ñòð. 47 |

be represented as a linear function of either its own lags, lags of other varia

error terms. If the world is â€˜non-linearâ€™ then clearly the linear form can at

approximation to the true non-linear system. Some non-linear time series m

However, this approach has not as yet featured greatly in the economic analy

prices and will not be discussed further.

Temporal Stability

A key issue in using time series models is temporal stability in the parameters

danger when investigating a VARMA or VAR system that it is â€˜too smallâ€™ to

â€˜trueâ€™ constant parameters out there in the real world. Suppose, for the sake o

that the VARMA equation (20.44) for y l , has stable parameters when cons

function of lagged values of y l r , ˜ 2 However, suppose that equation (20.45

˜.

some unstable parameters (even one will do). When we substitute for y2r from

(20.44) and reduce the system to a univariate form (either AR or MA) then th

system for y l r has â€˜newâ€™ parameters that incorporate the parameters of equat

for yzr. Hence if the latter are unstable, then the parameters of all â€˜smaller sy

equation 20.47) will also be unstable.

To give a concrete example of the above, suppose y l r and yzr are prices

rates, respectively. Without loss, let us assume the errors in (20.44) and (20.45

white noise, that is consist of E l r and ˜2˜ only. Equation (20.45), for interest ra

interpret as the monetary authorities reaction function. If the authorities have

rule whereby an increase in the price level causes the authorities to raise in

over successive time periods then we have:

+ h 2 Y 2 r - 1 + E2r

= $21Ylr-1

Y2r

with $21, $22 > 0. Suppose that after some time the monetary authorities dec

a feedback rule for the interest rate and simply try and gradually lower in

Equation (20.48) now becomes:

+ E2r

= h 2 Y 2 r -1

Y2r

with 0 < 4)22 < 1. Hence the parameters of the equation for y2r, the interest ra

varying over the two different monetary regimes. A researcher who estimates

system for either ylr or over the whole data set would find that the par

unstable. If he ignores this temporal instability then his estimates of the para

be biased and he will not have â€˜discoveredâ€™ the true constant parameters of

in the two distinct regimes. Any tests based on the â€˜smaller systemâ€™ (e.g.

on the parameters, forecast tests) will be incorrect. On the other hand the

equation (20.44) for y l r has stable parameters (although that for does not)

danger in representing the â€˜real worldâ€™ by time series models which involve

small number of variables is that the â€˜omitted variablesâ€™ which have been

outâ€™ to yield the smaller system of equations may have non-constant parame

the smaller system will also have non-constant parameters.

To represent a series either as an infinite VAR or AR model, or an infin

MA model, is impossible in a finite data set. One approximates such models

a finite order. There are diagnostic tests (e.g. tests for serial correlation in the

Structural and Statistical Models Revisited

This section draws some comparisons between a structural economic mode

derived from economic theory (or theories) and the purely time series repr

discussed above. As we shall see a structural economic model can be re

purely time series representation. The transformation to a time series representat

implies restrictions on the parameters but a pure time series modeller would i

and merely estimate an unrestricted time series representation.

A macroeconomic model usually consists of a set of equations that may re

behaviour of economic agents, identities or technical identities or equilibrium

Shown below is a stylised model with three endogenous variables

ylt = rate of wage inflation (percent per annum)

y2t = rate of domestic price inflation in the UK (percent per annum)

= percentage deviation of output from its long-run trend (i.e. from

˜3˜

rateâ€™ or non-accelerating rate of inflation (NAIRU))

y3t is called â€˜cyclical outputâ€™ from now on. The exogenous variables in the m

xlr = percentage change in import prices (in domestic currency)

xzt = trades union power

= government expenditure

˜3˜

The first equation in the model is a wages version of the expectations augmen

curve. Wage inflation ylr is assumed to depend on price inflation y2t, cycl

y3r and trades union power xzt. The second equation models price inflation

a cost mark-up equation. Price inflation ˜2˜ depends on wage inflation yl,

price inflation x l t . (These two equations are used in the chapter on exchange

final equation expresses equilibrium in the goods market. Real output y3t in

real income rises and hence is positively related to wage increases ylr but it is

