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Expected returns and expected dividend growth: time
to rethink an established empirical literature
a
b
Jun Ma & Mark E. Wohar
a
Department of Economics, Finance and Legal Studies, Culverhouse College of Commerce &
Business Administration, University of Alabama, Tuscaloosa, AL 35487–0024, USA
b
Department of Economics, University of Nebraska at Omaha, Mammel Hall 332S, Omaha,
NE 68182–0286, USA
Published online: 07 Apr 2014.
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To cite this article: Jun Ma & Mark E. Wohar (2014) Expected returns and expected dividend growth: time to rethink an
established empirical literature, Applied Economics, 46:21, 2462-2476, DOI: 10.1080/00036846.2014.899674
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Applied Economics, 2014
Vol. 46, No. 21, 2462–2476, http://dx.doi.org/10.1080/00036846.2014.899674
Expected returns and expected
dividend growth: time to rethink an
established empirical literature
Jun Maa and Mark E. Woharb,*
Downloaded by [Deakin University Library] at 15:18 21 May 2014
a
Department of Economics, Finance and Legal Studies, Culverhouse College of
Commerce & Business Administration, University of Alabama, Tuscaloosa, AL
35487–0024, USA
b
Department of Economics, University of Nebraska at Omaha, Mammel Hall
332S, Omaha, NE 68182–0286, USA
This article examines various state-space and VAR model specifications to
investigate the contributions of expected returns and expected dividend growth
to movements in the price-dividend ratio. We show that both models involve
serious inference problems that need to be dealt with carefully. We propose
procedures that offer more reliable inference results, and the corrected inferences
indicate that the aggregate data of dividends and returns alone do not provide
strong enough evidence to support the notion that the expected returns dominate
the stock price variation. However, we show that an alternative measure of cash
flows termed the net payout by Larrain and Yogo (2008) appears to lend strong
support to the notion that the expected cash flow explains a large fraction of the
firm value variation. This finding remains robust in both state-space and VAR
decompositions with the corrected inference.
Keywords: stock price decomposition; state-space model; weak identification
JEL Classification: G12; C32; C58
I. Introduction
An important inquiry in finance over the last three decades
has been the extent to which expected returns and
expected cash flows contribute to movements in the
price-dividend ratio. The vector autoregression (VAR)
decomposition methodology has been employed, using
both the traditional definition of dividends as well as a
broader measure (termed net payout), and have found that
expected returns contribute to most of the movement in the
price-dividend ratio when traditional dividends are
employed, while expected net payout contributes to most
of the movement in the price-net payout ratio when this
broader definition of dividends is employed. When a
state-space methodology is employed, the results also
have expected returns as being the primary contributor to
movements in the price-dividend ratio when dividends are
used as the cash flow measure.
This article makes an important contribution by illustrating that the earlier empirical work in this area is subject
to severe inference problems which make their findings
unreliable. This article revisits this subject by focusing on
the important inference problems that have plagued the
extant literature. In particular when we apply proper inference measures we find that the aggregate returns and
dividends data cannot provide sufficient statistical evidence to support the notion that it is expected returns
that explains the majority of the fluctuation in the
*Corresponding author. E-mail: [email protected]
2462
© 2014 Taylor & Francis
2463
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Expected returns and expected dividend growth
price-dividend ratio when dividends are used as the cash
flow measure. This is true whether we use the VAR
decomposition approach or the state-space modelling
approach. This work also extends Ma and Wohar (2013)
in a number of dimensions by exploring different model
specifications of the state-space model and by using additional measures of the cash flows. In particular, we find
that when we employ the broader measure of dividends
(net payout) and employ proper inference measures, we
find that expected net payout growth contributes most to
movements in the price-net payout ratio and expected
returns contribute nothing.
The objective we set out in this article is challenged by
the fact that neither expected cash flows nor expected
discount rates are observable. Two primary approaches
have been proposed to capture these unobserved expectations of future variables: VAR and the state-space model.
Remarkably, applications of the two approaches using
dividends as the cash measure have reached the same
conclusion: almost all aggregate stock price variations
are driven by discount rate news and almost none by
cash flow news. Notable papers among the large body of
literature that adopts the VAR methodology include
Campbell and Shiller (1988a, 1988b), Campbell (1991),
Shiller and Beltratti (1992), Cochrane (1992), and
Campbell and Ammer (1993). Examples of work that
apply the state-space methodology are Balke and Wohar
(2002) and Binsbergen and Koijen (2010).
As a preview of our work, we first estimate the state
space model of dividend growth and price-dividend as in
Binsbergen and Koijen (2010). Although the point estimates indicate that the expected return is very persistent
and explains the bulk of the price-dividend variations,1
such estimated state space model are plagued by a small
expectation shock relative to the corresponding realized
shock in the return process. The small signal-to-noise ratio
of the state-space model implies weak identification (in the
sense of Nelson and Startz (2007)) and that the standard
inference tends to underestimate the uncertainty of the
contribution estimate. The corrected inference based on
Ma and Nelson (2012) suggests that the contribution of
expected returns to movements in the price-dividend ratio
is statistically insignificant. This finding is consistent with
the intuition that when the predictable component of
returns is small the uncertainty of its contribution is
large. We show that a stylized VAR decomposition
equipped with nonparametric bootstrapped inferences
yields a similar result.
To bring to bear more information to this central issue in
asset pricing, we consider an alternative measure of the
cash flows. Recently, Larrain and Yogo (2008) construct a
1
broader measure of the cash flows termed the net payout
that is the sum of dividends, interest, equity repurchases
net of issuance, and debt repurchase net of issuance, and
they find that the ratio of net payout to asset value is
primarily explained by the expected future cash flows
growth. We extend our methodology to their unique data
set and show that the finding that the expected future cash
flows dominate the firm value variations remains robust in
both of the estimated state-space model and VAR decomposition with corrected inference.
Finally, one important contribution of our article is to
present a framework and method to deal with the problem
of weak identification in state-space models which results
in seemingly significant test statistic while actually the
parameters are being estimated with little precision. We
show how one might determine whether a particular process is weakly identified and then present methods which
yield proper inferences. These are important as the statespace methodology has become more and more prevalent
in the literature.
II. The State-Space Model of Stock Price
Decomposition2
Campbell and Shiller (1988a, b) derive the following wellknown log-linear approximation for the log return:
rtþ1 " κ þ Δdtþ1 þ ρ # pdtþ1 $ pdt
(1)
where Δdtþ1 is the log dividend growth, pdtþ1 the log
price-dividend ratio, and the two constants ρ and κ depend
on the steady-state log price-dividend ratio. Forward iteration gives rise to the result that today’s price-dividend ratio
is the sum of expectations of (discounted) future dividends
growth and future returns:
pdt ¼
1
X
!
