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+("4yt-1'I2- w2]

[

1

1

+a ( y t - l - -2

1 = -- ln [w -

w

2 t

The log-likelihood can therefore be expressed as a non-linear function of th

parameters w, a and p and the data series y t . Standard optimisation routine

be used to maximise the likelihood. In particular, one must ensure that the ti

variance is always positive. In this case this is easily done by replacing w and a

by w2 and a2 which are always positive and hence a in (20.64) will always

:

+,

GARCH Models

The ARCH process in (20.64) has a memory of only one period. We could gen

process by adding lags of E ; - ˜ :

+ a,&;+ +

2

at+,= w a2E;-, ***

but the number of parameters to estimate increases rapidly. A more parsimonio

of introducing a 'long memory' is the GARCH(1,l) process

+ a&;+ pa:

at+l= w

2

The GARCH process has the same structure as an AR(1) model for the error t

the AR(1) process applies to the variance. The unconditional variance den

constant) is:

+

a2 = w/[l - (a p)]

+

for (a p ) < 1. However, (20.69) can be rearranged to give

+ a&;

(1 - pL)a:+l = w

+l

= ˜ ( /3 + /!I2 + + a(l + BL + (pL)2+

.) .)E;

** **

Hence a can be viewed as an infinite weighted average of all past squar

:

+

,

errors. The weights on E;-, in (20.71) are constrained to be geometrically decl

a little manipulation (20.71) may also be rewritten

+

(a,,, - a2 ) = a(&;- a*) / ( ;- a 2 )

2

?a

Using either (20.69) or (20.72) we can take expectations of both sides and

Et$ = a we have (for equation (20.69))

:

+ (a+ p>a:

=w

0;+1

+ +

values of 0 f + j depends on the size of a /?.If a /? is close to unity then

+

time t will persist for many future periods. For a /? = 1 then any shock w

permanent change in all future values of hence shocks to the conditional v

+

â€˜persistentâ€™. For a /I = 1 we have what is known as an integrated GARCH p

IGARCH). For IGARCH the conditional variance is non-stationary and the un

variance is unbounded. The statistical/distribution properties of IGARCH pr

currently the focus of much research in this literature.

One can generalise (20.69) to a GARCH(p, q) process where there are p

and q lags of 0;.This allows the conditional variance to have an infinitely lo

(because of the c ˜ ; - ˜terms) but doesnâ€™t constraint the response of a,:+j to a

to have geometrically declining weights. Given the acute non-linearities of th

function and loss of degrees of freedom as the number of parameters in th

process increases, researchers have often used fairly low-order GARCH( p , q)

these have been found to fit the time varying volatility in the forecast erro

prices fairly well.

The likelihood function for (20.63) plus the GARCH(1,l) model is of the fo

However, an arbitrary starting value for 0; is required to generate the terms in

likelihood function. ˜1 is given by (y1 - Q) and this together with a starting v

+ +PO:. Values of Ef+l = y,+l - \I, for

be used to generate 0 = w :

+ +

= w a l ˜ ; /?c$(t = 2 , 3 , . . .) by recursive

can be used to generate

for 0;.

ARCH-M and GARCH-M Models

Suppose we now extend our economic model of asset pricing so that the exp

depends positively on the perceived riskiness of the asset or portfolio. Supp

riskiness can be adequately represented by the (conditional) â€˜ownâ€™ variance of

errors of returns o:+˜.This model of expected returns can be represented by

in meanâ€™ equation:

Yf+l = Qo + Q10?+1 + Ef+1

+

Thus the expected return Efyf+lis given by (Qo Q ˜ O ; + ˜ where is t

)

IQ,)

(conditional) variance (strictly we should write this as E(CT:+˜ but do no

notational ease). The forecast error squared is

2

= (Yf+l - Qo - QlCJf+l)

&;+I

We assume that there is persistence in the variance of the conditional forecast e

is either ARCH( p) or GARCH( 1 , l )