related to domestic price inflation yzr.Government expenditure ˜3˜ directly adds

and hence influences real output. The three-equation economic model is then

Lagged values have been included of all the basic variables of the system in eac

because it is probable that ylr,y2r and ˜3˜ react to changes in the RHS var

time, rather than instantaneously. The above three-equation system (20.50) is k

structural simultaneous equation system. The simultaneous aspect is due to th

one endogenous variable Yit depends on another endogenous variable Y j t at ti

â€˜identifiedâ€™. Space constraints prevent us from discussing the concept of iden

detail. However, unless the system is identified, estimates of the parameters ar

less since any linear combination of the three equations is equally valid statis

clearly an arbitrary linear combination would not in general conform to oneâ€™s

priors). We will assume that the three-equation system in (20.50) is identified

not a problem here.

Notice that in a three-equation system we can only solve (algebraically

unknowns (yl,, y2,, y3,) in terms of the â€˜knownsâ€™ (XI,, x2,, ˜ 3 ˜ The econo

).

a priori which of the variables are not determined within the model: these

exogenous variables. Finally, note that economic theory would usually imply

are explained by the RHS variables except for an additive white noise process

we can relax this assumption and assume the Ejr are ARMA processes and w

would still be valid.) The differences between the â€˜structural economic model

and the VARMA time series model are:

(i) the presence of current dated variables on the RHS of the structural m

and xir,

(ii) the presence of lagged exogenous variables x i t - j ( i = 1,2,3),

(iii) some â€˜exclusion restrictionsâ€™ on the variables in the structural model (i.e.

appear in equation (20.50)),

(iv) the economic model involves a structure that is determined by the econo

under consideration.

Taking up the last point we see, for example, that agents alter yl, only in

changes in y2,, y3r or x2, (and their lags) and not directly because of changes i

Economic theory would usually suggest that the behavioral parameters should

over time.

From a Structural Model to a V M A or VAR System

The structural model in (20.50) can be compactly written in matrix notation

+ A2 (L)Yr- + A3 (L)Xr- +

&Yt = A1X, 1 1 61

or

+ A2(L)Yt-i + A3(L)Xt-i +

Yt = &â€˜(AiX, 6,)

Equation (20.51), which is known as the â€˜final formâ€™, expresses Y,as a functio

values of Y,and current and lagged values of X,. We know that any covarianc

series may be given a purely statistical representation in the form of a VAR

If we apply this to X, we have

+ e(L)v,

X, = B(L)Xr-1

If X, is stationary all the roots of B(L) lie outside the unit circle and hence

X, = ( I - B(L))-â€™B(L)V,

a VARMA model of the form

where the final term is a moving average of the white noise errors E , and vt. He

VARMA statistical representation of the stationary series X,, any structural si

equations model can be represented as a VARMA model. It follows from

discussion that the structural model can be further reduced to the â€˜simpler

ARMA models outlined in the â€˜summaryâ€™ above.

In general, the fact that the Ai matrices of the structural model have restric

by economic theory (i.e. some elements are zero) often implies some rest

the derived VARMA equations. These restrictions can usually be tested. H

we ignore such restrictions then any VARMA model may be viewed as an u

representation of a structural economic model. Note that the VARMA model

will usually depend on lags of all the other yj,(i # j ) variables even though th

equation for Yjr might exclude a particular yjr.

Some modellers advocate starting with a structural model based on som

theory, which usually involves some a priori (yet testable) restrictions on the

One can then analyse this model with a variety of statistical tests. Others (e.g. S

feel that the apriori knowledge required to identify the structural model (e.g

restrictions) are so â€˜incredibleâ€™ that one should start with an unrestricted VAR

and simplify this model only on the basis of various statistical tests (e.g. exclus

tions and Granger causality tests, see below). These â€˜simplification restriction

would not be suggested by economic theory but one would simply trade off

parsimony on purely statistical grounds (e.g. by using the Akaike information

Expectations Variables Added

If we add an expectations variable for one particular variable, for example E,y

q = 1, or 2, or 3, etc.) to the structural economic model then we have additiona

of interpretation and identification (the latter are particularly problematic, e.g.

