance on [0,1] and on its sub-intervals
as B and p-d im Browni. an nlotions as B
P
1
1
1
Si..i larly, integrals such as I B (r) d r
I B(r) dr, etc,
I r (B) d r,
/i
o
o
1
1
l
are written simply as I B, I r B, I B, etc. All limits given dre
O
/i
0
as the sample size T -> m.
S
are
,
2,
Unit Roots and Recursive Estimation
Consider
the
Cdse
where,
under
the
univariate pr ocess is postulated ,t o have a
'
root;
one
possible
alter'native
hypothesis
cl.nd
a
non-'zero
is
s tation ary autoregressive process of ordet� p
null
+
lhat
hypothesis,
a
dr'ift and a unit
the series
is
Cl
1 with a non-zero dr'i ft
linear trend. ottll;�r albH�native hypoth,�ses consid'�n�d in this
paper' are thos{:! of stationar'ity of the autot"egressive pr'ocess about a
brok��n trend or
Cl
broken dri rt or of the series being d riven by a unit
root in o n e pa.rt of the sample and by
3
the other' ,
Cl
tr'end-stationar'y pr'ocess in
-9 -
The
estimate
null hypothesis can be tested using either the point
of the
t-statistic.
f i rst autor'egressive
coefficifmt
or
its
regre s s ion
As it is well known the distributions of both statistics,
for a given finitQ sample si.ze and asymptotically, are non-nonnal and
have
been tabulated
by
Fuller
(1976) and Dickey and Fuller ( 1 979 ) .
4
Specifically, the mQ(l�! underlying the test for the unit root is
(1)
with
in
capital GNP.
this
paper,
representing
the
logarithm
of
r'eal
per'
is a unit r'oot then cr. ::;;. 0, and this is the
1
.
.
.
2.
null hypothesis. We take ' t a b e 11
. d Wl th
varIance a .
t
,
If
there
Se now turn to a discus sion of the recursive and sequ<-mtial
testing procedur'es whi<.:h we shall use to test for
a
unit r'oot in the
N····P and F·-$ series .
2 . 1 . 8ecursive Tests
The
o-Z T
{ }
•
t
.
1S
o
sequence
of
r'ecurs ive
statistics
and
.
computed accord 1n9
to a stand ard a 1 gorithm ( see , e.g . , Harvey
( 1 9 8 1 »; {�1} is the sequence of OLS estimates of the coefficients, {�1}
is
the
estimator
t-statistic
of
the
un
is
standard
the
smallest sample used for estimation ( i . e . ,
initialization
will give us
of
the
recursive
evidence abuut
the
sequence
of
an
e
's
and t
is the
o
t '
the sample 5 ize used for
procedure s ) .
Examination
of
{� }
1
the possibility thdt the time se�' i e s is
gover'ned by a unit r'oot in s ome par't of the sample while it follows a
stationary process in a.nother pdr't of the ::sample.
-lU -
We will show that although,
t-stati stics
be
a
have
different
for any given sample size, the
5
Dickey-Fuller distribution , there will
the usual
critical
value
t-statistics over the sample;
probability
s tandard
that
the
the
maximum of
in any giv@n
minimum
Dickey-Fuller
for
of
cri tic.al
thi s
values
the
sequence of t·-ratios the
wi 11
sequence
is
minimum of
greater
lie below
thdn
the
the
nominal
size of the test. Thus the use of the Dickey-Ful l e r critical values is
prone to l,�ad to exce s s ive rejection of the unit root hypothe sis. BLS
( 1 989)
tabulate some critical values by Monte Carlo s imulation, which
are repr'oduced in section 3 below.
2.2. Sequential Te s t s
A . This tes t u s e s the following T-2k regressions
o
� Y = � + at +
t
a
1 Yt-l
Y r ( k ) + Y r2 ( k)
l 1t
2
t
+
tJ(t�k),
indicator function, for t
=
+
P
E
a.
]
j=2
J ( t)k),
k, k+l, , , . , T a nd k
=
6
�Y . + e
t-]
t
being
J( . )
k ' k + l,
o
o
(2)
,
,
.
,
T.- k
o
an
'
Model (2) represents ctnd tri.:"!nd and mean discontinuous break
at
6=0,
time
since
Under
under
the value of s"
HO'
the
the
1 imiting
The sequential
and
a =Y =Y 2=0,
1 1
are invariant to
hypothe sis,
null
d i s tribut ions
test therefore consists of computing
the
F-statistics
and/or
F
for
F
Yl=Y2=O
al=a=Yl=Y2=O
each regres sion, where each regre s s ion differs from dny other only in
Y
and/or Y
are
1
2
don indication of a s t ructur'al br'eak in the data series (Y
for a
2
mean break, Y
for a break in the trend). Again, while for any
1
given break at date k (pos tulated independently of the data) the
the
postulated
break
date .
Non-·zero
F-statis tic has
the
values
of
the s tandard F-density, in any given sequt:mce of FIs
max
exceeding the s tandard 5% ( 1 0%) c ritical
probabil ity of F
values is in excess of 5% ( 10%).
-
11
-
B . Sequential Test for' Shift in Trend (BL. S ( 1989))
This
te.t
considers
the
model
as
a
representing
r
( k)=(t-k)l(t�k)
and
r ( k )=O,
Zt
1t
trend with a kink at date k .
In the spirit of Christiano's test,
we estimate the T.-Zk
O
k=k
to
o
Thus
fur
fur
computed
null
<I
=Y �Y =0 .
1 Z
is
the
T-Zk
regressions
the
F-statistic
F
o
Y 1=YZ=l=O
l1lax
and F
, the maximum of this sequence of F-·statistics is
Under
the
hypothesis,
with
continous
regressions given by mov ing the trend-shift sequentially
k=T-k '
O
each
of
but
(Z)
in
1
compared with the appropiate critical values given below .
