arima models, the steady state of economic

BANCO DE ESPAÑA
ARIMA MODELS, THE STEADY STATE OF
ECONOMIC VARIABLES AND THEIR ESTIMATION
Antoni Espasa and Daniel Peña
SERVICIO DE ESTUDIOS
Documento de Trabajo nº 9008
ARIMA MODELS, THE STEADY STATE OF
ECONOMIC VARIABLES AND THEIR ESTIMATION
Antoni Espasa
Daniel Peiia (*)
(*)
We would like to express our gratitude to Agustin Maravall and Juan Jose Dolado for their comments on a draft
version of this work and to Juan CarIos Delrieu and M. de los Llanos Matea for carrying out the computations
of section five.
Banco de Espaiia. Servicio de Estudios
Documento de Trabajo n." 9008
ISBN: 84-7793·059·7
Dep6sito legal: M. 30892·1990
Imprenta del Banco de Espana
-3-
Summar'Y
This paper
forecast
function
presents a
of
an
procedure
ARIMA
model
to breakdown the
in
terms
of
its
permenent and transitory components.
The result throws
ARIMA
models
economic
and
terms.
its
some ligth on the structure of
interpretarion
Indeed,
the
forecasting function for a
estimate of the
which
t;
the
proves
permanent
base
useful
component
of
in
the
period t ·is a consistent
equilibrium level or' steady state path at
corresponding variable
is
tending
at each
time
and the transitory component describes how the approach
towards the permanent component tends to arise.
Here
permanent
th�
it
is
component
factors
which
important
of
the
to
dis tingui sh
forecasting
within
function
the
between
depend on the initial conditions of the
system and those which are deterministic.
Frequently
the
function
forecasting
permanent
is
logarithmic transformation
factors,
with
condi tions.
-called
a
Accordingly,
inertia
in
the
straight
of the
parameters
component
variable,
depending
the
line,
slope
paper-
the
gives
the
on
the
plus
seasonal
the
initial
on
of
of
straight
the
line
univariante
expectations for medium term growth. The study of how this
changes
parameter
over
time
is
useful
in
short-term
economic analysis.
The
and
gives
papel calculates the
examples
-for
Spanish
consumer
priceson
which
calculaled
for
the last few
diagnosis is presented.
formula for the inertia
imports
the
and
inertia
years,
and
exports
and
values
are
the
resulting
-5-
1. INTRODUCTION
A
disadvantage
frequently
attributed
to
ARIMA
models
is the difficulty involved in interpreting them in
terms
of
the
components
Though
classical
trend,
(Chatfield ,
it is well
forecasting
1977;
known
function
seasonal
Harvey
(Box and
of
a
and
and
Todd,
Jenkins,
seasonal
irregular
1983) .
1976) that the
multiplicative
ARIMA
model can be represented as a combination of an adaptative
trend
and
Pierce
a
and
seasonal
Newbold
component,
( 1987) ,
no
until
the
simple,
work
direct
of
procedures
had been developed for determining these components.
authors
use
with
signal
the
series
application
the
eventual
forecasting
extraction theory
into
for
its
the
to
components
IMA
model
function
perform
and
( 1. 1)
x
Box,
These
together
a
breakdown
to
detail
( 1. 1) .
of
its
commonly
known as the airline model.
In
obtain
a
permanent
this
breakdown
term,
non-stationary
the
one
work
these
of
the
produced by
are
generalised
forecasting
which is the
structure,
ideas
and
one
a
function
produced by
transitory
the
term,
into
to
a
model's
which is
the stationary operators. In seasonal
series the permanent term can be easily broken down into a
trend component and a seasonal component.
these
Calculating
advantages:
it
is
it
a
interventions
and it offers
makes
useful
which
components has
interpretation
diagnostic
may
affect
a means of
three important
of the
tool
for
model
easier;
identifying
trend or seasonal nature;
comparison
between
ARIMA models
and models in state representation space in their Bayesian
version
(Harrison and Stevens,
(Harvey and Todd,
1983) .
1977)
or the structural one
-
The
the
work
permanent
funct�on
-
is structured
and
of an
6
transitory
as follows:
components of
ARIMA model are defined ,
in
a
Section
2
prediction
and the breakdown
of the permanent component into trend and seasonal factors
is
described.
In
section three
an economic interpretation
of the components of the forecasting function is given.
section
of the
linear
four these
ARIMA
are determined on the basis
model' s predictions,
equations,
applications.
components
and
In
section
by solving a system of
five
presents
some
-
2. SEASONAL MODELS,
7
-
THEIR FORECASTING FUNCTIONS AND THEIR
COMPONENTS
2.1. Seasonal models and their forecasting functions
As it is well
every
linear
according to Wold' s theorem
known,
stationary
stochastic
without
process
deterministic components can be represented by:
(2. 1)
2
where �(L)=I+� L+� L + . is an infinite convergent
I
2
polynomial in the lag operator L, and a
is generated by
t
a
white
noise
stochastic
process. Approximating
this
.
polynomial
by
means
polinomials the
.
of
a
ratio
of
two
finite
order
result is the ARIMA representation,
(2.2)
where �(L)=[�(L)]
roots
outside
stationary.
the
seasonal
a(L), and the operator �(L) has all the
the
The
non-stationary
of
-l
unit
circle
previous
processes,
operator
�(L)
processes,
to
so
that
formulation
by
allowing
lie
on
Box-Jenkins
is
process
extended
one or
the
( 1970)
the
unit
is
to
more roots
circle.
simplify
(2.2)
For
by
factoriSing the polynomials in two operators one on L and
s
another on L , where s is the seasonal period. These two
contributions
are
backed
by
the
factorisation
properties
of polynomial operators which as we shall see, are crucial
in determining the structure of the model.
8
-
In general.