(1987)). However, since expectations are formed with information available a

earlier then it must be true that

We can therefore (assuming suitable transversality conditions hold, see Cuthb

Taylor (1987) and Pesaran (1987)) substitute out for E,y,+, in terms of c

past observable values of the variables in the system. Hence the structural m

expectations can be reduced to one without expectations and is of the ge

(20.51). In fact as we have noted, the assumption of RE usually implies some

restrictions on the Ai matrices of the structural model. In general, therefore, t

of expectations variables still allows us to express any set of time series v

as a multivariate VARMA, VAR or VMA model or as a univariate ARMA,

represent ation.

parameters. If either of these conditions does not hold then the resulting line

time series model for Yjr will be misspecified and have unstable parameters.

On the other hand suppose the world is linear but the apriori restrictions

sion restrictions) imposed on the structural model by the economic theorist a

â€˜in realityâ€™: in this case the VARMA representation may provide a superio

representation of the data.

The aim of the above is to point out the relationship between these

approaches to modelling time series. Both have acute potential problems, b

an enormous amount of judgement in deciding which approach is â€˜reasona

given circumstances. It is certainly this authorâ€™s view that economic theory ou

some role in this decision process but there are no clear-cut infallible â€˜rule

apply: â€˜beauty and truthâ€™ in applied economics are usually in the eye of the b

Stationarity and Non-Stationarity in Systems: Cointegration

This section deals with the issue of cointegration and discusses how non-statio

can yield â€˜spurious regressionsâ€™ and how one can test to see if individual v

non-stationary. Having found that a set of variables is non-stationary it is no

to outline how the Johansen procedure can be used to determine whether a

stationary series are â€˜linked togetherâ€™ in the long run, that is cointegrated.

relationship between the Johansen procedure, the Box Jenkins methodology, e

tion models and Granger causality is briefly discussed.

We have noted that any single series can be classified as stationary or non

We now consider possible relationships between a set of non-stationary var

series xr and yr might be â€˜highly trendedâ€™ because of a deterministic time trend

+

alt E r ) or because of a â€˜stochastic trendâ€™ (e.g. random walk with drift, yr = a

E t ) . Cointegration deals with data that have stochastic trends. In general two

stochastic trends will not be statistically related. For example, consider x, = PO

+ +

and y = a 0 yr-l Er where and vr are statistically independent (white n

r

In the â€˜trueâ€™ model there is no relationship between yr and x,. However, if w

on x, in a sample of data then standard statistics (e.g. R2, t statistics) will s

they are linearly related:

+ ilxr

y = 50

r

This is usually referred to as the â€˜spurious regressionâ€™ or â€˜nonsense regressio

(Granger and Newbold, 1974). The R2 and t statistics from such regr

misleading. The t statistics for Si are not distributed as a Studentâ€™s t distribution

be used for testing hypotheses on the parameters SO,&. The R2 is often bimod

and Newbold noted that in these â€˜spurious regressionsâ€™, R2 > DW (DW = Dur

statistic). The DW was â€˜lowâ€™ indicating positive serial correlation in the resid

Cointegration seeks to provide a correct method of estimating equations

a set of variables, some of which have stochastic trends. Such stochastic

frequently found in economic time series (e.g. stock prices, interest rates and

the random walk) are said to be integrated of order I , that is I(1).There are

tests available to ascertain whether an individual series is 1(1).We will only c

Dickey-Fuller (DF) and augmented DF tests. The AR(1) model for any time

+ crlYf-1 + Er

y , = a0

where we take E˜ to be a stationary white noise series. If a < 1 (we take a to 1

1

then y , is a stationary I ( 0 ) series. However, if a = 1 then y , is I(1) since it

1

differenced once to yield a stationary series:

Thus A y t is stationary given that is stationary. Rearranging (20.57) we ha

E,

where 8 = a - 1. If a1 < 1 then 8 < 0. Hence a test for stationarity is a tes

1

Dickey and Fuller (1979) show that the t statistic on 8 in the OLS regression

be used to test for 8 < 0. However, the critical value of the t statistic is not

Studentâ€™s t distribution and requires special tables of critical values. The DF c

for testing 8 < 0 is about 2.85 for reasonable sample sizes. Hence for y , to

require 8 < 0 and It1 > 2.85 where t = t statistic on 8. If y , is found to be I(

series is differenced once and the DF test applied to the A y , series to see if it

augmented Dickey -Fuller test includes additional lagged difference terms of

n

i= 1

to remove any serial correlation that may be present in E,. (A deterministic tim

also be included in (20.59).) Having ascertained that a set of variables are all in

the same order (we only consider I(1)series) then we can proceed to see if the

move together in the long run (i.e. have common stochastic trends). We consi

variable case first.

In general a linear combination of I(1) series that is q, = y , - Pâ€™x, is al

therefore non-stationary. However, it is possible that the linear combination q, i

and in this case y and x are said to be cointegrated with a cointegration para

q, is a stationary I ( 0 ) variable then we can say that the stochastic trend in y , is

byâ€™ the stochastic trend in Pâ€™x,. Hence the two series move together over tim

q, between them is finite and the gap doesnâ€™t grow larger over time.

If we have two variables which are cointegrated then the cointegrating vecto

+

However, for any Y 1 variables there can be up to Y unique cointegrating v

illustrative purposes let us consider a three-variable system, where y l , , y2,

I(1).Let us suppose that there are two unique cointegrating vectors, which w

of generality we normalise on y l , and y2,. The Engle and Granger (1987) rep

theorem states that cointegration implies that there exists a statistical represent

+ (terms in lagged Aylr-j, Ay2r-jV Ay31-j) + ˜3˜

The interesting features of the ECM are

(i) The two cointegrating vectors may, in principle, appear in all the equati

(3 x 3) system. The cointegrating parameters are 81 and 82.

(ii) All the variables in the error correction system are stationary I ( 0 ) var

yir (i = 1, 2, 3) are I(1) by assumption, hence Ayir must be I(0). T

(ylr-l - 81y3r-1) and (˜2r-1- 8 2 ˜ 3 ˜ - 1are stationary because the I(1)

)

cointegrated.

In fact (20.60) is nothing more than VAR where the non-stationary I(1) var

been â€˜transformedâ€™ into stationary series (i.e. into difference terms or co

vectors) so that a VAR representation is permissible. The error terms Eif(i =

given by:

Eir = (Ayit - â€˜linear combination of stationary variablesâ€™)

and hence are stationary. Since the error terms are stationary the usual statistic

be applied to the aij parameters of a VAR model in error correction form.

Before cointegration came on the scene the so-called Box- Jenkins metho

been used in analysing statistical time series models. This methodology im

any non-stationary I(1) series be differenced before estimating and testing th

VARMA) model. Hence a VAR model which contains only the first differences

variables is used in the Box-Jenkins methodology. This ensured that the error t

equation were stationary and hence conformed to standard distribution theory

statistical tests using â€˜standard tablesâ€™ of critical values could then be used

cointegration analysis indicates that a VAR solely in first differences is miss

there are some cointegrating vectors present among the I(1) series. Put ano

VAR solely in first differences omits potentially important stationary variab

error correction, cointegrating vectors) and hence parameter estimates may

omitted variables bias. How acute the omitted variables bias might be depe

correlation between the included â€˜differenced onlyâ€™ terms of the Box- Jenkin

the omitted cointegration variables (yjt - Sjyjr). If these correlations are low

the omitted variables bias is likely to be low (high).

The parameters of the error correction model (20.60) can be estimated jo

the Johansen procedure. This procedure also allows one to test for the numbe

cointegrating vectors in the system which involves non-standard critical valu

done for interest rates in Chapter 14.) One may be able to simplify the error

system by testing to see if any of the weights on the error correction terms

system. For any variables that are 1(1) one can apply the Johansen procedure

provide a set of unique cointegrating vectors (if any) and these variables are th

in the structural model in the form (yjt-1 - 8;zr-1).

