ñòð. 11 
Cuthbertson (1991)).
15.2 RECENT EMPIRICAL RESULTS
One can see that the VAR methodology is conducive to testing several variants
basic idea, namely RE crossequation restrictions. Recent studies recognise the
of the cointegration literature when formulating the variables to include in the V
not only include difference variables in the VAR but also the â€˜levelsâ€™ or cointe
variables such as f p r and ( r  r*), in the above examples). Recent studies a
unanimous in finding rejection of the VAR restrictions when testing the FRU
and the UIP hypothesis  under the maintained hypothesis of a time invarian
risk premium. The rejection of FRU and UIP is found to hold at several ho
3, 6, 9 and 12 months), over a quite wide variety of alternative information
different currencies and over several time spans of data (see, for example, Hak
Baillie and McMahon (1989), Levy and Nobay (1986), MacDonald and Ta
1988) and Taylor (1989˜)).
Term Structure of Forward Premia
Some studies have combined tests of covered interest parity with the EH o
structure of interest rates applied to both domestic and foreign interest rates
methodology is again useful for multiperiod forecasts. By way of illustratio
where dj" = r;''  t,?i) (i = 3,6). If the pure expectations hypothesis of in
holds in both the domestic and foreign country then
+ Efd!:\)/2
dj6' = (dj3'
+
where the subscript t 3 applies because monthly data is used. The above equa
a term structure of forward premia which involves forecasts of the forward p
Equation (15.38) is conceptually the same as that for the term structure of sp
zero coupon bonds, discussed in Chapter 14. Clearly, given any VAR involvin
fpj3' (and any other relevant variables one wishes to include in the VAR) t
will imply the, by now, familiar set of crossequation restrictions.
There have been a number of VAR studies applied to (15.38) and they usua
ingly reject the pure expectations hypothesis of the term premia in forward
MacDonald and Taylor (1990)). Since there is strong independent evidence t
interest parity holds for most time periods and most maturities, then rejec
restrictions implicit in the VAR parameters applied to (15.38) is most like
failure of the expectations hypothesis of the term structure of interest rates to
domestic and foreign countries.
The Spot Rate, Fundamentals and the VAR Approach
In our analysis of the flexprice monetary model FPMM in Chapter 13 we s
spot rate may be represented as a forward convolution of fundamental variab
Subtracting xt from both sides of (15.39)
i= 1
+
where 8, = p/(1 B), EtAxt+l = Erxr+l  xt and ErAxt+j = Er(xt+i  X r + i  1
Equation (15.41) is now in a similar algebraic form to that outlined for the EH
structure in Chapter 14. We may somewhat loosely refer to q = sr  xr as th
r
rate spread'. Hence, we can apply the same test procedures to (15.41) as in
analysis.
@e2â€™Aiz,= e2â€™8A(I  8A)â€™zt
elâ€˜z, =
i= 1
Hence the Wald restrictions implied by this version of the FPMM are:
elâ€™  e2â€™8A(I  8A)â€™ = 0
or

elâ€™(1 @A) e2â€˜0A = 0
Equation (15.44) is a set of linear restrictions which imply that the RE foreca
the spot rate is independent of any information available at time t other than
by the variables in x,. We can also define the â€˜theoretical exchange rate sprea
qi = e2â€™8A(I  ˜A)â€™z,
and compare this with the actual spread 4,. To implement the VAR methodolo
to have estimates of y, y* to form the variable qr = st  x, and estimates of /
3
8. An implication of the FPMM is that
is a cointegrating relationship (given that all the variables are Z(1)). Es
( y , y * , P I , /3*) may be obtained by a (multivariate) estimation procedure kn
Johansen procedure (see Chapter 20). The variables qr = sf  xr and Axr c
constructed and used in the VAR.
MacDonald and Taylor (1993) provide a good illustration of the implem
this procedure. Note that by using (15.41) we take explicit account of the po
stationarity in the data. Hence one avoids possible spurious regression probl
in earlier tests of the FPMM which used levels of trended variables. We also
specification error found in earlier studies that used only first differences of th
and ignored any cointegrating relationships among the I ( 1) variables. MacD
Taylor use monthly data January 1976December 1990 on the DM/$ rate
Johansen procedure they find they do not reject the null that the coefficients
money supplies and relative income are unity for the â€˜homeâ€™ (Germany) an
(USA) variables and that the interest rate coefficients are nearly equal and op
They take /3 = P* = 0.05 and the variable denoted x, above is taken to be

m*), ( y ,  y;). MacDonald and Taylor find that the Wald tests in (15.43) a
decisively reject the RE crossequation restrictions and there is excess volat
spot rate. The variance of the actual spread qr exceeds the variance of the
spread qi given by (15.45) by a factor greater than 100 (when /3 = 0.05) (a
standard errors are given).
+
O.XAsf2  0.42A2Amr  0.79Ayr  0.008A2r:  0.
As, = 0.005
(0.003) (0.07) (0.23) (0.34) (0.003) (0.
where R2 = 0.14, SE = 3.2 percent, (  ) = standard error. Equation (15.48) pa
usual diagnostic tests, although note that the R2 of 14 percent indicates that
variation in Asf is not explained by the equation. They then perform a â€˜rolling
and use (15.48) to forecast over different horizons as the estimated parameters a
They find that the RMSE for the ECM are slightly less than those from the ra
â€˜no changeâ€™ forecasts for horizons of 1, 2, 3, 6, 9 and 12 months. This is
to some of the earlier results in Meese and Roghoff (1983). However, note t
the statistical explanation in (15.48) appears to be due to the adhoc dynami
little of the statistical explanation seems due to the longrun error correction
forecasts are likely to be dominated by the difference terms, which probably a
a random walk themselves, hence the reason why the reported RMSEs are app
the same as those for the random walk modelâ€˜? The MacDonald and Tay
a valiant attempt to test correctly a sophisticated version of the FPMM but
is barely an improvement on the random walk in predicting monthly movem
DM/$ spot rate.
15.3 SUMMARY
Tests of risk neutrality and RE based on FRU and UIP over multiperiod foreca
are easily accomplished using the VAR methodology. The VAR methodolog
be applied to forwardlooking models of the spot rate based on economic fu
such as relative money supply growth. The main results using this methodo
FOREX market suggest
the FOREX market is not efficient under the maintained hypothesis of ris
0
and RE since both FRU and UIP fail the VAR tests
the FPMM applied to the $/DMrate indicates that this model does not
0
particularly well, the VAR restrictions do not hold, the spot rate is excessiv
(relative to its â€˜theoreticalâ€™ counterpart) and the model performs little b
random walk. However, as noted in Chapter 13 â€˜monetary fundamentalsâ€™ pr
predictive content for long horizon changes in the exchange rate
These results imply that the FOREX market is inefficient under risk neutral
This may be because of a failure of RE or because there exists a sizeable time
premium. The latter is examined in Chapter 18. Alternatively, there could b
of irrational behaviour in the market caused by â€˜fadsâ€™ or by the mechanics of
process itself. However, relatively sophisticated tests described in this chapt
rescued the abysmal performance of â€˜formalâ€™ fundamentals models of the sp
efficiency in the FRU and UIP relationships is rejected.
indicative that sr may not be cointegrated with the fundamentals (see Ba
(1993)).
L 2
Stock Price Volatility
It was seen in Chapter 6 that a definitive interpretation of the results from se
of variance bounds test on stock prices is dogged by the stationarity issue,
appropriate method of â€˜detrendingâ€™ the series for the actual price Pr and
foresight price P:. However, in the previous two chapters we have noted th
procedure tackles this problem head on by explicitly testing for stationarity in t
and it also allows several alternative metrics for assessing the validity of the
It. would be useful if we could apply the VAR methodology to stock prices.
the rational valuation formula (RVF) is nonlinear in dividends and the re
of return (or discount factor); however, a linear approximation is possible.
approximation also results in a transformation such that the â€˜newâ€™ variables a
be stationary. Hence the cointegration methodology can be used in setting up t
applying the VAR methodology to stocks a comparison can be made of a time
â€˜theoreticalâ€™ (log) stock price p: (or log dividend price ratio a;)(â€™) with the ac
the) stock price pr to ascertain whether the latter is excessively volatile. Resu
VAR methodology can be compared with earlier tests on excess volatility whic
var(P,) with the variance of the perfect foresight price var(P:). The VAR m
also gives rise to crossequation parameter restrictions, similar to those found w
the EMH in the bond and FOREX markets.