"
κ
ρj Δdtþ1þj $ rtþ1þj
þ Et
1$ρ
j¼0
(2)
In order to study which of the two components (future
dividends growth and future returns) on the right-hand
side of Equation 2 is more responsible for the pricedividend variations on the left-hand side, one has to have
an estimate the expectations of future dividend growth and
future returns. One standard approach in the literature is to
rely on the VAR model to extract these unobserved expectations (see, for example, Campbell and Shiller (1988a)).
Another line of research such as by Balke and Wohar
(2002), and Binsbergen and Koijen (2010) instead
Balke and Wohar (2002) find the more general result that the factor with the largest degree of persistence is the factor that contributes
most to movements in the price-dividend ratio.
2
The methodology in this section draws heavily from Ma and Wohar (2014).
2464
J. Ma and M. E. Wohar
employed the state-space approach to directly model and
estimate the expectation processes. An alternative to the
VAR model, the state-space approach attempts to model
expectations directly as latent factors. One other notable
advantage of the state-space approach is its ability to
capture potentially long-lasting serial correlations that a
VAR model with finite number of lags cannot do, because
the state-space model usually implies moving averaging
terms in its reduced form.
Et
pdt ¼
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In principle, we can decompose dividend growth and
returns into two components: the expected values and
any difference between the expectations and the actual
values:
Δdtþ1 ¼ gt þ εdtþ1
(3)
rtþ1 ¼ μt þ εrtþ1
(4)
where gt ¼ Et ½Δdtþ1 ' is the conditional expectation of the
dividend growth and εdtþ1 is the difference between the
expected value and the actual one, i.e. the so-called news
shock; likewise, μt ¼ Et ½rtþ1 ' denotes the expected value
of returns and and εrtþ1 is the news shocks. The news
shocks are assumed to be i.i.d, respectively. However,
the expectations may contain important dynamics and
thus are modelled as stationary AR processes:
fg ðLÞ # ðgt $ cg Þ ¼ εgt
(5)
fμ ðLÞ # ðμt $ cμ Þ ¼ εμt
(6)
where fg ðLÞ and fμ ðLÞ are lag polynomials with p lags.
The expectations shocks εgt and εμt are assumed to i.i.d,
respectively.
To facilitate calculations of the components in Equation
2, vectorize the state variables to obtain the companion
form of (5) and (6):
g
þ Vtg
Ztg ¼ Ag # Zt$1
(7)
μ
þ Vtμ
Ztμ ¼ Aμ # Zt$1
(8)
0
Ztμ ¼
where
Ztg ¼ ð gt $ cg ; # # # ; gt$pþ1 $ cg Þ ,
!
"0
μt $ cμ ; # # # ; μt$pþ1 $ cμ , Vtg ¼ ð εgt ; # # # ; 0 Þ0 ,
0
Vtμ ¼ ð εμt ; # # # ; 0 Þ ; Ag and Aμ are the corresponding
loading matrix. It is then easy to derive:
1
X
!
"
!
"$1
cg
0
ρj Δdtþ1þj ¼ e1 # I $ ρ # Ag #Ztg þ
1
$ρ
j¼0
(10)
where e1 is a column vector that has 1 as its first element
and zero elsewhere.
Substituting in (9) and (10), we rewrite Equation 2 as:
Model estimation and the variance decomposition
Et
1
X
!
"
!
"$1
cμ
0
ρj rtþ1þj ¼ e1 # I $ ρ # Aμ #Ztμ þ
1
$ρ
j¼0
!
"$1
cg $ cμ
κ
0
þ
þ e1 # I $ ρ # Ag
1$ρ
1$ρ
!
"$1
0
g
# Zt $ e1 # I $ ρ # Aμ #Ztμ
(11)
As Cochrane (2008b) pointed out, the identity (2) implies
a restriction that all shocks in Equations 3–6 have to
satisfy. Such restriction in our general context is given
below:
"
!
"$1
0
g
r
d
# ρVtþ1
εtþ1 ¼ εtþ1 þ e1 # I $ ρ # Ag
!
"$1
μ
$ I $ ρ # Aμ #ρVtþ1
#
(12)
One practical implication of the above restriction is that
one only needs to select two observed variables out of the
three, namely, dividend growth, returns and price-dividend ratio, to estimate the model and the rest of the
model can be backed out. To be comparable, we first
follow Binsbergen and Koijen (2010) to choose the pair
½ Δdtþ1 pdtþ1 '0 with one lag to estimate the model. The
resulting state-space model consists of the following measurement equation:
#
Δdtþ1
pdtþ1
$
¼
#
$
#
0
cg
þ
A
B2
1
0
0
$B1
2
3
$ gtþ1 $ cg
7
1 6
6 gt $ c g 7
0 4 μtþ1 $ cμ 5
εdtþ1
(13)
The transition equation that describes the dynamic evolution of the state variables is:
3 2
gtþ1 $ cg
fg;1
6 gt $ c g 7 6 1
6
7 6
6
7¼6
4 μtþ1 $ cμ 5 4 0
2
εdtþ1
0
0
0
0
0 fμ;1
0
0
2 g 3
εtþ1
6 0 7
6
7
þ6 μ 7
4 εtþ1 5
0
32
3
gt $ c g
0
6
7
07
76 gt$1 $ cg 7
76
7
0 5 4 μt $ c μ 5
εdt
0
εdtþ1
(9)
(14)
2465
Expected returns and expected dividend growth
where the constants implied by one
cg $cμ
κ
1
1
þ 1$ρ
; B1 ¼ 1$ρf
; B2 ¼ 1$ρf
.
A ¼ 1$ρ
μ;1
lag
are:
g;1
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To make the model general, we allow for possible
correlations between shocks (see Morley et al. (2003)).
However, Cochrane (2008b) and Rytchkov (2008) show
that one correlation in this particular model is not identifiable. Since Binsbergen and Koijen (2010) choose to
restrict σ dg ¼ 0, we follow the same strategy to make our
results comparable. Finally, the variance–covariance
2 2
3
σd
5.
matrix of three shocks is given by 4 σ dμ σ 2μ
2
σ dg σ μg σ g
We retrieve the CRSP market indices of NYSE/AMEX/
NASDAQ stocks as the market portfolio and follow
Hansen et al. (2008) aggregation procedure to construct
the annual equity return and dividend growth series for the
period 1946–2011.3 Using this data we estimate the above
state-space model using the maximum likelihood estimation by following the procedure in Kim and Nelson
(1999). Table 1 reports the estimation results.4
We note that most of the estimations results in Table 1
are similar to Binsbergen and Koijen (2010).5 What is of
particular importance is the very persistent process for the
expected return: the AR(1) parameter estimate is not only
^ ¼ 0:9233) but its SE appears quite small, givlarge (f
μ;1
ing rise to a very tight 95% confidence interval [0.8361,
1.0105]. On the other hand, the expected dividend growth
is far less persistent. We also conduct a test again the null
hypothesis fμ;1 ¼ fg;1 , and the LR test statistic clearly
rejects the null of equal persistence.6
We employ the approach of Cochrane (2008a) to
decompose the price-dividend variation into two covariance components: the covariance of the price-dividend
ratio with future dividend growth and future returns:
Varðpdt Þ ¼ Cov
1
X
ρj Δdtþ1þj ; pdt
j¼0
þ Cov $
1
X
j¼0
Table 1. State-space estimation result (Model 1: dividend
growth and log price-dividend, one lag)
!