C . Sequential Test for Jump in Trend (BlS (1989))
This
with
final
class
of tests is computed
r
dS
in B above but
( k)=J(t�k)
and
r (k)=O,
since
we
are testing for a
1t
Zt
shift in the mean of the series without change of slope in the tr·end .
3 . Asymptotic Distributions and Critical Values
3.1. Recursive Tests
The
and
of
test
the
analysis
statistics
unit
transformation
root
focuses
in
under
suggested
model
on
(1)
the properties of the estimators
under
the
H ' it is ver'y
O
in Sims, Stock and
null
<I
1
convenient
Watson
=�=O .
to
( 1990),
Because
use
so
the
that
under H ' (Z) can be rewritten as
O
(3)
where
Z!
=(/\ yt-1 - it, .
.
. , /\ Yt-p-ii)',
Zt =l
.
Z � = ( Yt-iit) and
zt
=
t
-12 -
such that
Also
l'
=
P
-1
�b with b -(1·- E a..)
1
2
1
E ZI Z ,
t
t
let
combinations
of
the unconditional In"an under HO'
The
transformed
P
original regressors
=
O .
the
regressors
with
are
different
linear
stochastic
orders of convergence. So define the scaling matrix I
:.= diag (Tl/2 lp'
T
1
3
T /2,
T, 1 /2)
and
6,
Since
the
partitioned
conformably
with
Z
t
rec.ursive
regressions
k(=, ko"'"
T),
t:.c:"'Ike
define
the coefficient vector is
[15]
A
6(5) -
where ° <
5 �
0
5
[ E
1
� 1.
Z
place
5=k T,
/
Z
t·-1
ov�r
and
t-1
,]-
the
1
[
the
samples
recursive
going
OLS
fr'om
1
estimator
[15]
to
of
( 4)
E
1
rhus
(5)
where
V
(5)
T
=
=,
Ther'e
are
analogous
computed Wald statistics
the
the
hypothesis
q
hypoUwsis
hypothesis
coefficients
m�xt
and
are
that
are
orden�d
on
restriccions
and
so
F (5)
T
=,
1
Z
t
Suppose
wher'e,
so
involve
forth,
for
a
the
Wald
[15]
E
1
genar'al
n�cursively
and for the Dickey-Fuller t-·statistic testing
a. "O,
1
R6=r
expressions
-1
T
T
(and
The
(R6(5)-r')'
U,,,,t
without
the
test
[R(
fir's"t
perhaps
coefficients
loss
on
statistics
of
2
Z
t'
statistic
generality,
restrictions
Z�
that
Z3
t'
4
(and
we
Z )
t '
tests
the
involve
perhaps
will
consider
(6)
(7)
-13-
&2 (0)
where
the
is
•
e stimator
er
of
2
computed
e
the
residuals
the
through
e stimated
[To]-th
ij
-1
denotes the (i, j ) element o f V (O)
.
V (o)
T
T
Then
using
observation
and
the asymptotic behaviour of the recursive estimators
and test-statistics are summarized in the following Theorem (proofs in
Appendix) .
Theorem 1: Suppose that y t is generated by model (1), then
b) Under the null R(e)
c) Under the null �
1
-.
=
r
0
where V(O) and �(o) are partioned conformably with
Vll
V
22
2
=: (J
=
e
0
0
p'
0, V
23
V
=er
Table
critical
test
minimal
values
(0= 1 ) ,
ij
=
0 (j
-
TT
and
2, 3 , 4 )
O
2
b r Bdr, V
= 0 /2
0
24
1,
taken
for
the
four
form BLS
statistics:
the
reproduce s
full
Dickey-Fuller
min
) and
statistic ( t
maximal
Dickey-Ful ler
( 1 989),
sample
statistic
the
spn�ad
the 5% ( 10%9
Dickey-Fuller
max
(t
),
the
betweQn
the
-14 -
max imal
and
m i n imal
experimentation
in
BI_S
d iff
(t
The
Monte
Carlo
stat i s t i c s
).
d iff
show that t
i s more powerful but its s ize
i s substantially larger than the level,
so that care i s needed in the
i nterpretat ion of the results with this stat i stic ,
3 . Z Sequential Tests
The
mod e l
can
3
Z
t
The
considered
be
is
(2.)
which
repaf'ameterised
defined
as
under
as
above,
correspond i ng
the
null that
in
and
scal i ng
are,
matrices
accord i ng to each case:
A
A) T
T
B) T
B
T
C
C) T
T
=
=
=
d iag (T
d iag (T
diag ( r
1/Z
1/Z
1/Z
I '
p
r
I ' T
p
1/Z
3/Z
3/Z
1/Z
, T
, T, T
, r
)
1IZ
3/Z
3/Z
, T, T
)
, T
3/Z
1/Z
1/Z
I ' r
, r
, r, r
)
p
The estimators and test stat i stics are computed us ing the
full
r
obser'vat i ons,
for
k=ko'"
.
r-kO'
For
k iT
->
0,
the
proce s s of these sequential stat i stics i s given by
(9)
where all the sums go from Z to T and 0< 0o� 00'-00 < 1
As
in
the
recur s i ve
stati stics
case,
the
asymptotic
representat ion of these proce s s e s can be summarized in the follow ing
theorem.
- 15 -
Theorem
Suppose
2:
that
!:l'",- ='Y ='Y =0, then
1 1 2
�
a ) In case A, y (e(5)-e)
'1'(5)
U
11
=
2
6(1), bcr/2[6 (1)-1], 6(1)-I
O[Q
0"
2
•
Q
P
; U
1l
.
