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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 non-linear sum o
variances and covariances. For given values of CT, a,, the variance of P,
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 non-linear. However, Campbell (1991) makes use of the log-linea
the one-period holding period return h f + l .
+ 6, - &+1 + Adr+l

Solving (16.58) forward we obtain the linearised version of the RVF in terms

- Adt+j+l) - k / ( l - PI
6, = d(ht+j+l

From (16.58) and (16.59) the surprise or forecast error in the one-period expe
can be shown to be (see Appendix 16.1):

ht+l - Etht+l = [Et+1 - Et] dAdr+j+l - [Er+1 - Er] P™hr+l+j
j= 1

which in more compact notation is:
d h
= q t + l - qr+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 ˜revisions
tions™. 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 ex-post 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 one-period ahead expected re
where is the innovation at time t

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 one-period ahead expected retur
may be interpreted as follows. Using (16.63) we see that an innovation
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

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)

+ 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 Campbell™s Ph statistic in (16
We can now use equation (16.67) to demonstrate that even if one-period 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 one-period 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 short-term Treasury bill rate and its one-year backward moving a
moving average element ˜detrends™ the 1(1)interest rate series.) The VAR for
variables zf = (h,, S f , rr, ) in companion form is:

+ Wf+l

where wf+l is the forecast error (zf+l - Etzf+l). can now go through the
hoops™ 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:
= P,+1 Er1
V,+l Pjhf+l+j

pjAiwr+l = p(1- pA)-™el™Awf+l
= el™

Since v:+˜is the first element of wf+l, that is el™w,+l, we can calculate $™+1
q:+l = (el™ p ( I - pA)-™el™A)w,+l
qr+l = $+1
Given estimates of the A matrix from the VAR together with the estim
variance-covariance 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 A™wt+1 )/a(el˜Aw,+ I ) = (A™@A)/ (el™A9A˜el)

where A˜ = p ( I - pA)-lel™A and a - ) indicates the standard deviation of th

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
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.)
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)
(iii) rr,+1 0.013 -0.017 0.739
(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
the current stock price and rrr is the one-month Treasury bill rate minus a one-year 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 R-squared for the returns equation should
For the whole sample 1927-1980 the latter results are strongly violate
they are not rejected for the post-war 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
post-1950s data.
Cuthbertson et a1 (1995) repeat the Campbell analysis on UK annual data 191
the value-weighted 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 post-1950s data, stock returns in a multivariate V
do appear to be (weakly) predictable and reasonably persistent. News concern
one-period returns does influence future returns and hence the variability in s
These time varying, predictable stock returns in the post-war 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).

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 one-period
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 one-period 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 risk-free 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.

This appendix does three things. It shows how to derive the Campbell-Shiller 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 one-period, real holding period return is:

where P is the real stock price at the end of period t and D,+1is the real dividend
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:
P =P/(P
and therefore p is a number slightly less than unity and k is a constant.
Using (3) and ( )
+ +
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:

pj[ht+j+l- ˜ d r + j + 1 1k/(l- PI
6, = -

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 ex-ante 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

2. Variance Decomposition
+ 1 as:
To set the ball rolling note that we can write (10)for period t


From (8) we have:

Substituting from (10)and (11) and- rearranging we obtain:
ffi ˜x)

i=O i=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:
- 4:+1
= %+l

unexpected returns news about future news about future
[ ][
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 one-period ahead expected
to be the innovation at time t 1 in the one-period ahead expected return

and the measure of persistence of expected returns is defined as:

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

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