ρj rtþ1þj ; pdt
!
Parameter estimates
cg
fg;1
cμ
fμ;1
σd
σμ
σg
ρdμ
ρμg
Log-likelihood value
Implied parameters
σr
ρdr
ρμr
ρgr
κ
ρ
ZILC
' parameters
σμ
'σ r
σg
σd
0.0568 (0.0110)
0.2889 (0.1253)
0.0824 (0.0144)
0.9233 (0.0445)
0.0030 (0.0080)
0.0160 (0.0072)
0.0633 (0.0055)
−0.9736 (0.0345)
0.2283 (0.1470)
116.8797
0.1567 (0.0138)
0.9448 (0.0405)
−0.8450 (0.0636)
0.3277 (0.1167)
0.1342
0.9701
0.1022
21.2698
Notes: Annual data from 1946 to 2011 is used; numbers in
parenthesis are SEs. The identification condition is ρdg ¼ 0. cg
is the average dividend growth rate; fg;1 is the AR(1) parameter
in the expected dividend growth process; cμ is the average return;
fμ;1 is the AR(1) parameter in the expected return process; σ d is
the size of the news shock to the realized dividend growth in
Equation 3; σ r is the size of the news shock to the realized return
in Equation 4; σ g is the size of the shock to the expected dividend
growth in Equation 5; σ μ is the size of the shock to the expected
return in Equation 6; ρdμ is the correlation between the news
shock to realized dividend growth and the shock to the expected
return; ρμg is the correlation between the shock to the expected
return and the shock to the expected dividend growth.
growth. With the specification of one lag in our model, we
further obtain:
(15)
These two covariance terms reflect the ability of the pricedividend ratio to predict future returns and future dividend
Cov
1
X
j¼0
2
j
ρ Δdtþ1þj ; pdt
!
3
Covðgt ; μt Þ
6 Varðgt Þ
&%
&7
¼ 4%
5
&2 $ %
1 $ ρfg;1 1 $ ρfμ;1
1 $ ρfg;1
(16)
3
Hansen, Heaton and Li’s data sources and procedures to compute return and dividend series are available on Nan Li’s website: http://
www.bschool.nus.edu.sg/staff/biznl/ . Note that this way of backing out and aggregating dividends is different from the way in Ma and
Wohar (2013) that backs out the annual dividends directly from the annual returns as in Cochrane (2011).
4
We also estimate the model using quarterly data and obtain similar results. These results are available upon request.
5
We focus on the estimation of the cash-reinvestment dividends model since the market-reinvestment dividends model yields similar
variance decomposition of the price-dividend variation.
6
We also estimate most of our models using the longer sample period starting from the year 1927. Except for some notable differences of
some correlation estimates, the major result that the returns contribution dominates remains the same. These results are available upon
request.
2466
J. Ma and M. E. Wohar
Cov $
2
1
X
j¼0
ρj rtþ1þj ; pdt
!
.3
Covðgt ; μt Þ
6 Varðμt Þ
&%
&7
¼ 4%
5
&2 $ %
1 $ ρfg;1 1 $ ρfμ;1
1 $ ρfμ;1
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.2
3
.1
(17)
The above results highlight the fact that the relative size of
the two contributions in (15) depends critically on the
persistence of the expectations processes through their
1
1
loadings
2 and
2 , besides the variances
ð1$ρfg;1 Þ
ð1$ρfμ;1 Þ
and covariances. It is also important to notice that a rise of
the persistence parameters fg;1 and fμ;1 increases both the
contributions loadings and the variances themselves. The
impact of the increasing persistence on such decomposition will become extremely pronounced given the nonlinear feature of the loading functions. A similar feature is
explored in the long-run risk literature (see Ma (2013)).
Therefore, the much persistent expected return process
tends to imply a very large contribution to the stock
price variations. In particular, we present the variance
decomposition results based on the above state-space
model in Table 2. Since a rise in the price-dividend ratio
typically predicts a decline in future returns, we report the
negative of that covariance as a normalization to facilitate
comparison. The results reveal that most of the pricedividend variation seems to come from its comovement
with the expected future returns.
We plot the filtered estimates of the expected dividend
growth in Fig. 1, together with the actual dividend growth
series. It is interesting that since the estimation results end
up revealing a very small amount of noise relative to the
signal, the expected dividend growth turns out to be
almost identical to the actual dividend growth series.
When we plot the difference between these two series,
essentially the estimated news shock ^εd , in Fig. 2, the
difference becomes visible. The small difference between
these two series is consistent with a very small amount of
noise relative
to the signal, or the large signal-to-noise
'
ratio σ g σ d ¼ 21:2698, as reported in Table 1. When we
plot the filtered estimates of the expected return together
Table 2. State-space decomposition of price-dividend ratio
(Model 1: dividend growth and log price-dividend, one lag)
Variance decomposition of price-dividend ratio (%)
Covariance of dividend and price-dividend
Negative of covariance of returns and price-dividend
8.72
91.28
Notes: Annual data from the period 1946 to 2011 is used. The
identification condition is ρdg ¼ 0. By construction contributions
add up to 100%.
.0
–.1
–.2
–.3
50
55
60
65
70
75
80
85
90
95
00
05
10
Realized Dividend Growth
Expected Dividend Growth
Fig. 1. State-space model 1: realized dividend growth and
expected dividend growth (both demeaned)
Note: Annual data from the period 1946 to 2011 is used.
.006
.004
.002
.000
–.002
–.004
–.006
–.008
50 55 60 65 70 75 80 85 90 95 00 05 10
Realized Dividend Growth - Expected Dividend Growth
Fig. 2. State-space model 1: difference between the realized
dividend growth and expected dividend growth (both
demeaned)
Note: Annual data from the period 1946 to 2011 is used.
with the actual return series in Fig. 3, however, the picture
is quite different. In particular, the expected return is very
persistent, and there is a substantial difference between the
two. This pattern, interestingly, reflects the
' other extreme,
that is, a small signal-to-noise ratio σ μ σ r ¼ 0:1022, as
again reported in Table 1. We present the historical decompositions of the price-dividend ratio in Fig. 4. The pricedividend ratio appears to be primarily explained by the
expected return contribution, which is consistent with the
overall variance decomposition in Table 2. The primary
reason that the expected return tracks the price-dividend
movements so well is its high persistence.