-
a seasonal multiplicative ARIMA model
is represented by :
(2.3)
where
A=l-L
is
operator.
difference
regular
the
s
is
is
A = 1_L
operator. �
the
difference
seasonal
s
the
of
the
stationary
series. � (L)
and
6 (L)
mean
P
q
are
finite operators (with roots outside the unit circle)
s
s
the
lag
operator
L
and
t (L ) .6 ( L )
in
are
p .
Q
s
the seasonal operators on L
also with stationary
roots. Calling
*
�
( L)
r
r=p+d+s (l+P)
*
c =
=
� (L)
�
and X t< It) the prediction of X t+lt from the origin t. we
have that this prediction is given by:
X
t
where
(It)
=
the
r
*� X (R.-i)
I
i t
i= 1
+
m
6 a
I
j t+R._j
j.. l
'l<'t (R.-i)
predictions
+
coincide
c
(2.4)
wi th
the
values
observed when the horizon is negative and the disturbances
are
zero if R.>j
t+lt_j
estimated values if j>R..
a
for
effect
R.>m
on
the
MA
part
prediction.
and
of
they
the
coincide
model
Consequently.
for
with
will
a
have
the
no
relatively
-9 -
far-off
time
horizon
the
so-called
eventual
forecasting
function is obtained in which:
(2.5)
The
provides
obtain
the
this
solution
of
structure
of
solution
we
this
the
are
difference
forecasting
g oing
to
use
equation
function.
the
To
following
theorem.
Theorem
Let the homogenous difference equation be
A (L)X
t
=
0
(2.6)
k
is
a
finite
polynomial
A (L)= 1+a L+. . . . +a L
k
1
the lag operator which can be factorised as:
in
where
A(L) �P(L)
Q(L),
(2.7)
where the polynomials P (L) and Q(L) are prime (they do not
have
common
rools.
Then ,
the
general
solulion
lo
this
equation can always be written as:
(2.8)
and q
are
P
t
t
each prime polyomial, that is to say:
where
the
sequences
P (L) P
t
=
Q(L) q
t
=
0,
the
solutions
to
(2.9)
-10 -
The proof of this theorem is given in the appendix.
To
apply
this
theorem
let
us
note
forecasting function can be written.
that
eventual
the
for l,m:
(2.10)
*
s
where � (l)=� (l) . (l ) and the l operator acts on
the
p
p
index 1 and t. the origin of the prediction. is fixed.
*
The
stationary
operator
has
all
the
roots
� (l)
d
has a unit
outside the unit circle. the operator ( 1-l)
root
repeated
d
times.
while
the
s
operator
(l-l ) can
be
written:
s
(l_l ) =
(l-l) Sell
where:
(2. 1 1)
S-l
Sell = ( 1 + l +. . . + l
This
operator
circle.
other
If
s
has
is
even.
s-2 complex
distributed
s-l
roots.
these
all
s-l
)
of
them
roots
in
include
the
l=-l
unit
and
conj ugated roots with a unit module and
sy n�etrically
in
the
unit
circle.
stationary
operator·s
�*(l)
and
the
d
non-stationary
ones
/l /l
have
no
root
in· common
s
and the euentual forecasting function can always be broken
Consequently.
the
down into two components:
where
P (l) is the
t
forecast. which is
( 1)
permanent
determined
component
only
by
of
the
long-term
the non-stationary
part of the model and is the solution to:
(2. 12)
-11-
(2)
t (i)
is
the
transitory
component,
which
is
t
determined by the stationary autoregressive operator. This
component
defines
component
tends
how
to
the
be
approach towards the permanent
produced.
The transitory
component
is defined by means of the equation:
S
41(L) t(L ) t (i)
t
Now
we
basis
will
41 (L)t (i)
t
form
the
ARIMA
the
of
how to
study
*
=
of
model and in
(2. 13)
0
=
on
components
these
section four
the
analyse
we
calculate them.
2. 2 The transitory component
The
forecasting
general
transitory
function
is
solution to this
supposing
that
to
solution
the
homogenous
the n=p+P.s roots
the
of
component
of
eventual
(2. 13).
The
difference equation,
the polynomial
41*(L)
are different , is
(2. 14)
-1
-1
, .. . , G
are
1
n
autoregressive
polynomial
where
G
depending
upon
the
the
operator
outside
the
unit
terms
G
are
j
Consequently, :
all
and
origin
hypothesis,
be
the
of
lhe
3
are
or, which
module
lim G
�
i-)CD
�
0
of
less
is
the
coefficients
prediction.
AR is stationary,
circle
in
b
roots
t)
its
Since,
roots will
equivalent,
than
by
the
the
unit.
(2. 1S)
-12-
and
the
term.
transitory
This
exist ,
same
component
will
be
zero
in
the
long
reasoning is ualid when h identical roots
since in that case the term associated with those h
equal roots , G , will be:
et)
et).
[b
'"
+b
2
1
+. . .+
(t).h- 1
Jt
]G
b
'"
h
h
which will tend once more to zero when Jt->oo if
Consequently , the
how
the
transition
transitory
towards
the
IG l<l.
h
component
permanent
specifies
component
is
produced and disappears for high prediction horizons.
2. 3.
The permanent component
By
component
using
of
the
the
factorisation
long-term
(2. 1 1)
forecasting
the
permanent
function
can
be
written (2.12) as follows:
�
According
to
solution to
two
terms
and S ( l),
d+ 1
Sell
the
p t (Jt)
theorem
=
� .
of
the
(2. 17)
preuious
section
the
this equation can in turn be broken down into
d+1
associated with
the
prime polynomials �
the first of which we will call trend component ,
T ' and will be the solution of:
t
(2. 18)
where
c
=
component,
�/s and the
second term we will call
seasonal
E ' and is the solution of:
t
(2. 19)
- 13 -
It
can
be
satisfy
immediately
the equation
checked
(2. 17).
that
these
components
In the follollJing section its
properties are analysed.
