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(i) The choice of alternative models for expected one-period 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 ˜me
such as the predictability of one-period 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 one-period 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 ˜r,™ is replaced by one of the above alternatives. Note tha
usual assumption of no-risk 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, doesn™t appea
is assumed constant) we have:
h, = 0.14161 - 0.012Adf
(0.057) (0.12)
1871-1986, 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 ˜correct™ 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 one-period 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 cross-equation
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 one-period returns r,+j and hence approximates a long-horizon
strong rejection of the Wald test is therefore consistent with the Fama and Fre
where non-predictability is more strongly rejected for long rather than sh
returns. We return to this issue below.
The results also make clear that even though one-period 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 one-period 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 ˜fully™ (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
˜fundamental™ variables Zt+l = (&+I * Ad,+l er+l but the basic VAR analy
)™
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 short-term 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 1901-1986 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 one-period (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.078d1-1
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 ˜smooth™ moving
.E
a
3 -3
B
a
.c
0
P
-I
-3.5



1910 1920 1930 1950 1960 1970 1980
1940

Figure 16.1 Log-Dividend 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, one-period 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 1927-1986 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 fundamentals™ 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, 1965-1993 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 non-linear rest
variance ratio between 6, and 6: and their correlation coefficient are consiste
RVF, when ˜volatility™ 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 1960-1980 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 10-year 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 10-ye
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 long-horizon return. Hence
of P , - P: on the information set 52, should yield zero coefficients if long-hor
are unforecastable. Fama and French use actual long-horizon 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
one-period 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 ˜long-horizon™ results in a multivariate
First, we define a sequence of one-period 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 two-period compound return from t to t + 2 is defined as


+ i is
hence in general the i-period return from t to t


+ H )we
Using lower case letters to denote logarithms and letting h: = ln(1
i-1


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:
i-1
hi,, = PJhl,r+j+l
j=O
Using (16.44) and the identity (16.4) for one-period returns hl,+l we have@
i-1
+k)
- P a t + l + j - Adr+j+l
hi,, = P™(&+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 doesn™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 'long-horizon', 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
one-period 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 post-m
by (I - pA)-'(I - piAi)we see that (16.49) also holds algebraically for any
manifestation of the fact that if one-period returns are unforecastable then so a
returns. Campbell and Shiller for the S&P index 1871-1987 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 (log-linearisation 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 Shiller™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 log-linear 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 one-period 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 ˜too variable™ 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 Shiller™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. Long-horizon returns (i.e. 3-5 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 regression-based tests on long-horizon 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 one-period 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 one-period 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 one-period 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 Campbell™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 = 2™3™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 ˜good news™ about dividends (i.e. v,+1 > 0) will ther
a substantial rise in current price if a! % 1: this is ˜persistence™ again. It is p
˜good news™ about dividends (i.e. v,+l > 0) may be accompanied by ˜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 non-linear 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+m-1+
/3>I + Er+m +- - -
hr+m = [/30/<1- B2Er+m-2

+ av,+m--l + a2 + - .-
+
Dt+m = [a0/(1- a)] Vt+m ˜t+m-2

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,+m™ o t + m I Q t ) = o E v ( l + *



= p,va,av(l + a!g + + . . .)

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