2467
Expected returns and expected dividend growth
particular the persistence parameter. Intuitively, small signal indicates a large amount of uncertainty around the
parameter estimates. However, Nelson and Startz (2007)
show that the standard inference is often misleading and
unreliable under this condition that they named the zeroinformation-limit-condition or ZILC. Recall that in
Table 1 the large persistence estimate of the expected
return is accompanied by a very small SE. This is counter-intuitive and needs to be corrected.
We first show explicitly that the ZILC holds in this
context. The state-space representation for return process
implies a reduced-form ARMA process:
.4
.2
.0
–.2
–.4
–.6
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50 55 60 65 70 75 80 85 90 95 00 05 10
Realized Equity Return
Expected Equity Return
Fig. 3. State-space model 1: realized equity return and
expected equity return (both demeaned)
Note: Annual data from the period 1946 to 2011 is used.
%
& !
"
1 $ fμ;1 L # rtþ1 $ cμ ¼ εrtþ1 $ fμ;1 # εrt þ εμt
(18)
Granger and Newbold’s (1986) theorem implies an
ARMA(1,1) representation:
%
& !
"
1 $ fμ;1 L # rtþ1 $ cμ ¼ ð1 $ θLÞ # utþ1
(19)
Assuming uncorrelated shocks, the mappings between the
parameters in the ARMA and state-space models are done
by computing and equating the variance and auto-covariance (γ0 ; γ1 ) of the right-hand sides of Equations 18
and 19:
1.2
0.8
0.4
0.0
γ0 : ð1 þ f2μ;1 Þσ 2r þ σ 2μ ¼ ð1 þ θ2 Þσ 2u
(20)
–0.4
γ1 : fμ;1 σ 2r ¼ θσ 2u
(21)
The unique MA parameter θ that implies invertibility is
given by:
–0.8
–1.2
50 55 60 65 70 75 80 85 90 95 00 05 10
Log Price–Dividend Ratio
Historical Contribution of Dividend Growth
Historical Contribution of Returns
Fig. 4. State-space model 1: historical decomposition of the
log price-dividend ratio (all demeaned)
Note: Annual data from the period 1946 to 2011 is used.
Weak identification and corrected inference
One important feature about the above state-space model
based on the rational expectation concept is that the time
series dynamics of the actual series and its expectation can
be drastically different as long as the signal-to-noise ratio
is small. This was pointed out as early as in Nelson and
Schwert (1977). However, under this circumstance, the
small signal-to-noise ratio raises a serious issue about the
statistical uncertainty around the parameter estimates, in
θ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
2
>
1þf2μ;1 ÞþS 2 '$ ð1$f2μ;1 Þ þS 4 þ2ð1þf2μ;1 Þ#S 2
>
½
ð
<
>
>
:
2#fμ;1
0;
; for fμ;1 ! 0
for fμ;1 ¼ 0
(22)
.
where the signal-to-noise ratio is denoted by S ¼ σ μ σ r.
When the signal-to-noise ratio becomes arbitrarily small,
the AR and MA roots become arbitrarily close to each
other, resulting in a near root cancellation of the ARMA
process:
%
&
lim fμ;1 $ θ ¼ 0
S!0
(23)
As Nelson and Startz (2007) show, the ARMA model with
near root cancellation is subject to ZILC and under this
circumstance the estimated SE for either AR or MA estimate will appear too small, leading to a large size
J. Ma and M. E. Wohar
distortion of the standard t-test. Even worse, the persistence estimate often tends to be upward biased in this
situation, reflecting the ‘pile-up’ issue. We implement a
simple Monte Carlo simulation to give readers some idea
about the size distortion of a standard t-test of the persis^ when the return process is weakly identent estimate f
μ;1
tified. In this experiment, we generate data from the
estimated state-space model using the values of parameter
estimates in Table 1, except for the persistence parameters
for both expected dividend growth and expected returns
that we specify at a low level, i.e. fμ;1 ¼ fg;1 ¼ 0:1 as the
null hypothesis. For each simulated set of data, we estimate the model and calculate the t-statistic of the persistence estimates. We find that the standard t-test for the
return persistence parameter fμ;1 is indeed oversized and
rejects the null about 42% of the time using the 5% critical
^ and its estimated SE in
value. Furthermore, we plot f
μ;1
Fig. 5. There is clearly a negative correlation between the
estimated SE and the absolute size of the persistence
estimate on both ends of the graph. This pattern is in
particular troublesome as it suggests that when the persistence parameter is estimated to be high, it is very likely the
time that its estimated SE is too small.
Ma and Nelson (2012) propose a valid testing strategy
in the presence of a small signal-to-noise ratio based on a
linear approximation to the exact test of Fieller (1954) for
a ratio of regression coefficients, and they show that such
test is also in the LM principle of Breusch and Pagan
(1980) since the test is evaluated under the null. They
find that this test has much improved finite sample
estimated standard error of expected return persistence estimate
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2468
performance compared with the standard t-test when the
model is weakly identified. Furthermore, they also evaluate the power performance of this test under the alternative
and find that this test has a good power to detect a departure from the null in the DGP.
The valid testing strategy for fμ;1 involves two steps.
First, estimate the following reduced-form VARMA for
the dividend growth and price-dividend under the
null fμ;1 ¼ f0 :
"
#
#"
ð1 $ fg;1 LÞ
0
Δdtþ1
0
ð1 $ fg;1 LÞð1 $ fμ;1 LÞ
pdtþ1
"
#
#"
ð1 $ θ1 LÞ
0
u1tþ1
¼
0
ð1 $ θ2 LÞ u2tþ1
(24)
where the shocks ½ u1tþ1 u2tþ1 '0 are allowed to be correlated. In the second step, compute the t-statistic for the null
λ ¼ 0 in the following regression:
~ LÞpdtþ1 ¼ τ #
ð1 $ f
g;1
1
X
f0i$1 ~u2tþ1$i
i¼1
1
X
þλ#
i¼2
ði $ 1Þ # f0i$2 ~u2tþ1$i
(25)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
expected return persistence estimate
0.6
0.8
1
Fig. 5. State-space model 1: scatter plot of the expected returns persistence estimates and their estimated SEs in the Monte
Carlo simulation
2469
Expected returns and expected dividend growth
2.5
2
Reduced-form test
95% Critical Value
95% Critical Value
1.5
reduced-form test
1
0.5
0
–0.5
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–1
–1.5
–2
–1
–0.8
–0.6
–0.4
–0.2
0
πµ1
0.2
0.4
0.6
0.8
1
Fig. 6. State-space model 1: the 95% confidence interval for expected return persistence based on the reduced-form test
Note: Annual data from the period 1946 to 2011 is used.
