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Another simple time series representation of yr is the moving average proce
1, MA(1):
+ +
yr = Er h t - 1 = (1 U)&,

a realisation of which is shown in Figure (20.3). An equivalent representatio
+ AL)-'y, = E,
(1
l l < 1) may be represented as an infinite autoregressive process. For 1A1 < 1
k
process is said to be â€˜invertibleâ€™. Similarly, comparing (20.3) and (20.6˜) AR
an
may be represented as an infinite moving average process (for 1/31 < 1).

ARIMA Process
If a series is non-stationary, then differencing often produces a stationary
example, if the level of yr is a random walk with drift then A y r is stat
equation (20.11)). Any stationary stochastic time series yr can be approximated
autoregressive moving average (ARMA) process of order (p, q), that is ARM

where 4 (L) and 8 (L) are polynomials in the lag operator:
- . . . - 4pLP
# ( L ) = 1 - 41L - &L2 - 4&3
+ + 8 , +˜. . .˜+ 0 q ˜ 4
#(L) = 1 e l L
The stationarity condition then is that the roots of t (L) lie outside the uni
\$
all the roots of \$ (L) are greater than one in absolute value). A similar condit
I
placed on 8 ( L ) to ensure invertibility (i.e. that the MA(q) part may be writt
of an infinite autoregression on y , see equation (20.14)).
If a series needs differencing d times (for most economic time series
is sufficient) to yield a stationary series then A d y r can be modelled as an AR
process or equivalently the level of yr is an ARIMA(p, d, q ) model. For examp
(1966) demonstrates that many economic time series (e.g. GDP, consumptio
may be adequately represented by a purely statistical ARIMA(O,l,l) model:

Autocorrelation Function: Correlogram
The properties of univariate time series models may be summarised by t
tion) autocorrelation function and its sample analogue, the correlogram. The (
autocorrelation between yr and Y r - r ( t = 0, f l , f 2 , . . .) is

where yr is the autocovariancefinction at lag t.

1
Yr = Cov(Yr Yr-r
9

YO = var(yr 1

By definition po = 1. There is a value of P r for each lag â€˜c, and the autocorr
dimensionless. The sign of Pr indicates whether yr and yr-5 are positively or
A plot of 6, against t is called the correlogram. The correlogram is useful in
classify the type of ARIMA model that might best characterise the data. A corre
approaches zero as r increases indicates a stationary series, whereas a â€˜flatâ€™ c
indicates that the series must be differenced at least once to yield a stationary
example, if yr is a random walk with drift, equation (20.10),then yr and the firs
of this series, Ay,, have correlograms like those of Figure 20.4. The corre
the stationary AR(1) model (with 0 < / < 1) and the MA(1) model (with
I
as represented by equations (20.3) and (20.13), respectively, have the distinc
shown in Figure 20.5. Hence a researcher wishing to model yr as a univariate
could immediately identifi from the correlograms in Figure 20.5 what typ
to estimate. When yr is generated by a more complex ARMA(p, q ) proce
shape of the correlogram can only be taken as indicative of the type of AR
to estimate. The information given by the shape of the correlogram and the
choice of univariate â€˜time seriesâ€™ model forms the basis of Box-Jenkins or
modelling.

Forecasts and the Variance of Forecast Errors
This section demonstrates the relationship between unconditional forecasts and
forecasts and their associated forecast error variances using a simple statio
model
+
Yr = a + Syr-1 IS1 < 1
Er

Random Walk with Drift ( y t )

Figure 20.4 Correlogram for Random Walk with Drift (yf) and for Ayf
L 1
1 I
I
L
2 4
3
1

+ c l ) and MA(1) (yt = cI +
Figure 20.5 Correlogram for AR(1) (yt = 0.5y,-l

is uncorrelated with y+1. For I/?I < 1 we have f
where Et is N ( 0 , a 2 )and Er
by back substitution:

+ + + .) + Bmyt-m + + B E t - 1 + +t
yt = (a! ˜ ˜ ˜ -
Et
** ***

a!
+ + pet-1 + B2E,-2 + - - -
Yt = - Et
1-B
where for < 1, the term Bmyt-m approaches zero. At time t the uncondi
of yr from (20.20) is:
E f =P
Y
where p = a/(l - B) and we have used Et&+, = 0 ( j 2 0). We can interpret
ditional (population) mean of yf as the best forecast of yt when we have no
about previous values of y. Alternatively, it is the â€˜average valueâ€™ around wh
lates in the population. Now let us calculate the unconditional variance o
(20.20) and (20.21) we can easily see that

+ But-1 + +
- a/(l - p) = /?(yr-1 - p )
(Yr - P ) = Et Et

Hence

For a stationary series var(yt) = var(yt-l) and because Et is uncorrelated with
the last term is zero, hence the unconditional variance of yt is:

var(y,) = a2/(1- p 2 )
It is also useful to derive this variance from (20.20)
The unconditional variance is the â€˜best guessâ€™ of the variance without any kn
recent past values of y. In the non-stationary case /3 = 1 and the unconditiona
variance are infinite (undefined). This is clear from equations (20.21) and
is the mathematical equivalent of the statement that â€˜a random walk series
anywhere â€™.

