# Valuation Duration of the Stock Market††thanks:  We thank Nick Barberis,
Geert Bekaert, Jules van Binsbergen, Zhi Da, Kent Daniel, Joost Driessen,
Stefano Giglio, Will Goetzmann, Benjamin Golez, Hui Guo, Bryan Kelly, Ralph
Koijen, Lars Lochstoer, Alan Moreira, Riccardo Sabbatucci, Tano Santos, José
Scheinkman, Jessica Wachter, and Michael Weber for in-depth discussions.

Ye Li University of Washington Foster School of Business. E-mail<EMAIL_ADDRESS>Chen Wang University of Notre Dame Mendoza College of Business. E-mail:
<EMAIL_ADDRESS>
(October 10, 2023)

###### Abstract

At the peak of the tech bubble, only 0.57% of market valuation comes from
dividends in the next year. Taking the ratio of total market value to the
value of one-year dividends, we obtain a valuation-based duration of 175
years. In contrast, at the height of the global financial crisis, more than
2.2% of market value is from dividends in the next year, implying a duration
of 46 years. What drives valuation duration? We find that market participants
have limited information about cash flow beyond one year. Therefore, an
increase in valuation duration is due to a decrease in the discount rate
rather than good news about long-term growth. Accordingly, valuation duration
negatively predicts annual market return with out-of-sample $R^{2}$ of 15%,
robustly outperforming other predictors in the literature. While the price-
dividend ratio reflects the overall valuation level, our valuation-based
measure of duration captures the slope of the valuation term structure. We
show that valuation duration, as a discount rate proxy, is a critical state
variable that augments the price-dividend ratio in spanning the (latent) state
space for stock-market dynamics.

[TO-DO]

##  1 Introduction

We define valuation duration of the stock market as the ratio of total market
capitalization to the price of the dividend in the very next year (the one-
year dividend strip price in Binsbergen, Brandt, and Koijen, 2012). In March
2000, the U.S. stock market has a valuation duration of 175 years, which
suggests that at the peak of the dot-com bubble, the U.S. stock market derives
99.4% of value from dividends beyond the next year. Duration varies
significantly over time, with a monthly standard deviation of 1.6 years in the
last three decades. At the market bottom during the global financial crisis,
the stock market duration is only 46 years in March 2009 with more than 2.2%
of market value coming from dividends paid within the next year.

What drives the relative valuation of long- vs. short-term dividends? An
increase in market duration represents a steepening of the valuation term
structure, with the long-term dividends commanding a higher valuation relative
to dividends in the near term. While duration represents the slope, the
standard price-dividend ratio captures the overall level of market valuation.
What information can we extract from the dynamics of slope and level of the
valuation term structure?

We find that market duration and the price-dividend ratio span the state space
of equity market. Stock prices are determined by the conditional expectations
of future returns and dividends, so a model of the aggregate stock market
requires at least two state variables for return and cash-flow dynamics (e.g.,
Lettau and Wachter, 2007; Binsbergen and Koijen, 2010; Kragt, de Jong, and
Driessen, 2020). In other words, the lower bound of the state-space dimension
is two. Our analysis shows that two is also the upper bound. Market duration
and the price-dividend ratio as the two state variables contain sufficient
information for forecasting returns and future dividends. Moreover, market
duration is the state variable that corresponds closely to the conditional
expected return. We show that this strong connection between market duration
and the discount rate is due to the striking difference in cash-flow
predictability at short and long horizons.

Identifying state variables is an important task in asset pricing and macro
finance as it lays the foundation for various research topics.111Previous
studies on macro dynamics and asset pricing based on consumption, production,
behavioral biases, and intermediation frictions have motivated a variety of
state variables (e.g., Sims, 1980; Cochrane, 1991; Campbell and Ammer, 1993;
Barberis, Shleifer, and Vishny, 1998; Campbell and Cochrane, 1999; Barberis,
Huang, and Santos, 2001; Lettau and Ludvigson, 2001; Ang and Piazzesi, 2003;
Bansal and Yaron, 2004; Campbell and Vuolteenaho, 2004; Kaltenbrunner and
Lochstoer, 2010; Lettau and Wachter, 2011; Gabaix, 2012; He and Krishnamurthy,
2013; Wachter, 2013; Kelly and Pruitt, 2013; Brunnermeier and Sannikov, 2014;
Barberis, Greenwood, Jin, and Shleifer, 2015; Muir, 2017; Campbell, Giglio,
Polk, and Turley, 2018, 2020; Cieslak and Pang, 2021). Our findings show that
beyond the standard price-dividend ratio, valuation duration is another key
state variable for the equity market, and it is particularly powerful in
revealing the discount-rate dynamics. Next, we summarize the key steps of our
analysis and our contributions to the asset pricing literature in three areas,
state-space modeling, return predictability, and cash-flow expectations.

Our analysis starts with a characterization of the equity-market state space.
We explore information directly from the prices of dividend strips rather than
select macro or market variables. The logarithm of dividend strip prices at
different maturities scaled by the realized dividend are linear functions of
(latent) state variables in an exponential-affine setup (e.g., Lettau and
Wachter, 2007) or through log-linearization (e.g., Campbell and Shiller, 1988;
Binsbergen and Koijen, 2010). Therefore, this set of valuation ratios of
dividend strips fully reveals the state variables as long as the number of
dividend strips is at least as large as the number of state variables and the
valuation ratios have linearly independent state-variable coefficients.

We find that two principal components drive the dynamics of valuation ratios
of dividends at different maturities. Importantly, when forecasting market
return and dividend growth, two valuation ratios are sufficient to maximize
the predictive power. While different pairs of valuation ratios possess
different predictive power, the pairs with the strongest predictive power
perform just as well as combinations of three or more valuation ratios. Our
findings suggest that the state space is two-dimensional. Next, we show that
the log market duration ($dr$), i.e., the logarithm of total market value
scaled by one-year dividend price, and the log price-dividend ratio ($pd$)
possess a predictive power for both return and dividend growth that is at
least as strong as any combination of dividend valuation ratios and thus can
serve as the pair of state variables.

A striking finding from our forecasting exercises is that $dr$ alone is
sufficient for forecasting returns. Augmenting $dr$ with $pd$ or other
valuation ratios of dividend strips does not improve the predictive power. In
fact, the return predictive power of $dr$ not only subsumes that of $pd$ and
other valuation ratios but also surpasses other predictors in the literature,
such as those summarized in Goyal and Welch (2007) and more recent studies.
Specifically, $dr$ demonstrates remarkable in-sample and out-of-sample (OOS)
return predictability of annual market returns, reporting substantial $R^{2}$
of 25% and 15%, respectively. The OOS $R^{2}$ is in sharp contrast to the
almost zero OOS $R^{2}$ from $pd$. Moreover, the predictive power of $dr$
survives the Hodrick (1992) adjustment for standard errors, the Stambaugh
(1999) adjustment for small-sample bias, and tests for out-of-sample
predictive power such as encompassing (ENC) and Clark-West (CW) tests.

These findings are critical for understanding both the discount-rate dynamics
and the fluctuation of market duration. If the conditional expected return is
a univariate function of $dr$, then we can rearrange the equation and solve
$dr$ as a univariate function of the conditional expected return, which
implies that the variation in duration is fully discount-rate driven. To
analyze the connection between $dr$ and the discount rate, we set up a two-
dimensional state space model. Without loss of generality, the (latent) state
variables are the conditional expected return and expected dividend growth
rate that follow AR(1) processes (Lettau and Wachter, 2007; Binsbergen and
Koijen, 2010). $dr$ and $pd$ are bivariate linear functions of state
variables, and vice versa.

In our model, a necessary and sufficient condition for the conditional
expected return to be a univariate function of $dr$ (and vice versa) is that
the market does not contain information on cash-flow growth beyond the very
next year. Intuitively, under this condition, the price of next year’s
dividends exhausts all the information about future cash flows, so $dr$, which
is the log market value minus the log price of next year’s dividend, teases
out cash-flow information and only contains information about the discount
rate, thus becoming a univariate function of the conditional expected return.
Given that the expected dividend growth rate follows an AR(1) process, this
condition translates into a zero autoregressive coefficient of the expected
dividend growth rate: All the relevant information about future cash flows—the
current expected growth rate and the history of realized shocks—is not
propagated into the future beyond the next year.222Given a two-dimensional
state space, adding more lags to the autoregressive processes of expected
return and expected dividend growth is not necessary. If AP(p) or ARMA(p,q)
models were required or if other (macroeconomic or market) variables feed into
the dynamics of expected return and expected cash-flow growth rate, we should
have found the state-space dimension to be higher than two because to span the
state space, we must augment the current conditional expectations of next-
period return and cash flow growth rate with lagged conditional expectations.

To estimate the autoregressive coefficient (persistence) of the expected
dividend growth rate, we use analyst forecasts to proxy for the expected cash-
flow growth and estimate two econometric models of belief dynamics that take
advantage of, respectively, analysts’ short-term and long-term forecasts. As
an alternative method, we also fit a state-space model to dividend data to
estimate the persistence of expected dividend growth. The message from our
findings is consistent: Cash-flow growth expectation lacks persistence. This
provides an explanation of the strong connection between market duration and
expected return.

We also show analytically that return prediction errors should comove with the
value of the autoregressive coefficient of cash-flow growth expectations. To
test this prediction, we conduct a rolling-window estimation. In each window,
we estimate the autoregressive coefficient of the expected cash-flow growth
and the forecast error from predicting returns with $dr$. Both the in-sample
and out-of-sample forecasting errors comove with the value of the
autoregressive coefficient, suggesting that when the market contains limited
information about cash-flow growth beyond one year, the mapping between $dr$
and the expected return strengthens.

In our model, a zero autoregressive coefficient of the expected cash-flow
growth rate fully captures the fact that the market contains limited
information about cash-flow growth beyond the very next year. The emphasis on
this autoregressive coefficient is warranted because the expected dividend
growth rate follows an AR(1) process. While our analysis of the state-space
dimension supports this model specification, we still go beyond our model to
characterize market participants’ cash-flow expectations. First, we show that
dividend and earnings growth within one year are highly predictable,
especially by analysts’ forecasts.333This can be explained by firms offering
forward guidance on near-term performance. The $R^{2}$ is 73% from forecasting
the near year’s earnings growth. In contrast, cash-flow predictability beyond
one year is weak. For growth from the next one to two years, the in-sample
$R^{2}$ of our best prediction model is only 8%. It declines to $7\%$ for
growth between the second and third years.

Our findings on the striking difference between short- and long-term cash-flow
predictability lend support to $dr$ as a discount-rate proxy. Cochrane (2011)
points out that the price-dividend ratio, $pd$, may predict future returns if
cash flows are not predictable. We show that short-term cash flows are in fact
highly predictable but long-term cash flows are not. This finding then
suggests that the discount-rate proxy should be market duration, $dr$, which
captures the relative valuation of long- vs. short-term cash flows and
reflects the slope of the valuation term structure, rather than $pd$, which
reflects the overall level of valuation. Consider a market timing strategy
based on the return predictive power of $dr$. Because $dr$ has a negative
predictive coefficient, the strategy reduces market exposure when duration
increases and increases market exposure when duration decreases. The strategy
delivers a Sharpe ratio of $0.58$, achieving an improvement of 55% over the
buy-and-hold strategy (Campbell and Thompson, 2008). It bets against the
valuation of long-term cash flows rather than the overall level of valuation
that is reflected in $pd$.

Beyond time-varying discount rate, return predictability typically entertains
an alternative interpretation based on mispricing. The profitability of
$dr$-based market timing strategy can be explained by market participants’
lack of information on long-term growth and the associated over- or under-
valuation of cash flows at long horizons. Moreover, the market timing strategy
also avoids betting against the valuation of short-term cash flows that market
participants are informed about. When the valuation of long-term cash flows
rises, the resultant increase in market duration signals exuberance, while
when the valuation of long-term cash flows declines, market participants may
be overly pessimistic about long-term growth. This interpretation is
consistent with the emphasis in Bordalo, Gennaioli, La Porta, and Shleifer
(forthcoming) on market participants’ errors in forecasting long-term growth
as a key driver of asset prices.

#### Literature.

Asset prices are typically modeled as functions of state variables. One
approach to selecting state variables is through economic theories (e.g.,
Campbell and Cochrane, 1999, Bansal and Yaron, 2004). Our paper contributes to
the alternative approach of state space modeling (e.g., Duffee, 2002; Dai and
Singleton, 2003). Using the valuation ratios of dividend strips, we
empirically analyze the dimension of the state space and provide support to
the widely adopted assumption of a two-dimensional state space (e.g., Lettau
and Wachter, 2007; Binsbergen and Koijen, 2010; Kragt, de Jong, and Driessen,
2020). Moreover, we show that our valuation-based measure of duration and the
commonly used price-dividend ratio map out the state space of stock market.
Our work builds upon the previous research on measuring the prices of dividend
strips (Binsbergen, Brandt, and Koijen, 2012, 2013; Binsbergen and Koijen,
2017; Cejnek and Randl, 2016, 2020; Cejnek, Randl, and Zechner, 2021; Golez
and Jackwerth, 2023; Giglio, Kelly, and Kozak, forthcoming). While this
literature emphasizes characterizing the term structure of equity risk
premium, our paper instead focuses on the term structure of equity valuation
ratios and how to use valuation ratios to map out state variables.444There is
an extensive literature on the term structure of equity risk premium (e.g.,
Lettau and Wachter, 2007, 2011; Hansen, Heaton, and Li, 2008; Croce, Lettau,
and Ludvigson, 2014; Belo, Collin-Dufresne, and Goldstein, 2015; Ai, Croce,
Diercks, and Li, 2018; Backus, Boyarchenko, and Chernov, 2018; Bansal, Miller,
Song, and Yaron, 2021; Gonçalves, 2021a; Gormsen, 2021; Boguth, Carlson,
Fisher, and Simutin, 2022).

