<< . .

. 48
( : 51)



. . >>





+("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
+
˜persistent™. 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 doesn™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) ˜own™ variance of
errors of returns o:+˜.This model of expected returns can be represented by
in mean™ 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 ˜ ˜ l rEr+l

+ a&;+ po; + y™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 investors™
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:
- “q™xr,o;+,)
Yr+lIQr

where the variables that influence yr+l are subsumed in x,. The conditional mea
q ™ 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 Student™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
˜observables™ 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 ˜good news™ 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( 1™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 one™s ˜instinct™ 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 investors™
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 ˜ARCH (or GARCH) in mean™ model assumes that expected returns d
investors™ 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 ˜errors in variables™ 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 ˜full™ 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 ˜white noi
˜innovation™, 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(&&™) 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,™,,
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+l™xr+l )-l (xr+1™4r)
From (20.101):

+ plim(w,+i™wt+l ) / T
plim(˜r+l™xt+l/ T = plim($+l™xf+l ) / T
)
on rewriting this more succinctly:

+ aw

<< . .

. 48
( : 51)



. . >>