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are usually restricted in some way (e.g. using low order rather than high orde
The theoretical model of the bond market outlined below incorporates bonds
maturities and also allows for time varying risk premia. The basic model used is
and hence time varying premia are modelled via time varying covariances. In
covariances are modelled by an ARCH or GARCH process. For a bond of
+
the CAPM RE implies that the excess (HPY) yield y!:\ = [H,':', - r,] is gi



where h is the market price of risk. For the marketportfolio the excess HPY is p
to the market price of risk and the variance of the excess yield on the marke



-
where we assume ˜ 1 " + ˜N ( 0 , o;,,,). From (19.25) the conditional variance of
excess HPY, var(y'")Q,), is equal to E , [ U : ˜ ] *= E , o ˜ , + Hence (19.25) can be
˜.



and from (19.25) [y!:! - Ey,'?!] = u z 1 .
From (19.24) [y!:: - Ey:$] =
conditional covariance of Hi:', and RF+l is given by cov(Hj[ti, R;,) = E,[u
0 " We can therefore rewrite (19.24) as:
;.
requires expressions for the variance and covariances of the error terms. A G
model implies that agents forecast future variances/covariances (i.e. ˜risk™) o
of past variances and covariances. For an n-period bond its own variance (
covariance with the market a may be represented by the following two G
:,
equations:
+ a10,2,+ a2(&:I2
Gn,+l= a0
2




The GARCH variance for the market forecast errors 0; is



The above equations are just a replication of the GARCH equations examined
Each equation implies that the conditional variance (or covariance) is a weigh
of the previous period™s conditional variance (covariance) and last period™s act
error (or covariance of forecast errors with the market). The degree to whic
believe that a period of turbulence will persist in the future is given by the size
+ +
(or 81 8 2 , etc.). If a1 a = 0 then the variance G,˜ = a and it is not ti
2 0
:+
If a + a = 1 then a forecast error at time t, E: will influence the investor™s
2
1
+
of the amount of risk in all future periods. If 0 < (a1 a2) < 1 then shocks
only influence perceptions of future risk, for a finite number of future period
The illustrative empirical application of the above model is provided by
(1992) who estimate the above model for bonds of several maturities and for
different countries. For each country, they assume that the market portfolio c
of domestic bonds (i.e. does not contain any foreign bonds) and HPYs are me
one month. For the marketportfolio of bonds there is evidence of a GARC
y1, y2 are non-zero. However, the conditional variance a $ 1 does not appear
#+
statistically at least, much of the variation in the excess market yields E , R z
in equation (19.26) A is only statistically different from zero for Japan and
we can accept A = 0 for the UK, Canada, the USA and Germany). In genera
also find that the covariance terms do not influence HPYs on bonds of mat
n = 1-3, 3-5, 5-7, 7-9, 10-15, and greater than 15 years) in a number of co
the UK, Japan, the USA and Germany).
Hall et a1 also test to see whether information at time t , namely lagged e
(H - r)t-j and the yield spread [R!“™ - r,], influence the excess yield E,Y:+,
(19.28). This is a test of whether the CAPM with time varying risk premia to
RE provides a complete explanation of excess yields. In addition, Hall et a1
conditional (own) variance of the n-period bond a alongside the covarianc
:
in equation (19.28). If the CAPM is correct the own-variance a should n
:
the HPY.
In the majority of cases they reject the hypothesis that the lagged HPY a
spread are statistically significant. Hence Mankiw™s (1986) results where the
rejecting the CAPM.
There are many potential candidates to explain the rather mixed results acros
and countries found in the Hall et a1 study. Candidates include:

(i) an approximation to HPYs is used,
(ii) holding periods other than one month are not investigated and the on
˜static™ CAPM is not invariant to the choice of the length of the holdin
the market portfolio is taken to be all domestic bonds (e.g. no foreign b
(iii)
equities are included),
(iv) there are a large number of parameters to estimate from a relatively l
set and therefore Hall et a1 assume that the parameters in the differe
equations for variances and covariances are equal (i.e. a;= 6; = y,, fo
This saves on degrees of freedom and mitigates possible difficulties in
a highly non-linear likelihood function but may impose invalid restrictio
even when these restrictions are imposed the likelihood function may
determined (i.e. may be rather ˜flat™).

