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Estimation and Model Specification for

Econometric Forecasting

2015-3

Manuel Sebastian Lukas

PhD Thesis

DEPARTMENT OF ECONOMICS AND BUSINESS

AARHUS UNIVERSITY � DENMARK

ESTIMATION AND MODEL SPECIFICATION FOR

ECONOMETRIC FORECASTING

By Manuel Sebastian Lukas

A PhD thesis submitted to

School of Business and Social Sciences, Aarhus University,

in partial fulfilment of the requirements of

the PhD degree in

Economics and Business

August 2014

CREATESCenter for Research in Econometric Analysis of Time Series

PREFACE

This dissertation was written in the period from September 2010 to August 2014

while I was enrolled as a PhD student at the Department of Economics and Business

at Aarhus University. During my PhD studies I was affiliated with the Center for

Research in Econometric Analysis of Time Series (CREATES), that is funded by the

Danish National Research Foundation. I am grateful to the Department of Economics

and Business and to CREATES for providing an inspiring, supportive, and friendly

research environment, and for the financial support for attending conferences and

courses.

Parts of this dissertation were written during my research stay at the Rady School

of Management at the University of California San Diego (UCSD) from August 2012

to January 2013. I thank Allan Timmermann for making this academically, profes-

sionally, and personally rewarding experience possible and I thank the Rady School

of Management of the hospitality. I am grateful to the Aarhus University Research

Foundation (AUFF) and the Department of Business and Social Science at Aarhus

University for their financial support in connection for my stay at UCSD. I thank Jack

Zhang for all the help and the hospitality during my stay in the United States, and for

introducing me to the UCSD graduate student life.

I am thankful to all people who have supported me in my research with their

advice, comments, and suggestions. My main supervisor Bent Jesper Christensen and

my co-supervisor Eric Hillebrand have supported me with guidance, expertise, and

encouragement for both my independent research and our joint research projects.

I wish to thank all fellow PhD students at Aarhus University for the excellent team

spirit, both in academic and in (very) non-academic matters, which has made the past

four years a great experiences. I especially wish to thank Rasmus, Heida, Kasper O.,

Andreas, and Anders L., who have accompanied me in the challenging and exciting

transition from Master’s to PhD student. During my PhD studies if have enjoyed

many welcome breaks from research during coffee breaks, social events, and floorball

matches with many of my colleagues, in particular Niels S., Juan Carlos, Anne F.,

Jonas E., Jonas M., Martin S., Niels H., Mark, Simon, Rune, Anders K., Laurent, Stine,

Morten, and Christina. A big thanks goes to Johannes for sharing the LATEX template

that is used for this dissertation.

I am very grateful to CREATES, especially the Center Director Niels Haldrup and

i

ii

the Center Administrator Solveig Sørensen, for creating a great research environment

and for the many interesting PhD courses that were organized by CREATES during

my studies. I also wish to thank Niels Haldrup for allowing me to participate three

times in the Econometric Game in Amsterdam for Team Aarhus University.

I am indebted to my family and friends in Switzerland for their patience, their

visits to Denmark, and their amazing hospitality on my visits back home. Last but

not least, I am grateful to my girlfriend Tanja for supporting me during the busy and

challenging time as PhD student.


Aarhus, August 2014

UPDATED PREFACE

The predefence took place on September 30, 2014. The assessment committee con-

sists of Asger Lunde, Aarhus University, Allan Timmermann, University of California,

San Diego, and Christian Møller Dahl, University of Southern Denmark. I wish to

thank the members of the committee for their detailed comments. After the prede-

fence the dissertation has been revised to incorporate the changes required by the

committee. Additionally, the committee has suggested improvements, some of which

are incorporated in this revised version of the thesis.