Our approach of extracting state variables from disaggregated valuation ratios
is similar in spirit to that in Kelly and Pruitt (2013) but differs in several
crucial ways. Kelly and Pruitt (2013) decompose the stock market into
individual stocks and filter out state variables from the valuation ratios of
individual stocks. We decompose the market by the horizon of aggregate cash
flows.555There is a large literature on measuring the cash-flow duration for
the aggregate market based on accounting information (Binsbergen, 2021; Golez
and Koudijs, 2023) and for individual stocks based on accounting information
(Dechow, Sloan, and Soliman, 2004; Da, 2009; Chen, 2011; Weber, 2018;
Gonçalves, 2021b; Walter and Weber, 2022) or derivative prices (Gormsen and
Lazarus, 2023). Our paper differs by constructing a valuation-based measure of
duration rather than a measure of duration based on the term structure of
corporate cash flows. By avoiding idiosyncratic noise in firm-level valuation
ratios, we do not rely on statistical filtering to extract state variables.
The valuation ratios of dividend strips, our measure of market duration, and
the price-dividend ratio are already combinations of state variables. Since
our two state variables can be measured in real time and directly from market
prices, they are less prone to estimation error and thus offer a real-time
characterization of the state of equity market.

An application of state variables in the finance literature is to predict
asset returns. We contribute to the voluminous literature on return
predictability by offering a novel predictor based on a relative valuation of
short- and long-term cash flows (e.g., Fama and French, 1988; Campbell and
Shiller, 1988; Lettau and Ludvigson, 2001; Baker and Wurgler, 2000; Lewellen,
2004; Goyal and Welch, 2007; Cochrane, 2007; Ang and Bekaert, 2007; Lettau and
Van Nieuwerburgh, 2007; Campbell and Thompson, 2008; Kelly and Pruitt, 2013;
Rapach, Ringgenberg, and Zhou, 2016; Martin, 2017; Johnson, 2019; Chen, Da,
and Huang, 2022; Kelly, Malamud, and Zhou, forthcoming; Bordalo, Gennaioli, La
Porta, and Shleifer, forthcoming). The construction of our predictor is
simple, and it outperforms the other stock-market return predictors across
various metrics. Moreover, we provide an intuitive explanation on the
connection between valuation duration and the conditional expected return
based on the term structure of cash-flow predictability.

We show that short-term cash flows are highly predictable but long-term cash
flows are not, and this is key to identifying a discount-rate proxy. Cochrane
(2011) suggests that the price-dividend ratio, $pd$, predicts future returns
and serves as a discount-rate proxy if cash flows are not predictable. But our
findings on the striking difference in cash-flow predictability at short- vs.
long horizons suggest that the proper discount-rate proxy and the state
variable driving the conditional expected return should be market duration,
$dr$. Our findings on cash flow predictability at different horizons
contribute to the literature on cash-flow predictability (Larrain and Yogo,
2008; Binsbergen and Koijen, 2010; Koijen and Van Nieuwerburgh, 2011;
Binsbergen, Hueskes, Koijen, and Vrugt, 2013; Chen, Da, and Zhao, 2013;
Jagannathan and Liu, 2018; Pettenuzzo, Sabbatucci, and Timmermann, 2020; Golez
and Koudijs, 2023; Sabbatucci, 2022; Pruitt, 2023)

Characterizing the dynamics of expected cash-flow growth rate has always been
an important topic in the asset pricing literature (Bansal and Yaron, 2004;
Beeler and Campbell, 2012; Belo, Collin-Dufresne, and Goldstein, 2015; Collin-
Dufresne, Johannes, and Lochstoer, 2016). Recently a growing body of work
focuses on analyzing market participants’ expectation formation. Adam and
Nagel (2022) summarize the latest developments.666There is a fast-growing body
of literature on firm-level cash flow expectations (La Porta, 1996; Dechow and
Sloan, 1997; Copeland, Dolgoff, and Moel, 2004; Da and Warachka, 2011;
Piotroski and So, 2012; Bordalo, Gennaioli, La Porta, and Shleifer, 2019;
Bouchaud, Krüger, Landier, and Thesmar, 2019; Binsbergen, Han, and Lopez-Lira,
2022) and expectations of aggregate cash-flow growth (Chen, Da, and Zhao,
2013; De La O and Myers, 2021; Gao and Martin, 2021; McCarthy and Hillenbrand,
2021; Nagel and Xu, 2022; Charles, Frydman, and Kilic, 2023; Bordalo,
Gennaioli, La Porta, OBrien, and Shleifer, 2023; Schmidt-Engelbertz and
Vasudevan, 2023). Our findings are most related to Bordalo, Gennaioli, La
Porta, and Shleifer (forthcoming), who document that market participants’
errors in forecasting long-term growth are key to understanding asset-price
fluctuations. Also related, Da and Warachka (2011) find that the disparity
between long- and short-term earnings expectations drives the cross-sectional
difference in average stock returns. Our contribution is to show that, first,
there exists a striking difference in short- vs. long-term cash flow
predictability, and second, such difference guides us towards finding a
discount-rate proxy. Moreover, we show that the term structure of cash-flow
predictability is tied to the persistence of agents’ cash-flow expectations.
Our findings echo the recent studies on the importance of agents’ perceived
persistence of key state variables in explaining belief formation and asset
prices (Gabaix, 2019; Wang, 2020).

##  2 Valuation Ratios and the State Space

We consider a dynamic economy where the information filtration is driven by a
Markov process. Specifically, the state of an economy at time $t$ is
summarized by $X_{t}$, a $K$-by-$1$ vector of state variables. We assume that
$X_{t}$ evolves as a first-order vector autoregression

$X_{t+1}=\Pi X_{t}+\sigma_{X}^{\top}\epsilon_{t+1},$ (1)

where $\epsilon_{t+1}$ is a $N$-by-$1$ vector of shocks that capture all the
news at $t+1$ and are independent over time with normal distribution
$N\left(\mathbf{0},\Sigma\right)$. Note that since any higher-order vector
autoregression can be written as a first-order vector autoregression by
expanding the number of state variables, the AR(1) specification is without
loss of generality. The autoregressive coefficients are given by $\Pi$, a
constant $K$-by-$K$ matrix, and $\sigma_{X}$ is a $N$-by-$K$ matrix of shock
loadings.

The growth rate of dividend from $t$ to $t+1$ has a $N$-by-$1$ shock-loading
vector $\sigma_{D}$,

$\ln\left(\frac{D_{t+1}}{D_{t}}\right)=g_{t}+\sigma_{D}^{\top}\epsilon_{t+1},$
(2)

where the time-varying expected dividend growth rate is given by

$g_{t}=\phi^{\top}X_{t}+\overline{g}-\frac{1}{2}\sigma_{D}^{\top}\Sigma\sigma_{D}.$
(3)

We allow the state-variable loadings, $\phi$, to be any $K$-by-$1$ vector. If
$g_{t}$ does not depend on a state variable, the corresponding element in
$\phi$ is zero; likewise, if a shock does not affect the growth rate of
aggregate dividend, the corresponding element in $\sigma_{D}$ is zero.

No arbitrage implies the existence of a stochastic discount factor

$M_{t+1}=\exp\left\\{-r_{f}-\frac{1}{2}\lambda^{\top}_{t}\Sigma\lambda_{t}-\lambda_{t}^{\top}\epsilon_{t+1}\right\\},$
(4)

where $r_{f}$ is the one-period risk-free rate and the $N$-by-$1$ vector of
risk prices, $\lambda_{t}$, is given by

$\lambda_{t}=\overline{\lambda}+\theta^{\top}X_{t}.$ (5)

We do not impose any restrictions on the state-variable loadings of the risk
prices in $\lambda_{t}$.

Let $P^{n}_{t}$ denote the time-$t$ price of the dividend paid at $t+n$. The
no-arbitrage pricing functional gives a recursive equation for the prices of
dividend strips: for $n\geq 1$,

$P^{n}_{t}=\mathbb{E}_{t}\left[M_{t+1}P^{n-1}_{t+1}\right],$ (6)

with the boundary condition $P^{0}_{t}=D_{t}.$ The log price-dividend ratio of
the dividend strip with maturity $n$ is given by

$s^{n}_{t}\equiv\ln\left(\frac{P^{n}_{t}}{D_{t}}\right)=A\left(n\right)+B\left(n\right)^{\top}X_{t},$
(7)

where $A\left(n\right)$ and $B\left(n\right)$ are deterministic functions of
$n$ given by a system of recursive equations with initial conditions
$A\left(0\right)=0\text{, and }B\left(0\right)=0$ (see equations (33)-(34) and
derivation in Appendix I).

Given $K$ log price-dividend ratios of strips, $\left\\{s^{n_{i}}_{t}:\forall
i\in\\{1,2,...,K\\},n_{i}\in\\{1,2,...,n,...\\}\right\\}$, with a full-rank
loading matrix,
$\mathbf{B}\left(\left\\{n_{i}\right\\}_{i=1}^{K}\right)\equiv\left[B\left(n_{1}\right),B\left(n_{2}\right),...,B\left(n_{K}\right)\right]^{T}$,
the state space is recovered by

$X_{t}=\mathbf{B}\left(\left\\{n_{i}\right\\}_{i=1}^{K}\right)^{-1}\left[s^{n_{1}}_{t}-A\left(n_{1}\right),\,...,\,s^{n_{K}}_{t}-A\left(n_{K}\right)\right]^{\top}$
(8)

When the rank condition fails, these valuation ratios can still recover part
of the state space. Let $J$ ($<K$) denote the maximum number of log price-
dividend ratios with linearly independent loadings $B\left(n\right)$ and
$\left\\{n_{i}\right\\}_{i=1}^{J}$ denote the corresponding set of maturities.
We can write (8) as follows

$\mathbf{B}\left(\left\\{n_{i}\right\\}_{i=1}^{J}\right)X_{t}=\left[s^{n_{1}}_{t}-A\left(n_{1}\right),\,...,\,s^{n_{J}}_{t}-A\left(n_{J}\right)\right]^{\top}.$
(9)

The stock market acts as a linear mapping, i.e.,
$\mathbf{B}\left(\left\\{n_{i}\right\\}_{i=1}^{J}\right)$, that compresses the
$K$-dimensional state space of $X_{t}$ into a $J$-dimensional space generated
by the log price-dividend ratios. In sum, a collection of log price-dividend
ratios of dividend strips (partially) span the state space. For information
embedded in the state variables, we analyze these valuation ratios.

Let $P_{t}$ denote the total stock market capitalization (i.e., the market
price of dividends across all maturities). Following the same method, we solve
the log price-dividend ratio of the market

$pd_{t}=\ln\left(P_{t}/D_{t}\right)=A+B^{\top}X_{t}\text{,}$ (10)

where $A$ and $B$ are constants defined in Appendix I. Stock market duration
is defined as the log ratio of total market capitalization to the price of
dividends paid in the next year.

$\displaystyle dr_{t}$
$\displaystyle\equiv\ln\left(P_{t}/P^{1}_{t}\right)=\ln\left(P_{t}/D_{t}\right)-\ln\left(P^{1}_{t}/D_{t}\right)=pd_{t}-s^{1}_{t}$
$\displaystyle=A-A(1)+\left(B-B(1)\right)^{\top}X_{t}.$ (11)

Therefore, just like the valuation ratios of dividend strips, $pd_{t}$ and
$dr_{t}$ are also driven by the state variables $X_{t}$. By having different
coefficients of $X_{t}$, these two variables reveal different information
about the state space. Next, we analyze the dimension of the state space using
$dr_{t}$, $pd_{t}$, and the valuation ratios of dividend strips with different
maturities. After showing empirically that the state space is two-dimensional,
we argue that $pd_{t}$ and $dr_{t}$, span the state space and contain
sufficient information for forecasting stock market returns and cash-flow
growth.

##  3 The Dimension of State Space

###  3.1 Variable construction and summary statistics

#### Constructing dividend strip prices.

Let $P^{n}_{t}$ denote the price of the dividend paid in year $n$. First, we
calculate $P^{n+}_{t}$, the price of dividends that are paid after the first
$n$ years. Under the risk-neutral measure,

$P^{n+}_{t}=e^{-nr_{f}}\mathbb{E}^{RN}_{t}\left[\sum_{\tau=1}^{+\infty}e^{-\tau
r_{f}}D_{t+n+\tau}\right]=e^{-nr_{f}}\mathbb{E}^{RN}_{t}\left[\mathbb{E}^{RN}_{t+n}\left[\sum_{\tau=1}^{+\infty}e^{-\tau
r_{f}}D_{t+n+\tau}\right]\right],$ (12)

where the expectation operator, $\mathbb{E}^{RN}_{t+n}\left[\cdot\right]$, was
inserted under the law of iterated expectations. Note that the (ex-dividend)
stock price at $t+n$ is

$S_{t+n}=\mathbb{E}^{RN}_{t+n}\left[\sum_{\tau=1}^{+\infty}e^{-\tau
r_{f}}D_{t+n+\tau}\right],$ (13)

so we have

$P^{n+}_{t}=e^{-nr_{f}}\mathbb{E}^{RN}_{t}\left[S_{t+n}\right].$ (14)

The first component, $e^{-nr_{f}}$, is ${ZCB}^{n}_{t}$, the price of a zero-
coupon bond with maturity $n$. The second component is the risk-neutral
expectation of stock price, i.e., the futures price, $F^{n}_{t}$ (Duffie,
2001).

