Out‐of‐sample equity premium prediction: A scenario analysis approach

AuthorFeifang Hu,Xiaoxiao Tang,Peiming Wang
Date01 August 2018
Published date01 August 2018
DOIhttp://doi.org/10.1002/for.2519
Received: 2 August 2016 Revised: 21 December 2017 Accepted: 31 January 2018
DOI: 10.1002/for.2519
RESEARCH ARTICLE
Out-of-sample equity premium prediction: A scenario
analysis approach
Xiaoxiao Tang1Feifang Hu2Peiming Wang3
1Olin Business School, Washington
University in St. Louis, MO, USA
2Department of Statistics, George
Washington University,Washington, DC,
USA
3Department of Finance, Business School,
Auckland University of Technology,
Auckland, New Zealand
Correspondence
Peiming Wang,Department of Finance,
Business School, Auckland University of
Technology,Auckland 1142, Private Bag
92006, New Zealand.
Email: peiming.wang@aut.ac.nz
Funding information
National Science Foundation (USA) ,
Grant/AwardNumber: DMS-1612970
Abstract
We propose two methods of equity premium prediction with single and multi-
ple predictors respectively and evaluate their out-of-sample performance using
US stock data with 15 popular predictors for equity premium prediction. The
first method defines three scenarios in terms of the expected returns under the
scenarios and assumes a Markov chain governing the occurrence of the scenar-
ios over time. It employs predictive quantile regressions of excess return on a
predictor for three quantiles to estimate the occurrence of the scenarios over
an in-sample period and the transition probabilities of the Markov chain, pre-
dicts the expected returns under the scenarios, and generates an equity premium
forecast by combining the predicted expected returns under three scenarios
with the estimated transition probabilities. The second method generates an
equity premium forecast by combining the individual forecasts from the first
method across all predictors. For most of predictors, the first method outper-
forms the benchmark method of historical average and the traditionalpredictive
linear regression with a single predictor both statistically and economically,
and the second method consistently performs better than several competing
methods used in the literature. The performance of our methods is further exam-
ined under different scenarios and economic conditions, and is robust for two
different out-of-sample periods and specifications of the scenarios.
KEYWORDS
combination forecast, equity premium, Markov chain, quantile regression, scenarioanalysis
1INTRODUCTION
Equity premium prediction is important in finance
because it is related not only to important investment
decisions on asset allocation made by practitioners, but
also to some important issues in finance such as market
efficiency and asset pricing models in which academics
in finance are interested. While many studies report evi-
dence of US equity risk premium predictability based
on predictors such as valuation ratios and interest rates,
there are mixed results from the research of equity pre-
mium predictability, particularly for out-of-sample fore-
cast.1For example, Goyal and Welch (2008) examine the
out-of-sample predictability of the traditional predictive
linear regression of return on either a single or multiple
predictors for a list of popular predictors fromthe literature
with respect to the benchmark method of historical aver-
age. They find that the benchmark method of historical
averagehas better out-of- sample performance than the tra-
ditional predictive linear regressions. On the other hand,
1See Cochrane (2011) and Rapach and Zhou (2013) for recent surveys.
604 Copyright © 2018 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting. 2018;37:604–626.
TANG ET AL.605
Campbell and Thompson (2008) show that forecasts based
on predictive linear regression with some restrictions on
regression coefficients and predicted returns perform bet-
ter out-of-sample than the historical average forecast. In
addition, using the same list of predictors as investigated
in Goyal and Welch (2008), Rapach, Strauss, and Zhou
(2010) provide empirical evidence that the combination
forecast of the individual forecasts from the traditional
predictive linear regression with a single predictor has bet-
ter out-of-sample performance than the historical average
forecast.