P

+ CaiE:-j

=w

ARCH

i=O

+

af+l = w +

GARCH (;YE: /?U:

that are

variables zlr in the expected return equation or additional variables

influence the conditional variance. For a GARCH(1,l) process this would co

following two equations:

+

+ *10:+1+

yr+l = q o ˜ 2 â€˜ ˜ l rEr+l

+ a&;+ po; + yâ€™zzr

=w

0;+1

For example, the APT suggests that the variables in zlr could be macroeconom

such as the growth in output or the rate of inflation. Often the dividend price ra

to influence returns and this might also be included in (20.78). Economic th

particularly informative about the variables ˜2˜ that might influence investorsâ€™

of future volatility but clearly if volatility in prices is associated with new

arriving in the market then a market turnover variable might be included in zz

variable for days of the week when the market is open (i.e. zzr = 1 for m

and zero otherwise) can also be incorporated in the GARCH equation. It is a

that elements of ˜2˜ also appear in the equation for yr+l (and a dummy variab

returns and the weekend effect are obvious candidates). So far we have assum

forecast errors Er+l are normally distributed, so that:

- â€œqâ€™xr,o;+,)

Yr+lIQr

where the variables that influence yr+l are subsumed in x,. The conditional mea

q â€™ x r and the conditional variance is given by o?+˜.However, we can assume a

tion we like for the conditional moments. For example, for daily data on stoc

changes in exchange rates (i.e. the yr+l variable) the conditional distribution oft

is the Studentâ€™s t distribution which has slightly fatter tails than the normal d

The likelihood is more complex algebraically than that given in equation (20

normal distribution but the principle behind the estimation remains unchanged

in q?and Er in the (new) likelihood equation are replaced by (non-linear) func

â€˜observablesâ€™ yr,xr and the unknown parameters Q and (w, a,p) of the GARC

The likelihood can then be estimated in the usual fashion (e.g. see the m

options in the GAUSS, LIMDEP or RATS programmes) which often involve

(rather than analytic) techniques.

Although the unconditional distributions for many financial variables

changes in stock prices, interest rates or exchange rates) may be leptokur

be that the conditional distribution based on a model like the GARCH-M

+

(20.78) (20.79) may yield a distribution that is not leptokurtic.

Covariances and ARCH Models

To motivate the application of ARCH models to incorporate covariances c

excess return ylr+l on a particular portfolio (e.g. the return on a portfolio of

in various chemical firms). The CAPM plus RE implies that

Hence the return on the market portfolio is determined by the conditional vari

*:

market portfolio 0 The term should equal zero and is the market pr

.

:

(Note that appears in both (20.81) and (20.82) according to portfolio th

shocks that influence the return on the market portfolio are also likely to in

return on asset 1. For example, this is because â€˜good newsâ€™ about the econom

an effect on the return on the stocks in portfolio 1 as well as all other stocks in

portfolio. Hence E l r and E,, are likely to be correlated (i.e. their covariance is

If we assume a GARCH( 1â€™1) process for the covariance we have

There will also be an equation of the form (20.83) to explain the conditiona

for both o:r+l and o:r+l(see equation (20.79)). The parameters a and /I in t

covariance equation and in the two GARCH variance equations need not b

+

Hence the degree of persistence (i.e. the value of a /I) can be different in ea

specification and generally speaking oneâ€™s â€˜instinctâ€™ would be that the degre

B)

tence might well be different in each process. In practice, however, the (a,

are often assumed to be the same in each GARCH equation because otherwis

cult to get precise estimates of the parameters from the likelihood function whi

non-linear in the parameters.

Clearly the GARCH-M model with variances and covariances involves a (2

of error terms ( E l r , E m r ) and the likelihood function in (20.66) is no longer

However, the principle is the same. Given starting values at t = 0 for elm,

the GARCH equations can be used recursively to generate values for these

t = 1 , 2 . . . in terms of the unknown parameters. Similarly equations (20.81)

provide a series for ˜ 1 and Emt to input into the likelihood, which is then

,

using numerical methods.