We construct the price of dividend strips using zero-coupon bond prices and
stock index futures prices. First, we calculate $P^{1}_{t}$, the price of the
dividend paid in the next year,

$P^{1}_{t}=P_{t}-P^{1+}_{t},$ (15)

as the difference between the price of all dividends, i.e., the current stock
price $P_{t}$, and the price of dividends paid after the next year. Following
the same method, we calculate the price of dividends paid in the next six
months, $P^{0.5}$ from $P_{t}-P^{0.5+}_{t}$. In our empirical analysis, we use
the valuation ratios of dividend strips with maturity $1$ and $0.5$, i.e.,
$s^{1}=\ln(P^{1}/D_{t})$ and $s^{0.5}=\ln(P^{0.5}/D_{t})$, and the valuation
ratio of dividends paid beyond one year, $s^{1+}=\ln(P^{1+}/D_{t})$. Our
analysis in the previous section shows that these valuation ratios are
different linear combinations of state variables.

#### Data and summary statistics

For futures prices, we use S&P 500 index futures, which are the most actively
traded stock futures. We obtain S&P 500 futures prices from Datastream.777We
obtain the daily settlement prices for the S&P 500 futures. For return and
cash-flow prediction at the monthly frequency, we use the settlement price of
the last trading day of each month. The maturities of the traded futures
contracts vary over time, so to obtain futures prices with constant maturity,
we apply the shape-preserving piecewise cubic interpolation to complete the
futures curve. The results using linear interpolation are similar.  We obtain
the zero-coupon bond prices from the Fama-Bliss database. The return and level
of the S&P 500 index are obtained from CRSP. The dividend data is from S&P
Global and obtained via the updated dataset of Goyal and Welch (2007). Our
final dataset is from January 1988 to December 2019. The sample starts in 1988
to have high-quality dividend data from S&P Global and, importantly, a
sufficiently liquid futures market without structural changes.888Wang,
Michalski, Jordan, and Moriarty (1994) identify structural changes of
liquidity in the S&P 500 futures market in the pre-1987 period, during the
market crash, and in the post-1987 period. After the market crash of October
1987, regulators overhauled several trade-clearing protocols.999The stock
market crash in October 1987 reveals anomalous trading in the futures market
that was primarily driven by portfolio insurance (Brady Report (1988)).
According to the New York Stock Exchange: “In response to the market breaks in
October 1987 and October 1989, the New York Stock Exchange instituted circuit
breakers to reduce volatility and promote investor confidence. By implementing
a pause in trading, investors are given time to assimilate incoming
information and the ability to make informed choices during periods of high
market volatility.” Lastly, Fama-French factors at the monthly frequency are
obtained from Ken French’s website.

Table 1: Summary Statistics This table reports the number of observations, mean, standard deviation, minimum, maximum, quartiles, and monthly autocorrelation ($\rho$) of the main variables in this paper, including our main return predictor, $dr$ (“duration”), the price-dividend ratio $pd$ of the S&P 500 index, the filtered series for demeaned expected returns and dividend growth following Binsbergen and Koijen (2010) $\mu^{F}$ and $g^{F}$, the single predictive factor extracted from 100 book-to-market and size portfolios from Kelly and Pruitt (2013) $KP$, short-term dividend strip price to dividend ratio (0.5 year and 1 year) $\log(P^{0.5}/D)$ and $\log(P^{1}/D)$, long-term dividend strip price to dividend ratio (beyond 1 year) $\log(P^{1+}/D)$, one-month and one-year log returns of the S&P 500 index $r^{S\&P}_{t+1/12}$ and $r^{S\&P}_{t+1}$, one-month and one-year log market returns from Fama-French market portfolio $r^{MKT}_{t+1/12}$ and $r^{MKT}_{t+1}$, and the 1-year dividend growth rate of S&P 500 index and the Fama-French market portfolio $\log(D_{t+1}/D_{t})$ and $\log(D^{MKT}_{t+1}/D^{MKT}_{t})$. Our sample is monthly observations 1988:01–2019:12.  | obs | mean | std | min | 25% | 50% | 75% | max | $\rho$
---|---|---|---|---|---|---|---|---|---
$dr_{t}$ | 384 | 4.027 | 0.494 | 2.952 | 3.727 | 4.044 | 4.208 | 6.632 | 0.919
$pd_{t}$ | 384 | 3.883 | 0.289 | 3.239 | 3.656 | 3.930 | 4.047 | 4.524 | 0.985
$\mu^{F}_{t}$ | 384 | -0.039 | 0.024 | -0.091 | -0.051 | -0.041 | -0.024 | 0.010 | 0.991
$KP_{t}$ | 384 | -0.504 | 0.073 | -0.725 | -0.562 | -0.482 | -0.450 | -0.378 | 0.955
$s_{t}^{0.5}\equiv\log(P^{0.5}/D)$ | 384 | 0.095 | 0.157 | -0.568 | 0.046 | 0.126 | 0.187 | 0.429 | 0.929
$s_{t}^{1}\equiv\log(P^{1}/D)$ | 384 | 0.009 | 0.041 | -0.184 | -0.015 | 0.013 | 0.034 | 0.108 | 0.022
$s_{t}^{1+}\equiv\log(P^{1+}/D)$ | 384 | 3.863 | 0.297 | 3.204 | 3.629 | 3.913 | 4.030 | 4.521 | 0.985
$r^{S\&P}_{t+1/12}$ | 384 | -0.142 | 0.280 | -2.241 | -0.210 | -0.098 | 0.016 | 0.393 | 0.766
$r^{S\&P}_{t+1}$ | 384 | -0.819 | 0.281 | -2.629 | -0.883 | -0.768 | -0.666 | -0.280 | 0.604
$r^{MKT}_{t+1/12}$ | 384 | 0.009 | 0.042 | -0.187 | -0.016 | 0.014 | 0.036 | 0.108 | 0.051
$r^{MKT}_{t+1}$ | 384 | 0.096 | 0.159 | -0.554 | 0.036 | 0.128 | 0.194 | 0.440 | 0.924
$\Delta d_{t+1}\equiv\log(D_{t+1}/D_{t})$ | 384 | 0.059 | 0.070 | -0.237 | 0.025 | 0.068 | 0.112 | 0.168 | 0.994
$\Delta d^{MKT}_{t+1}\equiv\log(D^{MKT}_{t+1}/D^{MKT}_{t})$ | 384 | 0.058 | 0.081 | -0.207 | 0.018 | 0.051 | 0.107 | 0.262 | 0.962
$g_{t}^{F}$ | 384 | 0.019 | 0.059 | -0.233 | -0.002 | 0.031 | 0.056 | 0.132 | 0.939

We construct market duration, $dr_{t}$, as the logarithm of the ratio of S&P
500 market capitalization to the price of dividends paid in the next year. The
traditional price-dividend ratio is the logarithm of the ratio of S&P 500
market capitalization to the realized dividend in the last year. Table 1
reports the summary statistics of $dr_{t}$, the traditional price-dividend
ratio $pd_{t}$, the valuation ratios ($s^{0.5}$, $s^{1}$, and $s^{1+}$), the
monthly return of S&P 500 ($r^{S\&P}_{t+1/12}$ where in the subscript $1/12$
denotes one month or $1/12$ of a year), the annual return of S&P 500
($r^{S\&P}_{t+1}$), and for comparison, the monthly and annual returns of the
Fama-French market portfolio (MKT) ($r^{MKT}_{t+1/12}$ and $r^{MKT}_{t+1}$).
Our sample includes monthly observations until 2019, i.e., before the extreme
market fluctuations during the Covid-19 pandemic. Our baseline analysis
focuses on the returns and dividends of the S&P 500 index because we construct
the strip prices using the S&P 500 futures data.101010Previous studies of
return predictability (e.g., Ang and Bekaert, 2007) also use S&P 500 Index as
a market proxy. For robustness, we also report results using Fama-French
market portfolio returns and dividends. Note that we include $\mu^{F}$ and
$KP$, the return predictors from Binsbergen and Koijen (2010) and Kelly and
Pruitt (2013), respectively, because these variables are also constructed to
extract information on state variables. To highlight our contribution, we will
compare our analysis with that in Binsbergen and Koijen (2010) and Kelly and
Pruitt (2013).

Figure 1: Spectrum and Cross-spectrum of Duration $dr$ and Price-Dividend
Ratio $pd$. The left panel shows the estimated spectral densities of $dr_{t}$,
$pd_{t}$, and the residuals of $dr_{t}$ after projecting on $pd_{t}$
($\epsilon^{pr}_{t}$). The integral of spectral density is equal to the
variance. The horizontal line starts from zero and ends at $\pi$, but is
labeled with the corresponding length of a cycle. The right panel shows the
cross-spectral density between $dr_{t}$ and $pd_{t}$. The integral of cross-
spectral density is equal to the covariance.

The mean of stock market duration $dr_{t}$ is $4.027$, which translates into
$56=\exp(4.027)$ years, meaning that the total market value is $56$ times the
valuation of dividends in the next year. $dr_{t}$ has a wide range of
variation, with a minimum of $2.952$ (i.e., 19 years) in November 1988 right
before the 1990-1991 recession and a maximum of 6.632 (i.e., 759 years) near
the end of the dot-com boom (November 2000). $dr_{t}$ has a lower monthly
autocorrelation (“$\rho$”) than $pd_{t}$.

$dr_{t}$ and $pd_{t}$ are correlated but contain distinct information. As
shown in the cross-spectrum in Figure 1, the correlation of 0.87 is mainly
from low-frequency movements. Panel A of Figure 1 shows the spectrum of
$dr_{t}$, $pd_{t}$, and $\epsilon^{dr}_{t}$ (the residual from linearly
projecting $dr_{t}$ on $pd_{t}$). The area under the spectrum curve is the
variance, so the figure provides a variance decomposition in the frequency
domain. On the horizontal axis, instead of showing the frequencies from zero
to $\pi$, we mark the corresponding length of the cycle for easier
interpretation. Once orthogonalized to $pd_{t}$, $dr_{t}$’s residual varies
mainly at annual or higher frequencies. Panel B plots the cross-spectrum of
$dr_{t}$ and $pd_{t}$. The integral is the covariance between $dr_{t}$ and
$pd_{t}$. The correlation between $dr_{t}$ and $pd_{t}$ is mainly from low
frequencies. This indicates that it is the high-frequency variation in
$dr_{t}$ that brings information distinct from that revealed by the
traditional price-dividend ratio. This is consistent with the findings in
Kragt, de Jong, and Driessen (2020) about different state variables
fluctuating at different frequencies. Figure A.1 in the Appendix shows the
spectrum analysis based on daily data with similar results.

As shown in Section 2, $dr_{t}$ and $pd_{t}$ are essentially different
combinations of state variables. Our state-space approach is closely related
to Binsbergen and Koijen (2010). Binsbergen and Koijen (2010) use the realized
returns and dividends to estimate a latent-state model and filter out the
conditional expected return, $\mu^{F}_{t}$, and the conditional expected
dividend growth rate, $g^{F}_{t}$. These filtered variables are also
combinations of state variables. We replicate the analysis of Binsbergen and
Koijen (2010), and throughout our analysis in this paper, we compare our
state-space representation via valuation ratios with the information about
state space in $\mu^{F}_{t}$ and $g^{F}_{t}$. Kelly and Pruitt (2013) also
take a state-space approach and use the cross-section of market-to-book ratios
of individual stocks to extract the conditional expected return of the
aggregate stock market. We have also replicated Kelly and Pruitt (2013) and
include $KP_{t}$ for comparison.

###  3.2 Analyzing the dimension of the state space

A Variance Explained

B Variable Loading

Figure 2: Principal Component Analysis of Valuation Ratios This figure reports
the PCA results for $dr$, $pd$, $s^{0.5}_{t}$, $s^{1}_{t}$, and $s^{1+}_{t}$.
Panel A plots the variance explained by each principal component. Panel B
plots the loading of each variable on the first two principal components.

We are interested in determining the dimension of the state space.
Traditionally, a variety of macroeconomic and financial-market variables have
been incorporated into different analytical frameworks as the state variables
that generate the relevant information for asset pricing (e.g., Cieslak and
Pang, 2021). For example, the market price-dividend ratio, $pd_{t}$, has been
commonly selected as a state variable (Muir, 2017). However, there has been
limited evidence on the dimension of the state space, i.e., how many state
variables are needed in an asset pricing framework. Among the theoretical
studies, such as Lettau and Wachter (2007) and Binsbergen and Koijen (2010), a
common assumption is that two state variables are sufficient for
characterizing the stock market dynamics and, in particular, for predicting
returns and cash-flow growth. In these models, these two state variables are
often directly specified as the conditional expectation of return and the
conditional expectation of cash-flow growth.

According to our analysis in Section 2, one way to determine the dimension of
the state space is to analyze the collection of valuation ratios, such as
those of dividend strips, $pd$, and our measure of market duration $dr$. In
Panel A of Figure 2, we report the results from principal component analysis
(PCA) of $dr$, $pd$, and valuation ratios of dividend strips with maturities
of six months, one year, and of dividends paid beyond one year. The first two
components account for 96.3% of total variance. In correspondence with our
theoretical analysis in Section 2, we show in Panel B of Figure 2 that these
valuation ratios have different loadings on the two principal components,
labeled as Dim1 (dimension 1) and Dim2 (dimension 2).