Equation 25 is obtained by taking a first-order Taylor
expansion of the second equation of (24) around the null
~
fμ;1 ¼ f0 . Here, τ ¼ fμ;1 $ θ2 , λ ¼ τ # ðfμ;1 $ f0 Þ, f
g;1
and ~
u2tþ1 are the restricted estimates under the null from
the first step. The reduced-form test statistic for the null
fμ;1 ¼ f0 is a standard t-statistic for testing the null λ ¼ 0.
We use the same simulated data in the above Monte
Carlo experiment to compute the reduced-form test statistic for fμ;1 and find that the reduced form test rejects the
null of low persistence, i.e. fμ;1 ¼ 0:1 about 18% of the
time at the 5% level, which, although not perfect, is much
better than that of the standard t-test which rejects the null
about 42% of the time. Furthermore, Ma (2013) finds that
this test has good power in detecting a departure from the
null in the bivariate VARMA model even when the model
is weakly identified.
However, we would like to emphasize that the weak
identification does not condemn all state-space models.
Specifically, a particular process of the state-space model
becomes weakly identified only when the signal-to-noise
ratio of that process is small. For processes that have large
signal-to-noise ratio, the issue of weak identification is not a
practical concern, and the standard inference likely remains
valid. For example, in our estimated Model 1, the dividend
growth
process has a signal-to-noise ratio as large as
'
σg
¼
σ d 21:2698 (see Table 1). Therefore, the standard
t-test for the persistence estimate of the expected dividend
growth should function just fine. In our simulation experiment, the standard t-test for the persistence parameter of the
expected dividend growth fg;1 rejects the null only about
14% of the time using the 5% critical value, which is indeed
about as good as the reduced-form test.
Numerically inverting the above reduced-form test statistic, we obtain the 95% confidence interval for the persistence parameter fμ;1 in Fig. 6. The reduced-form test
indicates a much wider confidence interval than that of the
standard test. This casts a serious doubt on the large
contribution of the expected returns to the price-dividend
variation, given that such decomposition critically
depends on the large persistence estimate.
III. Robustness of the State-Space Model:
Alternative Specifications
Although in theory any combination of the two variables
out of Δdtþ1 ; pdtþ1 ; rtþ1 should produce identical results
up to the Campbell–Shiller approximation errors, different
state-space models may implicitly impose different nonlinear restrictions and thus produce different decomposition results. In this section, we explore alternative
combinations of the variables and present the resulting
variance decomposition.
The combination of returns and price-dividend ratio
In this model, we use returns along with the price-dividend
ratio. The state-space model consists of the measurement
equation:
2470
#
J. Ma and M. E. Wohar
rtþ1
pdtþ1
$
#
$ #
0
cμ
þ
¼
A
$B1
1
0
0
B2
2
3
Table 3. State-space estimation result (Model 2: returns and
log price-dividend, one lag)
$ μtþ1 $ cμ
7
1 6
6 μt $ c μ 7
0 4 gtþ1 $ cg 5
εrtþ1
(26)
And the transition equation:
2
μtþ1 $ cμ
2
3
fμ;1
6 μ $ cμ 7 6 1
6 t
7 6
6
7¼6
4 gtþ1 $ cg 5 4 0
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εrtþ1
0
0
0
0
0
fg;1
0
0
μ 3
εtþ1
6 0 7
6
7
þ6 g 7
4 εtþ1 5
2
0
32
3
μ t $ cμ
0
6
7
07
76 μt$1 $ cμ 7
76
7
0 5 4 gt $ c g 5
εrt
0
εrtþ1
where the constants are given by A ¼
B1 ¼
1
1$ρfμ;1
; B2 ¼
(27)
κ
1$ρ
1
1$ρfg;1
þ
cg $cμ
1$ρ
;
, and the variance–covariance
2 2
3
σr
5.
matrix of the three shocks is 4 σ rg σ 2g
σ rμ σ μg σ 2μ
To impose the same identification restrictionσ dg ¼ 0,
we use the nonlinear relationship (12) with one lag
specification:
%
&$1
εgtþ1 $ ρ
εrtþ1 ¼ εdtþ1 þ ρ # 1 $ ρfg;1
%
&$1
εμtþ1
# 1 $ ρfμ;1
%
&$1
σ 2g þ ρ
σ dg ¼ σ rg $ ρ # 1 $ ρfg;1
%
&$1
σ μg
# 1 $ ρfμ;1
(28)
(29)
Table 3 presents the estimation results of this model,
and Table 4 presents the variance decomposition
results. It is noticeable that the results of this model
specification are similar to the baseline model. In
particular, the expected return is very persistent but
the expected dividend growth is not. As a result, the
expected returns appear to dominate in explaining the
price-dividend variations.
The combination of dividend growth and returns
Using the combination of dividend growth and returns, we
set up the state-space model that consists of the measurement equation:
Parameter estimates
cg
fg;1
cμ
fμ;1
σr
σμ
σg
ρrμ
ρμg
Log-likelihood value
Implied parameters
σd
ρdr
ρdμ
ρrg
κ
ρ
ZILC
' parameters
σμ
'σ r
σg
σd
0.0586 (0.0113)
0.3080 (0.1174)
0.0900 (0.0138)
0.9346 (0.0413)
0.1598 (0.0140)
0.0143 (0.0067)
0.0619 (0.0055)
−0.8473 (0.0610)
0.2342 (0.1429)
115.0644
0.0069 (0.0079)
0.9481 (0.0392)
−0.9722 (0.0342)
0.3179 (0.1156)
0.1342
0.9701
0.0895
8.9272
Notes: Annual data from the period 1946 to 2011 is used;
numbers in parenthesis are SEs. The identification condition
is ρdg ¼ 0. cg is the average dividend growth rate; fg;1 is the
AR(1) parameter in the expected dividend growth process; cμ
is the average return; fμ;1 is the AR(1) parameter in the
expected return process; σ d is the size of the news shock to
the realized dividend growth in Equation 3; σ r is the size of
the news shock to the realized return in Equation 4; σ g is the
size of the shock to the expected dividend growth in
Equation 5; σ μ is the size of the shock to the expected return
in Equation 6; ρrμ is the correlation between the news shock
to realized return and the shock to the expected returns; ρμg
is the correlation between the shock to the expected return
and the shock to the expected dividend growth.
Table 4. State-space variance decomposition (Model 2:
returns and log price-dividend, one lag)
Variance decomposition of price-dividend ratio (%)
Covariance of dividend and price-dividend
Negative of covariance of returns and price-dividend
1.64
98.36
Notes: Annual data from the period 1946 to 2011 is used. The
identification condition is ρdg ¼ 0. By construction all contributions add up to 100%.