Conditional Mean and Variance: Stationary Series
In calculating the conditional mean and variance we have to be precise abou
of the information set 52. If we have information at t or earlier (i.e. on yr-j or
then the conditional mean of yr is denoted E(yf(Qr)or Eryr for short. From
conditional mean in the AR( 1) model for different forecast horizons is obtained
substitution:

The conditional variance, given information at time t , for different forecast ho
defined as
var(yr+m152t) = E [yr+m - E ( Y ˜ I Q , ) I ˜
By successive substitution:

and

+ BYr+2 + Er+3
Yr+3 = a
+ B + B2) + B3Yf + (P2Er+1 + BEr+2 + E f + 3 )
=
m- 1 m-1

i=O
i=O

Using (20.6), (20.7) and (20.8) the conditional variances at various forecast e

var(Yt+l 152,) = Q2
+ B2 +
var(y,+3IQr) = (1 @IQ2

var(yr+,1Qr) = (1 + p2 + p4+ - - .˜ ˜˜-˜)
We can immediately see by comparing (20.25) and (20.29) that the conditional
always less than the unconditional variance at all forecast horizons. Of course
Conditional Mean and Variance: Non-Stationary Series
The above analysis is repeated with j9 = 1. After â€˜mâ€™ periods the expected valu

+ am
EtYr+m = yt

Since yt is a fixed starting point (which we can set to zero), the (conditiona
+
value of yr+* is a deterministic time trend â€˜amâ€™ Yr = U bt). However, th
(cf.
behaviour of the RW ( p = 1) with drift (a# 0) is given by

and is qften referred to as a stochastic trend. This is because in addition to the d
trend element â€˜amâ€™, there is a stochastic moving average error. For the ra
without drift (a = 0) the best conditional forecast of all future values of y is
current value y r .
Unlike the stationary case, the influence of past errors on yr+m does not
m -+ 00 since the CEt+i terms are not â€˜weightedâ€™ by < 1 (see (20.31)). The
variance of the forecasts can be obtained from (20.29) by setting p = 1 and i

var(yt+m IQ r ) = ma2
As m increases the variance increases and approaches infinity as the forec
m 00. The conditional variance for a random walk series is explosive.

Deterministic and Stochastic Trends
A deterministic trend is given by

+ pt + Er
yr = a
where t takes the values 1, 2, 3 . . ., etc. The dependent variable yr in (20.
stationary since its mean rises continuously over time. The mean of yt is
hence is independent of other economic variables, even when considering a f
the distant future. The conditional forecast error is a2 and it does not increase
the forecast horizon. Our stochastic trend for the RW with drift may be writte

+ Bt + + +- +
Yr = yo El)
(Et Et-1 **

Hence the stochastic trend has an infinite memory: the initial historic value of
(i.e. yo) has an influence on all future values of y. The implications of determ
stochastic trends in forecasting are very different as are the properties of te
applied to these two types of series. We need some statistical tests to discrimina
these two hypotheses. Appropriate â€˜detrendingâ€™ of a deterministic trend involv
sion of yt on â€˜timeâ€™ but for a stochastic trend (i.e. random walk) â€˜detrendin
taking first differences of the series (i.e. using Ayr).
no deterministic time trend present (this is often referred to as the spurious
problem). Also if a random walk series is detrended using a deterministic tren
the new detrended series is yf - ) t then the autocorrelation function will
indicate positive correlation at low lags and a cyclical pattern at high lags.

Summary
It may be useful to summarise briefly the main points dealt with so far, these
A stationary series is one that has a mean, variance and autocovariances i
lation (of data) that are independent of time and are finite. The populatio
variance are constant and the covariance between yr and Yr-m depends onl
length â€˜mâ€™ (and is constant for any given lag length). Since the populatio
(i.e. mean, variance and covariance) are constant, they can usually be
estimated by their sample analogues.
A graph of a stationary series (for the population of data) has no discern
upward or downward trend, the series frequently crosses its mean value
variability of the series around the mean value is, on average, a constan
(which is finite).
A non-stationary series is one that either has its population mean or
or its autocovariances which vary over time. In some special cases the
(unconditional) mean, variance (or even covariance) may approach plu
infinity (i.e. be undefined).
The simplest form of non-stationary series is the random walk with d
Gaussian error)
+ pyr-l where /I=1
yt = a +Er