The results in Figure 2 indicate that the state space, mapped out by the
valuation ratios, is likely to be two-dimensional, and that we may pick any
two valuation ratios, for example, the pair of $pd$ and $dr$, to span the
state space. However, as pointed out by Kelly and Pruitt (2015), a shortcoming
of PCA analysis is that the information embedded in the principal components
may not be the information most relevant for objects of interest, which, in
asset pricing studies, are cash-flow and return dynamics. Therefore, we also
take a predictive regression approach to analyze the state space. The expected
return and expected dividend growth rate are driven by the state variables. By
projecting future returns and dividend growth rates on the valuation ratios,
we are able to evaluate how many valuation ratios are needed to achieve the
highest predictive power and thereby analyze the dimension of the underlying
state space based on information relevant to cash-flow and return dynamics.

Figure 3: $R^{2}$ from Dividend Growth Predictive Regressions with
Bootstrapped Confidence Interval. This figure reports $R^{2}$ from one-year
S&P 500 Index dividend growth predictive regressions with bootstrapped
confidence interval. The predictors include our main predictor – a linear
combination of ‘duration’ and 1-year dividend strip price to dividend ratio
$dr+s^{1}$, the price-dividend ratio $pd$, short-term (0.5- and 1-year)
dividend strip price to dividend ratio ($pd^{0.5}$ and $pd^{1}$), long-term
(beyond 1-year) dividend strip price to dividend ratio ($pd^{1+}$), the
filtered series for demeaned dividend growth following Binsbergen and Koijen
(2010) $g^{F}$, the single predictive factor extracted from 100 book-to-market
and size portfolios from Kelly and Pruitt (2013) $KP$, and all combinations of
$pd$, $pd^{0.5}$, $pd^{1}$ and $pd^{1+}$. Each bar corresponds to one separate
predictive regression. The range of the bar represents 95% confidence
intervals of the predictive $R^{2}$ obtained using the bootstrap method.

In Figure 3, we report the $R^{2}$ of predicting the annual dividend growth of
the S&P 500 index in the next year with different combinations of valuation
ratios. We report the detailed regression results in Table A.1 in the
Appendix. Our predictive regression is run on monthly observations. In the
first specification, the predictors include $dr$ and $pd$, which achieve the
highest adjusted $R^{2}$. In the next specification, for comparison, we show
the forecasting performance of $g^{F}$, the predictor from Binsbergen and
Koijen (2010) who develop a latent state model and filter out the conditional
expectation of growth rate $g^{F}$.

In the third specification in Figure 3, we include our market duration measure
$dr$ alone. Interestingly, $dr$ shows limited return predictive power in
comparison with the combination of $dr$ and $pd$. This is surprising as one
would expect that when $dr$ increases, i.e., the stock market assigns a higher
valuation to long-run dividends than the short-run dividends, the expected
growth rate of dividends should rise accordingly. In the fourth specification,
we show that the traditional price-dividend ratio also has limited cash-flow
predictive power.

The weak standalone predictive power of $dr$ and $pd$ stands in contrast with
the strong predictive power of $dr$ and $pd$ combined in the first
specification. This again indicates that to fully capture the information
embedded in the state variables, two valuation ratios are needed. In the
subsequent specifications, we show that different pairs of valuation ratios
exhibit different cash-flow predictive power. This shows the importance of
taking a predictive regression approach in determining the dimension of the
state space rather than simply relying on PCA of valuation ratios. Any given
pair of valuation ratios fully spans the two principal components, as
indicated in the linearly independent principal-component loadings of
different valuation ratios in Panel B of Figure 2. However, different pairs of
valuation ratios may still contain different information about return and
cash-flow dynamics.

In the last five specifications in Figure 3, we show that three or four
valuation ratios do not outperform two valuation ratios in forecasting
dividend growth. The adjusted $R^{2}$ we report is in-sample $R^{2}$, which
does not reflect poor out-of-sample performance due to potential overfitting
from adding more predictors; in other words, when competing with $dr$ and
$pd$, the three or four valuation ratios do not have a mechanical disadvantage
due to our choice of performance metric. Overall, our results indicate that
two valuation ratios (in particular, the combination of $dr$ and $pd$) are the
necessary minimum and sufficient for forecasting dividend growth.

Figure 4: $R^{2}$ from Return Predictive Regressions with Bootstrapped
Confidence Interval. This figure reports $R^{2}$ from one-year S&P 500 Index
return predictive regressions with bootstrapped confidence interval. The
predictors include our main predictor ‘duration’ $dr$, the price-dividend
ratio $pd$, short-term (0.5- and 1-year) dividend strip price to dividend
ratio ($pd^{0.5}$ and $pd^{1}$), long-term (beyond 1-year) dividend strip
price to dividend ratio ($pd^{1+}$), the filtered series for demeaned expected
returns following Binsbergen and Koijen (2010) $\mu^{F}$, the single
predictive factor extracted from 100 book-to-market and size portfolios from
Kelly and Pruitt (2013) $KP$, and all combinations of $pd$, $pd^{0.5}$,
$pd^{1}$ and $pd^{1+}$. Each bar corresponds to one separate predictive
regression. The range of the bar represents 95% confidence intervals of the
predictive $R^{2}$ obtained using the bootstrap method.

Next, we examine the dimension of the state space by forecasting returns. In
Figure 4, we report the $R^{2}$ of predicting the annual return of S&P 500
with different combinations of price-dividend ratios. Our predictive
regression is run monthly. We report the detailed regression results in Table
A.2 in the Appendix. The conclusion is similar to that in cash-flow
prediction. Having three or more price-dividend ratios does not improve
predictability relative to the best performance of combinations of two price-
dividend ratios. In particular, combining $dr$ and $pd$ achieves the best
forecasting performance. These results again suggest that the state space is
two-dimensional, and to capture the information embedded in state variables,
we need $dr$ and $pd$ for the purposes of analyzing both return and cash-flow
dynamics.

Interestingly, the predictive power of $dr$ alone is comparable to that of two
or more valuation ratios. Moreover, in the second, third, and fourth
specifications in Figure 4, we find that $\mu^{F}$, the return predictor in
Binsbergen and Koijen (2010), $KP$, the return predictor in Kelly and Pruitt
(2013), and the traditional price-dividend ratio all underperform $dr$ in our
sample period. Overall, our results suggest a close connection between
$dr_{t}$ and the conditional expected return. This finding is critical for
understanding what drives the variation of market duration and, vice versa,
what drives the expected return. We will explore this topic in the next
section.

Before we move on to Section 4, we briefly discuss the rich information
content of the pair $dr$ and $pd$ beyond forecasting return and cash-flow
growth. In Table A.3 in the Appendix, we report the $R^{2}$ from forecasting a
variety of macroeconomic and financial-market variables with $dr$ and $pd$,
only $dr$, only $pd$, and for illustration purposes, $dr$ in combination with
$s^{1}$ (the log price-dividend ratio of one-year dividend strip), and only
$s^{1}$. The pair $dr$ and $pd$ demonstrate the strongest and most consistent
predictive power when forecasting variables related to financial
intermediaries’ balance-sheet capacity (with the $R^{2}$ ranging from 30% to
40%). When forecasting macroeconomic variables related to business-cycle
dynamics, $dr$ and $pd$ have an $R^{2}$ consistently above 20%. Moreover, the
$R^{2}$ from forecasting sentiment proxies is consistently above 10%. Finally,
combining $dr$ and $pd$ outperforms other specifications.

Overall, our results suggest that $dr$ and $pd$ together provide not only
relevant information for understanding asset-pricing dynamics but also other
key objects in macro-finance. Through our analysis, we hope to demonstrate the
importance of incorporating market duration, $dr$, as a key state variable in
macro-finance studies.

##  4 Duration and Expected Return

A striking finding from our forecasting exercises is that market duration
alone is quite sufficient for predicting returns. Augmenting $dr$ with $pd$ or
other valuation ratios does not significantly improve the performance.
Therefore, $dr$ captures the combination of state variables that drive the
conditional expected return. Next, we provide further evidence on the return
predictive power of $dr$. As shown in Section 2, the loadings of $dr$ on state
variables depend on various parameters. We find that $dr$ and the conditional
expected return coincide in their state-variable loadings.

###  4.1 Univariate return prediction

We run the standard predictive regression to forecast market return in the
next twelve months:

$r_{t+1}=\alpha+\beta dr_{t}+\epsilon_{t+1}\text{,}$ (16)

Because we use overlapping monthly data, we adopt Newey and West (1987)
standard errors with 18 lags to account for the moving-average structure
induced by overlap (Cochrane and Piazzesi, 2005). We also calculate Hodrick
(1992) standard errors. Hodrick (1992) shows that GMM-based autocovariance
correction (e.g., Newey and West, 1987) may have poor small-sample properties.
Under the serial correlation in the error term, another concern is the bias
induced by the persistence of the predictor.111111The persistence of a return
predictor can cause small-sample bias (Nelson and Kim, 1993; Stambaugh, 1999)
and spurious regression (Ferson, Sarkissian, and Simin, 2003). While $dr_{t}$
has an autocorrelation below that of the traditional price-dividend ratio,
$pd_{t}$, we still report the adjusted estimate of $\beta$ following Stambaugh
(1999). In the appendix (Table A.4), we also report the IVX-Wald test of
predictive power (Kostakis, Magdalinos, and Stamatogiannis, 2014) that
explicitly accounts for the predictor persistence.

The adjusted $R^{2}$ measures in-sample fitness. Several studies raised
concerns over the out-of-sample performance of return predictors (Bossaerts
and Hillion, 1999; Goyal and Welch, 2007). To address these issues, we report
the out-of-sample $R^{2}$ and two formal tests of out-of-sample performance.
We calculate out-of-sample forecasts as a real-time investor, using data up to
time $t$ in the predictive regression to estimate $\beta$, which is then
multiplied by the time-$t$ value of the predictor to form the forecast. Out-
of-sample forecasting starts from December 1997, when we have at least ten
years of data. Out-of-sample $R^{2}$ is

$R^{2}_{OOS}=1-\frac{\sum_{t}\left(r_{t+1}-\hat{r}_{t+1}\right)^{2}}{\sum_{t}\left(r_{t+1}-\overline{r}_{t}\right)^{2}}\text{,}$

where $\hat{r}_{t+1}$ is the forecast value and $\overline{r}$ is the average
of twelve-month returns (the first is January-December 1998). The out-of-
sample $R^{2}$ lies in the range $\left(-\infty,1\right]$, where a negative
number means that a predictor provides a less accurate forecast than the
historical mean.

We report the p-value of two out-of-sample performance tests, “$ENC$” and
“$CW$”. $ENC$ is the encompassing forecast test derived by Clark and McCracken
(2001), which is widely used in the literature. We test whether the predictor
has the same out-of-sample forecasting performance as the historical mean and
compare the value of the statistic with critical values calculated by Clark
and McCracken (2001) to obtain a p-value range. Clark and West (2007) adjust
the standard MSE t-test statistic to produce a modified statistic ($CW$) that
has an asymptotic distribution well approximated by the standard normal
distribution, so for $CW$, we report the precise p-value.

Table 2: Annual Return Prediction This table reports the results of predictive regression (equation (16)). The left-hand side variable is the return of the S&P 500 index in the next twelve months. We consider four right-hand side variables (i.e., predictors), $dr_{t}$, $pd_{t}$, filtered series for expected returns following Binsbergen and Koijen (2010) $\mu^{F}$, and the single predictive factor extracted from 100 book-to-market and size portfolios from Kelly and Pruitt (2013) $KP$. The $\beta$ estimate is reported, followed by Newey and West (1987) t-statistic (with 18 lags), Hodrick (1992) t-statistic, the coefficient adjusted for Stambaugh (1999) bias, and the in-sample adjusted $R^{2}$. We run the regression monthly. Starting from December 1997, we form out-of-sample forecasts of return in the next twelve months by estimating the regression with data up to the current month and use the forecasts to calculate out-of-sample $R^{2}$, ENC test (Clark and McCracken, 2001), and the p-value of CW test (Clark and West, 2007). Our monthly sample is 1988:01–2019:12.  | $r_{t+1}$
---|---
|  (1) |  (2) |  (3) |  (4) |  (5)
$dr_{t}$ |  -0.156 |  |  |  |  -0.228
Hodrick t |  [-3.354] |  |  |  |  [-2.924]
Newey-West t |  (-4.499) |  |  |  |  (-3.517)
Stambaugh bias adjusted $\beta$ |  -0.146 |  |  |  |
$pd_{t}$ |  |  -0.199 |  |  |  0.141
|  |  [-2.367] |  |  |  [1.721]
|  |  (-2.747) |  |  |  (1.209)
|  |  -0.189 |  |  |
$\mu^{F}_{t}$ |  |  |  2.584 |  |
|  |  |  [2.313] |  |
|  |  |  (2.804) |  |
|  |  |  2.594 |  |
$KP_{t}$ |  |  |  |  0.895 |
|  |  |  |  [2.960] |
|  |  |  |  (2.857) |
|  |  |  |  0.905 |
$N$ |  384 |  384 |  384 |  384 |  384
$R^{2}$ |  0.248 |  0.138 |  0.156 |  0.149 |  0.264
OOS $R^{2}$ |  0.146 |  0.004 |  -0.032 |  0.041 |  0.180
ENC |  2.968 |  0.833 |  0.651 |  2.978 |  5.985
$p(ENC)$ |  $<$0.05 |  $>$0.10 |  $>$0.10 |  $<$0.05 |  $<$0.01
$p(CW)$ |  0.022 |  0.200 |  0.303 |  0.031 |  0.021

Table 2 presents the results. Column (1) shows that the market duration, $dr$,
demonstrates a striking degree of return predictive power. The in-sample
estimation generates a predictive $R^{2}$ reaching $24.8\%$.121212Foster,
Smith, and Whaley (1997) discuss the potential data mining issues that arise
from researchers searching among potential regressors. They derive a
distribution of the maximal $R^{2}$ when $k$ out of $m$ potential regressors
are used as predictors and calculate the critical value for $R^{2}$, below
which the prediction is not statistically significant. For instance, when
$m=50$, $k=5$, and the number of observations is 250, the $95\%$ critical
value for $R^{2}$ is 0.164. Out-of-sample forecasts deliver an $R^{2}$ of
$14.6\%$, significantly outperforming the historical mean as shown by the
p-values of $ENC$ and $CW$.131313In our calculation of out-of-sample $R^{2}$
starts from Dec. 1997 (after the first ten years of data). Figure A.8 in the
Appendix reports the out-of-sample $R^{2}$ for different start dates and
compares the OOS $R^{2}$ of $dr$ with that of $pd$. The predictive coefficient
is also large in magnitude, indicating high volatility of the conditional
expected return. A decrease of $dr_{t}$ by one standard deviation adds 7.7% to
the expected return. Both Newey-West and Hodrick t-statistics are significant
at least at the $1\%$ level.