=
0 (j
U55
=
case
b) In
'1'(5)
=
=
�/2
->
=
2
(1-5 )/2; U 6
2
=
1-0
2
(1-5 )/2
=
2
(1-5 )/2; U66
6 •
B, YT(8(5)-e)
O[Q
1/2; U 5
2
3
(1-5 )/3; U 6
4
3
(1-5 )/3; U56
�,
1
6(5)-I 6, 6(1)-6(5)]
6
2,3..., )
=
=
- 113; U 5
4
with
(2)
1
U(5)- '1'(5), wher'e
->
� 6,
�/26(1)"
=
model
by
generated
is
Y
t
=
1-5
1
U(S)- '1'(5), where
�
1
2
6(1)', 8(1), bcr/2[B (1).-I], B(I)- I B, (1-5) B(1)- I B]
Clnd U i s a( 5x5) matrix with elements as i n case A except that
U 5
3
1
,," [(1-5) B(I)-I B]; U 5
6
4
.
=
C •
c) In case C, Y (8(6)-e)
T
'1'(5)
dnd
=
U
o[Q
is
_.>
�/2 B(I)', B(1),
dS
in
case A ,
matr'ix owe eliminated.
3
3
(1-5 )/3 - (0·-6 )/2
=
U(5)
-1
'1'(5), wher·e
�
2
bcr/2[B (1)-I], B(I)-I B, (1) - B(o)]
exct:!pt
l:hC'tt
the 5
lh
column clnd
row
of
the
-
The
result
provides
16
-
joint
uniform
convergence
of
all
estimators and test statistics, including Wald te'3ts as in Theorem 1 .
Accordindly we use simulations to compute the cr'itical v alues of the
max
distribution of F
in the sequence of regressions given by model
(2), The critical values, in case A, are giv\!n in the second and third
columns in Table 2, for sample sizes of 50,
100 and -150 in their 2 an
4 d. f vers ions and are cons is b!rI t wl th those repor'ted by Chri 5 tiano
(1988) for- a sample si.ze of 1 5 2 , using a bootstrap method, This table
agedn
reflects
lhe relative insensibility of the critical values to
changes in the sample size. The cr'itieal values. in cases B and C, an�
presented in the third and foud.:h columns in Table 2 respectively and
again
we
n:!ly
upon
the
insensitivity
of
these
critical
values
in
applying lhem to both F-S a.nd N--P series,
Not�
ar'e
stated
\='Y2=O
finally,
for
in
lhe
Theor'em
that
although
null
model
the
2,
the results in both theor-ems
in
results
"' �O d"d Il�-o (also
(
sufficiently gen�I"al to
which
are
handle the case where
'''' '<1, eto ('Y to, 'Y tO) as noted by B LS (1989) ,
1
1
2
In this case natur'ally the result on t (6) does no longer hold,
OF
since its distribution is standard.
4. Empirical Results
Figures
Fr'iedHlan--Schwar-!;z
inspection
of
and
1
r'(�al
these
P(H'
2
plot
capi Lal
figures,
the
GNP
two
the
Nelson-Plosser
s,!ries,
series
As
are
is
cl(�d.r
well,
but
and
from
not
perfectly. corTeldl�d over the periods in which they overlap, We begin
our analysis of these ser'ie. by repor-ting the full-sample estimates of
the Augmented Oicke Y--Fuller regression applied to each (wi th one lag):
Nelson--Plosser (19 11---1970)
IlY t
=-
1.278
(3 , 05)
+
0:0035t - 0,182Y t _l + 0 , 410 IlYt-l
(3 , 02)
(3 , 05)
(3 , 3 9)
s = 0 , 059; LM(5) = 2 , 52; N(2) = 3,61; ARCH (5)
_
(11)
1,94; H(6) = 10_8 1
-17 -
F r i edman-Schwar'l;z ( 1 8 71-1975)
II Y t
s
=
=
+
1. 474
(3 .93 )
0.0037 t - 0;226 Yt_ 1
(3.8 9 )
(3.8 1 )
=
0.0 6 1 ; LM(5)
=
8.50; N ( 2 )
+
0.247I1 Y t._ 1
(2.55)
24.35; ARCH (5) - 6. 79; H ( 6 )
Absolute values o f t-rat ios are i n parentheses;
error of the regr'es s ion; LM ( q )
(12)
5
=
12.15
denotes the s tandard
i s the Lagrange Mult iplier test against
an AR ( q ) or MA ( q ) i n the d i s turbance ( see Godfrey (1978»; N(2) i s the
Jar·que·-Ber·a (1980) normality test; ARCH (g) i s
th
aga inst
8
order
ARCH
d i strubances;
H(.)
Engle's
is
(1982)
White's
test
(1980)
hetero s k eda s t i c i ty test. In Qach case the s tat i s t i c i s asymptotically
2
d i s t r i buted x w i th the number of degrees of freedom b racketed.
U s i ng Fuller's c r i t ic"l values of ·-3.50 and ·-3.45 for the
on it
t-ratios
in (4) and (5) respec t i vely , we can reject the null
1
hypothe s i s of a un i t root on the full sample for the F-S series , but
not for the s hor'ter N -P series. There are, however,
several featur'es
of these results w h i c h lead us to consider a further exam ination of
th.. data.
F i rst,
sUbstantially,
as
F i gures
parti cularly
s o that the
inference w i th
change
Lhe sample.
over
3
in
and
4
s how,
the
t-ratios
change
the v i c inity of lhe Great Depress ion ,
respect t o the null o f a un i t root would
One could pos s i bly argue,
following Perron,
that a s tructur'al break could account .for the low absolute values of
these s ta t i s t i c s
the
in the v i c in i ty of
case of the F-S series
res iduals
values
w i ll
computed
errors .
exc<>ed
is
rejected.
not
using
Since
the
5"
be
is
very
Th i s
sens i t ive
prec i s ely
real i sed
Moreover,
to
a
in
suggests that the D i ckey-Fuller c r i t ical
appropriate,
process
stat i s t i c
for
theSe!
which
the
c r i t ical value by a large marg i n ,
deemed conclus i ve.