It was noted that Fama and French (1988) and others found that long hori
(e.g. over several years) are â€˜more forecastableâ€™ than short horizon returns
month or one year). Using the linearisation of the RVF it is possible to derive a
long horizon returns and the VAR methodology then provides complementa
to that of Fama and French.
The RVF assumes that stock prices change only on the arrival of new inf
news about â€˜fundamentalsâ€™: that is the future course of either dividends or dis
An interesting question is whether the observed volatile movements in stock
due solely to news. To answer this question a key element is whether onepe
are persistent. â€˜Persistentâ€™ means that the arrival of news about current re
strong influence on all future returns and hence on all future discount rates. If
is high then it can be shown that news about returns can have a large effe
prices even if oneperiod returns are barely predictable. This theme is develo
in the next chapter when the degree of persistence in a time varying risk
investigated. However, in this chapter the key aims are to:
based on the perfect foresight price
show that although oneperiod returns are hardly predictable this may n
0
imply that stock prices deviate significantly and for long periods from
mental value, that is there is excess volatility in stock prices
examine the relationship between stock price volatility and the degree of
0
in one period returns
Some of the analysis in deriving the above results is rather tedious and p
intuitively appealing than either the VAR methodology applied to the term
interest rates or volatility tests on stock prices. A attempt has been made to add
n
ition where possible and deal with some of the algebraic manipulations in App
However, for the reader who has fully mastered the VAR material in the previo
there are no major new conceptual issues presented in this chapter.
16.1 THEORETICAL ISSUES
We begin with an overview of the RVF and rearrange it in terms of the div
ratio, since the latter variable is a key element in the VAR approach as applied
market. We then derive the Wald restrictions implied by the RVF and the rati
tations assumption and show that these restrictions are equivalent to a key pro
the EMH, namely that oneperiod excess returns are unforecastable.
16.1.1 Linearisation of Returns and the RVF
We define Pr = stock price at the end of period t , Dr+l = dividends paid du
+
t 1, Hr+l = one period holding period return from the end of period t to
+
period t 1. All variables are in real terms. Define hr+l as
The oneperiod return depends positively on the capital gain (f'r+1 /Pr ) and on t
yield ( D f + l / P r ) .Equation (16.1) can be linearised to give (see Appendix 16.
_ _
+
where lower case letters denote logarithms (e.g. pr = In Pr), p = P / ( P 0 )is
tion parameter and empirically is calculated to be around 0.94 from the sample
k is a linearisation constant (and for our purposes may be largely ignored). Equa
is an approximation but we will treat it as an accurate approximation. Note that
equation (16.2) implies that the (approximate) oneperiod holding period yie
positively on the capital gain (i.e. pr+l relative to p t ) and the level of (the lo
dividends dt+l. If we define the (log) dividend price ratio as(*)
terribly intuitive. It is an (approximate) identity with no economic conten
implies that if we wish to forecast one period returns we need to forecas
+
dividend price ratio &+I and the growth in dividends during period t 1. It
to forecast h,+l then 6, and Ad, must be included in the VAR. The observant r
also notice that (16.4) is a forward difference equation in ˜5˜ and since 6, = d
be solved forward to yield an expression for the (logarithm of the) price
stock: a sort of RVF in logarithms. Before we do this however it is worth un
brief digression to explain how the dividend price ratio is related to the disco
the growth in dividends in the â€˜usualâ€™ RVF of Chapter 4. This will provide
we need to understand the RVF in terms of 6, which appears below, in equa
The RVF in the levels of the variables is
through by D,to give a price dividend ratio. Next w
where we have
for example:
where we have used the logarithmic approximation for the rates of growth o
From (16.5) and (16.6) we see that the RVF is consistent with the following:
(i) the current pricedividend ratio is positively related to all future grow
dividends Ad,+,,
(ii) the current price dividend ratio is negatively related to all future discount
We can, of course, invert equation (16.5) so that the current dividend pri
dividend yield) is qualitatively expressed as:
Since (16.7) is just a rearrangement of the RVF formula then intuitively we
(i) when dividends are expected to grow, ceterisparibus, the current price
and hence the dividend price ratio will be low,
(ii) where future discount rates are expected to be high, the current price
and hence the current dividend price ratio will be high.
We can obtain the same â€˜rearrangementâ€™ of the RVF as (16.7) but one which
the parameters, by solving (16.4) in the usual way using forward recursion (an
a transversality condition) (see Campbell and Shiller 1988):
of the RVF which, of course, does embody economic behaviour. From Chap
be recalled that if we assume the economic hypothesis that investors have a r
desired) expected oneperiod return equal to rr then we can derive the RVF (1
time varying discount rate (equal to r,). Similarly suppose the (log) expected
of return required by investors to wiZZingly hold stocks is denoted rf+j then(3
Eth,+, = rf+j
Then from (16.4)
&+,+ Ad;+l + k = rf+,
6, 
a,+,).
(note the superscript â€˜e7on Solving (16.10) forward
M
Equation (16.11) is now the linear logarithmic approximation to the RVF give
and it can be seen that the same negative relationship between 8, and Ad,+, an
positive relationship between 6, and rF+j holds as in the â€˜exactâ€™ RVF of (16.5
the same result is obtained if one takes expectations of the identity (16.4)
recursively to give:
00
6, = E P j [ h ; + j + ,  Ad;+,+ll  k / ( l  P )
j=O
where E161 = 6, as 6, is known at time t . The only difference here is that hf
interpreted as the expected oneperiod required rate of return on the stock. In
ical work to be discussed below much of the analysis concentrates on using
equations for the â€˜fundamentalsâ€™ on the RHS of (16.12), namely oneperiod
dividend growth, and then using (16.12) to give predicted values for the divi
ratio which we denote 6:. The predicted series 6; can then be compared with
values 6,. A forecast for the (log of the) stock price is obtained using the iden
namely pi = d ,  6: and the latter can then be compared with movements in
stock price.
Gordonâ€™s Growth Model
Further intuitive appeal can be provided for equation (16.11) for the dividend
version of the RVF by going back to Gordonâ€™s dividend growth model (see C
If the required rate of return r, and expected growth in dividends g are both co
the usual RVF gives the dividend price ratio (see Chapter 4) as:
(DIP)= r  g
Equation (16.12) can therefore be seen to be a dynamic version of Gordonâ€™s m
the required rate of return and dividend growth varying, period by period. In t
and the discount rate are allowed to vary over time. So the question that w
examine is whether the EMH can explain movements in stock prices based o
when we allow both time varying dividends and time varying discount rates.
draw a parallel with our earlier analysis of the expectations hypothesis (EH)
structure.
Spreads, Dividend Yields and All That
A useful intuitive interpretation of (16.12) can be obtained by comparing it wi
expression from the expectations hypothesis (EH) of the term structure. The
= Rtcn)  Rjm)is an optimal linear forecast and should Gr
that the spread
future changes in interest rates
where wi is a set of known constants. Equation (16.11) is the equivalent rela
stock prices. The spread is replaced by the (log) dividend price ratio 6, and fut
in interest rates are replaced by ( r f + j Adt+j).