Column (2) of Table 2 reports the results for $pd_{t}$. The predictive power
of $pd_{t}$ is much weaker than $dr_{t}$ in all aspects. Its in-sample $R^{2}$
is almost half of that of $dr_{t}$ and $pd_{t}$ barely exhibits any out-of-
sample predictive power with $R^{2}$ equal to $0.4\%$. In both $ENC$ and $CW$
tests, $pd_{t}$ fails to beat the historical mean with any statistical
significance. Its coefficient is smaller in magnitude than that of $dr_{t}$. A
decrease in $pd_{t}$ by one standard deviation leads to an increase of
expected return by 5.8%, implying a less volatile expected return than the one
from $dr_{t}$. The IVX-Wald test of Kostakis, Magdalinos, and Stamatogiannis
(2014) in Table A.4 in the Appendix also supports the significant predictive
power of $dr_{t}$ while rejecting the predictive power of $pd_{t}$.

Next, we compare $dr_{t}$ with two return predictors that are conceptually
related. Binsbergen and Koijen (2010) extract information about state
variables that drive the conditional expected return and expected cash-flow
growth by estimating a latent-state model. Our approach differs as we do not
estimate or filter the state variables but instead rely on $dr_{t}$, $pd_{t}$,
and valuation ratios of dividend strips. In particular, the construction of
our preferred return predictor, $dr_{t}$, only requires the total market
capitalization and the price of dividends paid in the next year, which are
directly observed from the market rather than estimated. In Column (3) of
Table 2, we follow the procedure in Binsbergen and Koijen (2010) to construct
their return predictor, $\mu^{F}_{t}$. While $\mu^{F}_{t}$ slightly
outperforms $pd_{t}$, its predictive power is weaker than that of $dr_{t}$
across different metrics. Note that Binsbergen and Koijen (2010) conduct
analysis with a time period different from ours. Our sample period is
constrained by the availability and reliability of index futures data and ends
before the Covid-19 era of extreme downward and upward market movements.

While Binsbergen and Koijen (2010) use the aggregate data on realized returns
and dividend growth, Kelly and Pruitt (2013) demonstrate another filtering
method that utilizes the cross-section of market-to-book ratios of individual
stocks. Individual stocks’ market-to-book ratios map out the state variables
that drive the aggregate market but, as we show in Appendix I.2, these
valuation ratios contain noise that is orthogonal to the expected market
return. Kelly and Pruitt (2013) use the method of partial least squares to
reduce the noise. Our approach differs as we avoid dealing with such noise.
$dr_{t}$, $pd_{t}$, and the valuation ratios of dividend strips of the
aggregate market differ in their loadings on the aggregate state variables but
do not contain idiosyncratic noise. In our approach, the challenge is to find
the valuation ratio or a combination of valuation ratios whose state-variable
loadings coincide with those of the conditional expected return and thereby
achieve the highest predictive power. In the next subsection, we characterize
the necessary and sufficient condition for $dr_{t}$ to be such a return
predictor. Following the procedure in Kelly and Pruitt (2013), we construct
their return predictor, denoted by $KP_{t}$. In column (4) of Table 2, we
report the prediction results. $KP_{t}$ significantly outperforms $pd_{t}$ but
still underperforms $dr_{t}$ across different metrics, such as Newey-West
t-statistic, Hodrick t-statistic, in-sample $R^{2}$, out-of-sample $R^{2}$,
$ENC$, $CW$, and, IVX-Wald test reported in Table A.4 in the Appendix.

Table 3: Correlation between Return Predictors This table reports the correlation matrix of four main return predictors. $dr$ is our main return predictor, “duration.” $pd$ is the price-dividend ratio of the S&P 500 index. $\mu^{F}$ is the filtered series for demeaned expected returns, following Binsbergen and Koijen (2010), $KP$ is the single predictive factor extracted from 100 book-to-market and size portfolios from Kelly and Pruitt (2013). Data sample: 1988:01–2019:12.  | $dr$ | $pd$ | $\mu_{F}$ | $KP$
---|---|---|---|---
$dr$ | 1 |  |  |
$pd$ | 0.873 | 1 |  |
$\mu_{F}$ | -0.892 | -0.967 | 1 |
$KP$ | -0.565 | -0.496 | 0.468 | 1

While these return predictors are correlated as shown in Table A.12, their
predictive power differs significantly with market duration outperforming the
rest. In the appendix, we demonstrate the robustness of our results by
repeating the analysis with different forecasting targets. In Table A.5, we
replace the S&P 500 annual return with the excess S&P 500 annual return, and
in Table A.6 and A.7, we consider the Fama-French market portfolio return and
excess return, respectively. Finally, we also show that market duration
demonstrates superior return predictive power at a monthly horizon. Our
baseline results are reported in A.8, and see Table A.9 for results on
predicting monthly S&P 500 excess return. Table A.10 and Table A.11 report the
results on predicting the monthly Fama-French market portfolio return and
excess return, respectively.

Figure 5: Comparison with Alternative Return Predictors. This graph compares
the 1-year return predictive power between $dr_{t}$ and other commonly studied
predictors in our sample period. Panel A reports the in-sample adjusted
$R^{2}$. Panel B reports the out-of-sample $R^{2}$. Negative out-of-sample
$R^{2}$ indicates that the predictive power is below the historical mean.
Panel C reports the absolute values of Newey and West (1987) t-statistic (with
an 18-month lag). Panel D reports the absolute values of Hodrick (1992)
t-statistic. Most predictors are from Goyal and Welch (2007) and include the
price-dividend ratio (pd), the default yield spread (dfy), the inflation rate
(infl), stock variance (svar), the cross-section premium (csp), the dividend
payout ratio (de), the long-term yield (lty), the term spread (tms), the
T-bill rate (tbl), the default return spread (dfr), the dividend yield (dy),
the long-term rate of return (ltr), the earnings-to-price ratio (ep), the book
to market ratio (bm), the investment-to-capital ratio (ik), the net equity
expansion ratio (ntis), the percent equity issuing ratio (eqis), and the
consumption-wealth-income ratio (cay). SII is the short interests index from
Rapach, Ringgenberg, and Zhou (2016) (1988-2014). SVIX is an option-implied
lower bound of the 1-year equity premium from Martin (2017) (1996-2012). KP is
the single predictive factor extracted from 100 book-to-market and size
portfolios from Kelly and Pruitt (2013). BK is the filtered series for
expected returns following Binsbergen and Koijen (2010).

So far, we have compared the return predictive power of $dr_{t}$ with that of
$pd_{t}$, which together with $dr_{t}$ span the state space. We also
considered $\mu^{F}_{t}$ and $KP_{t}$ that are conceptually related. Figure 5
compares $dr$ with other predictors proposed in the literature, including the
price-dividend ratio (pd), the default yield spread (dfy), the inflation rate
(infl), stock variance (svar), the cross-section premium (csp), the dividend
payout ratio (de), the long-term yield (lty), the term spread (tms), the
T-bill rate (tbl), the default return spread (dfr), the dividend yield (dy),
the long-term rate of return (ltr), the earnings-to-price ratio (ep), the book
to market ratio (bm), the investment-to-capital ratio (ik), the net equity
expansion ratio (ntis), the percent equity issuing ratio (eqis), the
consumption-wealth-income ratio (cay), the short interests index (SII), and
the option-implied lower bound of 1-year equity premium (SVIX).141414Note that
the dividend yield (dy) is not the inverse of price-dividend ratio ($pd$)
because in the denominator of dy is the lagged market value (not the current
value). Most predictors are studied in a return predictability survey by Goyal
and Welch (2007), and others are proposed more recently, such as the short
interest index, “SII” in Rapach, Ringgenberg, and Zhou (2016), and SVIX
(Martin, 2017). In the comparison, we also include $pd_{t}$, $\mu^{F}_{t}$,
and $KP_{t}$ from Table 2.151515When constructing $KP_{t}$, we estimate the
partial least squares model following the procedure in Kelly and Pruitt
(2013). In Figure A.5, We report the comparison of return predictive power
using the authors’ original parameter estimates rather than our own estimates.

In Figure 5, we report in-sample (“IS”) $R^{2}$, out-of-sample (“OOS”)
$R^{2}$, the absolute values of Newey-West, and Hodrick t-statistics. $dr$
outperforms other predictors in all aspects. Among the alternatives, $KP$,
$pd$, and the book-to-market ratio (“$bm$”) deliver the most consistent
performance across the four metrics while $\mu^{F}$ exhibits strong
performance except for OOS $R^{2}$. The performance of other return predictors
is not quite consistent across different metrics.

In Table A.12, we report the correlation between $dr$ and these predictors.
Besides $pd$, $\mu^{F}$, and $KP$, all the other predictors with correlation
above 50% or below -50%, are all valuation ratios, such as the market-to-book
ratio and price-earnings ratio. This is consistent with the intuition in Kelly
and Pruitt (2013) and with our emphasis on using valuation ratios to capture
information embedded in the state variables. In the appendix, we compare $dr$
with other predictors for alternative forecasting targets, such as S&P 500
excess annual return (Figure A.2), Fama-French market portfolio annual return
(Figure A.3), and Fama-French market portfolio excess annual return (Figure
A.4). As shown in Figure A.5, market duration exhibits the strongest
predictive power across the four metrics and such performance is consistent
across all forecasting targets.

###  4.2 A two-dimensional state space model

The findings on return predictability are critical for understanding the
structure of both the conditional expected return and market duration. Next,
we show that the conditional expected return is a univariate function of $dr$
(and vice versa) when the market does not contain information on future cash-
flow growth beyond the next year. Intuitively, under this condition, the price
of next year’s dividends exhausts all information about future cash-flow
growth, so $dr$, which is the logarithm of market value minus the logarithm of
the price of next year’s dividend, teases out cash-flow information and only
contains information about the discount rate.

Stock prices are determined by the conditional expectations of future returns
and dividends over different horizons (Campbell and Shiller, 1988). Therefore,
a model of the aggregate stock market requires at least two state variables
that capture the conditional expected return and dividend growth; in other
words, the lower bound of the state-space dimension is two. Our analysis so
far shows that two is also the upper bound. Guided by our findings in Section
3, we simplify the general model in Section 2 by specifying a two-dimensional
state space. As in Lettau and Wachter (2007) and Binsbergen and Koijen (2010),
one state variable drives the conditional expectation of the dividend growth
rate, and the second state variable drives the conditional expected return
through the price of risk. Specifically, $g_{t}$ is given by

$g_{t}=z_{t}+\overline{g}-\frac{1}{2}\sigma_{D}^{\top}\Sigma\sigma_{D},$ (17)

where $z_{t}$ has the following law of motion

$z_{t+1}=\rho_{z}z_{t}+\sigma_{z}^{\top}\epsilon_{t+1}\,.$ (18)

The second state variable, $y_{t}$, with a law of motion

$y_{t+1}=\rho_{y}y_{t}+\sigma_{y}^{\top}\epsilon_{t+1}\,,$ (19)

drives the price of risk $\lambda_{t}$, so equation (5) becomes

$\lambda_{t}=\overline{\lambda}+y_{t},$ (20)

and the stochastic discount factor (SDF) is given by

$M_{t+1}=\exp\left\\{-r_{f}-\frac{1}{2}\lambda_{t}^{2}(\sigma_{\lambda}^{\top}\Sigma\sigma_{\lambda})^{2}-\lambda_{t}\sigma_{\lambda}^{\top}\epsilon_{t+1}\right\\}.$
(21)

In a two-dimensional state space where one state variable, $z_{t}$, is
assigned to drive the expected dividend growth rate, the price of risk,
$\lambda_{t}$, must be unidimensional, driven by $y_{t}$, and accordingly,
shocks enter into the SDF as a scalar, i.e., the linear combination given by
$\sigma_{\lambda}$. Any shock can be priced. The price of risk for the $n$-th
shock is $\lambda_{t}\sigma_{\lambda}(n)$, where $\sigma_{\lambda}(n)$ is the
$n$-th element of $\sigma_{\lambda}$.