Second,
the null hypothe s i s of normal ity of the
a data gener'at i on
the
1 93 0 in each series .
values hav i ng been
incorporatQd
F--S
series
nor'mal
does
not
lhe test cannot be
as t h i s skewness-kurto s i s normality test
few
especially
large
dev i ations.
it
is
-18 -
conceivOl.blc
a str'uctur'al
that
also account for the n or 'ma l i t y
for
str'uctur'bl.l
BLS
take
we
the
('Titical
lninimlJm DickGY dnd Fuller � tat istl c s
sampl e s .
SHt
N-·P
and
size
The samj:)ll�
may
also
se r-ies,
of
n�aliably
th (�
since
is
100
values
for'
to
conduc;t
a r'e
t(�!St
succ��ss ive
of th(�
F·-S
data
thH :> ho r' b�r'
inFer'ence in
not
\.\J(�
maximum and
th(�
o bse t 'v � d ov��r' a :>et uf
values
cr-itical
Lhe � )(� n�dsons,
r-ur'
close to the si.ze
us (� d
be
se r'i ,�� could
shift in the
result.
te,:;t
br'eaks bel ow .
From
tr'end
bn.�bl.k or'
sensitive
vet'y
to
cha.nges in the �a11lple size .
sample,
On the N--P
on the
in 1924;
mini.mum in
t hese
dr<�
1.930
ver-y
non-nol"mality
the minimum of the t--statistics is --3.80
series the l1l inimum
F-··S
Fr 'om th(�
of -3,76,
close to Lhe
in
c
se r i es
F-·f,
the
10%
va lw� s in
the
hy pothes is
that
However, Lhe maximum
whereas th(�
bo th
valUt�s of
cases non rejecting the null
f,
fo r
the trond
N-P series,
sign i f i ca n t
model
with
and
6
in
at eVl-!n the
no
plot
1930
break,
10%
and
for
for-
br ea k .
In
bn�ak test
trend
is
trend ) .
a
we
o r'd(� r'
bet te r
miqht
nut
wdnt
an�
d , f. test under' the null
local
in
the
mind
application
d.
not
finlt
ar'e
of
n!jec;tion.
and
·--1,36
2..44 and 2.,21
·-1.68
r'esp(�ctively,
F-statistics
in
The maxima an� 8.19
ttle
F-S
irl
jo int
alternative
the
1939 for'
the
uF
of
a
data
the
trend
and
of
str-uc;t\Jr-cd
thdt Lhe stochd.stic
than
statiol"lOl.r'y
is
mt!an
d.
de t e r'm ini st i (;
compute the critical values for the
hy pothe � i s
2.
r'Cmdl)m walk
r'esult of this
to be cer' tain
of
with
series. Neither' of these
under the null
poss i ble
to
but
Christiano--type
chanl1.ctet"isation
al:w
cl
it is ch!ar- that
liter-al
t '-S Ldt i stic s
to b(� confident in
( s i nc e it is
lh
hd-ve sOllle suggestive Qvidence
root,
level,
a
wi
hy poth esis.
c\nd m(�an break,
and 6.07
I,
Tabh�
makes
lIlay
lhe
differ'ence t-·statistic�
Figur'es 5
d,
uni . t
Cl
of
in 19.15,
r i ti c al values, flear'ing
these critical valu\�s hdzardous, \oJe
against
is ,-3.89
d(�vi.at.i.(Hls
fr'olll
2
a
-19 -
deterministic trend (again following Christiano). We do so in Table 3.
The for'm of nu 11 hypothes is chosen is based upon the reg,'ess ions (3)
(4) :
and
to
these
=
Y
t
results
a
fair
approximation
is
1 +0.003t + 0.8Y _ +e
t
t l
and to those of models which drop the term in
8Y _ ' on both da.ta sets.
t l
to be a fixed constant.
Again,
the
The
initial
observation
is
again
chosen
test statistics just given are lower than the
critical values. For neilh�r of the null models, then, Cdn we conclude
that there is a structural break.
tests
r'epor'tad
The t-statistics in the unit root
Q"arlier Cdn the�'efore be interpreted in the usual way;
a structural break is not a persuasive explanation for any failures to
reject the unit root model.
Similarly,
computed,
Christiano-·type
F-statistics
with
4
d.f.
are
with maximal values of 13.75 "nd 14.10 for the l\I-P ,md F-S
series, both values being highly non significant.
S�quential statistics for the altEH'native of a tr'end br'eak
against the
null
of
no break in each of the series are plotted in
Figur'es 7 and 8. The maxima of lhe F-statistics are 5.73 in 1932 for
N-P, and 6.61, also in 1932., for F-S. Again, neither is significant at
even
the
2 above).
10%
level,
Finally,
hypotyhesis
of
a
using lhe BLS
sequential
Ill��a.n
bt'(,�ak
an�
cd tical
values (reproduced in Table
statistics
repot'ted
for
in
the
Figu?'e
9
alternative
and
10. The
noaxima a,"e 6.87 in 1938 (N-P) and 6.63 in 1938 (F-S). Once again, the
statistics offer no statistically significant Qvidence of such a break.
5. Conclusion
In summary, we ar'e able to reject the nu 11 hypothesi s of a
unit ,'<lot only
for Lhe full F-S sample,
although we give r·ea.ons for
- 20 -
in ter'preting this rejection cautiousl Y ;
the recursive test statistics
cOlne close to rejecting this null as well. There is much less eviden<:::e
against the null hypothesis of no structut�O\l break, whether' or' not the
also
null
incorporab�s
process for GNP;
a
diFference-stationary
although
the
their la"gest values
have
reach even our
10% Monte
finding
us
allows
to
or
trend-stationary
statistics used for testing the b,'"eak
en the neighbourhood
Car'lo cr'itical values
interpr'ct
Lhe
uf
1930's, they do not
for these tests. This
recursive
t-statistics
face
dt
value, putting aside the possi.bility of a break,
In
classical
statistics,
the
�nvesti9ator
is
vicwed
as
using data to test an hypothesis formed independently of the data on
which
it is to be tested.
bas,!d
on the data,
will
it is clearly relatively easy
to
find
on(� which
not be rejp-cb�d on the given sample.