Despite some restrictive assumptions in the Hall et a1 study it does sugges
varying risk premia (which exhibit persistence) do exist in bond markets b
rather difficult to pin down empirically and the effects are not uniform acro
different maturities and different countries. The study does highlight the di
using the ARCH procedure to obtain a tractable model of the CAPM with ti
variances and covariances. The number of parameters to be estimated and the n
of the maximisation procedure on a limited (and perhaps somewhat poor) dat
that any results are likely to involve wide margins™of error and they may not
robust to slight specification changes.


19.3 INTERACTION BETWEEN STOCK AND BOND MA
The CAPM determines the expected returns on each and every asset if agents
them in their ˜equilibrium™ portfolio. In the CAPM, all agents hold the market
all assets in proportion to market value weights. However, we know that most
diversification may be obtained by holding around 20 assets. Given transactions
of collecting and monitoring information, as well as the need to hedge project
of cash against maturing assets, then holding a subset of the market portfolio m
for some individuals and institutions. An investor might therefore focus his a
the choice between blocks of assets and be relatively unconcerned about the
of assets within a particular ˜block™. Thus the investor might focus his deci
returns from holding a ˜block™ of six-month bills, a ˜block™ of long bonds an
of stocks. Over a three-month holding period the return on the above assets i
The above simplification is used by Bollerslev et a1 (1988) in simultaneously
the HPY on these three broad classes of asset using the CAPM. They also util
+ w 2 cov[H˜”, H ˜ 2 ™ ] + w3 cov[H˜”,
= w1 cov[H˜”H“™]

+ W2012r + W3013r
= W l a lI f
where wi = value weight of each asset in the market portfolio. Thus we ca
˜single™ CAPM covariance term on the LHS of (19.32) into the above three c
The estimating equations for the excess HPYs on the three assets yit+l(i = 1 ,
the form:
r3 1


where the wj are known and h is the market price of risk, which according to
should be the same across all assets. Bollerslev et a1 estimate three excess HPY
of the form (19.32): one for six-month bills, another for 20-year bonds, and
stock market index. They model the three time varying variance and covari
using a GARCH(1,l) model (but with no restrictions on the parameters of th
procedure) of the form:
+
+
a i j r + l = aij Bijaijr YijEirEjr

The broad thrust of the results are:
(i) The excess holding period yield on the three assets does depend on the ti
conditional covariances since h = 0.499 (se = 0.16).
(ii) Conditional variances and covariances are time varying and are adequatel
using the GARCH(1,l) model. Persistence in conditional variance is
bills, then for bonds and finally for stocks (a1+a;!equals 0.91, 0.62
respectively).
(iii) Although the CAPM does not imply that conditional covariances sho
all of the movement in ex-post excess HPYs (because the arrival of ˜n
a divergence between yt+l and E,y,+l) one can see that actual HPYs
more than the expected HPY given by the prediction from (19.32) (see F
for bills (the graphs for bonds and stocks are qualitatively similar).

The CAPM with GARCH time varying volatilities does reasonably well em
explaining HPYs and when the own variance from the GARCH equations
(19.32) it is statistically insignificant (as one would expect under the CAPM)
the CAPM does not provide a ˜complete™ explanation of excess HPYs. For exa
the lagged excess HPY, for asset i, is added to the CAPM equation for asset i
to be statistically highly significant - this rejects the simple static CAPM f
Also when a ˜surprise consumption™ variable is added to the CAPM equa
usually statistically significant. This indicates rejection of the simple one-fac
and suggests the consumption CAPM might have additional explanatory pow
I
I I 1 I I 1
I
1959 1963 1967 1971 1975 1979 1983 1987 1991

Figure 19.1 Risk Premia for Bills. Source: Bollerslev et a1 (1988). ˜A Capital Asset P
with Time Varying Covariances™, Journal of Political Economy, 96( l), Fig. 1, pp.
Reproduced by permission of University of Chicago Press

The Bollerslev et a1 model of expected returns in equation (19.32) is very
that of Thomas and Wickens (1993) described in Chapter 18. The difference b
two approaches is that Thomas and Wickens consider a wider set of assets (e.g
foreign bonds and stocks), they allow the shares w; to vary and they test (and
additional restriction that the elements 0 ; j are equal to the variance-covarianc
error terms. However, the less restrictive model of Bollerslev et a1 does app
better empirically.