1 . STOCK PRICE VOLATILITY AND THE VAR
62
METHODOLOGY
Even when measured in real terms, stock prices and dividends are likely
stationary but the dividend price ratio and the variable (r,+j  Ad,+,) are mo
be stationary so that the VAR methodology and standard statistical results may
to (16.12). If the RVF is correct we expect 6, to Granger cause (r,+,  Ad,+j)
the RVF equation (16.12) is linear in future variables we can apply the variety
(16.11) used when analysing the EH with the VAR methodology. This can no
The vector of variables in the agents information set is taken to be
where rdr = r,  Ad,. Taking a VAR lag length of one for illustrative purpos
+ Wf+l
=
Zf+l
Defining e l = (1,O)â€™ and e2â€™ = (0, 1)â€™it follows that
If (16.18) is to hold for all zf then the nonlinear restrictions (we ignore the co
since all data is in deviations from means) are:
f(a) = el'  e2'A(I  pA)' = 0
Postmultiplying by (I  pA) this becomes a linear restriction
el'(1 pA)  e2'A = 0
These restrictions can be evaluated using a Wald test in the usual way(4).Thus
formula is true and agents use RE we expect the restrictions in (16.19) or (16.
The restrictions in (16.20) are
that is
1 = Pall +a21
0 = Pal2 4 a22
One Period Returns are not Forecastable
There is little or no direct intuition one can glean from the linear restrictions
it is easily shown that they imply that expected oneperiod real excess return
rf+l)are unforecastable or equivalently that abnormal profit opportunities do
the market. Using (16.4) and ignoring the constant:
From (16.16)
and
Hence given the VAR forecasting equations, the expected excess oneperio
predictable from information available at time t unless the linear restrictio
the Wald test (16.21) hold(5).The economic interpretation of the nonlinear W
discussed later in the chapter.
˜ j ( r f + + ˜
˜ AdF+j+l)= e2â€™A(I  PA)â€™Zt
6; =
j=O
+
Under the RVF RE, in short the EMH, we expect movements in the actu
price ratio 6, to mirror those of 8: and we can evaluate this proposition by (i)
6, and a:, (ii) SDR = a(S,)/a(G;) should equal unity and (iii) the correlation
corr(6,, 6;) should equal unity. Instead of working with the dividend price ra
use the identity pi = dr  8; to derive a series for the theoretical price level g
EMH and compare this with the actual stock price using the metrics in (i)(ii
Further Implications of the VAR Approach
The constructed variable p: embodies the investorâ€™s best forecast of the DP
dividends using time varying rates of return (discount) and given the info
assumed in the VAR. It is therefore closely related to the expected value of
foresight price EtP: in the original Shiller volatility tests. The difference b
two is that P,* calculated without recourse to specific expectations equation
is
fundamental variables but merely invokes the RE unbiasedness assumption (
+
ErDt+j qt+ j ) . Put another way, EP; is an unconditional expectation and does
an explicit model for the behaviour of dividends whereas p : is conditional on
statistical model for dividends.
It is worth briefly analysing the relationship between the (log of the) perfe
price p : = ln(P:) and the (log of the) theoretical price pi, in part so that w
about the distinction between these two allied concepts. In so doing we are a
out some of the strengths and weaknesses of the VAR approach compared with
variance bounds tests and regression tests based on P that we discussed in
T
The loglinear identity (16.2) (with ht+l replaced by rr+1) can be rearranged
Solving (16.25) by recursive forward substitution gives a loglinear express
DPV of actual future dividends and discount rates which we denote pf.
00 00
Equation (16.26) uses actual (expost) values and is the loglinear version
(16.24) in Chapter 6 and hence pT represents the (log of the) perfect foresig
Under the EMH we have the equilibrium condition
Under RE agents use all available information Qt in calculating EtPT but the
price pi only uses the limited information contained in the VAR which is ch
econometrician. Investors â€˜in realityâ€™ might use more information than is in the
is more, even if we add more variables to the VAR we still expect the coeff
to be unity. To see this, note that from (16.24)
zr = [a,, rdr]
6; = e2â€™f(a)
f ( a ) = A(I  pA)â€™
For VAR lag length of one, f(a) is a (2 x 2) matrix which is a nonlinear fun
a i j S of the VAR(7).Denote the second row of f(a) as the 2 x 1 vector [f21(a)
e2â€™f(a) where f 2 1 and f 2 2 are scalar (nonlinear) functions of the aij param
from (16.28) we have
+
6: = f21(8)8, f22(a)rd,
Since under the EMH = 8: we expect the scalar coefficient fzl(a) = 1 a
f22(a) = 0. These restrictions hold even if we add additional variables to th
that happens if we add a variable xf to the VAR system is that we obtain an add
f3(a)xf and the EMH implies that f3(a) = 0 (in addition to the above two r
Thus if the VAR restrictions are rejected on a limited information set they
be rejected when a â€˜largerâ€™ information set is used in the VAR. The latter is true
of logic and should be found to be true if we have a large enough sample of d
sense the use of a limited information set is not a major drawback. However, in
set we know that variables incorrectly omitted from the regression by the econ
yet used by agents in forecasting, can result in â€˜incorrectâ€™ (i.e. biased or in
parameter estimates. Therefore a larger information set may provide additional
on the validity of the hypothesis.
Thus while Shiller variance bounds tests based on the perfect foresight
suffer from problems due to nonstationarity in the data, the results based o
methodology may suffer from omitted variables bias or other specification
wrong functional form). Although there are diagnostic tests available (e.g. tes
correlation in the error terms of the VAR, etc.), as a check on the statistical val
VAR representation it nevertheless could yield misleading inferences in finite
We are now in a position to gain some insight into the economic interpreta
Wald test of the nonlinear restrictions in (16.19), which are equivalent to thos
for our twovariable VAR. If the nonlinear restrictions are rejected then this m
to f24a) # 0. If so, then rd, influences
j=O
To the extent that the (weighted) sum of oneperiod returns is a form of multipe
then violation of the nonlinear restrictions is indicative that the (weighted) re
long horizon is predictable. This argument will not be pushed further at this
it is dealt with explicitly below in the section on multiperiod returns.
We can use the VAR methodology to provide yet another metric for as
+
validity of the RVF RE in the stock market. This metric is based on splittin
+ a:,
6, = 62,
where:
61, = e3â€™A(I  pA)â€™z,
 pA)lz,
62, = e2â€™A(I
Hence, we expect corr(6,  a&,, S i r ) = 1. If this correlation coefficient is subst
than one then it implies that the variation in real expected returns 6 is not
;
variable to explain the movements in the dividend price ratio, corrected for th
of future dividend forecasts 6,  6&,.
As we shall see in Section 16.4 a variance decomposition based on (16
useful in examining the influence of the persistence in expected returns on t
price ratio 6, and hence on stock prices ( p , = dr  6,). The degree of pe
expected returns is modelled by the size of certain coefficients in the A m
VAR. We can use (16.31) to decompose the variability in 6, as follows:
+ + 2cov(6&,,8;)
var(&,) = var(6&,) var(6Lt)
where the RHS terms can be shown to be functions of the A matrix of the VAR
this analysis will not be pursued here and in this section the covariance ter
appear since we compare 6,, with 6r  a&,.
Summary
We have covered rather a lot of ground but the main points in our application
methodology to stock prices and returns are:
(i) We can obtain a linear approximation to the oneperiod return which m
solved forward to give an expression for the current (log) dividend pric
terms of expected future dividend growth rates Ad,+j and a sequence
required oneperiod returns on the stock (denoted r,+j or h,+j)
(ii) Using a simple transformation of the dividend price ratio, namely p , =
can obtain the linearised version of the RVF:
I
00 00
[ G pjdt+j+l C pjrr+j+l+ k / ( l 
Pr = E , (1  P ) P)
j=O
j=O
(iii) Analysis of the dividend price ratio using (16.35) or the price level us
is equivalent but use of (16.35) in estimation enables one to work wi
excess returns are unforecastable and that RE forecast errors are inde
information at time t .
From the VAR we can calculate the â€˜theoreticalâ€™ dividend price ratio
+
theoretical stock price pi(= d ,  6;). Under the null of the EMH RE
6, = 6; and p , = p : . In particular, the coefficient on 6, (or p , ) from the
weighted appropriately (see equation (16.30)) should equal unity, wi
variables having a zero weight.