Summary

ARCH and GARCH models provide a fairly flexible method of modelling ti

conditional variances and covariances. Such models assume that investorsâ€™

of risk tomorrow depend on what their perception of risk has been in earl

ARCH models are therefore autoregressive in the second moment of the d

An â€˜ARCH (or GARCH) in meanâ€™ model assumes that expected returns d

investorsâ€™ perceptions of risk. A higher level of risk requires a higher level

returns. ARCH models allow this risk premium to vary over time and henc

equilibrium returns also vary over time.

20.4 RATIONAL EXPECTATIONS: ESTIMATION ISS

Over the last 10 years the role of expectations formation in both theoretical

financial economics has been of central importance. At the applied level rel

pure expectations hypothesis of the term structure, the long rate on bonds

expectations about future short-term interest rates. The current price of stocks

expected future dividends and under risk neutrality the current forward rate is

predictor of the expected future spot rate of exchange.

In general the efficient markets literature is concerned with the proposition

use all available information to remove any known profitable opportunities in

and this usually involves agents forming expectations about future events.

propositions outlined above we need a framework for modelling these un

expectations.

The literature on estimating expectations models is vast and can quickly b

complex. An attempt has been made to explain only the main (limited i

methods currently in use. We shall concentrate only on those problems int

expectations variables and will not analyse other econometric problems that

arise (e.g. simultaneous equations problems). The reader is referred to intermed

metrics text books for the basic estimation methods (e.g. OLS,IV, 2SLS, GL

analysing time series data (e.g. Cuthbertson et a1 (1992) and Greene (1990)).

The rational expectations (RE) hypothesis has featured widely in the literat

begin by discussing the basic axioms of RE which are crucial in choosing an

estimation procedure. We also examine equations that contain multiperiod ex

In the next section, we discuss the widely used â€˜errors in variablesâ€™ method

estimating structural equations under the assumption that agents have ration

tions. The use of auxiliary equations (e.g. extrapolative predictions) to generat

proxy variable for the unobservable expectations series give rise to two-step

and the pitfalls involved in such an approach are also examined.

Problems which arise when the structural expectations equation has serially

errors will then be highlighted. The Generalised Method of Moments (GMM

of Hansen (1982) and Hansen and Hodrick (1980) and the Wo-Step Wo-S

Squares estimator (Cumby et al, 1983) provide solutions to this problem.

20.4.1 The Economics of Expectations Models and the RE Hypothesis

This section analyses the various ways in which expectations variables are uti

applied literature and the implications of the economic assumptions for the

issues discussed in a later section.

Usually the applied economist is interested in estimating the structural pa

a single equation or set of equations containing expectations terms which form

of a larger model. (In a â€˜fullâ€™ Muth-RE model (Muth, 1961) we would have

the whole model.) The simplest structural expectations equation can be repre

where:

of exchange is an unbiased estimate of the future spot rate then ylr = fr

the expected future spot rate. In the absence of data on Ex,+, (e.g. quantita

data) we must posit an auxiliary hypothesis for Ex,+,. Whatever expectations

choose, of key importance for the econometrics of the model are

(i) the forecast horizon.

(ii) the dating and content of the information set used in making the foreca

(iii) the relationship between the forecast error and the information set.

To develop these issues further it is useful to discuss the basic axioms of RE

Basic Axioms of RE

If agents have RE they act as if they know the structure of the complete mod

a set of white noise errors (i.e. the axiom of correct specification). Forecasts a

on average, with constant variance and successive (one-step ahead) forecas

uncorrelated with each other and with the information set used in making t

Thus, the relationship between outturn x,+l and the one-step ahead RE fo

using the complete information set 52, (or a subset A,) is:

+

= r$+1

&+l W,+l

where

The one-step ahead rational expectations forecast error @,+I is â€˜white noi

â€˜innovationâ€™, conditional on the complete information set 52, and is orthogona

of the complete information set (A, c 52,).