#
Δdtþ1
rtþ1
$
¼
#
$ #
cg
1 0
þ
cμ
0 1
1
1
0
B2
2
3
gt $ c g
$ 6 μt $ c μ 7
7
0 6
6 εdtþ1 7
6
7
$B1 4 r
εtþ1 5
εμtþ1
(30)
2471
Expected returns and expected dividend growth
And the transition equation:
2
gt $ cg
3
2
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0
fg;1
6μ $c 7 6
fμ;1
6 0
6 t
μ7
6 d
7 6
6 εtþ1 7 ¼ 6 0
0
6
7 6
6 r
7 6
ε
0
0
4 tþ1 5 4
μ
εtþ1
0
0
2
3
0
6 0 7
6
7
6 d 7
7
ε
þ6
6 tþ1 7
6 g 7
4 εtþ1 5
εμtþ1
0
0
1
0
0
0
0
0
0
0
Table 5. State-space estimation result (Model 3: dividend
growth and returns, one lag)
32
3
0
gt$1 $ cg
76
7
1 76 μt$1 $ cμ 7
76
7
6
7
07
εdt
76
7
76
7
εrt
0 54
5
0
εμt
κ
where the constants are given by A ¼ 1$ρ
þ
(31)
cg $cμ
1$ρ
;
1
1
; B2 ¼ 1$ρf
, and the variance–covariance
B1 ¼ 1$ρf
μ;1
g;1
2 2
3
σd
5. We again
matrix of three shocks is 4 σ dμ σ 2μ
2
σ dg σ μg σ g
impose the same identification restriction σ dg ¼ 0.
Table 5 reports the estimation results of this model. The
most interesting result is that now the persistence parameter for the expected returns becomes less persistent
(fμ;1 ¼ 0:8530). Furthermore, although the expected
returns remain primarily responsible for the price-dividend variation (see Table 6), its contribution becomes
much smaller compared with the first two models.
Parameter estimates
cg
fg;1
cμ
fμ;1
σd
σμ
σg
ρdμ
ρμg
Log-likelihood value
Implied parameters
σr
ρdr
ρμr
ρgr
κ
ρ
ZILC
' parameters
σμ
'σ r
σg
σd
0.0559 (0.0094)
0.2958 (0.1146)
0.1029 (0.0112)
0.8530 (0.0610)
0.0007 (0.0077)
0.0276 (0.0101)
0.0630 (0.0055)
−0.9583 (0.5869)
0.2850 (0.1362)
116.4949
0.1551 (0.0135)
0.9633 (0.5838)
−0.8472 (0.0580)
0.2678 (0.1185)
0.1342
0.9701
0.1779
96.9542
Notes: Annual data from the period 1946 to 2011 is used; numbers in parenthesis are SEs. The identification condition is
ρdg ¼ 0. cg is the average dividend growth rate; fg;1 is the
AR(1) parameter in the expected dividend growth process; cμ is
the average return; fμ;1 is the AR(1) parameter in the expected
return process; σ d is the size of the news shock to the realized
dividend growth in Equation 3; σ r is the size of the news shock to
the realized return Equation 4; σ g is the size of the shock to the
expected dividend growth in Equation 5; σ μ is the size of the
shock to the expected return in Equation 6; ρdμ is the correlation
between the news shock to realized dividend growth and the
shock to the expected return; ρμg is the correlation between the
shock to the expected return and the shock to the expected
dividend growth.
IV. VAR Decompositions
The VAR model return decomposition has been widely
adopted in the literature to study the contributions of cash
flow news and discount rate news to stock price variations.
In this section, we aim to offer some formal statistical
inference in the VAR return decomposition.
This body of work typically chooses the predictive
variables (for example, the price-dividend ratio) in the
VAR framework to estimate expectations of future market
fundamentals. Using the Campbell and Shiller’s
(1988a, 1988b) log linear return approximation, this
work decomposes asset prices into the expectations of
future returns and future dividend growth.
There are a number of limitations and pitfalls in such
VAR decompositions. For example, the finding in the
extant literature that expected returns dominate dividend
growth in explaining movements in the price-dividend
ratio, is sensitive to the time period (Chen, 2009), and to
the choice of predictive variables (Goyal and Welch, 2008;
Chen and Zhao, 2009).
Table 6. State-space variance decomposition (Model 3:
dividend growth and returns, one lag)
Variance decomposition of price-dividend ratio (%)
Covariance of dividend and price-dividend
Negative of Covariance of returns and price-dividend
26.71
73.29
Notes: Annual data from the period 1946 to 2011 is used. The
identification condition is ρdg ¼ 0. By construction all contributions add to 100%.
In this article, we focus on the formal statistical inference of the contributions in the VAR decomposition.
Specifically we construct the inference using a nonparametric bootstrap procedure. We work with the following
stylized VAR specification:
#
rtþ1
Xtþ1
$
¼F#
#
$
rt
þ Vtþ1
Xt
(32)
2472
J. Ma and M. E. Wohar
Table 7. VAR variance decomposition (four lags)
Variance decomposition of price-dividend 95% Critical values
ratio (%)
(%)
Covariance of dividend and
9.80
7.19, 196.56
price-dividend
Negative of covariance of returns 90.20 −96.56, 92.81
and price-dividend
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Notes: Annual data from 1 the period 946 to 2011 is used; the
contributions are normalized to sum up to 100%. Note that for
covariance contributions the critical values are two-sided, i.e.,
the left one is 2.5% percentile and the right one is 97.5%
percentile.
where Xtþ1 is a vector containing variables that predict
returns. In this work, the only predictive variable is the
price-dividend ratio; F is the companion matrix that
allows for a general VAR(p) specification. Given the parameter estimates, we can compute first compute the contribution of the expected returns to the price-dividend
variations. Then the contribution of the expected dividend
growth can be backed out using the Campbell–Shiller
return identity. The contribution of the expected returns
is given by:
Et
1
X
j¼0
ρj rtþ1þj ¼ e01 # F # ðI $ ρF Þ$1 #Zt
(33)
where Zt ¼ ½ rt Xt '0 and e1 is the selection vector.
We apply this method to our annual data from the period
1946 to 2011.7 We present the price-dividend decomposition in Table 7 using four lags. As typical in this literature,
the VAR decomposition reveals that most of the stock
price variations are primarily explained by the expected
returns. In particular, the decomposition shows that about
90% of the price-dividend variation comes from its
comovement with the expected returns.