In
per capita
eX
When the hypoth esis undHr test is chosen
the
GNP,
case
of at least some hypotheses F'(�lating to real
par'ticularly those involving structur'al breaks,
it is
little difficult to imagine an investigator in the purely classical
position;
the
gener'al
hypothesis such as HO:
f��atures
of
the
dclLa
a structu,'al b,'eak
i. n
are
GNP
well
known,
took place c,
An
1930
seems very likely to have been influenced by knowledge of the data, It
is with this in mind lhat sequential tests are applied here,
Clearly
an investigator in the classical position, with the
a priori hypothesis that there was a structur'al break in
GNP
c,
19 30,
would reject a null of no break; the corr'esponding F-·statistics exceed
S% critical values by a comfortable mar'gin.
mo"e ,'ealistic test,
that
However,
for' the p��rhaps
of the hypothesis that a br'eak took place
somewhere on the sample ... - so thctt we do not use our' knowl(�dge of the
data C·pr·e .... test··
of
the
spit(�
examination)
to choose the br'eak point
-
... rejection
null of no bn�ak is not possible dot conventional levels,
of the app(�ar'(il,neE:! of game fair'ly large statistie g. 1hus
relatively
(1 988) and
long
BLS
dclLa
( 1 989)
sets
fo,'
conf .l,r-m
the
results
post--war dat.. sets,
found
by
I
in
these
Christiano
- 21 -
Appendix
for
convenience
!
is
iid(O,,, ),
being
drawn
martingale
Both
fr'om
it
is
assumed
assumptions
a
(MSO)
sequence
wi th
yo �O
and
that
generalised,
be
di.str'ibution
stationary
difference
can
that
at
and
to
to
e
being a
t
bounded fourth
least
moments, at the cost of complicating the algebra,
Proof of Theorem 1
a) Without
loss
of
generality
it
is
assumed
that
r
and define the sequence of par'ti"l s<,ems S (O)= E
T
1
Central Umit Theorem implies that S (o)
T
C(l)
�
-
(1
P
i -1
E '''; l )
and write Y
t
2
has all its
Lemma:
with
in
Let
yt
-2
2
E Y
T
t
ii)
T
-1
E Y
to
C (l),
,/e
e
t-1
f
t
t
+
Then
C�'(l)
l
C(
_.>
Because
uniform
be an 1(1)
->
-3/2
E t
iii) T
is
(0,1) ,
given by Y =C (l) S
t
t
i)
=
b
there
V
T
and
T
(1987):
C(l) S
t
roots outside the unit circle,
->
o'bB(I)
Y
r
k (k:;::ko'" TL
�
�
-> "
/
the
the
e"
let
The functional
1
+
C*(L) e where C*(l)
t
-1/2
It follow th",t T
sample
of
following
size
each
�
1
and
1
lemma
in
to
of'
Stock
ti.me series with a Wold representation
et'
then
2
I B
2
2
C(l) 0 12 [B (1)-1]
e
-) <1
Vll'
is
element
e
1
[B (1) - I B)
0
obtain all the stochastic limiting distributions in
except
r=[T5]
B (o), Then let
convergence
apply
I'�O,
intet-changing
the
(0,1)
integt-al
the text,
limits
by
- 22-
1
4
) ( .., it follows
£, is a MSD wi th sUP(z
it_..1 et
1
0 2 B (I». i. e a p-dim. Bt�ownian motion with
a
that �
lT(l» -) e P
1
2 0
2
1 P
z.
'
:;;:; a
deterministic
Finally the
cova,'iance (j E Z
t
t
e
e
P
V
emu
ar'e obtained a. J d .,
components such as V , V
24
22
44
2
J .d. and J • d. with integral limits between 0 and I).
(0,1». Because
b
&
t-1
2
c) Given the conve"\lence re.U 1 t. in (a) ",nd that � (I»
limiting
dist,·ibution.
Stock ",nd Watson (1990).
follow
directly
f,'om
1heorem
2
-) (J
both
•
2 of Sims,
a) The pr'oof is s imi.lar to the proof of Theor'em 1 except that now the
siz e
sdomple
f2t (k)
T
such
goes
a.
-·3/2 T
E t J(t�k)
£t
1
Fr'om
'¥5
-
1
to
and
-3/2 T
T
Et
k
T.
"'6
£t
The tar'IllS
follow
involving fl (k) and
t
di,'ectly. For example
wich implies integr'al
1 imits
from I) to 1. The dete'"Illinbtic component. such as U55 (o)'=T
•
T
tend in limit to
b
&.
J!
.
2
··-3
d. etc.
c) The argument fot� both cases is similar to that for' case 1.
going
,.
E[tJ (t�k) ]2
1
- 23 -
Footnotes
1 , These
are
the
( 1 869-1975)
Nel son
Friedlll..n
and
( a .,odif ied
and
Plos ser
(constructed by
annual
serles
from
the US Opt .
annual
series
of Corl1mer'ce) and
fr'om
1 909
to
the National
1 9 70
Income
Only the f i r'St two, wh i c h we re fer to
and N- ,P , are used her'e; the NIPA s e" i e s star't. only in 1929 ,
F-S
�nd
( 1982)
ver'sion of the Kuznets ( 1 9 6 1 ) serie s , the
( 1982)
and Product Account series ,
as
Schwar'l;z
is
thereflJ r'e
Depession
rlut
when
long
a. l lowance
enoLlgh
is
made
Fur
eXCl.l1lination of the Gn':!d.t
for
i n i t i a l i zation
of
the
recurs i ve procedures.
