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It. would be useful if we could apply the VAR methodology to stock prices.
the rational valuation formula (RVF) is non-linear 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 ˜new™ 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
˜theoretical™ (log) stock price p: (or log dividend price ratio a;)(™) 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 cross-equation 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 ˜more forecastable™ 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 ˜fundamentals™: 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 one-pe
are persistent. ˜Persistent™ 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 one-period 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 one-period 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 one-period 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 one-period 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) one-period 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 ˜usual™ 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 price-dividend 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 ˜rearrangement™ 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 one-period 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 ˜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 ˜exact™ 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 one-period required rate of return on the stock. In
ical work to be discussed below much of the analysis concentrates on using
equations for the ˜fundamentals™ on the RHS of (16.12), namely one-period
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.

Gordon™s Growth Model
Further intuitive appeal can be provided for equation (16.11) for the dividend
version of the RVF by going back to Gordon™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 Gordon™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)™ and e2™ = (0, 1)™it follows that
If (16.18) is to hold for all zf then the non-linear restrictions (we ignore the co
since all data is in deviations from means) are:
f(a) = el' - e2'A(I - pA)-' = 0
Post-multiplying 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 one-period 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 one-perio
predictable from information available at time t unless the linear restrictio
the Wald test (16.21) hold(5).The economic interpretation of the non-linear W
discussed later in the chapter.
˜ j ( r f + -+ ˜
˜ AdF+j+l)= e2™A(I - PA)-™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 investor™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 log-linear identity (16.2) (with ht+l replaced by rr+1) can be rearranged


Solving (16.25) by recursive forward substitution gives a log-linear express
DPV of actual future dividends and discount rates which we denote pf.
00 00




Equation (16.26) uses actual (ex-post) values and is the log-linear 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 ˜in reality™ 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; = e2™f(a)
f ( a ) = A(I - pA)-™
For VAR lag length of one, f(a) is a (2 x 2) matrix which is a non-linear fun
a i j S of the VAR(7).Denote the second row of f(a) as the 2 x 1 vector [f21(a)
e2™f(a) where f 2 1 and f 2 2 are scalar (non-linear) 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 ˜larger™ 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 ˜incorrect™ (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 non-stationarity 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 non-linear restrictions in (16.19), which are equivalent to thos
for our two-variable VAR. If the non-linear restrictions are rejected then this m
to f24a) # 0. If so, then rd, influences


j=O
To the extent that the (weighted) sum of one-period returns is a form of multipe
then violation of the non-linear 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, = e3™A(I - pA)-™z,
- pA)-lz,
62, = -e2™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 one-period 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 one-period 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 ˜theoretical™ 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 ˜metrics™ 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:

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