19.4 SUMMARY
There appears to be only very weak evidence of a time varying term premium
term zero coupon bonds (bills). As the variability in the price of bills is in m
smaller than that of long-term bonds (or equities) this result is perhaps not too
When the short-term bill markets experience severe volatility (e.g. USA, 1979-
the evidence of persistence in time varying term premia and the impact of the
variance on expected return is much stronger.
Holding period yields on long-term bonds do seem to be influenced by ti
conditional variances and covariances but the stability of such relationship
to question. While ARCH and GARCH models provide a useful statistica
modelling time varying second moments the complexity of some of the param
is such that precise parameter estimates are often not obtained in empirical wo
studies the conditional second moments appear to be highly persistent. If, in ad
are thought to affect required returns then shocks to variances and hence to
can have a strong impact on bond prices. However, such results are sometim
be sensitive to specification changes in the ARCH process.
It was noted in Chapter 17 that generally speaking the above points als
empirical work on stock returns. However, there is perhaps somewhat stronge
ENDNOTES
This is most easily demonstrated assuming continuously compounded
fixed maturity value M on a zero coupon bond we have:
In Pj3) = In M - ( 1/4)rf
In pj6)= In M - ( 1 / 2 ) ˜ ,




The excess HPY is:
Hf+13 - (1/4)rr
and substituting from (3) in (4) we see that the expected excess HPY
2R, - Et˜r+13 r, as in the text.
-

Given that for continuously compounded rates on zero coupon bond
1nM - nR:")the absolute change in interest rates on an n-period bond is p
to the percentage change in the price of the bond.


FURTHER READING
The recent literature in this area tends to be very technical, covering a wid
ARCWGARCH-type models as well as non-parametric and stochastic volatil
Cuthbertson et a1 (1992) and Mills (1993) provide brief overviews of the us
and GARCH models in finance. Overviews of the econometric issues are giv
and Schwert (1990), Bollerslev et a1 (1992), with Bollerslev (1986) and E
being the most accessible of these four sources.
PART 7
1
I
Econometric Issues in Testing As
Pricing Models

This section of the book presents a brief overview of the key concepts
econometric analysis of time series data on financial variables. Since these c
techniques are widely used in the finance literature dealing with discrete time
material is included in order to make the book as self-contained as possible. Th
of these topics is fairly brief and concentrates on the use to which these tech
be put rather than detailed proofs. Nevertheless, it should provide a concise i
to this complex subject matter.
First, an analysis is given of univariate time series covering topics such as
sive and moving average representations, stationarity and non-stationary, cond
unconditional forecasts and the distinction between deterministic and stocha
Then the extension of these ideas in a multivariate framework (Section 20.2) a
tionship between structural economic models, VARs and the literature on co
and error correction models are considered. Section 20.3 presents the basic id
ARCH and GARCH models and their use in modelling time varying variances
ances. Attention will not be given to detailed estimation issues of ARCH mod
see Bollerslev (1986)) but on the economic interpretation of these models. S
outlines some basic issues in estimating models which invoke the rational e
assumption, a key hypothesis in many of the tests reported in the rest of
Again the aim is not a definitive account of these active research areas but to
overview which will enable the reader to understand the basis of the empir
reported elsewhere in the book.
This page intentionally left blank
L Economic and Statistical Mode
20.1 UNIVARIATE TIME SERIES
An ˜economic model™ can be defined as one that has some basis in econo
Economic theory usually yields ˜static equilibrium™ or ˜long-run™ relatio
example, if purchasing power parity (PPP) holds then the log of the nomina
rate y is linked to the price of domestic goods relative to the price of for
xr = In Pr - In Pr. If we assume instantaneous adjustment we have:

+ B2xr + Er
Y f = B1
where Er is a random error term which is often taken to be white noise (see bel
PPP, we expect 8 2 = 1. However, it may be possible to obtain a ˜good™ repre
the behaviour of a variable yr without recourse to any economic theory. For
purely statistical model of the exchange rate yr is the univariate autoregressiv
order 1, that is AR(1):
+ +
Yr = (2 BYr-1 Er