Copenhagen, January 2015

iii

CONTENTS

Summary vii

1 Bagging Weak Predictors 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Bagging Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Application to CPI Inflation Forecasting . . . . . . . . . . . . . . . . . 19

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Return Predictability, Model Uncertainty, and Robust Investment 332.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Investment and Confidence Sets . . . . . . . . . . . . . . . . . . . . . 36

2.3 Models and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Frequency Dependence in the Risk-Return Relation 613.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 The Empirical Risk-Return Relation . . . . . . . . . . . . . . . . . . . 65

3.3 Frequency Dependence in the Risk-Return Relation . . . . . . . . . 72

3.4 Frequency-Dependent Real-Time Forecasts . . . . . . . . . . . . . . 83

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

v

SUMMARY

This dissertation comprises three self-contained chapters with the theme of econo-

metric forecasting as their common denominator. We analyze methods for parameter

estimation and model specification of econometric models and apply these methods

to macroeconomic and financial time series. Turning to econometric forecasting we

shift the focus of econometric modeling from fitting all available data, testing for

statistical significance, and testing for correct specification towards fitting future data,

i.e., achieving good out-of-sample performance. Applying the classical econometric

toolbox for parameter estimation and model specification is not always appropriate

for forecasting because a statistically significant relation and good in-sample fit are

insufficient to ensure satisfactory forecasting performance. It is therefore important

to take into account the very aim of out-of-sample forecasting at the time when the

model is estimated and specified. The three chapters in this dissertation each deal

with some aspects of estimation and model specification for econometric forecast-

ing with empirical applications to inflation rates, equity premia, and the risk-return

relation.

The first chapter, "Bagging Weak Predictors", is joint work with Eric Hillebrand.

We propose a new bootstrap aggregation, bagging, predictor for situations where the

predictive relation is weak, i.e., for situations in which predictors based on classical

statistical methods fail to provide good forecasts because the estimation variance is

larger than the bias effect from ignoring the relation. In the literature on econometric

forecasting, it is often found that predictors suggested by economic theory do not

lead to satisfactory forecasting results. Successful forecasting with such predictors

requires prediction methods that reduce estimation variance. The bagging method of

Breiman (1996) is based on bootstrap re-sampling and it can improve the properties

of pre-test and other hard-threshold estimators by reducing the estimation variance.

Standard bagging estimators are based on standard t-tests for statistical significance.

A statistically significant relation is, however, not sufficient for successful out-of-

sample forecasting. We therefore base our new bagging predictor on the in-sample

test for predictive ability proposed by Clark and McCracken (2012). The null hypoth-

esis of this test is that the inclusion or the exclusion of a predictor in a forecasting

regression leads to equal forecasting performance. Thus, when the test is rejected,

we know whether or not to include the predictor. By using the test of Clark and Mc-

vii

viii SUMMARY

Cracken (2012), our predictor shrinks the regression coefficient estimate not to zero,

but towards the null of the test which equates squared bias with estimation variance.

We derive the asymptotic distribution in the asymptotic framework of Bühlmann and

Yu (2002) and show that the predictor has a substantially lower the mean-squared

error (MSE) compared to standard t-test bagging if a weak predictive relationship

exists. Because the bootstrap re-sampling for bagging can be computationally heavy,

we derive an asymptotic shrinkage representation for the predictor that simplifies

computation of the estimator. Monte Carlo simulations show that our predictor works

well in small samples. In the empirical application, we consider forecasting inflation

using employment and industrial production in the spirit of the so-called Phillips

Curve. This application fits our framework because inflation is notoriously hard to

forecast from other macroeconomic variables.

In the second chapter, "Return Predictability, Model Uncertainty, and Robust

Investment", the model uncertainty in stock return prediction models is analyzed.