As in Section 2, the $N$-by-$1$ shock vector $\epsilon_{t+1}$ contains all
information at $t+1$. The variables’ shock loadings may differ, for example,
$\sigma_{z}\neq\sigma_{y}$. $z_{t}$ and $y_{t}$ can be correlated through
their overlapping exposure to shocks. Also note that, in comparison with the
general model in Section 2, the state-variable loadings of $g_{t}$ and
$\lambda_{t}$, i.e., $\phi$ and $\lambda_{t}$ are set to one. This
simplification can be done as $g_{t}$ and $\lambda_{t}$ load on one state
variable, $z_{t}$ and $y_{t}$, respectively.

In Appendix I, we solve the log price-dividend ratio of the aggregate market

$pd_{t}=A_{pd}+B_{pd}y_{t}+C_{pd}z_{t},$ (22)

where $A_{pd}$, $B_{pd}$, and $C_{pd}$ are constant. And the log price-
dividend ratio of the dividend strip that matures in one year,

$s^{1}_{t}=A_{1}+B_{1}y_{t}+C_{1}z_{t}.$ (23)

Therefore, market duration is

$dr_{t}=pd_{t}-s^{1}_{t}=A_{pd}-A_{1}+(B_{pd}-B_{1})y_{t}+(C_{pd}-C_{1})z_{t},$
(24)

where $B_{pd}\neq B_{pd}-B_{1}$ and $C_{pd}\neq C_{pd}-C_{1}$. Since $dr_{t}$
and $pd_{t}$ have different loadings on $z_{t}$ and $y_{t}$, we can solve
$z_{t}$ and $y_{t}$ from $dr_{t}$ and $pd_{t}$. Therefore, market duration and
the traditional price-dividend ratio span the state space. In particular,
$dr_{t}$ and $pd_{t}$ contain all the necessary information for forecasting
return and dividend growth as we show in Section 3.

######  Proposition 1 (Spanning).

The state variables are functions of $dr_{t}$ and $pd_{t}$, and vice versa.

In Appendix I, we show $C_{pd}=1/(1-\kappa_{1}\rho_{z})$ and $C_{1}=1$ and
obtain the following result.

######  Proposition 2 (Discount rate and duration).

The conditional expected return is a function of $y_{t}$

$\mathbb{E}_{t}[r_{t+1}]=A_{er}+B_{er}y_{t},\,$ (25)

where $A_{er}$ and $B_{er}$ are constant. Under $\rho_{z}=0$,
$\mathbb{E}_{t}[r_{t+1}]$ is a univariate function of $dr_{t}$ and vice versa.

When $\rho_{z}$, the autoregressive coefficient of expected dividend growth
rate $z_{t}$, is zero, we have $dr_{t}=A_{pd}-A_{1}+(B_{pd}-B_{1})y_{t}$ as
the exposure of market duration to $z_{t}$ is muted. As $y_{t}$ drives the
conditional expected return, $dr_{t}$ and $\mathbb{E}_{t}[r_{t+1}]$ become
univariate functions of one another. Market participants’ information at time
$t$ about future dividends is summarized by $z_{t}$ which determines the
expected dividend growth rate from $t$ to $t+1$. Therefore, if $z_{t}$ lacks
persistence, the market participants do not have information about dividends
beyond $t+1$. By construction, $dr_{t}$ teases out information about dividend
growth up to $t+1$ by deducting $s^{1}_{t}$ from $pd_{t}$, so, intuitively, it
contains only the information about the discount rate or expected return under
$\rho_{z}=0$.

###  4.3 The persistence of cash-flow growth expectations

We estimate $\rho_{z}$, the persistence of expected dividend growth, using two
approaches. Our main analysis is based on analyst forecasts as a proxy for
market participants’ growth expectations. We estimate $\rho_{z}$ with two
econometric models. For robustness, we also estimate $\rho_{z}$ by fitting the
latent state model given by (17) and (18) to dividend data.

Our model does not require investors to have rational expectations. As a
matter of fact, even if the data-generating processes differ from equations
(17) and (18) in our model, as long as agents in this economy take these
equations as the stochastic processes that govern the aggregate dividends, our
results on stock and dividend strip valuation hold. Therefore, to estimate
$\rho_{z}$, the persistence of the expected dividend growth rate, we examine
market participants’ subjective expectations—the analyst forecasts obtained
from IBES.

A large literature has demonstrated that analyst forecasts reflect analysts’
beliefs as their compensation are tightly linked to forecast precision, and
that analyst forecasts are likely to represent the market participants’
beliefs more broadly (e.g., Mikhail, Walther, and Willis, 1999; Cooper, Day,
and Lewis, 2001; Bradshaw, 2004). Admittedly, analyst forecasts may be
distorted due to various incentive and institutional frictions (e.g., Gu and
Wu, 2003; Malmendier and Shanthikumar, 2007, 2014). However, the bias is
contained as long as such frictions do not vary over time systematically in a
way that correlates with the analysts’ true beliefs. Moreover, we will
supplement our estimate of $\rho_{z}$ based on analyst forecasts with an
alternative estimate from a standard filtering approach applied to the state
space model given by equation (17) and (18).

Recent studies using analysts’ cash-flow expectations have made substantial
progress in explaining a variety of phenomena in asset pricing.161616For
example, Bordalo, Gennaioli, La Porta, and Shleifer (2019) find that analyst
forecasts of firms’ long-term earnings growth overreact to news about
fundamentals, leading to predictable cross-sectional return variations. De La
O and Myers (2021) show that analyst short-term earnings forecasts, aggregated
to the S&P 500 index level, explain a substantial share of the price-dividend
ratio’s variation. Nagel and Xu (2022) propose a model where agents form
beliefs of cash-flow growth based on their experienced growth. Such beliefs
give rise to a sizable and counter-cyclical equity premium. Bordalo,
Gennaioli, La Porta, and Shleifer (forthcoming) incorporate analyst forecasts
of short- and long-term earnings of the S&P 500 index into a dividend discount
model with a constant discount rate, which generates a “synthetic” index price
that closely resembles the actual price and exhibits “excess” volatility.
Moreover, our focus on the persistence of agents’ growth expectations is
related to the theoretical and empirical works of Gabaix (2019) and Wang
(2020), who highlight the importance of agents’ perceived persistence of key
state variables in generating patterns of under and over-reaction in belief
formation and asset prices.

Table 4: Summary Statistics: Cash Flow Growth Forecasts This table reports the number of observations, mean, standard deviation, minimum, maximum, quartiles, and monthly autocorrelation ($\rho$) of various measures of cash flow growth expectations. $\mathbb{E}^{A}_{t}{\Delta e_{t,t+1}}$, $\mathbb{E}^{A}_{t}{\Delta e_{t+1,t+2}}$, and $\mathbb{E}^{A}_{t}{\Delta e_{t+2,t+3}}$ are forecasts of 1-year earnings growth for fiscal year 1, 2, and 3 provided by IBES Global Aggregate (IGA). IGA $\Delta e_{t}$ and Compustat $\Delta e_{t}$ are the actual 1-year earnings growth from IGA and Compustat, respectively. $\mathbb{E}^{A}_{t}{\Delta e_{t,t+1}}$ and $LTG_{t}$ are forecasts of 1-year and long-term earnings growth that we self-aggregate from the IBES Unadjusted US Summary Statistics File. Data sample: 1988:01–2019:12.  | obs | mean | std | min | 25% | 50% | 75% | max | $\rho$
---|---|---|---|---|---|---|---|---|---
$\mathbb{E}^{A}_{t}{\Delta e_{t,t+1}}$ | 384 | 0.103 | 0.096 | -0.167 | 0.056 | 0.103 | 0.154 | 0.425 | 0.897
$\mathbb{E}^{A}_{t}{\Delta e_{t+1,t+2}}$ | 384 | 0.134 | 0.043 | -0.069 | 0.104 | 0.127 | 0.157 | 0.269 | 0.830
$\mathbb{E}^{A}_{t}{\Delta e_{t+2,t+3}}$ | 384 | 0.130 | 0.036 | 0.052 | 0.100 | 0.122 | 0.159 | 0.217 | 0.953
IGA $\Delta e_{t,t+1}$ | 384 | 0.072 | 0.135 | -0.380 | -0.008 | 0.092 | 0.148 | 0.425 | 0.929
Compustat $\Delta e_{t,t+1}$ | 384 | 0.068 | 0.481 | -2.175 | -0.042 | 0.122 | 0.187 | 2.190 | 0.976
$LTG_{t}$ | 384 | 0.125 | 0.018 | 0.093 | 0.115 | 0.120 | 0.129 | 0.187 | 0.986

As the coverage of dividend forecasts started in 2003 in IBES and is too short
for our analysis, we follow the literature and proxy for analysts’ expectation
of dividend growth with their earnings forecasts (available since 1976). The
following accounting identity connects the earnings and dividends:
$D_{t}=\text{Earnings}_{t}\times(1-\text{plowback rate}_{t}).$ As documented
by Pástor, Sinha, and Swaminathan (2008) and Chen, Da, and Zhao (2013), the
plowback rate is quite stable. Therefore, the growth rates of dividends are
empirically close to those of earnings: For $k=1,\,2,\,$ and $3$ years,

$\displaystyle\Delta
d_{t+k}\equiv\ln\left(\frac{D_{t+k}}{D_{t+k-1}}\right)\approx\Delta
e_{t+k}\equiv\ln\left(\frac{\text{Earnings}_{t+k}}{\text{Earnings}_{t+k-1}}\right).$
(26)

Note that $\Delta d_{t+k}$ is the one-year growth rate from $t+k-1$ to $t+k$.
For example, for $k=2$, $\Delta d_{t+2}$ is the dividend growth rate from
$t+1$ to $t+2$. The same notation applies to earnings and earnings growth
expectations. Taking the analyst’s expectations,
$\mathbb{E}^{A}_{t}\left(\cdot\right)$, on both sides, we can proxy for
expected dividend growth using expected earnings growth:

$\displaystyle\mathbb{E}_{t}^{A}\left(\Delta
d_{t+k}\right)\approx\mathbb{E}_{t}^{A}\left(\Delta
e_{t+k}\right)\approx\ln\left(\frac{\mathbb{E}_{t}^{A}\left(\text{Earnings}_{t+k}\right)}{\mathbb{E}_{t}^{A}\left(\text{Earnings}_{t+k-1}\right)}\right),$
(27)

where, for $k=1$,
$\mathbb{E}_{t}^{A}\left(\text{Earnings}_{t+k-1}\right)=\text{Earnings}_{t}$
is the realized dividend in the current year.171717We switch the order of
$\mathbb{E}^{A}_{t}\left(\cdot\right)$ and $\ln\left(\cdot\right)$, assuming
Jensen’s inequality terms are negligible for the purpose of estimating
$\rho_{z}$. This is in line with the model in Section 2: The growth rates are
log-normally distributed, so the ignored terms contain constant variances,
which do not affect the estimates of $\rho_{z}$.

IBES Global Aggregates (IGA) provides a forecast of aggregate earnings growth
for the S&P 500 index based on firm-level earnings forecasts. The aggregation
procedure weighs individual companies by their market capitalization.181818To
deal with the fact that companies have different fiscal year-end, IGA
calendarizes all company-level data to a December calendar year before
aggregation. This approach follows the Compustat rule. Please refer to
“Thomson Reuters Datastream IBES Global Aggregates Reference Guide” for more
detail. To transform earnings forecasts to forecasts of growth rates, IGA
takes the procedure given by equation (27). The IGA data is available at a
weekly frequency. We consider both weekly and monthly frequencies. For monthly
frequency, we take the last weekly observation of each month. At time $t$,
analysts’ forecasts of earnings are available at three horizons: one, two, and
three years, i.e., $\mathbb{E}_{t}^{A}\left(Earnings_{t+1}\right)$,
$\mathbb{E}_{t}^{A}\left(Earnings_{t+2}\right)$, and
$\mathbb{E}_{t}^{A}\left(Earnings_{t+3}\right)$. Moreover, since IGA does not
aggregate firm-level long-term growth (LTG) forecasts, we follow the same
aggregation procedure to create an index-level LTG forecast that will allow us
to estimate $\rho_{z}$ with alternative methods. The firm-level LTG forecasts
are obtained from the IBES Unadjusted US Summary Statistics File. Table 4
provides summary statistics.