Since the ECM is a VAR involving lags of stationary variables it can, like

be transformed to give the following alternative representations. First, a â€˜sm

correction system (e.g. 3 x 3 to a 2 x 2 system), or a VMA system. Also it can

to a â€˜single-equationâ€™ ARMA, AR or MA model in exactly the same way a

above for the VAR with stationary variables. It follows that the â€˜dangersâ€™, pa

terms of parameter stability of the resulting representations, will again be of k

In general the choice between a â€˜largeâ€™ VAR/ECM or a â€˜smallerâ€™ system

off between efficiency and bias. The larger system is less likely to suffer fr

variables bias but the parameters may not be very precisely estimated (sinc

â€˜degrees of freedomâ€™ as the number of parameters to be estimated increases

the fixed sample of data). On the other hand, excluding some variables (i.e. pa

starting with a very â€˜smallâ€™ system may increase the precision of the estimated

but at a cost in terms of potential omitted variables bias and perhaps a loss of

accuracy. With any real world finite data set one can apply a wide variety

guide oneâ€™s choice but ultimately a great deal of judgement is required.

Granger Causality

There is one widely used and simple test on a VAR which enables a more pa

representation of the data. To illustrate this consider a (3 x 3) VAR system in th

variables ylr, y2r, ˜ 3 The . equation for ylr is:

˜

where sufficient lags have been included to ensure that E l r is white noise. One

to test the proposition that lags of ˜2˜ taken together have no direct effect o

restriction that all the coefficients in 82(L) are zero is not rejected then we ca

that y2 does not Granger cause yl. The terms in y2 can then be omitted from t

which explains yl. A similar Granger causality test can be done for y3 in equat

We can also apply Granger causality tests for the equations with and y3 a

variables. Granger causality tests are often referred to as â€˜block exogeneity

that Granger causality is a purely statistical view of causality: it simply tests t

of yjf have incremental explanatory power for Yir ( i # j ) .

In the VAR/ECM, lags of y also appear in the error correction term so th

must also include the â€˜weightsâ€™ ajj in the Granger causality test. Otherwise t

is the same as in the pure VAR case described above.

A good example of where economic theory suggests a test for Granger

the term structure of interest rates. Here the expectations hypothesis of the ter

suggests that the long rate Rf is directly linked to future short-term interes

( j = 1 , 2 , 3 , .. ., etc). Hence the theory would imply that the long rate shou

cause short rates. Similarly, according to the fundamental valuation equatio

The main conclusions of this section are as follows:

(i) A structural economic model can be represented as a purely vector

model and the latter can be reduced to a univariate model. In using V

the issue of the temporal stability of the parameter is of key importanc

(ii) Cointegration deals with the relationships between non-stationary I(1) v

a set of variables are cointegrated then any cointegrating vectors should

in the VAR representation. Hence the VAR system purely in first differen

variables may be misspecified.

(iii) The VAR cointegration framework has been extensively used in testing

EMH holds for speculative asset prices.

2 . SIMPLE ARCH AND GARCH MODELS

03

This section outlines the basis of models of autoregressive conditional hetero

(ARCH) and the generalised ARCH (or GARCH) approach in modelling ti

risk premia. The emphasis is on the intuitive economic reasons for using th

rather than the details of the estimation procedures and algorithms. We begin

simple ARCH model and then build up to a generalised ARCH in mean mode

conditional variances and covariances.

Simple ARCH Model

The basic idea behind ARCH models is that the second moments of the distri

have an autoregressive structure. Consider an asset return model where we

expected excess return E,y,+l is constant

=q

&Yf+l

Now assume RE

+ Ef+l

Yf+l = q

where E f + l = yf+l - Efyf+lis the RE forecast error. Many asset markets are ch

by periods of â€˜turbulence and tranquillityâ€™, that is to say large (small) forecas

whatever sign) tend to be followed by further large errors (small) errors. There

persistence in the variance of the forecast errors. The simplest formulation of

+

2

a,+, = var(Ef+11S2,)= w (;YE;

If a < 1 the unconditional variance of denoted a2,is given by

Er,

a* = w / ( l - a)

and is a constant. However, the conditional variance given by (20.64) varie

and E; can be used to predict the variance next period af+l.

+ (20.64) the likelihood may be expressed as:

In terms of (20.63)

ñòð. 47 |