To investigate the statistical significance of these
contributions, we offer an inference based on the nonparametric bootstrapping procedure. In our bootstrapping exercise, we use the coefficients estimates from
the above estimated VAR but set all coefficients in the
return forecasting equation to zeros.8 In doing this, we
assume no return forecastability as the null hypothesis
and also the benchmark case to study the empirical
distributions of relevant metrics. We implement the
nonparametric bootstrapping with replacement. For
7
each set of bootstrapped data, we estimate the VAR,
compute and record the price-dividend decomposition
results. We do 5000 replications. The 95% upper percentile of the empirical cumulative distribution of the
expected return contribution is 92.81%, which is larger
than the point estimate (see Table 7). As a result, the
large contribution of the expected returns is barely
significant at 5% level.
To offer more details about the empirical distributions
of these bootstrapped returns and dividends contributions
and explain how we calculate the bootstrapped critical
values, we plot the histograms of the two covariance
contributions in Figs 7 and 8. Since the distributions in
this case are two-sided, the 95% critical values are twosided, one representing the 2.5% percentile on the left and
one 97.5% percentile on the right. Notice that the point
estimate of the return contribution is smaller than the rightside critical value in Fig. 7 and as a result the null cannot
be rejected. Furthermore, Fig. 8 tells the other side of the
same story. The data is generated under the joint null of
0% return contribution and 100% dividend contribution.
The point estimate of the dividend growth contribution is
9.80% and is small, but it is still well above the left side
critical value, and thus, we cannot reject the joint null
hypothesis.
Distribution of covariance of pd and returns under the null:Cochrane
600
500
400
300
200
100
0
–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
Fig. 7. Empirical distribution of covariance contribution of
price-dividend and returns under the null
Notes: The graph is based on 5000 replications; the scale of yaxis is made the same as that of Fig. 8 to facilitate comparison;
the rightmost (leftmost) bar counts the percentage of those contributions greater (less) than 200% (–200%).
We also did the same exercise using the long span data of 1927–2011, and our major conclusion stays the same. These results are
available upon request.
8
We also follow Campbell (1991) and implement a bootstrap procedure by setting the price-dividend ratio to follow a unit root process
under the null. The resulting coverage of confidence intervals all becomes even wider, making it harder to reject the null hypothesis,
lending further support to our major findings. These results are available upon request.
2473
Expected returns and expected dividend growth
V. Larrain and Yogo’s Alternative Measure of
Cash Flows
Distribution of covariance of pd and dividends
under the null:Cochrane
600
In a very important study, Larrain and Yogo (2008) point
out the shortcomings of using the dividends and dividendyield in the stock price decomposition. They construct a
broader measure of cash flows, namely, the net payout that
includes dividends, interest, equity repurchase net of issuance and debt repurchase net of issuance. Using the VAR
decomposition, they show that the expected future cash
flows turn out to contribute most to the variation of the net
payout yield, the ratio of net payout to asset value. In this
section, we estimate both the state-space and VAR model
to decompose the variation of the net payout ratio and
offer further statistical inference of these contributions
estimates using the proposed methodology so far.10
500
400
300
200
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100
0
–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
Fig. 8. Empirical distribution of covariance contribution of
price-dividend and dividend growth under the null
Notes: The graph is based on 5000 replications; the scale of yaxis is made the same as that of Fig. 7 to facilitate comparison;
the rightmost (leftmost) bar counts the percentage of those contributions greater (less) than 200% (–200%).
Table 8. VAR variance decomposition (four lags)
Variance decomposition of price-dividend
ratio (%)
Covariance of dividend and
price-dividend
Negative of covariance of returns
and price-dividend
33.38
95% Critical
values (%)
41.87, 156.17
66.62 −56.17, 58.13
Notes: Shiller’s annual data from the period 1872 to 2009 is used;
the contributions are normalized to sum up to 100%. Note that
for covariance contributions the critical values are two-sided, i.e.,
the left one is 2.5% percentile and the right one is 97.5%
percentile.
The primary purpose of this section is to emphasize
a general inference procedure within the VAR framework that is applicable to decomposing the stock price
and evaluating the statistical significance of their contributions using any data set. For example, we also
applied this procedure to Shiller’s S&P 500 returns
and price-dividend data from the period 1872 to
2009.9 The results are presented in Table 8. It is
interesting that for Shiller’s long span of data the
returns contribution is not only large but also statistically significant.
9
The state-space variance decomposition
Larrain and Yogo (2008) show that the present value
relationship as laid out by (1) through (2) holds for the
net payout (Δd), asset returns (r) and the ratio of asset
values to net payout (pd). We select the observed variables
½ Δdtþ1 pdtþ1 '0 and setting lag p ¼ 1 to estimate the
state-space model as set up by (13) and (14). The identification restriction is σ μg ¼ 0.11 The state-space estimation
results and the variance decomposition results are reported
in Tables 9 and 10, respectively.
We find that the expected net payout growth is more
persistent than the expected returns. But the persistence of
the expected net payout growth is nowhere near that of the
expected returns when using the price-dividend data (see
Section II). Furthermore, our state-space model reports
consistent results with Larrain and Yogo (2008) that
when measuring cash flows using this broader term the
expected cash flows seem to contribute most to the asset
valuation fluctuations: the covariance contribution of the
expected net payout growth is 100.52% (see Table 10).
This is opposite to what one would find using the pricedividend and dividends data.
Certainly the small signal-to-noise ratio is a concern. We
apply the Ma–Nelson procedure and the valid inference for
^ , the confidence interval for the persistence estimate of
f
g;1
the expected net payout growth, turns out to have two
disjoint intervals: [–0.99, –0.09]U[0.36, 0.99], of which
the second interval appears to be consistent with the notion
of the predictable cash flows growth. In order to sharpen the
inference, we further restrict the dynamics of the expected
return process. When we restrict the AR parameter of the
expected returns process to be zero, the valid inference of
Ma–Nelson procedure produces a much tighter confidence
This data set is available on Robert Shiller’s website: http://www.econ.yale.edu/~shiller/data.htm
We thank Yogo for generously providing their data set.
11
We find that the model fails to converge for the identification restriction σ dg ¼ 0.