Next, we map the expectations of cash-flow growth to the model counterparts
and derive a system of equations that can be used to estimate $\rho_{z}$.
First, we acknowledge that the analyst forecasts may not perfectly capture
market participants’ expectations by adding a noise term between analysts’
expectation and the market participants’ expectation: For $k=1,2,3$,

$\mathbb{E}^{A}_{t}\left(\Delta e_{t+k}\right)=\mathbb{E}_{t}\left(\Delta
e_{t+k}\right)+\varepsilon^{A}_{t,k},$ (28)

where $\mathbb{E}_{t}\left(\cdot\right)$, is the market participants’
expectation. Equations (17) imply

$\displaystyle\mathbb{E}^{A}_{t}\left(\Delta e_{t+1}\right)$
$\displaystyle=g+z_{t}+\varepsilon^{A}_{t,1}$
$\displaystyle\mathbb{E}^{A}_{t}\left(\Delta e_{t+2}\right)$
$\displaystyle=g+\mathbb{E}_{t}\left(z_{t+1}\right)+\varepsilon^{A}_{t,2}=g+\rho_{z}z_{t}+\varepsilon^{A}_{t,2}$
$\displaystyle\mathbb{E}^{A}_{t}\left(\Delta e_{t+3}\right)$
$\displaystyle=g+\mathbb{E}_{t}\left(z_{t+2}\right)+\varepsilon^{A}_{t,3}=g+\rho_{z}^{2}z_{t}+\varepsilon^{A}_{t,3}.$

Using the first equation to substitute out $z_{t}$ in the second and third
equations, we obtain a system:

$\underset{\equiv\,\mathbf{y}^{A}_{t}}{\underbrace{\begin{bmatrix}\mathbb{E}^{A}_{t}\left(\Delta
e_{t+2}\right)\\\ \mathbb{E}^{A}_{t}\left(\Delta
e_{t+3}\right)\end{bmatrix}}}=\left(1-\rho_{z}\right)g+\rho_{z}\underset{\equiv\,\mathbf{x}^{A}_{t}}{\underbrace{\begin{bmatrix}\mathbb{E}^{A}_{t}\left(\Delta
e_{t+1}\right)\\\ \mathbb{E}^{A}_{t}\left(\Delta
e_{t+2}\right)\end{bmatrix}}}+\underset{\equiv\,\epsilon_{t}}{\underbrace{\begin{bmatrix}\varepsilon^{A}_{t,1}-\rho_{z}\varepsilon^{A}_{t,0}\\\
\varepsilon^{A}_{t,2}-\rho_{z}\varepsilon^{A}_{t,1}\end{bmatrix}}}.$ (29)

Therefore, we estimate $\rho_{z}$ by regressing $\mathbf{y}^{A}_{t}$ on
$\mathbf{x}^{A}_{t}$. The identification assumption is that under the
econometricians’ belief, the expectation of the disturbance $\epsilon_{t}$ is
zero conditional on $\mathbf{x}^{A}_{t}$. The deviations of analysts’
expectations from market participants’ expectations are allowed to be
correlated across the starting dates of annual dividend growth, i.e., $t$,
$t+1$, and $t+2$.191919The identification of $\rho_{z}$ is robust to the
correlation between these errors and $\epsilon_{t}$ in the model in Section 2,
i.e., the structural shocks to realized dividend, market participants’ beliefs
on cash-flow dynamics, and their price of risk.

Table 5: The Persistence of Expected Cash-Flow Growth from Analyst Forecasts This table reports the estimates of $\rho_{z}$, the autoregressive coefficient of expected cash-flow growth, based on equation (29). The estimation uses aggregate earnings growth forecasts of the S&P 500 Index obtained from IGA. Columns (1) and (3) report the estimates of $\rho_{z}$ using monthly data, while columns (2) and (4) report the estimates of $\rho_{z}$ using weekly data. columns (1) and (2) use earnings growth forecasts for 1,2 and 3 years ahead to estimate the two-equation system (29), while columns (3) and (4) only use earnings growth forecasts for 1 and 2 years ahead to estimate the first equation in (29). $t$-statistics based on Driscoll-Kraay standard errors with autocorrelation of up to 18 lags are reported in parentheses. Data sample: 1988:01–2019:12.  | (1) | (2) | (3) | (4)
---|---|---|---|---
$(1-\rho_{z})g$ | 0.122 | 0.129 | 0.141 | 0.133
| (16.906) | (13.995) | (15.536) | (16.745)
$\rho_{z}$ | 0.015 | 0.028 | -0.071 | -0.073
| (0.381) | (0.690) | (-1.379) | (-1.295)
$N$ | 1887 | 768 | 384 | 943
$R^{2}$ | 0.001 | 0.003 | 0.025 | 0.028
Sample | Monthly | Weekly | Monthly | Weekly
Periods | Y1:Y3 | Y1:Y3 | Y1:Y2 | Y1:Y2

We estimate equation (29) with both monthly (columns 1 and 3) and weekly
observations (columns 2 and 4) of analyst forecasts. The results are reported
in Table 5, Panel A. In columns (1) and (2), our estimation includes both
equations in (29), while in Column (3) and (4), we only include the first
equation, i.e., only using forecasts at one- and two-year horizons for better
data quality. Across the specifications, the estimate $\hat{\rho}_{z}$ is
statistically indistinguishable from zero.

Table 6: The Persistence of Expected Cash-Flow Growth from LTG Forecasts This table reports the estimates of $\rho_{z}^{LT}$, the regression coefficient in  $\log(1+LTG_{t})=\text{const}+\rho^{LT}_{z}\mathbb{E}_{t}^{A}\left[\Delta e_{t+1}\right]+\varepsilon_{t},$ where $LTG_{t}$ is the long-term growth forecasts (LTG) of the S&P 500 Index, self-aggregated from stock-level LTG forecasts from the IBES Unadjusted Summary File. The short-term forecast, $\mathbb{E}_{t}^{A}\left[\Delta e_{t+1}\right]$, is the IGA 1-year earnings growth forecast (IGA $\mathbb{E}_{t}^{A}\left[\Delta e_{t+1}\right]$). $t$-statistics based on Newey-West standard errors with autocorrelation of up to 18 lags are reported in parentheses. Our monthly observations are from 1988:01 to 2019:12.  | (1)
---|---
| $\log(1+LTG_{t})$
Intercept | 0.116
| (28.615)
$\mathbb{E}_{t}^{A}\left[\Delta e_{t+1}\right]$ | 0.017
| (0.711)
$N$ | 384
$R^{2}$ | 0.011

Next, we consider an alternative way to estimate $\rho_{z}$ by exploring the
relationship between forecasts of short-term and long-term earnings growth
(LTG). IBES provides firm-level forecasts of the annualized average growth
rate of earnings over the next three to five years and has been adopted in the
recent literature on expectation formation (e.g., La Porta, 1996 and Bordalo,
Gennaioli, La Porta, and Shleifer, 2019). Given the autoregressive structure
in equation (18), the expected growth rate from period $n$ to $n+1$ depends on
the expected growth rate over the very next period via a coefficient
$\rho_{z}^{n}$. If $\rho_{z}$ is zero, then such a coefficient is zero, which
implies that the average growth rate over three years and beyond does not
depend on the expected growth rate over the next year. Therefore, we regress
monthly observations of LTG forecast on $\mathbb{E}_{t}^{A}\left[\Delta
e_{t+1}\right]$, and denote the regression coefficient by $\rho_{z}^{LT}$. Our
estimate of $\rho_{z}^{LT}$ is statistically indistinguishable from zero.
Consistent with our findings in Table 5, this implies $\rho_{z}$.

Table 7: The Persistence of Expected Cash-Flow Growth from the State-Space Model This table reports the estimation results of 1) the unrestricted state-space model given by equations (17) and (18) in Section 2, 2) the restricted state-space model (i.e., $\rho_{z}=0$), and for comparison, 3) the MA(1) model ($\Delta d_{t+1}=g+\sigma_{D}\varepsilon_{t+1}+\chi\sigma_{D}\varepsilon_{t}$), and 4) the AR(1) model ($\Delta d_{t+1}=g+\gamma\Delta d_{t}+\sigma_{D}\varepsilon_{t+1}$) of the dividend growth ratres. Panel A uses the annual (non-overlapping) dividend growth of the S&P 500 index, and Panel B uses the annual (non-overlapping) dividend growth of the Fama-French market portfolio. The log likelihood (“LogL”), AIC, and BIC are reported. $t$-stats are in the squared bracket. | $\hat{\rho}_{z}$ | $\hat{g}$ | $\hat{\sigma}_{d}$ | $\hat{\sigma}_{z}$ | $\hat{\chi}$ | $\hat{\gamma}$ | LogL | AIC | BIC
---|---|---|---|---|---|---|---|---|---
Panel A: S&P 500
Unrestricted | 0.26 | 0.06 | 0.00 | 0.11 |  |  | 74.44 | -140.88 | -128.97
| [0.94] | [3.01] | [0.00] | [1.70] |  |  |  |  |
Restricted |  | 0.06 | 0.08 | 0.08 |  |  | 71.36 | -136.72 | -127.79
|  | [4.68] | [0.00] | [0.00] |  |  |  |  |
MA(1) |  | 0.06 | 0.10 |  | 0.41 |  | 76.41 | -146.82 | -137.89
|  | [3.38] | [13.45] |  | [6.11] |  |  |  |
AR(1) |  | 0.04 | 0.11 |  |  | 0.26 | 74.50 | -142.99 | -134.06
|  | [3.64] | [14.90] |  |  | [3.51] |  |  |
Panel B: MKT
Unrestricted | -0.08 | 0.06 | 0.00 | 0.15 |  |  | 43.96 | -79.92 | -69.8
| [-0.06] | [3.86] | [0.00] | [0.12] |  |  |  |  |
Restricted |  | 0.06 | 0.11 | 0.11 |  |  | 43.67 | -81.34 | -73.8
|  | [3.62] | [0.10] | [0.10] |  |  |  |  |
MA(1) |  | 0.06 | 0.15 |  | -0.09 |  | 44.00 | -82.00 | -74.4
|  | [3.94] | [6.99] |  | [-1.02] |  |  |  |
AR(1) |  | 0.06 | 0.15 |  |  | -0.08 | 43.96 | -81.93 | -74.39
|  | [3.89] | [6.98] |  |  | [-0.87] |  |  |

Our final method to estimate $\rho_{z}$ is to directly estimate the state-
space model given by equations (17) and (18) with the realized dividend data.
Using the standard Kalman filter, we obtain estimates of $\rho_{z}$. For
comparison, we report results for both the S&P 500 index and the Fama-French
market portfolio (“MKT”).202020We obtain dividend data for the Fama-French
market portfolio from the CRSP NYSE/NYSEMKT/Nasdaq Value-Weighted Market
Index. Since the model is set up at annual frequency, we use annual (non-
overlapping) dividend growth data. The sample spans 1926 to 2019.212121We also
used the longest available S&P 500 dividend series starting from 1872 and
obtained similar results. The results are available upon request. The results
are reported in Table 7, where Panel A and B are for S&P 500 and MKT,
respectively. In the row “Unrestricted” of Panel A and B of Table 7, the
estimates of $\hat{\rho}_{z}$ are statistically indistinguishable from
zero.222222The Kalman filter assumes that the shocks to realized and expected
dividend growth are uncorrelated. In the appendix, we demonstrate the
robustness of our estimate of $\rho_{z}$ by considering different values of
the correlation, from -0.9 to 0.9, while fixing the volatility of realized-
dividend shock at the estimate in Panel A. The estimated $\rho_{z}$ barely
moves with the value of shock correlations in $\left[-0.9,0.9\right]$ as shown
in Figure A.7. The restricted model with $\rho_{z}=0$ generates similar
likelihood and information criteria, indicating that allowing $\rho_{z}$ to be
a free parameter does not significantly improve the model fitness. We also
estimate MA(1) and AR(1) models for comparison and find that the estimates of
the autoregressive coefficient, i.e., $\chi$ and $\gamma$ for MA(1) and AR(1),
respectively, are statistically indistinguishable from zero. In sum, the
state-space approach delivers a similar message as the estimation based on
analyst forecasts: The persistence of growth expectation is close to zero.

Table 8: Time-varying $\rho_{z}$ Estimates This table reports the summary statistics of the rolling-window estimates of $\rho_{z}$ using weekly observations, where each rolling window spans three years. In each window, we estimate $\rho_{z}$ following equation (29), using aggregate earnings growth forecasts of the S&P 500 Index obtained from IGA. Data sample: 1988:01–2019:12.  | count | mean | std | min | 25% | 50% | 75% | max | $\rho$
---|---|---|---|---|---|---|---|---|---
$\hat{\rho}_{z,t}$ | 384 | 0.025 | 0.157 | -0.260 | -0.066 | 0.001 | 0.074 | 0.791 | 0.975

In the last exercise, we conduct a rolling-window estimation of $\rho_{z}$
following the method in Table 5. A rolling window contains three years of
weekly observations.232323The results are similar if we use alternative window
lengths from one to five years (available upon request). Our sample period is
1988–2019. The first estimate of $\rho_{z}$ uses three years of IGA data
starting in 1985. The summary statistics of the rolling-window estimates are
reported in Table 8. Naturally, the model of agents’ belief formation may
adjust over time, so the estimate of $\hat{\rho}_{z}$ fluctuates. Overall, the
results are similar to those in Table 5, and in particular, the mean and
median of the $\hat{\rho}_{z}$ series are close to zero. Our findings suggest
that the expected cash-flow growth rate lacks persistence ($\rho_{z}$ is close
to zero), which in our model implies a one-to-one mapping between market
duration, $dr$, and expected return. In the next proposition, we characterize
more tightly the connection between the value of $\rho_{z}$ and return
predictability. The proof is in the Appendix.

Figure 6: Rolling Estimate of Expected Growth Persistence and Return
Prediction Errors This figure plots the rolling estimate of the autoregressive
coefficient of expected cash flow growth, $\hat{\rho}_{z,t}$, and the return
prediction errors using market duration ($dr_{t}$) as the predictor.
$\hat{\rho}_{z,t}$ is estimated using analyst forecasts of S&P 500 aggregate
earnings in rolling regressions with a three-year window. This figure also
plots the predictive residuals (denoted by $\varepsilon_{t}$) from the
rolling-sample predictive regressions. The correlation between the two time
series is also reported on the graph. Our monthly sample is 1988:01–2019:12.

######  Proposition 3 ($\rho_{z}$ and return forecast errors).

Let $\nu_{t+1}$ denote the forecast error when predicting $r_{t+1}$ with
$dr_{t}$ and let ${\rho}_{z}$ denote the autoregressive coefficient of
expected cash flow growth $z_{t}$ in equation (18). If ${\rho}_{z}>0$, then
$\nu_{t+1}$ is positive. If ${\rho}_{z}<0$, then $\nu_{t+1}$ is negative.