The trend component
2.4.
The trend component
of the model is lhe solution
of (2. 18) IIJhich can be IIJritten:
(l)
(t)
(t) d * d+ 1
T (�)�c
� +c �
�+. . ,+c
+c
t
o
d
1
\1
IIJhere c * s(d+1)!
and
is
a
(2. 20)
'
polynomial
of
degree
d+1
lIJith
varying lIJith the origin of the prediction,
latter
IIJhich
lherefore,
there
are
is
of
no
an
polynomial
the
order
trend
ARIMA
seasonal
the
is
constant
is
d.
polynomial
and
model
equal
is
IIJhilst
in
When there is' a
is
of
degree
except for the
c*.
alllJays
differences
d-1,
to
\1�O,
the
same
seasonal
if
The
trend ,
polynomial:
and
d,
coefficients
if
the order of
if
11",0
difference
the
\1�O,
case
and
d+1
if
II�O
2.5. The seasonal component
'The
solution
of
seasonal
component
of
the
model
is
the
(2. 19) IIJhich is any function of period s,lIJith
values summing zero each s lags.
We lIJill call
It-1,
the
s
solutions
seasonal
of
the
coefficients
of
equation
the
. . . S
(2. 19),
forecasting
which
are
the
function.
It
should be noted that the seasonal coefficients observe the
restriction
-14 -
so that
when they
are unknown we
only
have (s-l) unknown
factors.
The
su perindex
t
in
the
seasonal
coefficients
indicates that these coefficients vary with the origin of
the prediction and at'e updated as
The
seasonal
coefficients
will
new data are received.
be
determined
from
the
initial conditions , as we shall see in section four.
2.6 The long-term forecasting function
As
component
we have seen ,
of
the
in the long term the transitor'y
eventual
forecasting
function
is
made
zero and only the permanent component remains, that is for
a very large l1,
where
T (l1,) is a
polynomial
trend
and
E (l1,)
is
t
t
seasonal component which is repeated every s periods.
the
-
15 -
3. ECONOMIC INTERPRETATION OF THE COMPONENTS OF THE
UNIUARIATE FORECASTING FUNCTION
In
this
section we attempt to analyse what type
of information is
the
previous
variable.
from
X
the
have
moment
the
for
corresponding
prediction
would
Consequently.
section
section
The
variable
provided by the forecasting function of
if
in
is
the
no
future
an
value
economic
which
the
type of innovation occurred
which
the
predictions
different
to
values
prediction
described
in
of
the
i
are
is
the
made.
previous
expectations
held in the moment t on the values of the variable in t+l.
t+2
.
.
.
.
•
.
It
must
constructed
by
be
noted
using
that
these
exclusively
expectations
information
are
on
the
history of the phenomenon in question.
Supposing that
are
known.
the
parameters of
can
value
be
the ARIMA model
broken
down
in
the
following way:
(3.1)
where
with
we
denote
the
prediction
error
which
is equal to
(3.2)
The
breakdown
(3. 1)
divides
the
observed
value
X
into two parts which are mutually independent:
/:>.
-X
:
t+i
the moment t;
eXE!ectation
for
t+i
which
we
have
in
:
effect
of
the
surE!rises
which occurred
t+i
t+l and t+i. which is obtained as a weighted sum
-e
between
X
t+i
of the corresponding innovations.
16
-
When
variable
if
lends.
innovations or
Therefore.
the
forecasting
equilibrium
if
constant
stable
we
stochastic
the
the system
were
� high and ever higher values
describes
the
conclude
equilibrium.
future
long-term
the
which
economic
of
forecasling
the
which affect
to
the
limit
will
function has
the
in
function
of
to
value
giving
path
the
infinite.
disturbances
zero.
Thus.
the
indicates
function
to
tends
�
-
variable
forecasting
that
while
the
the
on
in
question.
function
variable
the
lonq-term
is
a
to
a
tends
contrary.
if
this
no limit we will say that the variable tends
to a situation of steady state.
importance
of
fundamental
causes:
quantify
long-term
is
on
the
equilibrium
and
hand
one
on
the
path
it
this
is
given
enables
it
which
by
two
obeys
us
lo
expectations for a
other.
towards
economic
the
function
term univariate
phenomenon;
moving.
that
have
forecasting
the
the different
particular
we
conclusion
In
describes
this
the
the
phenomenon
trend
of
the
forecasting function.
In
adding
comment
in
defined
variables
Escribano
to
the
and
Engle
be differentiated
transformation
different
to
mathematical
zero
order
we
(h.
re presented
one
Granger
integrated
( 1987)
and
order (h.
zero.
say
has
If
expectation
will
a
in
mathematical
the
previous
of the stationary
that
0) .
In
general
by
(h.
m).
according
l)if it needs
h times to become stationary and the
stationary
is
of
( 1987) we will say that a variable g enerated by
an ARIMA model is integrated of
to
concept
the
the
where
to whether
variable
m
of
takes
the
case
the
transformation
is
order
expectation
integrated
of
integration
is
value
zero
or
the mathematical expectation of
the stationary transformation is nil or not.
from what has
17
-
been
-
seen in the previous section,
we have that the order
of integration fully describes the polynomial struc ture of
the trend of
the
order
the forecasting function,
max (o, h+m-l).
in the sense that
The
which will be of
trend is purely stochastic,
all its coefficients are determined by
the initial conditions of the system,
if m is zero, and it
is mainly deterministic if m is different to zero.
This definition of integration makes explicit the
presence
or
series due
otherwise
of
a
constant
to the importance which,
in
the
stationary
as we shall see,
this
parameter has.
If h+m adds up to zero or one the variable tends
to a stable equilibrium,
deterministic
if
h
is
the value of which will be purely
zero,
or it will be
determined by
the initial conditions if h is one.