2 , Th i s
section
re l i es
upon
results
in
BLS
( 1989)
",her'e
weaker
cond i t ions are as sumed for the d i sturbance e '
t
3 , Th i s i s , of course the a l ternat lve con s idered by Christ iano ( 1 988 )
and PerTon ( 1 9 89 ) ,
4 , Which may d i ffer' from the data gener'ation proces s i t se l f ,
? , Baner'jee et a ! .
critical
Values
( 1 990) provide an appr'oxilllate a l gorithm to compute
for
the
D i c key-Ful ler'
stat i s t i c
in
recur s i ve
sd.mples .
6 , Where
T
is
estimating
the
total
sample
minu s
the
" l:wo
lags.
This
sample
is
ohser'vat ions at each end .
two obser'vations
lost
further
by
tri mmed
in
two
- 24 -
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I
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Journal of
- 26-
Perron, P. ( 1 9 8 9 ) "The Great Crash, the Oil Shock and the Unit Root
Hypothes is " . Econometric .. 57, 1 3 6 1-1402.
Perron, P. and P.C . B . Ph illips ( 1 9 8 7 ) "Does GNP have a Unit Root? a
Re-evaluation" , Economics Letters 2 3 , 1 3 9-145 .
Perron, P . and P.C . B . Ph illips ( 1 9 8 8 ) "Testing for a Unit Root in T ime
Series Regress ion". �iometri:..ka 75, 3 3 5-246 ,
Ph illips , P.C.B. ( 19 8 7 ) IIT ime Series Regress ion with a Unit Rootl!
Econometrica 55, 277-· 3 02.
Plober'ger, W . , W. Kramer and K . Kontr'us ( 1 9 8 9 ) ,
Structural
Stability
in
the
"A New Test for
L i near
Regress ion
Model'· ,
.Journ.",l of Econometrics, 40, 307·-3 1 8 .
Quandt, R.E .
( 1 960) "Tests of the Hypothesis that a Linear Regress ion
Obeys
Two
S{�par'ate
Regimes " ,
Statist ical Association,
55,
Journal
of
the
Amer::ican
3 2 4-·3 30 .
Schwert ( 1 9 8 9 ) "Tests for Unit Roots: A Monte Carlo Inve s t igation",
;r.ournal of Business and Economic Statistics 7, 147-·159.
S ims, C . , Stock, J. and M. Watson ( 19 9 0 ) " Inference in Linear Time
Series Models with Some Unit Roots" gconometr ica, 58 .
Stock, J.H .
( 1 9 8 7 ) " Asymptotic Pr'operties of Least-Squares Est imators
of Cointegrating Vectors . Econometric .. 55, 1035-·1056.
Stock, J . H . and M.W. Watson ( 1 9 8 6 ) "Does GNP have a Unit Root?"
Economics L�tlers 2 2 , 1 47-1 51 .
:'o t b .
U. S. Oepar·tment of Commerce ( 1 9 7 3 ) Long Ter..m Eco'!2.'!!.i.£.J?.!w
Wash ington D. C.
- 27 -
Wh i te , H . ( 1 980) " A Heteroskedast i c i ty -Cons i stent Covariance Md trix
E s t i mator
and
a
D i rect
Econome t r i ca 48 , 8 1 7-8 3 9 .
Test
for
Heteroskedas t i c i ty " .
- 28-
TABLE 1
Recursive U n it Root Tes t s :
Cr i t i c al Values
( 2 . 000 R e p l i cations)
5� ( IOX) C r i t i c a l Values
T
t
DF
min
t
(I)
t
"
ma)(
diff
t
lOO
- 3 . 4 5(-3 . 1 5 )
)
- I . 9 3 (--2 2 1 )
3 . 3 7 ( 2 . 95 )
250
-3 . 43 (-3 . 1 3 )
-4 . 07 ( -3 . '0)
-I . 88 ( - 2 . 1 4 )
3 . 3 6 ( 2 . 9' )
500
- 3 . 42 (-3 . 1 3 )
- 4 . 1 0 (·· 3 . ' 2 )
- I . 8 8 (- 2. . 1 4 )
3 . 45 ( 3 . 0 1 )
Note :
Critical
values
by
values
for
- 4 . 13(-3 .
tor { l )
f o r t h e rema i n i ng
OYt
the nu l l model
=
takcm
fr'um Table
t-·stat i s t i c s
tt,
w i th 0 . 25
taken
$ 6 �
8.5.2
i n F u l ler
from lable
1.
1
( 19 76 ) ;
the
in B L S ( 1 989 ) ,
(; r i t i c a l
generated
,ABLE 2
Sequent i a l Tests for Trene and Mean Break (Case s A. 8, Cl:
E s t in�ted C r i t i c a l values (Nu l l
incorporat e s SI st o c h ast ic tre�
( 2 . 000 R p- p l i c a t i o n s )
5 X ( 10�) Critical
Values
C( 3 )
A(2)
A(4)
B(3)
50
1 2 . 1 1 ( 10 8 1 )
20 3 ' ( 1 ' . 1 3 )
1 7 6 3 ( 15 . 16 )
18 7 3 ( 1 6 . 0 2 )
lOO
1 2 . 32(10.91)
20 . 3 1 ( 1 8 . 1 1 )
1 6 . 74 ( 1 4 . 30 )
1 B . 40 ( 1 5 . 9 2 )
ISO
1 2 . 1 5 ( 1 0 . 90)
20 . 29 ( 1 8 . 1 7 )
1 6 . 3 9 ( 1 3 . 8 5)
1 ' . 50 ( 1 6 . 06 )
Note :
The
rur
�ntries
0 . 10
�
respe c t i v e l y ,
W�
o
�
computed
0 . 90 ,
whereas B ( .