10
2474
J. Ma and M. E. Wohar
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Table 9. State-space estimation result (Model 1: net payout
growth and log asset-payout ratio, one lag)
Parameter estimates
cg
fg;1
cμ
fμ;1
σd
σμ
σg
ρdμ
ρdg
Log-likelihood value
Implied parameters
σr
ρdr
ρμr
ρgr
κ
ρ
ZILC
' parameters
σμ
'σ r
σg
σd
0.0655 (0.0170)
0.7784 (0.0662)
0.0794 (0.0159)
−0.3392 (0.2404)
0.3497 (0.0280)
0.0454 (0.0483)
0.0889 (0.0258)
0.3433 (0.1203)
−0.9287 (0.0198)
25.3069
0.1124 (0.0257)
−0.1067 (0.1199)
0.7697 (0.4491)
0.4655 (0.0991)
0.0740
0.9859
0.4035
0.2543
Notes: Annual data from the period 1927 to 2004 in Larrain and
Yogo (2008) is used; numbers in parenthesis are SEs. The
identification condition is ρμg ¼ 0. cg is the average net payout
growth rate; fg;1 is the AR(1) parameter in the expected net
payout growth process; cμ is the average return; fμ;1 is the AR
(1) parameter in the expected return process; σ d is the size of the
news shock to the realized net payout growth in Equation 3; σ r is
the size of the news shock to the realized return in Equation 4; σ g
is the size of the shock to the expected net payout growth in
Equation 5; σ μ is the size of the shock to the expected return in
Equation 6; ρdμ is the correlation between the news shock to
realized net payout growth and the shock to the expected return;
ρdg is the correlation between the shock to the realized net payout
growth and the shock to the expected net payout growth.
Table 10. State-space variance decomposition (Model 1: net
payout growth and log asset-payout ratio, one lag)
Variance decomposition of price-dividend ratio (%)
Covariance of dividend and price-dividend
Negative of covariance of returns and price-dividend
100.52
−0.52
Notes: Annual data from the period 1927 to 2004 in Larrain and
Yogo (2008) is used. The identification condition is ρμg ¼ 0. By
construction all contributions add to 100%.
^ : [0.62, 0.91], which is quite consistent with
interval for f
g;1
the notion that the expected cash flow might contain some
predictable component that results in a large contribution to
the valuation ratio variations.
The VAR decomposition
We also apply the same bootstrap procedure as in Section
IV to Larrain and Yogo’s data set to provide a VAR
Table 11. VAR variance decomposition (four lags)
Variance decomposition of price-dividend
ratio (%)
Covariance of dividend and pricedividend
Negative of covariance of returns
and price-dividend
95% Critical
values (%)
85.92 −57.50, 48.98
14.08
51.02, 157.50
Notes: Annual data from the period 1927 to 2004 in Larrain and
Yogo (2008) is used. Note that for covariance contributions the
critical values are two-sided, i.e., the left one is 2.5% percentile
and the right one is 97.5% percentile. The null hypothesis is that
the cash flow has zero contribution to the payout ratio variations.
decomposition result. First of all, we do not attempt to
employ all three data and estimate the over-identified
model as in Larrain and Yogo (2008). Instead, we first
estimate the VAR using the returns and the asset payout
ratio, and then we estimate another VAR model consisting
of the payout growth and the asset payout ratio. We find
that our results are robust in both models and therefore we
report the decomposition results along with their bootstrapped confidence intervals for the VAR model that
consists of payout growth and the asset payout ratio in
Table 11. VAR decomposition gives rise to a similar result
of the state-space decomposition: the net payout growth
seems to explain a large fraction of the payout yield
variation. To investigate the statistical significance of
this result, we resort to the bootstrap procedure.
Notice here since now the cash flows contribution
becomes large, we want to test the null hypothesis of
zero contribution by the cash flows. Therefore, in our
bootstrapping, we use the coefficients estimates from the
above estimated VAR but set all coefficients in the net
payout growth forecasting equation to zeros. In doing this
we assume no payout forecastability as the null hypothesis
and also the benchmark case. The bootstrapped critical
values are reported in Table 11. Based on the 5% critical
values, the contribution of the expected payout growth to
the asset payout ratio is not only large (85.92%) but also
statistically significant. The point estimate exceeds the 5%
critical value by a large margin.
To offer more details about the empirical distributions
of these bootstrapped returns and net payout contributions, we plot the histograms of the two covariance contributions in Figs 9 and 10. Notice that the point estimate
of the net payout growth contribution is much larger than
the right-side critical value in Fig. 9 and as a result the null
is strongly rejected. Furthermore, Fig. 10 tells the other
side of the same story. The data is generated under the joint
null of 0% net payout growth contribution and 100%
returns contribution. The point estimate of the returns
contribution is 14.08% and is well below the left-side
critical value, and thus, we again strongly reject the joint
null hypothesis.
2475
Expected returns and expected dividend growth
VI. Conclusion
Distribution of covariance of pd and dividends
under the null:Cochrane
800
700
600
500
400
300
200
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100
0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Fig. 9. Empirical distribution of covariance contribution of
payout yield and payout growth under the null
Notes: The graph is based on 5000 replications; the scale of yaxis is made the same as that of Fig. 10 to facilitate comparison;
the rightmost (leftmost) bar counts the percentage of those contributions greater (less) than 200% (–200%).
Distribution of covariance of pd and returns
under the null:Cochrane
800
700
600
500
400
300
200
100
0
–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
Fig. 10. Empirical distribution of covariance contribution
of payout yield and returns under the null
Notes: The graph is based on 5000 replications; the scale of yaxis is made the same as that of Fig. 9 to facilitate comparison;
the rightmost (leftmost) bar counts the percentage of those contributions greater (less) than 200% (–200%).
In summary, the VAR decomposition results with bootstrapped confidence intervals provide strong evidence in
support of the finding by Larrain and Yogo that the cash
flows, when measured in a broader term, can contribute
most to the firm value variations.
The extant literature examining the factors that contribute
to movements in the aggregate price-dividend ratio have
yielded, to some, a puzzling result employing the standard
definition of dividends. Employing both VAR decomposition methodology as well as state-space modelling methods, the empirical results of the extant literature find that it
is expected returns that contribute most to movements in
the price dividend ratio rather than expected dividend
growth (which might be expected from finance theory).
When a broader definition of dividends (net payout) is
employed, one finds that it is the broader measure of cash
flows that contributes most to movements in the price-net
payout ratio. Nonetheless, it remains puzzling why
expected returns so strongly contribute to movements in
the price dividend ratio.
This article contributes to the existing literature by
illustrating that the earlier empirical work in this area is
subject to severe inference problems which make their
findings unreliable. This article revisits this subject by
focusing on the important inference problems that have
plagued the extant literature. We show that estimated statespace model is plagued by a small expectation shock
relative to the corresponding realized shock in the return
process. The small signal-to-noise ratio of the state-space
model implies weak identification and that the standard
inference tends to underestimate the uncertainty of the
contribution estimate. When we adjust for proper inference, we find that the aggregate returns and dividends data
cannot provide sufficient statistical evidence to support
the notion that it is expected returns that explains the
majority of the fluctuation in the price-dividend ratio.
We show that a stylized VAR decomposition equipped
with nonparametric bootstrapped inferences yields a similar result. However, the broader measure of dividends
(termed net payout) does hold up to scrutiny and is
found to significantly to contribute to movements in the
price-net payout ratio.
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