If
h+m
adds
up
to
more
does not tend to a stable value,
than
one,
the
variable
but evolves according to
a polynomial structure which accords it a steady state. In
this polynomial structure
the most important thing in the
long-term is the coefficient corresponding to the greatest
power,
since
negligible
contibution.
deterministic
path
will
compared
if
also
m
be
to
it
Now,
all
other
this
is
one,
in
so.
This
means
powers
coefficient
which
case
that
the
the
have
will
a
be
long-term
factor
which
contributes most to this path is not altered by changes in
the
conditions
of
the
system.
On
the
contrary,
if
m
is
zero all the parameters of the trend of the forecasting
function
In
such
depend on the initial conditions of the system.
cases
the
long-term law is determined by a time
polynomial of the order (h-l),
but the parameters of this
polynomial change as new disturbances reach the system.
- 18-
an
To
complete
the
economic
variable
we must specify the
uncertainty
by
infinite.
is
to
about
have
it.
long
term
of
magnitude of the
This
uncertainty
is
in
(3. 1) when lI. tends to
t+lI.
If the process is stationary h;O, the polynomial
expressed
!'(JI.)
we
des cription of the
the
which
enters
convergent
infinite
bearing
in
estimation
This
finite.
the
definition
variance
the
mind
of
the
in
and
is
e
term
the
of
result
is
uncertainty
parameters
(see
e
(3. 2)
of
associated
Box and
t+lI.
tends
lI.
when
t+lI.
certain
e
even when
with
Jenkins
the
( 1970)
appendix A7. 3).
In such a case we say that the uncertainty
regarding
future,
the
limited.
If
h
the
variance
that
we
say
bounded.
ARIMA
is
of
that
It
is
models
series,
is
however
not
zero,
far
!'(lI.)
off
does
it
not
may
be,
converge,
tends
to
infinity
with
t+lI.
uncertainty regarding the future
worth
pointing
generate,
for
out
the
that
case
the
of
and
JI.,
e
is
so
is not
fact
that
non-stationary
predictions regarding the future whose uncertainty
not
bounded
as
the
horizon
of
(lI.)
prediction
increases is not a disadvantage of these models, since the
nature
of
uncertainty
regarding
the
future
is
not
a
characteristic which indicates to us whether the model is
good
or
bad,
but
an
aspect
which
defines the
real
world
which we are attempting to make a model of.
In
regarding
Note
that
exogenous
models,
the
in
future
a
difference
variables
simply
is
are
hypothesis
not
limited
economic
generated by
predictions
variables
with
the
structural
long-term
endogenous
found
economics
with
respect
to
are
seems
model
also
ARIMA
uncer·tainty
acceptable.
(SEM)
where
non-stationary
non-bounded
the
that
generated
ARIMA
on
uncertainty.
predictions
in the fact that the uncertainty may
infinite more slowly and with a greater delay.
can
the
The
be
tend to
-19 -
of
charac lerislics
The
from the
models from the ARIMA
An
model with
deriving
path
long-lerm
the
models with the
most common values for h and m are shown in Table 1.
ARIMA
(h+m-2)
implies that in the
long term the level of the c orresponding variable tends to
infinite.
Such
una c ceptable
charac teristic
in
substituting
the
a
E conomics,
one
of
the
differentiations
but
positive
the
depends
way
on
will
in
will
not
facel,
the
be
in
c onsidered
note
unit
that
roots
by
fa ct
since
transitory
defined
case
component
in
(2.13).
have
canc elled
pra c lice,
the
it
a
out
included
of
the
As
this
t)
�
term
in
will nol
long
is
term
not
in
in
the
b
which is
slowly
economic
variable
that
prediction
c omponent
(0.99)i
medium
term
and
it
this
be able to be distinguished
E c onomic s
possible
available sample sizes,
simply
simply
become a stable equi1ibrium.­
from the first mentioned one in whi ch (h+m)
In
as
in which this stable equilibrium is a c hieved
t (Jt),
t
this
function
be
1
will be suffic ient for
(0.99)-
the law of the long-term to
But,
may
to
equalled two.
c annot be estimated,
discriminate,
with
the
between a fixed slructure and one
evolving.
Therefore,
follows
a
linear
when
we say
growth
path
that
we
an
mean
in the medium term it tends to follow such a
behaviour path.
From
Table
constants in the
the
1 it
ARIMA
charac terisation
follows
that
the
inc lusion
of
model means severe restrictions on
of
the
long
term
of
an
e c onomic
variable.
Having
forec asting
seen
func tion
that
of
an
the
parameters
ARIMA model,
the slope of the trend of the permanent
with
time,
it
is
important
to
of
the
and specifically
c ompo.nent
analyse
how
c hange
we
can
calculate them. The nexl section is devoted lo this topi c .
- 20 -
Table 1
Characteristics of the long ten. path derived from the ARt� model
corresponding to an econanic variable
tnfluence of initial
condi tions on the
h..... nature of the
long-te.. path
(h, ftI)
o. NIL LONG-TE� VAlUE
(0,0)
1
ESTA8I.E EQUILlIIIUIJI!
(0,1)
none
(1,0) they detenoi.. the
equlibrhln value
2
LINEAR GRIIITII
(a)
(h)
long-term path
Uncertainty regarding long-term
On the level
finite
none
Cl,Il they dete..l.. the ordenate
in the origin of the
on the growth rates
nil (growth Is zero)
finite
nil (growth Is zero)
infinite (it growth nfl (growth is zero)
lineary with)
Infinite (It growht
finite
Ii..ary)
straight Hne. but hawe no
influence
on
its slope
(2,0) they dete..i.. the boo
parawete" which define the
infinite (It growth Infinite (It growth
quadraticallYl
Ii..ary)
line
(a). h is the total _r of differentiations required bY the .arlable to _ stationary.