+
(1 - BL)yr = a Er

The non-stationary series yr is said to have a unit root in the lag polynom
(1 - BL) has /?= 1. If a = 0 the best conditional forecast of yr+m based on
at time t, for all horizons â€˜mâ€™, is simply the current value yr. For a #
conditional forecast of Yr+m is am.
Stochastic trends and deterministic trends have different time series pro
must be modelled differently.
It may be shown that any stationary stochastic series yr may be represented
nite moving average of white noise errors (plus a deterministic component
be ignored). This is Woldâ€™s decomposition theorem. If the moving average
is invertible then the yt may also be represented by an infinite autoregressi
lag ARMA(p, q) process may provide a parsimonious approximation to
lag AR or MA process.
The unconditional mean of a stationary series may be viewed as the lon
to which the series settles down. The unconditional variance gives a
20.2 MULTIVARIATE TIME SERIES MODELS
This section generalises the results for the univariate time series models discu
to a multivariate framework. In particular, it will show how a multivariate sys
reduced to a univariate system. Since the real world is a multivariate system the
concerning the appropriate choice of multivariate system to use in practica
will be discussed and a brief look taken of the relationship between a structura
model and a multivariate time series representation of the data. Finally a m
system where some variables may be cointegrated will be examined.
It is convenient at this point to summarise the results of the univariate ca
decomposition theorem states that any stationary stochastic series yt may be rep
a univariate infinite moving average of white noise errors (plus a deterministic
which we ignore throughout, for simplicity of exposition):

+ +
+
where 6(L) = (1 OIL 82L2 - - .). If 8(L) is invertible then yt may also be
as an infinite univariate autoregression, from (20.33):

A stationary series may also be represented as an ARMA (p, q ) model:

As yr is stationary the roots of 4(L) lie outside the unit circle and (20.3
transformed into an infinite MA model

or if 8(L) is invertible then (20.35) may also be transformed into an infinite lag
plus a white noise error
e-' (L)#(L)Yr= Er

Hence we have a number of alternative equivalent representations of any univ
series yr. By using a matrix formulation we will see that the above alternativ
tations apply to a multivariate system.
consider only three for illustrative purposes) the vector autoregressive mov
model (VARMA) is

- -) Hence ea
where \$ 1 1 , 4 2 2 , 4 3 3 are of the form \$11 = (1 - &)L - &)L2 -
is of the form

The above equations can be represented in matrix form as:
O(L)Y, = 8 ( L ) s r
where Y, = (ylt, y2,, y3) and 8 , = (El,, ˜ 2 t ˜, 3 and) O(L) and 8 ( L ) are c
˜
matrices which depend on the \$ij and 8ij parameters, respectively.
The VARMA equation system (20.39) is consistent with Wold's decomposit
if all the roots of Q ( L ) lie outside the unit circle, since (20.39) implies:
Y,= O-'(L)B(L)s,
Hence each Yir can be represented as an infinite moving average of current an
noise errors E r . Since any linear combination of ( E l l , czr,â‚¬31) can also be rep
terms of a MA of a single error, say, vjr, then each yir can be written:

It is also straightforward to see that if 8 ( L ) is invertible then:
e-'(L)Q(L)Y, = 61
and hence any set of variables yir(i = 1 , 2 , 3 ) may be represented as an inf
autoregression plus a linear combination of white noise errors E i r ( i = 1, 2,3).
Yjt, take ylr as an example, is of the form

where vlr is a linear combination of white noise errors at time t and hence is
noise. The above representation is known as a vector autoregression (VAR) an
notation is:
+
Y, = A(L)Y,-1 vt

Can we Reduce the Size of the VARMA System?
It can now be demonstrated how
To simplify the algebra consider reducing a simple 2 x 2 system to a univari
We begin with a VARMA (2 x 2) model:

+ 412Y2r-1 + @l(L)Elr
y1r = 4llYlr-1
+ 422Y2r-1 + @2(L)E2r
= 421Ylr-1
Y2r

and will only derive the univariate equation for ylr. From (20.45) we can obt
as a function of ylf-j (and the error terms):

+ @2(L)E2t]
= (1 - 422LI-l
Y2t [421Ylr-l

= f(Ylr-j, E24
Y2f

Substituting for from (20.46) in (20.44) we have
y2r-1

Equation (20.47) is a univariate ARMA model for yl,. As we have seen
model can be further reduced to an infinite autoregression or moving averag
stationarity and invertibility apply). The results of this section can be sum
follows. A set of k stationary stochastic variables ylr, yzr, . . . ykr can be repre

a (k x k) VARMA system,
(i)
a smaller (k - I ) x (k - I ) VARMA system,
(ii)
(iii) an infinite vector autoregressive VAR series with white noise errors B(L
that each yir depends only on lags of itself yir-, and lags of all the oth
yk,,-j (and a white noise error).
(iv) The set of k variables can be reduced to a set of univariate ARMA eq
each of the yir:
+(L)yir = @(L)Eir (i = 1,2, . . .k )

(v) The univariate ARMA representation can be transformed into an infin
average representation (Woldâ€™s decomposition theorem) or an infinite
autoregression (assuming invertibility and stationarity).