We can compare 6, and 6; (or p , and p i ) graphically or by using t
deviation ratio SDR = a(S,)/a(S;) or by looking at the correlation co
6, and 6:: these provide alternative â€˜metricsâ€™ on the success or other
+
RVF RE. 6, is a sufficient statistic for future changes in r,  Ad,
should, at a minimum, Granger cause the latter variable.
A measure of the relative strength of the variation in expected dividen
the variation in expected future returns (6;) in contributing to the v
stock prices (6;) can be obtained from the VAR.
16.3 EMPIRICAL RESULTS
The results are illustrative. They are not a definitive statement of where the
the evidence lies. Empirical work has concentrated on the following issues:
(i) The choice of alternative models for expected oneperiod holding retu
the variables r,+j (or hf+j).
(ii) How many variables to include in the VAR, the appropriate lag leng
stability of the parameter estimates.
(iii) How to interpret any conflicting results between the alternative â€˜me
such as the predictability of oneperiod returns in a single equation stu
correlation, variance ratio statistics and Wald tests of the VAR methodo
16.3.1 The RVF and Predictability of Returns
The first study we examine was undertaken by Campbell and Shiller (1988
annual data on an aggregate US stock index and associated dividends for
1871 1986. They use four different assumptions about the oneperiod requi
return r,+j which are:
(i) Required real returns are constant (i.e. r, = constant).
(ii) Required nominal (or real) returns equal the nominal (or real) Treas
commercial paper) rate rr = r:.
(iii) Required real returns are proportional to consumption growth (consumpti
plied by a constant which is a measure of the coefficient of relative ris
a),that is r, = aAc,.
Hence in the VAR â€˜r,â€™ is replaced by one of the above alternatives. Note tha
usual assumption of norisk premium, (iii) is based on the intertemporal co
CAPM of expected returns while (iv) has a risk premium loosely based on
for the market portfolio (although the risk measure used, o, equals squa
:
returns and is a relatively crude measure of the conditional variance of stock
Chapter 17).
Results are qualitatively unchanged regardless of the assumptions (i)(iv)
required returns and therefore comment is made mainly on results under assu
that is constant real returns (Campbell and Shiller, 1988, Table 4). The vari
and Ad, are found to be stationary I(0) variables. In a single equation re
(approximate) log returns on the information set 6, and Ad, (r, doesnâ€™t appea
is assumed constant) we have:
h, = 0.14161  0.012Adf
(0.057) (0.12)
18711986, R2 = 0.053 (5.3 percent), ( . ) = standard error.
Only the dividend price ratio is statistically significant in explaining annual (
real returns but the degree of explanatory power is low (R2= 5.3 percent).
In the VAR (with lag length = 1) using Zl+l = @,+I, Ad,+l) the variable 6,+
autoregressive, and most of the explanatory power (R2 = 0.515) comes from
(coefficient = 0.706, s.e. (6,)= 0.066) and little from Ad,. The change in rea
Adt+l is partly explained by Ad, but the dividend price ratio 6, is also
significant with the â€˜correctâ€™ negative sign (see equation (16.35)). Hence Gra
Adt, a weak test of the RVF. If we take the estimated A matrix of the VAR and
to calculate f21(a) and f22(a) of (16.30) then Campbell and Shiller find:
6; = 0.63661  0.097Adt
(0.123) (0.109)
+
Under the null of the RVF RE we expect the coefficient on 6, to be unity and
to be zero: the former is rejected although the latter is not. From our theoretic
we noted that if 6; # 6, then oneperiod returns are predictable and therefor
results are consistent with single equation regressions on the predictability of r
as those found in Fama and French (1988). The Wald test of the crossequation
is rejected as is the result that the standard deviation ratio is unity since
SDR = o(Sl)/a(G,) = 0.637 (s.e. = 0.12)
However, the correlation between 6, and 6; is very high at 0.997 (s.e. = 0.0
not statistically different from unity. It appears therefore as if 6, and 8; move
direction but the variability in actual is about 60 percent (i.e. 1/0.637 = 1
than its rationally expected value 6; under the EMH.That is the dividend pric
hence stock prices are too volatile to be explained by fundamentals even whe
dividends and the discount rate to vary over time.
of all future oneperiod returns r,+j and hence approximates a longhorizon
strong rejection of the Wald test is therefore consistent with the Fama and Fre
where nonpredictability is more strongly rejected for long rather than sh
returns. We return to this issue below.
The results also make clear that even though oneperiod returns are barely
nevertheless this may imply a fairly gross violation of the equality 6, = 6; (o
Hence the actual stock price is substantially more volatile than predicted b
even when oneperiod returns are largely unpredictable.
The correlation between (6,  82,) and a for VAR lag lengths that exce
,;
generally found to be low. The correlations are in the region zero to 0.6 unde
alternative assumptions about required returns investigated (e.g. required retu
tional to consumption growth, required returns equal the real interest rate, etc
tentative conclusion would be that expected future returns are not sufficiently
explain the variability in actual stock prices. Variability of the stock price is
to variability in expected dividends although even the latter is not sufficiently
explain stock price variability â€˜fullyâ€™ (i.e. var(6,) > var(6j) and var(p,) > var
In a second study Campbell and Shiller (1988, Chapter 8) extend the info
in the VAR to include not only 6, and Ad, but also the (log) earnings price ra

e, = e,  pr and Z = long moving average of the log of real earnings. The
r
for including er is that financial analysts often use forecasts of earnings in orde
future price movements and hence future returns on the stock. Indeed Ca
Shiller find that the earnings yield is the key variable in determining return
statistically it works better than the dividend price ratio. The VAR now includ
â€˜fundamentalâ€™ variables Zt+l = (&+I * Ad,+l er+l but the basic VAR analy
)â€™
unchanged.
Campbell and Shiller (1988) consider two hypotheses for expected or re
period real returns. First, they assume required returns are constant and second,
returns are constant. Excess returns equal h,+l  r, where r, is the shortterm i
There is therefore no time varying risk premium incorporated in the analysis.
broadly similar for both of the above assumptions about required returns. Th
and actual values for (a,? 6:) and ( p r yp i ) over the period 19011986 for an ag
stock price index are shown in Figures 16.1 and 16.2.
It is clear particularly after the late 1950s that there is a substantial divergen
+
the actual and theoretical series thus rejecting the RVF RE. There is exce
in stock prices and they often diverge substantially from their fundamental v
given by the RVF. The latter is the case even though actual oneperiod (log) re
the theoretical return h: are highly correlated (e.g. corr(h,, hi) = 0.915, s.e. =
Shiller (1989), Table 8.2). The reason for the above results can be seen by us
and pi = dt  6; to calculate the theoretical price:
+ +
pi = 0.256˜1 0.776er 0.046df  O.078d11
Hence pt only has a weight of 0.256 rather than unity in determining pi an
run movements in p : (in Figure 16.2) are dominated by the â€˜smoothâ€™ moving
.E
a
3 3
B
a
.c
0
P
I
3.5
1910 1920 1930 1950 1960 1970 1980
1940
Figure 16.1 LogDividend Price Ratio 6, (Solid Line) and Theoretical Counterpar
Line), 1901  1986. Source: Shiller (1989). Reproduced by permission of the Amer
Association
1030
1910 1920 1940 1950 1960 1970 1980
Figure 16.2 Log Real Stock Price Index p , (Solid Line) and Theoretical Log Real
pj (Dashed Line), 1901 1986. Source: Shiller (1989). Reproduced by permission of t
Finance Association
earnings er. However, in the short run p , is highly volatile and this causes pi t
volatile. By definition, oneperiod returns depend heavily on price changes, h
hi are highly correlated. (It can be seen in Figure 16.2 that changes in p t
correlated with changes in pi, even though the level of p , is far more volatil
The Campbell and Shiller results are largely invariant to whether required
or required excess returns over the commercial paper rate are used as the ti
discount rate. Results are also qualitatively invariant in various subperiods o
data set 19271986 and for different VAR lag lengths. However, Monte C
(Campbell and Shiller, 1989 and Shiller and Belratti, 1992) demonstrate tha
test may reject too often under the null that the RVF fundamentalsâ€™ model is
the VAR lag length is long (e.g. greater than 3). Notwithstanding the Monte C
Campbell and Shiller (1989) note that in none of their 1000 simulations are th
Using UK aggregate data on stock prices (monthly, 19651993 and annual 1
Cuthbertson and Hayes (1995) find similar results to Campbell and Shiller ex
case where returns depend on volatility and here they find stronger evidence tha
and Shiller (1988) in favour of the RVF. The Wald test of the nonlinear rest
variance ratio between 6, and 6: and their correlation coefficient are consiste
RVF, when â€˜volatilityâ€™ is included in the VAR.