..,c) 1",,11" that the ..tI_tlcal expectancy or the stationary series is not nil •
... 1 i""lI" that this .._tlcal expectancy Is not nil.
(b)
- 21-
4. THE DETERMINATION OF THE COMPONENTS OF THE FORECASTING
FUNCTION
4.1 General Approach
The
results
of
the
previous
sections
indicate
that the eventual forecasting function of a seasonal ARIMA
model can be written:
(4. 1)
to
simplify
D=I.
Then
However.
it.
analysis
n=p+P. s
and
we
the
are
assuming
equation
is
that
valid
�=O
for
and
i>q+sQ.
d+l+p+sP initial values are required to determine
therefore.from
already
The
the
be
related
coefficients
K>q+sQ-d-l-p-sP
among
each
the
other
predictions
according
to
will
(4.1).
of this
equation can be obtained through
two different procedures:
the first is to generate as many
predictions
system
c
of
as
parameters
equations.
s-1 seasonal
J
expressed as the
.
•
and
n
The
and
equation
parameters
sum
to
solve
the
resulting
(4.1) has d
parameters
(since
of the others
a
coefficient
can
be
with a changed sign)
coefficients
b . Therefore. we need to generate a
i
number of predictions equal to R=d+ l+s-l+n=d+s+n.
Calling
.
A
and the parameter vector .2the prediction vector �
t+R
we
can write from a certain moment
Jl
the
following
expression:
1
1
1
0
1
2
o
1
(t)
c
d
t)
se
1
1
1
R
(t)
s-1
o
S
1
t)
be
1
be
n
t)
- 22-
(t)
and the coefficient S
will be equal to
s
s-l
1:
j=l
Writing
=
M a
�
where
M
is
the
data
which
coefficients
�
matrix
which
the
multiply
contains
the
known
vector
parameter
we
a
-
can express 0 as:
,..
�
which
=
enables
�-l }St+r
all
the
'
(4.2)
.
parameters
for
the
eventual
forecasting function to be obtained.
The second procedure �s first to obtain a value r
high
out
enough
for
for
k>r.
the transitory
This
value
component
depends
on
to
the
be cancelled
roots
of
the
autoregressive polynomial and is determined .in such a way
r
G1
is
the
G
with
the
where
that
IG11",O,
i
highest absolute value. A simple way of checking whether
the
transitory
component
is
practically
nil
for
K>j . s ,
consists of taking the differences:
which
will be free of the seasonal effect,
whether
positive
a
such
a
difference
values of K.
prediction
practically
horizon
nil.
For
stays
and to observe
practically
constant for
In this case we shall say that from
j.s
the
example
transitory
this
component
implies
that
is
with
- 23-
mont.hly
dat.a
t.he
annual
di.fferences
between
the
monthly
predictions will be constant from a certain year onwards.
Thus
taking
the expression of the g eneral predictions and
eliminating from it the transitory component we can set up
a system of equations to determine the coefficients of the
trend equation and the seasonal coefficients,
general those of interest.
which are in
Let us see some specific case�.
4. 2 The airline model
A seasonal ARIMA model much used for representing
the euolution of monthly economic series is the one called
the airline model:
(4.3)
According
to
what
has
been
preuiously
discussed,
the
forecasting equat.ion of this model for k>O can be written;
and contains 13 parameters. (Remember that Es
By
equalling
with
the
the
model
predictions
(4. 3)
with
for
the
k=1,
f
t)
=O).
2, . .. , 13
structural
form,
obtained
we
haue:
"
X
1
t+1
1
1
0
·
.
.
0
b
et)
0
et)
1
et)
s
1
b
=
"
X
t+12
�t+13
1 12
0
0
·
.
.
1
1 13
1
0
·
.
.
0
et)
s
12
shall
- 24-
a
system
of
13
equations and 14 unknowns whic h wi th the
t)
�s�
;O
parameters
the
enables
J
coefficients
seas·onal
and
the
restri ction
t)
t)
be
be
1
0
� t) '
s
obtained.
to
be
J
equation from the last and
By
dividing
by
first
the
subtra cting
twelve,
we obtain
directly
(4.4)
By
adding
up
the
first
12
equations
the
seasonal
coefficients are can celled out and we obtain:
X
12
(t) + (t) 1+...+12
)
b
(
1: Xt (K ) ; bo
12
1
1
1
_
; _
t
12
,
which gives as a result
t)
be ; X
0
t
-
-1L b
2
(4. �)
1
Fynally the seasonal �oefficients are obtained by:
(4. 6)
It
must
be
noted
that
if
the
ARIMA
model
is
spec ified on the logarithmic transformation of X, then the
t)
coefficients b
can
be interpreted as growth rates
�
S. measure the seasonal
J
percentage of one on the level of the series.
and the
coefficients
nature as a
- 25-
4. 3. General models with a differen c e of ea c h type
Any
operators
the
ARIMA
nn
s
forecasting
trend
and
a
parameters
and
be
coefficients
K=si+j as high
negligible,
has
and
non-stationary
permanent
is
the
we
enough
as
the
c omponent.
measure
Sj'
has
a
whi c h
seasonal
which
seasonal
taking
�=O
which
function
stable
b,
model
sum
To
linear
will
use
c omponent
of
the
linear
determine
trend
the
and
fa c t
for the stationary
equalling
a
predictions
of
the
the
that,
terms to
to
the
permanent component:
(4. 7)
s + 1
)
2
(4.8)
(4. 9)
equations analogous to those of (4. 4) and (4. 6) , where now
.....
j( is the average of the s observations in the interval
(K+ 1,
K+s).
- 26-
S.