)
u s ing
dala
gnne ,·att!d
A(Z)
and
A(4)
refltr
and C( . ) have 3 d . f .
by
to
lhe
te � t s
nu l l
wilh
modQl
2
OY t
�nd
�
Et
d . f.
- 29 -
lABLE 3
Sequential Tests for Trend and Mean Break (eas e A): Est imated Critical Values
( N u l l incorporates a deterministic t rend)
( 2 . 000 RepliCations)
T/size
Note : The
Yt-l
entries were
9.
lOO
8.4
computed
for O . lO,8� O . 90
10�
50
51
I�
10 . 7
13.7
'.8
11.7
u s i ng data gene,"ated
by
the n u l l
model
Y t::=lrO. OO3t+O . 8
- 30 -
Figure 1
N-P: 1909-1970
8.5 .------,
8
"-
z
"
!!
'0.
•
0
7.5
�
c-
o;
e
�
7
'"
': t
1 860
1 880
1900
1 920
1940
1 960
1980
year
F-S: 1869-1975
Figure 2
8.5 ,------,
8
"-
z
"
7.5
!! '0.
•
0
�
c-
o;
e
�
7
'"
6.5
---L---__
__
__
__
__
__
�
__
__
�
�
�
__
__
__
�
6 �
1860
1880
1900
1920
1 940
1 960
1980
year
- 31 -
Figure 3
N-P: RECURSIVE ADF T-STATlSTlCS
4
3.5
"
.�
3
�
'0
•
,
..
2.5
>
•
'5
'0
•
�
•
2
1.5
t
1
1 860
1880
1900
1920
1 940
1 960
1 980
year
Figure 4
F-S: RECURSIVE ADF T-STATlSTlCS
4
3.5·
"
't;
.�
!
'0
•
,
..
>
S
3
2.5
2
'0
•
�
•
1.5
1
1 860
1880
1 900
' i920
year
1940
1960
- 32 -
Figure 5
N-P: F-TESTS FOR TREND AND MEAN BREAK
10
I>
8
0
:�
;;
6
"
U.
4
�
2
o L-----------�--
�
__
__
1 860
1880
1 900
1 920
�������
__
__
1940
1960
1 980
year
Figure 6
F-S: F-TESTS FOR TREND AND MEAN BREAK
10
r-------,
8
0
�
�U.
6
4
2
o
1860
1 900
1 920
year
1980
- 33 -
Figure 7
N·P: F·TESTS FOR TREND BREAK
5.8
�
5.6
5.4
u
5.2
.�
5
.�
••
�
4.8
4.6
4.4
f
4.2
4
1 860
1880
1 900
1 920
1 940
1 960
1 980
year
Figure 8
F·S: F·TESTS FOR TREND BREAK
5.8
5.6
f
5.4
u
�
.�
;;
"-
5.2
5
4.8
•. 6
4.4
4.2
4
1 860
1880
1 900
1920
year
1940
1960
1980
- 34 -
Figure 9
N-P: F-TESTS FOR MEAN BREAK
7.5
.
7
u
6.5
.�
.u.�
6
�
I
r
f-L
5.5
II
5 rr
;
4. 5
4
f-
L
'860
'880
, 900
'920
'940
' 960
' 980
year
Figure 10
F-S : F-TESTS FOR MEAN BREAK
7.5
r
7
u
.�
."
�
u.
6.5
�
I
r
,
6
�
5.5
�
5
,
,
r
i
r
�
4.5
i
rf
4
' 860
'880
'900
'920
year
' 940
1960
1980
- 35 -
DOCUMENTOS DE TRABAJO
(1):
850 1
Agustin Maravall: Prediccion con modelos de series temporales.
8503
Ignacio Mauleon: Prediccion multivariante de 105 tipos interbancarios.
8502
Agustin Maravall: On structural time series models and the cha racterization of cempo·
nents.
8504
Jose Vinals: El deficit publico y sus efectos macroecon6micos: algunas reconsideraciones.
8506
Jose Vifials: Gasto publico, estructu ra impositiva y actividad macroeconornica en una
8505
8507
Jose Luis Malo de Molina y Eloisa Ortega : Estructuras de ponderacion y de precios
relativos entre 105 deflactores de la Contabilidad Nacional.
economfa abierta.
Ignacio Mauleon: Una tuncion de exportaciones para la economia espanola.
8508
J. J. Dolado, J. L. Malo de Molina y A. Zabalza : El dese'llpleo en el sector industrial
8509
Ignacio MauleDn: Stability testing in regression models.
85 1 1
J. J. Dolado and J. L. Malode Molina: An expectational model of labour demand in Spanish
8512
J. Albarracin y A. Yago: Agregacion de la Encuesta Industrial en 105 1 5 sectores de la
8513
Juan J. Dolado, Jose Luis Malo de Molina y Eloisa Ortega : Respuestas en el deflactor del
8514
Ricardo Sanz: Trimestralizacion del PIS por ra-nas de actividad, 1 964-1 984.
8516
A. Espasa y R. Galian: Parquedad en la parametrizacion y o'llisiones de factores: el modelo
8510
8515
85 17
espanol: algunos factores explicativos. (Publicada una edicion en ingles con el mismo
numeral.
Ascension Molina y Ricardo Sanz: Un indicador mensual del consumo de energia electrica
para usos industriales, 1976-1 984.
industry.
Contabilidad Nacional de 1970.
valor anadido en la industria ante variaciones en los cestes laborales unitarios.
Ignacio Maule6n: La inversion en bienes de equipo: determinantes y estabilidad.
de las lineas aereas y las h ipotesis del census X-l 1 . (Publ icada una edicion en ingles con el
mismo numeral.
Ignacio Mauleon: A stabil ity test for simu ltaneous equation models.
8518
Jose Viiials: iAumenta la apertura financiera exterior las fluctuaciones del tipo de cambio?