An interesting disaggregated study by Bulkley and Taylor (1992) uses the
from the VAR, namely the theoretical price, in an interesting way. First, a VAR
recursively 19601980 for each company i and the predictions Pit are obtaine
year of the recursive sample, the gap between the theoretical value Pit and the
Pi, are used to help predict returns R; over one to 10year horizons (with cor
company risk variables) z k :
cm
+ vo(p:/p;)+
R; = a YkZk
k= 1
Contrary to the EMH, they find that ˜0 # 0. They also rank firms on the b
topbottom 20 and fopbottom 10, in terms of the value of ( P j / P i ) and forme
of these companies. The excess returns on these portfolios over one to 10ye
are given in Figure 16.3. For example, holding the top 20 firms as measured b
ratio for three years would have earned returns in excess of those on the S
of over 7 percent per annum. They also find that excess returns cumulated
years suggest mispricing of the top 20 shares of a (cumulative) 25 percent. Th
therefore rejects the EMH.
7
5
1 2 6 8 9 10
4
0 3
Holding Period (Years)
Figure 16.3 Excess of Portfolio Returns over Sample Mean. Source: Bulkley and Ta
Reproduced by permission of Elsevier Science
(where P: = perfect foresight price) is like a very longhorizon return. Hence
of P ,  P: on the information set 52, should yield zero coefficients if longhor
are unforecastable. Fama and French use actual longhorizon returns over N p
and find that these are predictable using past returns, particularly for retur
over a three to five year horizon. Fama and French use univariate AR models i
Campbell and Shiller (1988, Chapter 8) are able to apply their linearised
oneperiod returns to yield multiperiod returns and the latter can be shown to
equation restrictions on the coefficients in the VAR. Hence using the VAR m
one can examine the Fama and French â€˜longhorizonâ€™ results in a multivariate
First, we define a sequence of oneperiod holding period returns, H l , + j fo
+ + +
time periods t to t 1 , t 1 to t 2, etc.:
+ Hl.t+l = V f + l + D f + l ) / P f
1
+ H1,,+2 = W f + 2 + Df+Z)/Pt+l
1
The twoperiod compound return from t to t + 2 is defined as
+ i is
hence in general the iperiod return from t to t
+ H )we
Using lower case letters to denote logarithms and letting h: = ln(1
i1
j=O
Equation (16.43) is unbounded as the horizon i increases, so Campbell and S
to work with a weighted average of the i period log return:
i1
hi,, = PJhl,r+j+l
j=O
Using (16.44) and the identity (16.4) for oneperiod returns hl,+l we have@
i1
+k)
 P a t + l + j  Adr+j+l
hi,, = Pâ€™(&+j
j=O
Equation (16.45) is an (approximate) identity which defines the multiperio
from period t to t + i in terms of 6,, 6t+i and Adt+j. It doesnâ€™t have a great d
we have anchored d , with the term 6,(= d ,  p , ) it must therefore depend on
in dividends  this is the third term on the RHS of (16.45). The second
decreasing importance as the return horizon increases (since p'S,+i + 0 if S i
and 0 < p < 1). It appears because returns over a finite horizon depend on t
+
(selling) price at t i and hence on 6r+i.
We are now in a position to see how a multivariate forecasting equation base
may be compared with the Fama and French 'longhorizon', single equation reg
VAR in Zr = (a,, Ad,) can be used to forecast the RHS of (16.45) which is the
return over i periods denoted hiqr. can then compare the actual i period ret
We
hi, using graphs, the variance ratio test and the correlation between hi,, and h
Although (16.45) can be used to provide a forecast of hi, from a VAR b
and Adt, equation (16.45) does not provide a Wald test of restrictions on th
since hi,, is not in the information set. However, a slight modification can y
test for multiperiod returns. We again introduce the behavioural hypothesis th
oneperiod excess returns are constant'
E(hl,,  rr+d = c
It follows that
Taking expectations of (16.45) and equating the RHS of (16.45) with the RHS
we have the familiar difference equation in 6, which can be solved forward
dividend ratio model for i period returns
If we ignore the constant term, (16.48) is a similar expression to that obtai
(see (16.12)) except that the summation is over i periods rather than to infin
the dynamic Gordon model over i periods. Campbell and Shiller (1988) use
form a Wald test of multiperiod returns for different values of i = 1 , 2 , 3 , 5 ,
and also for an infinite horizon. For zt = (a,, rd,, er)' these restrictions are:
el'(I  piAi)= e2'A(I  pA)'(I  piAi)
For i = 1 (or i = 00) the above reduces to
el'(1 pA) = e2'A
which is the case examined in detail in section 16.2. If (16.50) holds then postm
by (I  pA)'(I  piAi)we see that (16.49) also holds algebraically for any
manifestation of the fact that if oneperiod returns are unforecastable then so a
returns. Campbell and Shiller for the S&P index 18711987 find that the W
for constant real and expected returns. But they find evidence that multiperiod
not forecastable when a measure of volatility is allowed to influence current
Perfect Foresight Price and Multiperiod Returns
It can now be shown that if the (loglinearisation of the) multiperiod return
castable then this implies that the Shiller variance bound inequality is also
violated. Because of the way the (weighted) multiperiod return hi,was define
it is the case that hi,f remains finite as i + 00:
+ k/( 1  p )
= In P:*  In P ,
The first term on the RHS of (16.51) has been written as InPT* because it is the
equivalent of the perfect foresight price Pf in Shillerâ€™s volatility tests. If
period return is predictable based on information at time t (a,), it fo then
(16.51) that in a regression of (InPT*  l n P , ) on 52, we should also find
+
statistically significant. Therefore In PT* # In P , Ef and the Shiller variance b
be violated. It is also worth noting that the above conclusion also applies
horizons. Equation (16.45) for hi,, for finite i is a loglinear representation of I
when PT, is computed under the assumption that the terminal perfect foresi
+
t i equals the actual price Pr+i. The variable InPT, is a close approxima
variable used in the volatility inequality tests undertaken by Mankiw et a1 (1
they calculate the perfect foresight price over different investment horizons. He
hi, for finite i are broadly equivalent to the volatility inequality of Mankiw et
in Chapter 6. The two sets of results give broadly similar results but with Ca
Shiller (1988) rejecting the EMH more strongly than Mankiw et a1 .
Summary
(i) An earnings price ratio e, helps to predict the return on stocks (hi,,)
for returns measured over several years and it outperforms the dividend
(ii) Actual oneperiod returns hl,, stock prices p r and the dividend price rat
too volatile (i.e. their variability exceeds that for their theoretical counter
and 6;) to conform to the fundamental valuation model, under rational e
This applies under a wide variety of assumptions for required expected
(iii) Although hl, is â€˜too variableâ€™ relative to its theoretical value hi,, neve
correlation between h l , and hi, is high (at about 0.9, in the constant d
case). Hence the variability in returns is in part at least explained by
in fundamentals (i.e. of dividends or earnings). Movements in stock
therefore be described as an overreaction but it seems to be an ove
fundamentals such as earnings or dividends and not to fads or fashions
evidence is broadly consistent with the recent ideas of intrinsic bubbles (
equivalent to a violation of Shillerâ€™s variance bounds tests for j + 0
tation of the hypothesis that multiperiod returns are not predictable (a
Fama and French (1988) and Mankiw et a1 (1991)).