APPLICATION Of THE CALCULATION Of THE TREND Of THE
UNIVARIATE FORECASTING fUNCTION TO SERIES ANALYSIS
Of THE SPANISH ECONOMY
In
certain
estimate
an
se c tion
this
is
for
made,
of growth rates in the
sequen ce of months,
a
trend
of lhe forecasting function of the following series of the
Spanis h
imports,
e c onomy:
The use of the above-mentioned rate in
index for servi ces.
a
relatively
and the consumer pri ce
exports
complete
short-term
analysis of
an
phenomenon
is put forward and desc ribed in Espasa
following
the
used
terminology
the
in
ec onomic
(1990).
above-mentioned
work,
we will call it inertia to the rate of growth of the
trend
of
model
whi c h
in
a univariant ARIMA
the forecasting function of
will
when
(4.7),
the
b , defined
1
spec ified on the logarithmic
by
given
be
model is
the
parameter
transformation of the variable.
Regarding
goods
the following
Spanish
foreign
univariale
trade
on
non-energy
monthly models can be used
to explain imports (M) and exports (X)
(5.4)
o
=
0'092
,
(5.5)
0 = 0'117
in which AIM and AIX are particular intervention analyses
requiring
slope
of
both
the
series
and
forec asting
we will ignore them.
which
have
function,
no effe c t upon the
therefore,
henceforth
- 27-
The
original
trends· are given
series
in Chart
with
1 and
their
corresponding
the inertias are show in
Chart 2.
The
1986,
inertias from January
the month in which Spain joined the EEC, to December
Thus,
1989.
happened
that
last chart shows these
this
chart
can
be
to Spanish overseas
date. Obviously,
trade,
which
an
variables
the
are
illustrate
what
in nominal terms from
since we are not using models
determining
analysis
to
on the basis of this description no
causal analysis can be made,
incorporating
used
can
variables
be
responsible,
trend changes. Nevertheless,
made
and
of
of
to
M
which
what
and
X
with
explanatory
extent,
for
the
the mere description of these
changes is in fact of interest in itself. However, it must
be pointed out that the trend evolutions shown in Chart 2
refer to the
and,
sale and purchase of
therefore,
prices
are
also
goods in nominal terms
influencing
these
very
same trend movements.
We
can
expectation
deduce
in
from
nominal
systematic ally throughout
that
1987,
then
this
minor osc illations,
1989,
when
it
Chart
that
2
imports
trend
was
growth
increasing
1986 and first three quarters of
expectation
has
stabilised,
with
at around 23% until the second half an
started
the
decrease
very
slowly.
As
a
result,
a worsening of perspectives for Spanish imports of
around
four
percentual
points
has
occurred
during
this
period.
With
expectation
expectation
exports
of
of
18%
around
there was a movement from a
at
the
14%
at
beginning
the
end
of
of
1986
that
growth
to
year
an
and
- 28-
CHART 1
SPANISH IMPORTS AND EXPORTS OF NON-ENERGY GOODS
(Original data and trend)
Imports
m.m.
1000 r------r--�r_--_r--_.--,_--_, 1000
800
800
200
1983
Originol dota
Trend
Forecasts
1984
1985
1986
1987
1988
1989
1990
Exports
m.m.
600
600
,
f\ P1J-
500
400
300
200
100
o
fir::.
�V
A.J
"""VI( If{
AA I
'V
r"
1982
1983
Originol doto
Trend
Forecasts
1984
1
/\"
A
r
AJ1 r'\AI-'"7
I"
J)�f--
500
400
300
200
100
1985
1986
1987
1988
1989
1990
o
- 29 -
CHART 2
SPANISH IMPORTS AND EXPORTS OF NON-ENERGY GOODS
(Original data and trend)
Imports
%
I
40
I
i
40
30
30
20
""'-
-
""'"
20
10
10
o
1986
I
1987
I
1988
1989
1
o
Exports
%
40
I
I
I
40
I
30
30
20
20
10
�
�
10
0
0
t
-10
1986
,
1987
I
1988
1989
-10
-30 -
during
Since
1987.
then
the
expectations
haue
remained
fairly stable.
can
we
conclusion
In
say
the
that
perspectiues
1986 and
for imports worsened (increased) progressiuely in
1987
taking
time
until
on
relatiuely
a
the
improuement
occurred.
exports, although
they
of
half
second
As
they have maintained a fairly
that
certain
a
when
1989
for
worsened
from
euolution
stable
expectations
(declined)
during
high level during
three years of the sample considered.
for
1986,
the last
If the evolutions of
imports and exports are compared in order to haue a better
understanding of of
trade
deficit,
that,
it
a
the
possible euolution of the Spanish
conclusion
is necessary,
in
quickly,
and,
as
both
drawn
to
the
effect
series
at least for the growth rates
to
equal
insofar as export
optimistic,
be
giuen that the level of imports is
higher than that of exports,
expected
can
giuen
the
each
growth can
level
of
other
fairly
be considered
world
commercial
activity and the relative level of Spanish prices compared
to
the
must
rest
of
per'force
the world,
require
a
to bring these rates together
significant
reduction
in
impor·t
growth.
. In
the
consumer
price
index
for
the
Spanish
economy the component referring to the prices of services,
which we shall call
behaviour
with
IPCS,
regard
has been showing fairly uneven
to
the
component
referring
prices for non-energy manufactured goods.
make
up
services
the
IPSEBENE,
and
the
non-energy
manufactured
represents 77.54% of the IPC,
on
which
it
is
worthwhile
or the inflationary trend.
consumer
to
the
Both components
price
index
goods,
for
which
and is an appropriate index
analysing
underlying inflation
- 31-
By
using
the sample
comprising
from
May
1984 to
January 1989 the following model has been estimated
(7. 6)
a
0'0014 ,
where
AIS
are interventions required
by
this
index.
t
interventions include a step effect which begins in
These
January
1986 and is due to the introduction of Value Added
Tax. The moving average coefficient is fixed at 0. 8�.
from
the
this
inertia
of
model
the
a
calculation
IPCS, (corrected
has
of
during the period comprising from January
These
1989.
calculations
are
shown
.in
been
made
of
interventions)
1986 to December
Graph 3.