The k-step RE forecast errors (k > 1) are serially correlated and are MA

demonstrate this in a simple case assume x, is AR(1).

From (20.88) it is easy to see that:

while the two-period ahead forecast error is:

are independent of (orthogonal to) the information set n, (or At). There is

property of RE that is useful in analysing RE estimators and that is the form

to expectations. The one-period revision to expectations

+ 1 and hence from

depends only on new information arriving between t and t

easily seen to be

[t+lxF+j- tx;+j] = @-'wr+l

The two-period revision to expectations

will of course depend on ot+l and and be MA(1): one can generalise th

0t+2

k-period revisions to expectations.

Direct Tests of RE

Direct tests of the basic axioms of RE may involve multiperiod expectatio

immediately raises estimation problems. For example, if monthly quantitative

is available on the one-year ahead expectation, txf+12, test of the axioms oft

a

a regression of the form:

PO+ Pl[tx;+121+ + qr

xr+12 = P2Ar

where

H o : PO = 82 = 0, =1

Under the null, qr+12 is MA(11) and an immediate problem due to RE is the

some kind of Generalised Least Squares (GLS) estimator if efficiency is to be a

course, for one-period ahead expectations where data of the same frequency

the error term is white noise and independent of the regressors in (20.93): OL

provides a best linear unbiased estimator.

An additional problem arises if the survey data on expectations is assu

measured with error. If the true RE expectation is txf+12 and a survey data

measure r.?f+12 where we assume a simple linear measurement model (Pesara

+ Er

+

= a0 Ql[rx;+121

tz;+12

Then substituting for r ˜ f + 1 2from (20.94) in (20.93):

where

able instrument set. However, it is not always simply the case that A t prov

instrument set for the problem at hand.

The E VM and Extrapolative Predictors

In order to motivate our discussion of the estimation problems in the next two

is useful at this stage to summarise some of the problems encountered when e

structural expectations model: problems that arise include serial correlation and

between regressors and the error term. For illustrative purposes assume th

model of interest is:

+ 82[rX;+21+

Yr = 61[rn;+,l ur

ur is taken to be white noise and xr is an exogenous expectations variable

assumption of RE we have

+ wt+j

= tx;+j

xr+j

A method of estimation widely used (and one of the main ones discussed in t

is the errors in variables method (EVM), where we replace the unobservable

realised value xr+,. This method is consistent with agents being Muth rationa

also be taken as a condition of the relationship between outturn and forec

invoking Muth-RE. Substituting from (20.97) in (20.96)

Clearly from (20.97)xt+ j and wt+j are correlated and hence plim [xi+j & r ]/T #

#

and E(&&â€™) 0:1 because of the moving average error introduced by the

errors Hence our RE model requires some form of instrumental variable

procedure with a correction for serial correlation. These two general problems

focus for this section.

20.4.2 The Errors in Variables Method EVM

The EVM is a form of IV or 2SLS approach. Under RE, the unobservable

variable rxf+j is determined by the full relevant information set 52,. In the EVM

the true information set Ar(C Q r )is sufficient to generate consistent estimate

first it is shown that OLS yields an inconsistent estimator.

One-Period Ahead Expectations: White Noise Structural Error

It is important to note that here we are dealing with a very specific expecta

The simplest structural model embodying one-period ahead expectations is:

+ ur

Yr = Bx,â€™,,

at (or A

and the RE forecast error wr+l is independent of the information set

E(Q:w+l) = 0

Substituting (20.101) in (20.99) we obtain

+ 4t

Yr = b r + l

4 = (U1 - B W r + l )

1

Consider applying OLS to (20.103) we have:

B = B + (xr+lâ€™xr+l )-l (xr+1â€™4r)

From (20.101):

+ plim(w,+iâ€™wt+l ) / T

plim(˜r+lâ€™xt+l/ T = plim($+lâ€™xf+l ) / T

)

on rewriting this more succinctly:

+ aw

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