On balance, the above results support the view that the EMH in the form of the
does not hold for the stock market. Longhorizon returns (i.e. 35 years) are
although returns over shorter horizons (e.g. 1 month 1 year) are barely predicta
theless it appears to be the case that there can be a quite large and persistent
between actual stock prices and their theoretical counterpart as given by the
evidence therefore tends to reject the EMH under several alternative models
rium returns. The results from the VAR analysis are consistent with those fro
work using variance inequalities (see Chapter 6). The rejections of the EMH
this chapter are also somewhat stronger than those found by Mankiw et a1 (1
variance bounds and regressionbased tests on longhorizon returns.
16.4 PERSISTENCE AND VOLATILITY
This section demonstrates how the VAR analysis can be used to examine the
between the predictability and persistence of oneperiod returns and their impl
the volatility in stock prices. We have noted that monthly returns are not very
and single equation regressions have a very low R2 of around 0.02. Persi
univariate model is measured by how close the autoregressive coefficient is to
section also shows that if expected oneperiod returns are largely unpredicta
persistent, then news about returns can still have a large impact on stock pric
using a VAR system we can simultaneously examine the relative contributio
about dividends, news about future returns (discount rates) and their interac
variability in stock prices.
We begin with a heuristic analysis of the impact of news and persistenc
prices based on the usual RVF, and then examine the problem using the Campb
linearised formulae for oneperiod returns and the RVF, first using an AR(1)
then using a VAR system. Finally, some illustrative empirical results usin
methodology are presented.
16.4.1 Persistence and News
Campbell (1991) considers the impact on stock prices of (i) changes in expe
discount rates (required returns) and (ii) changes in expected future dividen
element in Campbellâ€™s analysis is the degree of persistence in future div
discount rates. The starting point is the rational valuation formula
returns is represented by &,+I. If /3 is close to unity then an increase in
not only h,+l to increase but also causes all subsequent h f + j ( j = 2â€™3â€™4, . . .)
quite substantially. As PI depends inversely on allfuture values then a smal
&,+I may cause a very large change in the current price P I , if 1. Hence
in expected returns can cause substantial volatility in stock prices even if
current discount rates &,+I is rather small. To introduce a stochastic element in
behaviour assume an AR(1) model:
+m +
=a
Q+l 0 V,+l
Using (16.54) current â€˜good newsâ€™ about dividends (i.e. v,+1 > 0) will ther
a substantial rise in current price if a! % 1: this is â€˜persistenceâ€™ again. It is p
â€˜good newsâ€™ about dividends (i.e. v,+l > 0) may be accompanied by â€˜bad
hence higher discount rates (i.e. &,+I > 0). These two events have offsetting
P,. It is conceivable that the current price may hardly change at all if these
are strongly positively correlated (and the degree of persistence is similar). Pu
loosely, the effect of news on P, is a weighted nonlinear function which
revisions to future dividends, revisions to future discount rates and the covarian
the revisions to dividends and discount rates. The effect on P I is:
(i) positive, for positive news about dividends,
(ii) negative, for news which generates increases in future discount rates,
(iii) the effects from (i) or (ii) on the current price P, are mitigated to
that upward revisions to future dividends are offset by upward revision
discount rates (i.e. if R f + l and v,+l are positively correlated).
Let us now turn to the impact of the variability in dividends and discount r
variability in stock prices. We know that for 1/31 < 1 and lal < 1 we can rew
and (16.54) as an infinite moving average:
+ BEr+m1+
/3>I + Er+m +  
hr+m = [/30/<1 B2Er+m2
+ av,+ml + a2 +  .
+
Dt+m = [a0/(1 a)] Vt+m ˜t+m2
The variance of rf+m and Dr+m conditional on information at time t is therefo
+ p4+ . . .)
var(ht+,IQt) = a,2(1+ p2
= CT:(I + + + . . .)
var(D,+,IQr)
and the covariance is:
+ +  .>
cov(h,+mâ€™ o t + m I Q t ) = o E v ( l + *
= p,va,av(l + a!g + + . . .)
where we have assumed that E f + l and v,+l are contemporaneously correlated
lation coefficient pEV.From (16.52) the variability in PI is a nonlinear sum o
variances and covariances. For given values of CT, a,, the variance of P,
and
In case (iii) if pEV< 0 then the influence of positive news about dividends
is accompanied by news about future discount rates which reduces hr+m (i.e
Since P , depends positively on Dr+m and inversely on hr+m then two effec
each other and the variance of prices is large.
L inearisation
The problem with the above largely intuitive analysis is that the effects w
(16.52) are nonlinear. However, Campbell (1991) makes use of the loglinea
the oneperiod holding period return h f + l .
+ 6,  &+1 + Adr+l
=k
h+l
Solving (16.58) forward we obtain the linearised version of the RVF in terms
00
 Adt+j+l)  k / ( l  PI
6, = d(ht+j+l
j=O
From (16.58) and (16.59) the surprise or forecast error in the oneperiod expe
can be shown to be (see Appendix 16.1):
00
00
ht+l  Etht+l = [Et+1  Et] dAdr+j+l  [Er+1  Er] Pâ€™hr+l+j
j= 1
j=O
which in more compact notation is:
d h
= q t + l  qr+1
4+1
I=[ I[ news about future
unexpected returns news about future
+ dividend growth expected returns
in period t 1
The LHS of (16.61) is the unexpected capital gain Pt+l  E,p,+l (see appe
terms qf+l and $+1 on the RHS of (16.61) represent the DPV of â€˜revisions
tionsâ€™. Under RE such revisions to expectations are caused solely by the arriv
or new information. Equation (16.61) is the key equation used by Campbell. I
more than a rearrangement (and linearisation) of the expected return identity
alently of the rational valuation formula). It simply states that a favourable
the expost return ht+l over and above that which had been expected E,h,+l m
to an upward revision in expectations about the growth in future dividends
downward revision in future discount rates hr+j. If the revisions to expecta
either the growth in dividends or the discount rate are persistent then any ne
items will have a substantial effect on unexpected returns (h,+l  E,hr+l) an
the variance of the latter.
The RHS of (16.60) is a weighted sum of two stochastic variables Adf+j
The variance of the unexpected return var(v;+,) can be written:
returns and expected dividends vary through time. Campbell suggests a mea
persistence in expected returns:
+ 1 in the oneperiod ahead expected re
where is the innovation at time t
u,+l
so that ur+l is a revision to expectations over one period only. Ph is therefore
standard error of news about the DPV of all future returns
Ph =
standard error of news about oneperiod ahead expected retur
may be interpreted as follows. Using (16.63) we see that an innovation
Ph
period expected return ur+1 of 1 percent will lead to a Ph percent change i
discount rates r$+j and hence via (16.61) a Ph percent unexpected capital los
Univariate Case
It is useful to consider a simple case to demonstrate the importance of pe
explaining the variability in stock prices. Suppose expected returns follow
process:
The degree of persistence depends on how close is to unity. For this AR(
can be shown that
= p/(l  pB) % 1/(1  /?) (for p 1)
%
Ph
+ B)R2/(1  B)(1  R2>
vam;+1 I/ var(v;+,) = (1
where R2 = the fraction of the variance of stock returns ht+l that is pred
v;+˜ = ht+l  E,h,+l. For /? close to unity it can be seen that the P h statis
indicating a high degree of persistence. In an earlier study Poterba and Summ
found that = 0.5 in their AR univariate model for returns. They then calc
the stock price response to a 1 percent innovation in news about returns is app
2 percent. This is consistent with that given by Campbellâ€™s Ph statistic in (16
We can now use equation (16.67) to demonstrate that even if oneperiod st
h,+l are largely unpredictable (i.e. R2 is low) then as long as expected returns
tent, the impact of news about future returns on stock prices can be large. Tak
and a value for R2 in a forecasting equation for oneperiod returns as 0.025 we
(16.67) that:
var(qf+l = 0.49 var(˜;+˜
price movements.