There
it
can be seen that during these years the medium-term growth
expectations
It
can
of
also be
this
index
detected
have always remained above 7%.
that
during
this index increased. That is to say,
wi th
the
entry
is
in
the
on
unlike what occurred
Spain' s
EEC meant no improvement in expectations for
borne
greater
expectations
prices of non-energy manufactured goods,
the prices of services.
it
1986
in
This result
mind that entry
competitiveness
in
the
is not surprising if
scarcely
brought with it
Spanish
service
sector.
Graph 3 also shows that throughout 1987 there was a slight
improvement
completely
(fall)
in
1988,
in
the
IPCS
and in
services continued.
threat
the
IPC
since
for 34. 24% of this index.
which disappeared
1989 this deterioration in the
prices of
to
inertia,
the
All this represents a grave
services
component
accounts
- 32-
CHART 3
CONSUMER PRICE INDEX FOR SERVICES IN SPAIN
9,0
9,0
8,5
8,5
8,0
8,0
7,5
r
.-/
7,5
�
7,0
7,0
6,5
6,5
6,0
1986
1987
1988
1989
6,0
- 33 -
BIBLIOGRAPHY
Box, G.E. P.,
Y G.M. Jenkins ( 1976),
Holden Day.
Forecasting and Control,
Box, G.E.P., Pierce,
trend
and
D. and Newbold, P. 1967,
growth
J.A.S.A., 62,
Chatfield,
C. 1977,
in alasonal time series",
"Some recent developments in Time
Y C.W.J.
Error
rate
"Estimating
pp. 276-282.
series Analysis",
Engle, R. F. ,
Time series Analysis,
J. R. S.S., A,
Granger ( 1987),
Correction:
140, pp. 492-S10.
"Co-integration and
Representation,
Estimation
and
Testing" Econometrica, SS, pp. 2S1-76.
Escribano, A. ( 1987),
"Co-Integration, Time Co-Trends and
Error-Correction
CORE
Systems:
Discussion
Paper
an alernative Approach"
no.
87 1S,
Univer'site
Catholique de Louvain.
Espasa, A.
( 1990),
economic
"Univariate Methodology for short-term
analysis",
Banco
de
Espana,
Working
paper 9003, Madrid, Spain.
Harrison, P. J. and Stevens, C. F. , 1976 ,
forecasting" J.R.S.S.,
" Bayesian
B. , 38, pp. 20S-247.
Harvey, A.C. and Todd, P.H. J. , 1983 "Forecasting economic
Time
series
models",
299-31S.
J.
with
of
structural
Bus.
and
Eco.
and
Stat,
Box
1,
Jenkins
4.
pp.
-34 -
APPENDIX 1
Demonstration of the theorem in Section 2
It can be immediately
is
sufficient
and
that
proved that the condition
(2.3)
is
a
solution
of
(2.1).
Because of the commutability of the operators
Let
us
now
necessary,
that
written
in
(2. 3).
prime
two
Q(B)
as
are
that:
is,
prove
that
from
any
that
the
solution
Bezout's
polynomials
condition
of
(2. 1)
theorem,
exist
can
be
if P (B) and
T (B),
1
T (B)
2
therefore:
calling
(A. 1)
(A. 2)
is verified
is
such
- 35 -
both
multiplying
members
of
(A.l)
and
(A. 2) by
Q(B) and
P(B) respectively:
and
therefore
(2.3)
that
a.nd
the
any
(2. 4)
solution
indicated
breakdown
is
can
in
be
the
unique.
written
theorem.
let
us
in
the
let
us
assume
form
prove
another
breakdown:
Z
t =
where q'
t
q'
+ p'
and P'
analogously
P'
t'
t
it
t
t
verify
is
(2. 4).
proved
that
Then:
P
t
must
be
identical
to
- 37-
DOCUMENTOS DE TRABAJO
850 I
8502
(1):
Agustin Maravall: Prediccion con modelos de series temporales.
Agustin Maravall: On structural time series models and the characterization of components.
8503
Ignacio MauleO": Prediccion multivariante de los tipos interbancarios.
8505
Jose Luis Malo de Molina y Eloisa Ortega: Estructuras de ponderacion y de precios
8506
Jose Vifials: Gasto publico. estructura impositiva y actividad macroeconomica en una
8507
IgnacioMaule6n: Una funci6n de exportaciones para la economia espanola.
8504 Jose Viiials: El deficit publico y 5US efectos macroecon6micos: algunas reconsideraciones.
8508
relativos entre los deflactores de la Contabilidad Nacional.
economfa abierta.
J. J. Dolado. J. L Malo de Molina y A. Zabalza: El desempleo en el sector industrial
espanol: algunos factores explicativos. (publicada una edici6n en ingles con el mismo
8509
numera).
IgnacioMaule6n: Stability testing in regression models.
8510
Ascension Molina y Ricardo Sanz: Un indicador mensual del consumo de energia electrica
8511
J. J. Dolado andJ. LMalo deMolina: An expectational model of labour demand in Spanish
8512
J. Albarracin y A. Yago: Agregaci6n de la Eneuesta Industrial en los
para usos industriales. 1976-1984.
industry.
Contabilidad Naeional de
1970.
15
sectores de la
8513
Juan J. Dolado. Jose Luis Malo de Molina y Eloisa Ortega: Respuestas en el deflactor del
8514
Ricardo Sanz: Trimestralizaei6n del PIB por ramas de actividad. 1964-1984.
8516
A. Espasa y R. Galhin: Parquedad en la parametrizacion y amisiones de factores: el modelo
8515
8517
vat�r 8nadido en la industria ante variaciones en los costes laborales unitarios.