Empirical evidence suggests that stock returns are not completely unpredictable, see,

e.g., Lettau and Ludvigson (2010) for a comprehensive survey. Under stock return

predictability, investment decisions are based on conditional expectations of stock

returns. The choice of appropriate predictor variables is, however, subject to great

uncertainty. In this chapter, we use the model confidence set approach of Hansen,

Lunde, and Nason (2011) to quantify the uncertainty about expected utility from

stock market investment, accounting for potential return predictability, for monthly

data over the sample period 1966:01–2002:12 on the US stock market. We consider the

popular data set of Welch and Goyal (2008), which contains standard predictor vari-

ables used in this literature. For the econometric analysis we take the perspective of a

small investor with constant relative risk aversion (CRRA) utility and short-selling con-

straints. The model confidence set is then applied recursively and, for every month in

the out-of-sample period, it identifies the set of models that contains the best model

with a given confidence level. The empirical results show that the model confidence

sets imply economically large and time-varying uncertainty about expected utility

from investment. To analyze the economic importance of this model uncertainty we

propose investment strategies that reduce the impact of model uncertainty. Reducing

the model uncertainty with these strategies requires lower investment in stocks, but

return predictability still leads to economic gains for the small investor. Thus, we

conclude that although model uncertainty concerns reduce the share of wealth that

investors wish to hold in stocks, it does not prevent them from benefiting from return

predictability using econometric models.

The third chapter, "Frequency Dependence in the Risk-Return Relation", is co-

authored with Bent Jesper Christensen and considers a specification of the risk-return

relation that allows for non-linearities in the form of frequency dependence. The

risk-return relation is typically specified as a linear relation between stock returns and

some measure of the conditional variance, motivated by the intertemporal capital

ix

asset pricing model (ICAPM) of Merton (1973). Since the empirical analysis in Merton

(1980), empirical estimation of the risk-return relation has attracted much attention

in the literature. In this chapter we use the band spectral regression of Engle (1974)

with the one-sided filtering approach of Ashley and Verbrugge (2008) to allow for

frequency dependence in the risk-return relation, which is a feature that cannot

be accommodated by a linear model. The combination of one-sided filtering and

conditional variances constructed from lagged observations make our estimation

approach robust to contemporaneous leverage and feedback effects. For daily returns

and realized variances from high-frequency intra-daily data on the S&P 500 from

1995 to 2012 we strongly reject the null hypothesis of no frequency dependence.

This finding is robust to changes in the conditional variance proxy. In particular, the

rejection of the null hypothesis is strongest when we allow for lagged leverage effects

in the conditional variance. Although the risk-return relation is positive on average

over all frequencies, we find a large and statistically significant negative coefficient

for periods of around one week. Subsample analysis reveals that the negative effect

at these frequencies is not statistically significant before the financial crisis, but

becomes very strong after July 2007. Accounting for the frequency dependence in the

risk-return relation can improve the out-of-sample forecasting of stock returns after

2007, but only if the forecasting approach reduces in increased estimation variance

from the additional parameters of the band spectral approach.

References

Ashley, R., Verbrugge, R. J., 2008. Frequency dependence in regression model coeffi-

cients: An alternative approach for modeling nonlinear dynamic relationships in

time series. Econometric Reviews 28 (1-3), 4–20.

Breiman, L., 1996. Bagging predictors. Machine Learning 24, 123–140.

Bühlmann, P., Yu, B., 2002. Analyzing bagging. The Annals of Statistics 30 (4), 927–961.

Chernov, M., Gallant, R., Ghysels, E., Tauchen, G., 2003. Alternative models for stock

price dynamics. Journal of Econometrics 116 (1), 225–257.

Clark, T. E., McCracken, M. W., 2012. In-sample tests of predictive ability: A new

approach. Journal of Econometrics 170 (1), 1–14.

Corsi, F., 2009. A simple approximate long-memory model of realized volatility. Jour-

nal of Financial Econometrics 7 (2), 174–196.

Corsi, F., Reno, R., 2012. Discrete-time volatility forecasting with persistent leverage

effect and the link with continuous-time volatility modeling. Journal of Business

and Economic Statistics, 46–78.

x SUMMARY

Engle, R. F., 1974. Band spectrum regression. International Economic Review 15 (1),

1–11.