8519
Jose Viiials: Deuda exterior y objetivos de balanza de pagos en Espana: Un analisis de
largo plazo.
8520
(publicada una edicion en ingles con el ."ismo mjrnera).
Jose Marin Areas: Algunos indices de progresividad de la imposicion estata l sobre la renta
en Espana y otros paises de la OCDE.
860 1
Agustin Marayall: Revisions in ARI'v'IA signal extraction.
8603
Agustin Maravall: On minimum mean squared error estimation of the noise in unobserved
8602
8604
8605
8606
Agustin Marayall and Dayid A. Pierce: ... prototypical seasonal adjustment model.
c:Jmponent models.
Ignacio MauleDn: Testing the rational expectations model.
Ricardo Sanz: Efectos de variaciones en los precios energeticos sobre los precios sectoria­
les y de la demanda final de nuestra economia.
F. Martin Bourgon: Indices anuales de valor unitario de las exportaciones: 1972-1980.
8607
Jose Viiials: La politica fiscal y la restriccion exterior. (Publicada una edicion en inglE�s con
8608
Jose Viiials and John Cuddington: Fiscal policy and the current account: what do capital
8609
Gonzalo Gil: Politica agricola de la Comunidad Economica Europea y montantes campen­
el '11 ismo numeral.
controls do?
satorios monetarios.
8610
Jose Viiials: iHacia una menor flexibilidad de los tipos de cambio en el sistema monetario
870 1
Agustin Marayall: The use of ARIMA models in unobserved components estimation: an
8702
Agustin Maravall: Descomposicion de series temporales: especificacion, estimacion e
internacional?
appl ication to spanish monetary control.
inferencia (Con una aplicacion a la oferta monetaria en Espana).
- 36 -
8703
Jose Vhials y Lorenzo Domingo: la peseta y el sistema monetario europeo: un rnodelo de
tipo de cambio peseta-marco.
8704
Gonzalo Gil: The functions ofthe Bank of Spain.
8705
Agustin Maravall: Descomposicion de series temporales, con una apl icacion a la oferta
8706
P. L'Hotellerie y J. Viilals: Tendencias del comercio exterior espanol. t>.pendice estadistico.
monetaria en Esparia: Comentarios y contestacion.
8707
Anindya Banerjee and Juan Dolado : Tests ofthe Life Cycle-Permanent Income Hypothesis
8708
8709
Juan J. Dolado and TIm Jenkinson: Cointegration: A survey of recent developments.
in the Presence of Random Walks: Asymptotic Theory and Small-Sample Interpretations.
Ignacio Maul"';n: La demanda de dinero reconsiderada.
880 1
8802
Agustin Maravall: Two papers on arimB signal extraction.
8803
Agustin Maravall and Daniel Pena: l\/Ii55in9 observations in time series and the «dual»
8804
Jose Viiials: El 5istema Monetario Europeo. Espana y la politica macroeconomica. (Publi-
8805
Antoni Espasa : Metodos cua ntitativos y analisis de la coyuntura economica.
8806
8807
8808
890 1
8902
8903
8904
9001
9002
9003
9004
9005
9006
9007
Juan Jose Camio y Jose Rodriguez de Pablo: El conSU'TlO de ali mentos no elaborados en
Esparia: Analisis de la inforrnaci6n de Mercasa.
autocorrelation function.
cada una edicion en ingles con el mismo numero).
Antoni Espasa: El pertil de crecimiento de un fenomeno economico.
Pablo Martin Acena: Una estimaci6n de los principales agregados monetarios en Espaiia:
1 940-1962.
Rafael Repullo: Los efectos econo'T1icos de los coeficientes bancarios: u n analisis te6rico.
M.a de los Llanos Matea Rosa : Funciones de transferencia simultaneas del indice de
precios al consumo de bienes elaborados no energeticos.
Juan J. Dolado: COintegraci6n: una panoramica.
Agustin Maravall: La extracci6n de senales y el analisis de coyuntura.
E. Morales, A. Espasa y M. L. Rojo : Metodos cuantitativos para el analisis de la actividad
industrial espaiiola. (Publicada una edici6n en ingles con el mismo numeral.
Jesus Albarracin y Concha Artola: El crecimiento de los salarios y el deslizamiento salarial
en el periodo 1981 a 1 988.
Antoni Espasa. Rosa G6mez-Churruca y Javier Jareno: Un analisis econometrico de los
ingresos porturismo en la economia espaiiola.
Antoni Espasa : Metodologia para realizar el analisis de la coyuntura de un fenomeno
economico. (Publicada una edicion en ingles con el mismo numeral:
Paloma Gomez Pastor y Jose Luis Pellicer Miret: Informaci6n y documentacion de las
Comunidades Europeas.
Juan J. Dolado. Tim Jenkinson and Simon Sosvilla-Rivero: Cointegration and unit roots: a
survey.
Samuel Bentolila and Juan J. Dolado : Mismatch and Internal Migration in Spain, 1962-1 986.
Juan J. Dolado, John W. Galbraith and Anindya Banerjee: Estimating euler equations with
integ rated series.
9008
Antoni Espasa y Daniel Peila: Los modelos ARIMA, el estado de equilibrio en variables
9009
Juan J. Dolado and Jose
Vinals: lIJIacroeconomic policy, external targets and constraints:
.
the case of Spain.
9010
(1)
econ6micos y su estirnacion. (Publicada una edicion en ingles con el mismo numeral.
Anindya Banerjee, Juan J. Dolado and John W. Galbraith: Recursive and sequential tests
for unit roots and structural br.eaks i n long annual GNP series.
Los Docurnentos d e Trabajo anteriores a 1985 figuran en el cat.illogo de publicaciones del Banco de
Esparia.
Informaci6n: Banco de Esparia
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TelE'dono: 338 51 80
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