Campbell is able to generalise the above univariate model by using a multiv
system. If we have one equation to explain returns hf+l and another to explain
in dividends Ad,+l then the covariance between the error terms in these tw
provides a measure of the covariance term in (16.62). We can also include any
variables in the VAR that are thought to be useful in predicting either ret
growth in dividends. Campbell (1991) uses a (3 x 3) VAR system. The varia
monthly (real) return h,+l on the value weighted New York Stock Exchange
dividend price ratio S, and the relative bill rate rrf. (The latter is defined as the
between the shortterm Treasury bill rate and its oneyear backward moving a
moving average element â€˜detrendsâ€™ the 1(1)interest rate series.) The VAR for
variables zf = (h,, S f , rr, ) in companion form is:
+ Wf+l
=&
Zf+l
where wf+l is the forecast error (zf+l  Etzf+l). can now go through the
We
hoopsâ€™ using (16.69) and (16.60) to decompose the variance in the unexp
return v;+˜ into that due to the DPV of news about expected dividends and
expected discount rates (returns). Using the VAR it is easily seen that:
00
h

= P,+1 Er1
V,+l Pjhf+l+j
j=1
00
pjAiwr+l = p(1 pA)â€™elâ€™Awf+l
= elâ€™
j=1
Since v:+˜is the first element of wf+l, that is elâ€™w,+l, we can calculate $â€™+1
identity:
+
+
d
q:+l = (elâ€™ p ( I  pA)â€™elâ€™A)w,+l
qr+l = $+1
Given estimates of the A matrix from the VAR together with the estim
variancecovariance matrix of forecast errors @ = E(w,+lw;+,) we now h
ingredients to work out the variances and covariances in the variance decompo
variances and covariances are functions of the A matrix and the variance
matrix 9. The persistence measure Ph can be shown (see Appendix 16.1) to
Ph = a Aâ€™wt+1 )/a(elâ€˜Aw,+ I ) = (Aâ€™@A)/ (elâ€™A9Aâ€˜el)
(
where Aâ€˜ = p ( I  pA)lelâ€™A and a  ) indicates the standard deviation of th
(
parentheses.
16.4.2 Results
An illustrative selection of results from Campbell (1991) is given in Tab
monthly data over the period 1952(1)1988(12). In the VAR equation for re
AR(1) model with a low R2 of 0.0281.) Equation (ii), Table 16.1, indicates th
dend price ratio is strongly autoregressive with a lagged dependent variable o
is a near unit root which might affect statistical inferences based on tests wh
the dividend price ratio is stationary (see below). The relative bill rate, equa
determined by its own lagged value and by the dividend price ratio. The R2 fro
equation for monthly returns ht+l has a relatively low R2 of 0.065 compared w
dividend yield (R2 = 0.96) and the relative bill rate (R2 = 0.55). The persisten
is calculated to be Ph = 5.7 (s.e. = 1.5) indicating that a 1 percent positive in
the expected return leads to a capital loss of around 6 percent, ceterisparibus
persistence is smaller in the earlier 1927(1)1951(12) period with Ph = 3.2,
News about future returns var(qh) account for over 75 percent of the varian
pected returns with news about dividends accounting for about 13 percent (
This leaves a small contribution due to the negative covariance term of about
News about expected future returns are negatively correlated with news abou
thus amplifying the volatility of stock returns to new information about these
mentals'. This is similar to the 'overreaction to fundamentals' found in othe
the predictability of stock returns. However, here the effect is small and not
significant.
Campbell (1991) notes that these VAR results need further analysis and h
the sensitivity of the variance decomposition to the VAR lag length, possi
a b l e 16.1 VAR Results for US Monthly Stock Returns 1952(1) 1988(
Dependent hr rr,
Variable (s.e.)
(s.e.)
0.048 0.490 0.724
(0.060) (0.227) (0.192)
(ii) (D/f')r+l 0.001 0.980 0.034
(0.003) (0.009)
(0.011)
(iii) rr,+1 0.013 0.017 0.739
(0.058)
(0.012) (0.052)
hl+l is the log real stock return over a month, (DIP) the ratio of total dividends paid in the p
is
the current stock price and rrr is the onemonth Treasury bill rate minus a oneyear backward m
Standard errors and test statistics are corrected for heteroscedasticity.
Source: Campbell (1991, Table 1, Panel C, p. 166).
Table 16.2 Variance Decomposition for Real Stock Returns 1952( 1) 1988
0.065 0.127 0.772 0.101
[O,˜l (0.164)
(0.016) (0.153)
R2 is the fraction of the variance of monthly real stock returns which is forecast by the VAR
marginal significance level for the joint significance of the VAR forecasting variables. ( . ) =
The VAR lag length is one.
Source: Campbell (1991, Table 2, p. 167).
using monthly returns or returns measured over three months.
(ii) The variance of news about future returns is far less important (a
future dividend is more important) when the dividend price ratio is exc
the VAR.
(iii) Performing a Monte Carlo experiment with h,+l independent of any oth
and a unit root imposed in the dividend price equation has 'a devastatin
the bias in the variance decomposition statistics reported in Table 16.2. W
ficially generated series where h,+l is unforecastable then unexpected s
are moved entirely by news about future dividends. Hence, var(qp'+,
should equal unity and the Rsquared for the returns equation should
For the whole sample 19271980 the latter results are strongly violate
they are not rejected for the postwar period.
(iv) Because of the sensitivity of the results to the presence of a unit roo
tests the actual dividend data and is able to reject the null of a unit ro
notes that even when d , has a unit root, none of the Monte Carlo ru
a greater degree of predictability of stock returns than that found in
post1950s data.
Cuthbertson et a1 (1995) repeat the Campbell analysis on UK annual data 191
the valueweighted BZW equity index. They include a wide array of variables
the key ones being the dividend price ratio and a measure of volatility. Th
evidence that persistence in volatility helps to explain persistence in expec
The contribution of the news about future returns to the movement in curren
about four times that of news about dividends (with the covariance term being
insignificant). These results broadly mirror those of Campbell (1991) on US
Campbell (1991) and Cuthbertson et a1 (1995) have shown that there is som
to support the view that in post1950s data, stock returns in a multivariate V
do appear to be (weakly) predictable and reasonably persistent. News concern
oneperiod returns does influence future returns and hence the variability in s
These time varying, predictable stock returns in the postwar data imply th
with a constant discount rate is likely to be a misleading basis for examina
EMH. Of course, this analysis does not provide an economic model of wh
returns E,h,+l depend on dividends and the relative bill rate, but merely pr
of statistical correlations that need to be explained. However, although Cam
Cuthbertson et a1 (1995) results show that the variability in stock prices is
be solely due to news about future cash flows the relative importance of
dividends and news about returns is difficult to pin down precisely. The resu
variance decomposition depend on the particular information set chosen (a
dividends have a unit root).
Summary
A summary of the key results has already been provided, so briefly the main
are as follows:
Under a variety of assumptions about the determination of oneperiod
evidence strongly suggests that stock prices do not satisfy the RVF and t
tional efficiency assumption of RE. These rejections of the EMH seem con
more robust than those found in the variance bounds literature (see Chap
Although monthly returns are barely predictable, the VAR approach in
returns at long horizons are predictable. (Thus complementing the Fama
(1988) results).
There is some persistence in oneperiod returns so that although the latte
predictable, nevertheless news about current returns can have quite a stron
on future returns and hence on stock prices. Thus there is some influence
mentals such as dividends and returns on stock prices but the quantitati
this relationship is not sufficient to rescue the RVF.
This chapter has 'examined the RVF under the assumption of time varying di
which depend on the riskfree rate, the growth in consumption and the varian
prices. The latter two variables can be interpreted in terms of a time varying ris
However, the measure of variance used in this chapter is relatively crude, bein
ditional variance. The next chapter examines the role of time varying conditiona
in explaining asset returns.