It may be the case that some economic theory is consistent with equation (2
purely statistical or time series modeller would not be concerned or probably n
of this. Clearly then the ˜statistical modeller™ and ˜economic modeller™ might e
the same statistical representation of the data. However, their motivation and c
whether the representation is ˜adequate™ may well be different. Both the statist
economic modellers require that their models adequately characterise the data
the economic modeller will also generally require that his model is in confo
some economic theory.
Whether one is a statistical modeller or an economic modeller it is useful to
data one is using in various ways. To set the ball rolling consider the followin
autoregressive model of order 1, AR(l), for y f :
+ Er
yr = Byr-1
where Er is a zero-mean random variable with constant variance 02, Er is u and
with any other variable in the sequence { E r - j , j = f l , f 2 , . . .}
EEr = 0 Vt
var(Er) = E(&:) = o2 Vt

V j # 0 and t
=0
C O V ( E ˜ ,E 1 - j )
We can represent (20.3) as:
(1 - /3L)Yf = El

y,-m and (1 - /3L) is said to be a poly
L is the lag operator, such that L"y,
order 1) in the lag operator. A equivalent representation of (20.5) is:
n




yr is therefore an infinite geometrically weighted average of the error term E
also have obtained (20.6˜) repeated 'back substitution':
by

+Er-1) +
Yr = B(Byr-2 Et


Yr = P 2 ( / 3 Y f - 3 + E r - 2 ) + B E t - 1 + Er
++ +f
yr = /3"yr-n ˜ ˜ ˜ -* 2
/3Et-1+
Et * 6




As long as 1/31 < 1 then /Y + 0 as n + 00 and the term /Yy,-, become
cant. The lag operator is a convenient shorthand and is useful in manipula
expressions. For example
(1 - p L ) - l & , = (1 + / 3 L + / 3 2 L 2 + * * * ) E l


It is obvious that y, in (20.7˜)depends on current and all past values of E
however Y , is uncorrelated with future values of E f + j ( j 3 1).

Stationarity
If 1/31 < 1 then yl is a stationary series. Broadly speaking a stationary series ha
mean and variance and the correlation between values yr and Y r - j depends
time diflerence ' j ' . Thus the mean, variance and (auto-) correlation for any
independent of time. A stationary series tends to return often to its mean va
variability of the series doesn't alter as we move through time (Figure 20.1).
The condition for stationarity 1/31 < 1 can be seen intuitively by noting that
a starting value yo then subsequent values of y in periods 1, 2, . . . are Pyo
+ + +
˜ 2 ) ,(P2&1 / 3 ˜ 2 ˜ 3 ) etc. If 1/31 <
plus the random error terms ˜ 1 , ,
deterministic part of y , namely /3"yo, approaches zero and the weighted ave
E i S are also finite (and eventually they tend to cancel out as the E i S are rand
zero). Hence y,+":
Figure 20.1 Stationary Series.

remains finite. More formally a process is ˜weakly™ or ˜covariance™ stationary
E f =cL
Y
var(yt) = o2


where all the RHS population ˜moments™ are independent of time t and have f
If in addition y, is normally distributed then the process represented by (20.9a
is strongly stationary. However, the distinction between weak and strong sta
not important in what follows so ˜stationarity™ is used to mean ˜weak™ or ˜
stationary. A white noise error term Er is a very specific type of stationary s
the mean and covariance are zero.
All the usual hypothesis testing procedures in statistics are based on the
that the variables used in constructing the tests are stationary. For a non-statio
the distribution of ˜conventional™ test statistics may not be well behaved. Th
properties of tests on non-stationary series generally involve substantial change
of the ˜conventional™ tests (e.g. special tables of critical values). To illustrate
a non-stationary series consider:

+ Byr-1 +
=a Er
Yf

where /?= 1 and Er is white noise. This is known as a random walk with dri
parameter is a and the model is:
+
Ayr = a E t
The growth in yf (assume yf is in natural logarithms) is a constant (= a) p
noise error. The realisation of (20.10) is shown in Figure 20.2. Clearly yr h
which increases over time and hence the level of y, is non-stationary.
If a! = 0 and = 1 we have a random walk without drip. The realisation of
in Figure 20.2 and y, is non-stationary because the (unconditional) variance
larger as n + 00 and therefore is not independent of time.
+ Er + + ˜ t - 2+
yf = (1 - L ) - l E t = E , Er-1 ***
Figure 20.2 Non-Stationary Series. Random Walk with Drift(-), Random Wa
Drift( - - - - - )




˜




7me

Figure 20.3 MA(I) Series (yf = E, - 0.5cf-1).

Ey, = O




Hence var(y,) + 09 as n + 09.

Moving Average Process

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