The proposition implies that in a subsample where the estimate of $\rho_{z}$
is positive (negative), we would expect the return forecasting error to be
positive (negative). In Figure 6, we plot the rolling-window estimate of
$\rho_{z}$ estimates against the return forecasting residuals (denoted by
$\varepsilon_{t}$) from the corresponding rolling window with $dr_{t}$ as the
predictor. The two time series track each other closely, with a correlation of
0.42.242424In Figure A.6 in the Appendix, we plot $\rho_{z,t}$ against the
out-of-sample forecast errors and obtain a similarly positive correlation. We
also regress the rolling-window return prediction errors, both in-sample and
out-of-sample, on the rolling-window estimate of $\rho_{z}$ and find a
positive regression coefficient (see Table A.13 in the Appendix). This
provides further evidence of the connection between return predictability and
the lack of persistence in expected cash-flow growth.

Market duration $dr$ is the log price-dividend ratio minus the valuation ratio
of the one-year dividend strip, which, by construction, teases out market
information about near-term growth embedded in the price of the one-year
dividend strip. Therefore, $dr$ only contains information about the discount
rate, if the market participants possess limited information about growth
beyond the next year. In our model, the time-$t$ expectation of growth from
$t+2$ onward is a constant under $\rho_{z}=0$ so the market does not contain
information about cash-flow growth beyond the next year under $\rho_{z}=0$.
Next, we step outside of our model given by equations (17) and (18) and
consider more broadly whether the market contains information about long-term
growth.

###  4.4 Cash-flow growth predictability: Short horizon vs. long horizon

Table 9: One-year Dividend and Earnings Growth Prediction This table reports the results of dividend and earnings growth prediction. The dependent variables are the 1-year-ahead realized earnings growth from IGA (columns 1-3), realized dividend growth from Bloomberg (columns 4-6), and realized earnings growth from Compustat (columns 7-9). The independent variables are analyst forecasts of one-year earnings growth from IGA ($\mathbb{E}^{A}_{t}\left(\Delta e_{t+1}\right)$), duration $dr$, and the price-dividend ratio $pd$. $t$-statistics are calculated based on Newey-West standard errors with 18 lags and are reported in parentheses. Data sample: 1988:01–2019:12.  |  IGA $\Delta e_{t+1}$ | $\Delta d_{t+1}$ |  Compustat $\Delta e_{t+1}$
---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Intercept | -0.056 | -0.426 | -0.328 | 0.028 | -0.312 | -0.295 | 0.122 | 0.478 | 0.394
| (-4.127) | (-1.357) | (-2.413) | (1.300) | (-1.526) | (-1.663) | (0.814) | (0.374) | (0.352)
$\mathbb{E}^{A}_{t}\left(\Delta e_{t+1}\right)$ | 1.204 |  | 1.119 | 0.326 |  | 0.204 | -0.591 |  | -0.963
| (20.101) |  | (15.193) | (2.537) |  | (2.824) | (-0.629) |  | (-1.067)
$dr_{t}$ |  | -0.248 | -0.110 |  | -0.181 | -0.156 |  | -0.333 | -0.452
|  | (-2.328) | (-2.625) |  | (-3.535) | (-4.188) |  | (-1.129) | (-2.016)
$pd_{t}$ |  | 0.385 | 0.187 |  | 0.285 | 0.249 |  | 0.238 | 0.409
|  | (2.354) | (2.599) |  | (2.782) | (3.095) |  | (0.405) | (0.958)
$N$ | 372 | 372 | 372 | 372 | 372 | 372 | 372 | 372 | 372
$R^{2}$ | 0.731 | 0.199 | 0.769 | 0.193 | 0.386 | 0.454 | 0.014 | 0.053 | 0.086

In Section 4.3, our focus is on estimating $\rho_{z}$, the autoregressive
coefficient of expected cash-flow growth. In our model, $\rho_{z}=0$ implies
that the market participants do not have information about cash-flow growth
beyond the next year. While our analysis of state space in Section 3 supports
the model specification, we want to step outside of our model and provide more
evidence in this subsection on the fact that market information about long-
term growth is limited. We do so by characterizing cash-flow predictability at
different horizons. Intuitively, if the market participants have information
about cash-flow growth at a certain horizon, we should be able to predict such
cash-flow growth with information revealed in analysts’ forecasts and $dr$ and
$pd$ as state variables.

In Table 9, we show strong predictability of near-term growth. In column (1),
we simply regress the realized one-year growth rate of aggregate earnings from
firms covered by IGA on the ex ante analysts’ forecast. The $R^{2}$ is 0.73,
so analysts and market participants in general are able to forecast near-term
cash-flow growth very well. In column (2), we use our pair of state variables,
$dr$ and $pd$, to forecast cash-flow growth and obtain a $R^{2}$ of 0.20.
Combining the information in $dr$ and $pd$ with the analysts’ forecast in
column (3), the in-sample prediction $R^{2}$ rises to 0.77. In the other
columns of Table 9, we replace the forecasting target. Naturally, cash-flow
predictability declines in the other cases because the predictor (IGA analyst
forecast, in particular) targets the earnings growth of IGA-covered firms
rather than that of all firms covered by the Bloomberg database (columns 4-6)
or that of all Compustat firms (columns 7-9). Overall, our findings suggest
that market participants are informed of near-term cash-flow growth. One
explanation is that firms tend to provide forward guidance on their earnings
outlook, increasingly so in recent years.

Table 10: Weaker Earnings Growth Predictability beyond One Year This table reports results of regressions that predict earnings growth at various horizons. The dependent variables are realized earnings growth from IGA of next year (columns 1-3), between the first and second years (columns 4-6), and between the second and third years (columns 7-9). The independent variables are analysts’ forecasts of one-year earnings growth from IGA (IGA $\mathbb{E}^{A}_{t}\left(\Delta e_{t+1}\right)$), the self-aggregated long-term earnings growth forecasts ($LTG_{t}$) of the S&P 500 Index, the market duration $dr$ and the price-dividend ratio $pd_{t}$. $t$-statistics calculated based on Newey-West standard errors with 18 lags are reported in parentheses. Data sample: 1988:01–2019:12.  |  IGA $\Delta e_{t,t+1}$ |  IGA $\Delta e_{t+1,t+2}$ |  IGA $\Delta e_{t+2,t+3}$
---|---|---|---
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Intercept | 0.053 | -0.426 | -0.309 | 0.325 | -0.063 | -0.037 | 0.185 | 0.513 | 0.496
| (1.360) | (-1.357) | (-2.786) | (2.446) | (-0.211) | (-0.141) | (1.596) | (1.577) | (1.685)
IGA $\mathbb{E}^{A}_{t}\left(\Delta e_{t+1}\right)$ | 1.222 |  | 1.174 | 0.030 |  | -0.047 | -0.319 |  | -0.274
| (20.312) |  | (15.749) | (0.192) |  | (-0.348) | (-2.594) |  | (-1.947)
$LTG_{t}$ | -0.889 |  | -1.344 | -2.155 |  | -2.295 | -0.711 |  | -0.295
| (-3.171) |  | (-3.092) | (-2.186) |  | (-1.578) | (-0.768) |  | (-0.234)
$dr_{t}$ |  | -0.248 | -0.077 |  | -0.144 | -0.103 |  | 0.096 | 0.068
|  | (-2.328) | (-1.568) |  | (-2.878) | (-2.319) |  | (1.449) | (1.215)
$pd_{t}$ |  | 0.385 | 0.189 |  | 0.181 | 0.207 |  | -0.216 | -0.166
|  | (2.354) | (2.797) |  | (1.690) | (2.351) |  | (-1.666) | (-1.399)
$N$ | 372 | 372 | 372 | 360 | 360 | 360 | 348 | 348 | 348
$R^{2}$ | 0.74 | 0.20 | 0.78 | 0.08 | 0.08 | 0.14 | 0.07 | 0.07 | 0.11

Next, we examine the predictability of cash-flow growth at longer horizons. So
far, our analysis of growth expectations has been based on the autoregressive
model given by equations (17) and (18), and the focus has been on $\rho_{z}$,
the persistence parameter, and its connection with the return predictive power
of market duration $dr$. In this model, we have the time-$t$ expected growth
from $t+1$ to $t+2$,
$\mathbb{E}_{t}[z_{t+2}]=\rho_{z}\mathbb{E}_{t}[z_{t+1}]=\rho_{z}^{2}z_{t}$,
so market information about long-term growth is contained in the expected
growth rate over the next period. In the following exercises, we expand the
set of potential predictors of long-term growth. In particular, we consider
the LTG forecast, the aggregated long-term growth forecast over three to five
years.

In Table 10, we first predict earnings growth over the next year from columns
(1) to (3), and consistent with our findings in Table 9, predictability is
strong, which suggests that market participants are informed about short-term
growth. In columns (4) to (6), we use the same variables to predict earnings
growth from $t+1$ to $t+2$ and find that predictability declines dramatically.
Comparing columns (1) and (4), the predictive $R^{2}$ declines from 0.74 to
0.08. Our forecasting exercise in columns (7) to (9) delivers the same
message. The predictability of growth over an even longer horizon, i.e., from
$t+2$ to $t+3$, is even weaker. In Figure A.9 in the Appendix, we report the
results of alternative predictive models that take advantage of more valuation
ratios (and potentially richer information embedded in the state space). Our
conclusion remains robust.

Overall, our findings suggest that market participants are not informed about
long-term growth, but regarding short-term growth, market information, whether
from analysts’ forecasts or our state variables, is quite rich. These results
on the term structure of cash-flow predictability shed light on the strong
connection between market duration $dr$ and expected return. By construction,
$dr$ deducts the valuation ratio of the one-year dividend strip from the
price-dividend ratio $pd$ and thereby teases out market information about
near-term growth.

One may form a market timing strategy guided by our findings on $dr$ as a
return predictor. Specifically, when market duration increases, the strategy
shorts the market, and when market duration decreases, the strategy adds to
the long position. This strategy bets against the market valuation of long-
duration cash flows. An out-of-sample $R^{2}$ of $14.6\%$ in column (1) of
Table 2 implies that the Sharpe ratio of this market timing strategy is
$0.58$, which is much higher than the Sharpe ratio from other return
predictors, for example, $0.37$ in Campbell and Thompson (2008). In the
appendix, we show how to calculate the Sharpe ratio based on the out-of-sample
$R^{2}$.

##  5 Conclusion

We construct a valuation-based measure of stock market duration as the ratio
of total market capitalization to the price of dividends paid in the very next
year. An increase in market duration implies the valuation of long-term cash
flows increases relative to that of short-term cash flows. While market
duration ($dr$) represents the slope of the valuation term structure, the
traditional price-dividend ratio ($pd$) represents the level. After
establishing the connection between state variables and valuation ratios of
dividend strips, we demonstrate that the state space is two-dimensional.
Through return and cash-flow predictions, we find that $dr$ and $pd$ span the
state space and may serve as a pair of state variables for asset pricing and
in macro-finance studies.

Our findings not only establish market duration as a key state variable of
important applications but also shed light on what drives market duration. We
find a tight link between market duration and expected return. $dr$ exhibits
strong return predictive power, outperforming other predictors. Such a
connection between $dr$ and the discount rate emerges when market information
on long-term cash flows is limited, and empirically, we document the
supporting evidence of a sharp decline of cash-flow predictability once the
forecasting horizon goes beyond one year.

Cochrane (2011) points out how return predictability emerges from the lack of
cash-flow predictability and the price-dividend ratio should forecast future
returns. We find that cash flows in the near term are in fact highly
predictable, but cash flows over the long term are not. Therefore, our measure
of market duration is a better return predictor than the price-dividend ratio.
Our findings support a market timing strategy of betting against market
duration. When the valuation of long-term cash flows rises relative to that of
near-term cash flows about which market participants are more informed, it is
likely due to a lower discount rate or exuberance over long-term growth
(Bordalo, Gennaioli, La Porta, and Shleifer, forthcoming).

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## Appendix I: Derivation

### I.1 Solving the valuation ratios

The price-dividend ratio of the dividend strip with maturity $n$,
$P_{n,t}/D_{t}$, satisfies the following recursive equation

$\frac{P_{n,t}}{D_{t}}=\mathbb{E}_{t}\left[M_{t+1}\frac{D_{t+1}}{D_{t}}\frac{P_{n-1,t+1}}{D_{t+1}}\right].$
(30)

We conjecture that

$\ln\left(\frac{P_{n,t}}{D_{t}}\right)=A\left(n\right)+B\left(n\right)^{T}X_{t}.$
(31)

Substituting this expression and expressions of stochastic discount factor and
dividend growth into the recursive equation, we have

$\displaystyle\exp\left\\{A\left(n\right)+B\left(n\right)^{\top}X_{t}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}\left[\exp\left\\{-r_{f}-\frac{1}{2}\lambda_{t}^{\top}\Sigma\lambda_{t}-\lambda_{t}^{\top}\epsilon_{t+1}+g_{t}+\sigma_{D}^{\top}\epsilon_{t+1}+A\left(n-1\right)+B\left(n-1\right)^{\top}X_{t+1}\right\\}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}\left[\exp\left\\{g_{t}-r_{f}-\frac{1}{2}\lambda_{t}^{\top}\Sigma\lambda_{t}+A\left(n-1\right)+B\left(n-1\right)^{\top}\Pi
X_{t}+\left(\sigma_{D}-\lambda_{t}+\sigma_{X}B\left(n-1\right)\right)^{\top}\epsilon_{t+1}\right\\}\right]$
$\displaystyle=$
$\displaystyle\exp\left\\{g_{t}-r_{f}-\frac{1}{2}\lambda_{t}^{\top}\Sigma\lambda_{t}+A\left(n-1\right)+B\left(n-1\right)^{\top}\Pi