APPENDIX 16.1 RETURNS, VARIANCE DECOMPOSI
AND PERSISTENCE
This appendix does three things. It shows how to derive the CampbellShiller linearise
stock returns and the dividend price ratio. It then shows how these equations give rise t
variance decomposition and the importance of persistence in producing volatility in
Finally, it demonstrates how a VAR can provide empirical estimates of the degree of
1. Linearisation of Returns
The oneperiod, real holding period return is:
where P is the real stock price at the end of period t and D,+1is the real dividend
I
+
period t 1. (Both the stock price and dividends are deflated by some general pr
example the consumer price index.) The natural logarithm of (one plus) the real h
return is noted as hr+l and is given by:
If lower case letters denote logarithms then (2) becomes:
+D)
P =P/(P
and therefore p is a number slightly less than unity and k is a constant.
Using (3) and ( )
4:
+ +
h,+l = k PPf+l (1  P)d,+l  pr
Adding and subtracting d , in (6) and defining 6, = d ,  p, as the log dividend price ra
+ 6,  P4+1 + Ad,+]
hf+l = k
Equation (7) can be interpreted as a linear forward diference equation in 6:
Solving (8) by the forward recursive substitution method and assuming the transver
tion holds:
00
pj[ht+j+l ˜ d r + j + 1 1k/(l PI
6, = 
j=O
Equation (9) states that the log dividend price ratio can be written as the discounte
future returns minus the discounted sum of all future dividend growth rates less a c
If the current dividend price ratio is high because the current price is low then it mea
future either required returns h,+j are high or dividend growth rates Ad,+j are low, o
Equation (9) is an identity and holds almost exactly for actual data. However, it c
as an exante relationship by taking expectations of both sides of (9) conditional on the
available at the end of period t :
It should be noted that 6, is known at the end ofperiod t and hence its expectation
itself.
2. Variance Decomposition
+ 1 as:
To set the ball rolling note that we can write (10)for period t
=
&+l
From (8) we have:
Substituting from (10)and (11) and rearranging we obtain:
ffi â€˜x)
i=O i=O
j=O
j= 1 j=l
j= 1
Rearranging (14) we obtain our key expression for unexpected or abnormal returns:
j=O j=1
Equation (15) is the equation used by Campbell (1991) to analyse the impact of p
expected future returns on the behaviour of current unexpected returns hf+l  Erh,+l.
(15) can be written as:
d
 4:+1
= %+l
4+l
3
unexpected returns news about future news about future
3
[
[ ][
+
in period t 1 = dividend growth  expected returns
From (16) we have
The variance of unexpected stock returns in (17) comprises three separate components:
associated with the news about cash flows (dividends), the variance associated with the
future returns and a covariance term. Given this variance decomposition, it is possible
the relative importance of these three components in contributing to the variability o f s
Using (6) it is also worth noting that for p % 1 and no surprise in dividends:
pr+1  ErPt+l
h,+l  E,h,+l %
where P, is the stock price.
Campbell also presents a measure of the persistence of expected returns. This is d
ratio of the variability of the innovation in the expected present value offuture returns
error of q:+l) to the variability of the innovation in the oneperiod ahead expected
+
to be the innovation at time t 1 in the oneperiod ahead expected return
define
and the measure of persistence of expected returns is defined as:
Ph
Expected return follows an AR(1) process
The expected stock return needs to be modelled in order to carry out the variance de
(17) and to calculate the measure of persistence (19). For exposition purposes Campbe
and initially it is assumed that the expected stock return follows a univariate AR(1
calculation is then repeated using h,+l in the VAR representation. The AR(1) model
returns is:
E,+lhf+2= BE,hf+l + U f + l
(Ef+1  Er)hr+z = Uf+1
Leading (21) one period and taking expectations at time t we have:
EA+3 = BE,hf+2 = B2E1hf+l
and similarly:
+
Et+lhr+3 = BEr+lhr+Z = B(BErhr+l ur+l)
Subtracting (23) from (24) we obtain:
(E,+1  Et)hr+3 = B&+1
In general, therefore, we can write:
(Et+l  ˜ ˜ h t + j = lPi'ur+1
+
Using the definition of news about future returns, in (15) and (16) and using (25), we
j=l
Hence the variance of discounted unexpected returns is an exact function of the vari
period unexpected returns:
var(q;+l) = [P/U  PS,l2 var(u1+1)
Using (27), the measure of persistence Ph in (19) is seen to be
Ph = P/(l  p g ) % 1/(1 
Hence if B is close to unity which we can interpret as a high degree of persistence i
model, then Ph will also be large. Since pr+l Efpt+l = Q!+˜ (when qd = 0) and q
pB)]u,+l % [ l / ( l  /l)]u,+l then for the AR(1) case pl+l  E,P,+˜= Ph . u,+1. Hence
increase in uf+l leads to a Ph percent increase in qh and hence a P h percent unexpected
For the AR(1) case it can now be shown that even if we can only explain a sma
of the variability in oneperiod returns h,+l (i.e. returns are difficult to forecast), yet i
persistent, then news about returns can be very important in explaining stock price
short, the more persistent are expected returns, the more important is the variance of
future returns var(qf+l) in explaining unexpected returns v:+˜  E,h,+l (or unexpected
or losses, P f + l  EfPf+l).
R2 can be defined as the fraction of the variance of stock returns which is predictable
R2 = var(E,ht+l)/ var(h+1)
1  R2 = var(vF+,)/ var(h,+l)
R2/(1  R2) = var(E,hf+1)/var(vF+,)
Also from (20) the variance of E,h,+l is:
var(Erhr+1) = var(u,+1)/(I  B2)
Substituting (31) in (30) and solving for var(u,+l), we obtain:
var(u,+l) = (1  ˜ 2 ) v a r ( v ˜ + l ) R 2(R2)
/1
The lefthand side of (33) is one of the components of the variance decomposition
interested in and represents the importance of variance of discounted expected future re
to variance of unexpected returns (see equation (17)).
For monthly returns, a forecasting equation with R2 RZ 0.025 is reasonably represe
variance ratio (VR) in (33b) for B = 0.5 or 0.75 or 0.9 is VR = 0.08 or 0.18 or 0.49,
Hence for a high degree of persistence but a low degree of predictability, news about f
can still have a large (proportionate) effect on unexpected returns var(v;+,).
3. The VAR Model, Variance Decomposition and Persistence
The above univariate case neglects any interaction between news about expected retur
about dividends, that is the covariance term in (17). At a minimum an equation is requir
dividend growth. The covariance between the forecast errors (i.e. news) for dividend
those for returns can then be examined, and other variables in the VAR that it is th
help in forecasting these two fundamental variables can be included. It is possible th
the expected return along with some other forecasting variables in the context of a VA
carry out the variance decomposition for this multivariate case.
This section assumes the (m x 1) vector zr+l contains hr+l as its first element. The ot
+
in z1+1 are known at the end of period t 1 and are used to set up the following VAR
*
zr+1 = A Z r + Wf+l E(wf+lw;+l)=
where A is the companion matrix. The first element in wr+l is v:+˜.First note that:
+
= AJ+â€˜Zt AJw,+l
â‚¬r+lZf+j+l
E,z,+,+l = AJ+â€™z,
Subtracting (37) from (36) we get:
(Er+1  Er)zr+j+l = ˜ â€™ w t + l
Since the first element of zr is hr, if we premultiply both sides of (38) by elâ€™ (where el
row vector containing 1 as its first element with all other elements equal to zero) we
(Er+1  Er)hr+j+l = elâ€™Aâ€™wr+l
and hence:
j=l j=l
= elâ€™pA(1 pA)lwr+l = Aâ€™wr+l
where Aâ€™ = elâ€™pA(1 pA)â€™ is a nonlinear function of the parameters of the VAR. S
element of wr+l is v:+˜, using (16) and (40) we can write:
+ pAU  pA)â€™lwr+l
d
= ef[I =h
vr+1 +l
It can be seen from (40) and (41) that both unexpected future returns and unexp
dividends can be written as linear combinations of the VAR error terms where each
multiplied by a nonlinear function of the VAR parameters. Setting j = 1 in (39) an
we obtain:
ñòð. 11 