There are also other recent studies on alternatives to
the traditional predictive linear regression for equity pre-
mium prediction in the literature, which deliver better
out-of-sample performance than the historical average
forecast. For instance, Markov regime-switching models
for equity premium prediction outperform the historical
average forecast by modeling regime shifts in the state
of an economy according to a finite-state Markov chain
and different linear relationships between return and pre-
dictors for different regimes (e.g., Guidolin & Timmer-
mann, 2009; Henkel, Martin, & Nadari, 2011). The linear
dynamic models proposed by Dangl and Halling (2012)
improve out-of-sample predictive performance relative to
the benchmark method of historical average by allowing
the coefficients of the traditional predictive linear regres-
sion to be random and time varying. In addition, Meligkot-
sidou, Panopoulou, Vrontos, and Vrontos (2014) propose
the quantile regression method of equity premium pre-
diction that first generates individual forecasts from pre-
dictive quantile linear regression of return on a single
predictor and then produces an equity premium forecast
by combining these individual forecasts across all predic-
tors. They find that their method has better out-of-sample
performance than the benchmark method of historical
average.
In general, asset returns may depend on many different
factors, including market and economic conditions. That is
why scenario analysis is widely used in practice to predict
the expected return on an investment based on the returns
expected under a number of possible future scenarios for
the investment, such as the bad, normal, and good case sce-
narios for an investment. It is a what-if analysis in which
the likelihood of various scenarios and their investment
outcomes are evaluated, and the predicted expected return
on an investment is derived from the predicted expected
returns under different scenarios by combining them with
the probability that they will occur.
In this paper, we follow the principle of scenario analy-
sis to propose a method of equity premium prediction with
a single predictor. Specifically, we define three possible
scenarios for a stock investment in terms of the expected
excess returns achieved under each of the scenarios and
assume a first-order three-state Markov chain governing
the occurrence of the scenarios over time. In the specifi-
cation of the three scenarios, the expected excess return
under Scenario 1 (3) is below (above) the lower (upper)
quartile of excess returns, and the expected excess return
under Scenario 2 is equal to the median excess return.
We use predictive quantile linear regressions of excess
return on a predictor for the three quartiles to estimate
the occurrence of the three scenarios over an in-sample
period and the transition probabilities of the Markov chain
with in-sample data. Then we predict the one-step-ahead
out-of-sample excess returns expected under each of the
three scenarios as follows. The predicted expected excess
return under Scenario 1 (3) is the average of the pre-
dicted lower (upper) quartile from the predictive quan-
tile linear regression and the excess returns under that
scenario during the in-sample period, and the predicted
expected excess return under Scenario 2 is the predicted
median return from the predictive quantile linear regres-
sion. Finally, we generate a one-step-ahead out-of-sample
forecast of equity premium by combining the predicted
expected excess returns under the three scenarios with the
estimated transition probabilities, based on the scenario
that occurred in the end of the in-sample period.
Methodologically, our prediction method with a
single predictor is related to the work of Markov
regime-switching models for stock returns (e.g., Guidolin
& Timmermann, 2009; Henkel et al., 2011) because we
employ a Markov chain to model the switching between
scenarios. However, it differs from the previous studies
because of the differences in the specification of regimes,
the estimation of parameters and regimes, and the deter-
mination of the expected return conditional on regimes.
Our predition method with a single predictor is also related
to studies on the financial applications of quantile regres-
sion (e.g., Engle & Manganelli, 1999; Meligkotsidou et al.,
2014) because of the use of quantile regression in the anal-
ysis, but differs from these studies as quantile regression
is used differently in equity premium prediction.
We also propose a method of equity premiumprediction
with multiple predictors, which combines the individual
forecasts from our meth od with a single predicto r across
all predictors to form a single forecast of equity premium.
While studies show that many predictors such as valuation
ratios and term spreads can be used to detect changes in
factors associated with asset prices (e.g., Fama & French,
1989), different predictors could capture different factors
that have different relationships with asset prices. This
implies that individual forecasts based on different pre-
dictors could have a weak correlation between each other.
As pointed out by Rapach et al. (2010), the combination
forecast of individual forecasts like the average of individ-
ual forecasts based on different predictors should be more

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