Ignacio Maule6n: la inversion en bienes de equipo: deterrninantes y estabilidad.
de las lineas aereas y las hipotesis del census X-11. (Publicada una edicion en ingles con el
mismo numero).
Ignacio Maule6n: A stability test for simultaneous equation models.
8518
Jose Vinals: lAumenta la apertura financiera exterior las fluctuaciones del tipo de cambia?
8519
Jose Vifials: Deuda exterior y objetivos de balanza de pagos en Espana: Un analisis de
8520
JoseMarin Areas: Algunos indices de progresividad de la imposicion estatal sabre la renta
8601
Agustin Maravall: Revisions in ARIMA signal extraction.
8602
8603
8604
8605
8606
8607
8608
(Publicada una edici6n en ingles con el mismo numero).
largo plazo.
en Espana y otros paises de la aCDE.
Agustin Maravall and David A. Pierce: A prototypical seasonal adjustment model.
Agustin Maravall: On minimum mean squared error estimation of the noise in unobserved
component models.
IgnacioMaule6n: Testing the rational expectations model.
Ricardo Sanz: Efectos de variaciones en 105 precios energeticos sabre los precios sectoria­
les y de la demanda final de nuestra economia.
F. Martin Bourgon: Indices anuales de valor unitario de las exportaeiones: 1972-1980.
Jose Vifials: la politica fiscal y la restriccion exterior. (Publicada una edici6n en ingles con
.
el mismo numero).
Jose Vifials and John Cuddington: Fiscal policy and the current account: what do capital
controls do?
8609
Gonzalo GiI: Politiea agricola de la Comunidad Economica Europea y montantes compen­
8610
Jose Vifials: (Haeia una menor flexibilidad de los tipos de eambio en el sistema monetario
8701
Agustin Maravall: The use of ARIMA models in unobserved components estimation: an
8702
application to spanish monetary control.
Agustin Maravall: Descomposicion de series temporales: especificacion, estimacion e
satarios monetarios.
internacional?
inferencia (Con una aplicacion a la oferta monetaria en Espaiia).
�
8703
8704
8705
8706
8707
8708
8709
8801
8802
38
�
Jose Vifials y lorenzo Domingo: La peseta y el sistema monetario europeo : un modelo de
tipo de cambia peseta-marco.
Gonzalo Gil: The functions ofthe Bank of Spain.
Agustin Maravall: Descomposicion de series temporales, con una apl icaci6n a la oferta
monetaria en Esparia: Comentarios Y contestaci6n.
P. L'Hotellerie y J. Vinals: Tendencias del comercio exterior espanol. Apendice estad'stico.
Anindya Banerjee and Juan Dolado : Tests ofthe Life Cycle-Permanent Income Hypothesis
in the Presence of Random Walks: Asymptotic Theory and Small-Sample Interpretations.
Juan J. Dolado and TIm Jenkinson : Cointegration: A survey of recent developments.
Ignacio Maule6n: La demanda de dinero reconsiderada.
Agustin Maravall: Two papers on arima signal extraction.
Juan Jose Camio y Jose Rodriguez de Pablo: El consumo de ali mentos no elaborados en
Espafla: Analisis de la informacion de Mercasa.
8803
Agustin Maravall and Daniel Pena: Missing observations in time series and the «dual»
8804
Jose Vinals: El Sistema Monetario Europeo. Espana y la politica macroeconomica. (Publi-
8805
cada una edicion en ingles con el mismo numero).
Antoni Espasa : Metodos cuantitativQs y analisis de la coyuntura economica.
8806
8807
8808
autocorrelation function.
Antoni Espasa : El pertil de crecimiento de un fen6meno econ6mico.
Pablo Martin Aceiia: Una estimaci6n de 105 principales agregados monetarios en Espaiia:
1940-1962.
Rafael Repullo: Los efectos econ6micos de los coeficientes bancarios: un analisis te6rico.
8901
M_a de los Uanos Matea Rosa : Funciones de transferencia simultaneas del indice de
8902
Juan J. Dolado: Cointegraci6n: una panoramica.
8903
8904
precios a l consum� de bienes elaborados no energeticos.
Agustin Maravall : La extraccion de senales y el analisis de coyuntura.
E. Morales, A. Espasa y M_ L. Roja: Metodos cuantitativos para el analisis de la actividad
industrial espaiiola. (Publicada una edicion en ingles con el
misma numeral.
9001
Jesus Albarracin y Concha Artola: El crecimiento de 105 salarios y el deslizamiento salarial
9002
Antoni Espasa, Rosa Gomez-Churruca y Javier Jareiio: U n analisis econometrico de los
9003
Antoni Espasa: Metodologia para realizar el analisis de la coyuntura de un fen6meno
9004
9005
9006
en el periodo 1981 a
1988.
ing resos por turismo en la economia espanola.
econ6mico. (Publicada una edici6n en ingles con el mismo numerol.
Paloma Gomez Pastor y Jose Luis Pellicer Mire!: Informacion y documentacion de las
Comunidades Europeas.
Juan J. Dolado, TIm Jenkinson and Simon Sosvilla-Rivero: Cointegration and unit roots : a
survey.
9007
Samuel Bentolila andJuan J_ Dolado: Mismatch and Internal Migration in Spain. 1962-1966.
Juan J. Dolado, John W. Galbraith and Anindya Banerjee: Estimating euler equations with
9008
integrated series.
Antoni Espasa y Daniel Pena: Los model os ARIMA. el estado de equilibrio en variables
econ6micos y su estimaci6n. (Publicada una edici6n en ingles con el mismo numeral.
(1)
Los Documentos de Trabajo anteriores a 1985 figuran en el catalogo de publicaciones del Banco de
Esparia.
Informaci6n : Banco de Esparia
Secci6n de Publicaciones. Negociado de Distribuci6n y Gesti6"
Telefono: 338 51 80
Alcalit, SO. 28014 Madrid