Gouriéroux, C., Monfort, A., Renault, E., 1993. Indirect inference. Journal of Applied

Econometrics 8 (S1), S85–S118.

Hansen, P. R., Lunde, A., Nason, J. M., 3 2011. The model confidence set. Econometrica

79 (2), 453–497.

Lettau, M., Ludvigson, S., 2010. Measuring and modeling variation in the risk- return

tradeoff. In: Ait-Sahalia, Y., Hansen, L.-P. (Eds.), Handbook of Financial Economet-

rics. Vol. 1. Elsevier Science B.V., North Holland, Amsterdam, pp. 617–690.

Merton, R. C., 1973. An intertemporal capital asset pricing model. Econometrica,

867–887.

Merton, R. C., 1980. On estimating the expected return on the market: An exploratory

investigation. Journal of Financial Economics 8 (4), 323–361.

Welch, I., Goyal, A., 2008. A comprehensive look at the empirical performance of

equity premium prediction. Review of Financial Studies 21 (4), 1455–1508.

CH

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R

1BAGGING WEAK PREDICTORS

Manuel Lukas and Eric Hillebrand

Aarhus University and CREATES

Abstract

Relations between economic variables can often not be exploited for forecasting,

suggesting that predictors are weak in the sense that estimation uncertainty is larger

than bias from ignoring the relation. In this chapter, we propose a novel bagging

predictor designed for such weak predictor variables. The predictor is based on an

in-sample test for predictive ability. Our predictor shrinks the OLS estimate not to

zero, but towards the null of the test which equates squared bias with estimation

variance. We derive the asymptotic distribution and show that the predictor can

substantially lower the MSE compared to standard t-test bagging. An asymptotic

shrinkage representation for the predictor is obtained that simplifies computation of

the estimator. Monte Carlo simulations show that the predictor works well in small

samples. In an empirical application we apply the new predictor to inflation fore-

casts.

Keywords: Inflation forecasting, bootstrap aggregation, estimation uncertainty, weak

predictors.

1

2 CHAPTER 1. BAGGING WEAK PREDICTORS

1.1 Introduction

A frequent finding in pseudo out-of-sample forecasting exercises is that including

predictor variables does not improve forecasting performance, even though the pre-

dictor variables are significant in in-sample regressions. For example, there is a large

literature on forecast failure with economic predictor variables for forecasting infla-

tion (see, e.g., Atkeson and Ohanian, 2001; Stock and Watson, 2009) and forecasting

exchange rates (see, e.g., Meese and Rogoff, 1983; Cheung, Chinn, and Pascual, 2005).

Including predictor variables suggested by economic theory, or selected by in-sample

regressions, typically does not help to consistently out-perform simple time series

models across different sample splits and model specifications. Forecasting failure

can be attributed to estimation variance and parameter instability. In this chapter, we

focus exclusively on the former. These two causes of forecast failure are, however, of-

ten interrelated in practice. If we are unwilling to specify the nature of instability, it is

common practice to use a short rolling window for estimation to deal with parameter

instability. While a short estimation window can better adapt to changing parameters,

it increases estimation variance compared to using all data. In this sense, estimation

variance can result from the attempt to accommodate parameter instability, such

that our results are relevant for both kinds of forecast failure.

This chapter is concerned with reducing estimation variance by bagging pre-test

estimators when predictor variables have weak forecasting power. Modeling weak

predictors in the framework of Clark and McCracken (2012) leads to a non-vanishing

bias-variance trade-off. CM propose an in-sample test for predictive ability, i.e., a

test of whether bias reduction or estimation variance will prevail when including a

predictor variable. Based on this test, we propose a novel bagging estimator that is

designed to work well for predictors with non-zero coefficient of known sign. Under

the null of the CM-test, the parameter is not equal to zero, but equal to a value

for which squared bias from omitting the predictor variable is equal to estimation

variance. In our bagging scheme, we set the parameter equal to this value instead of

zero whenever we fail to reject the null. For this, knowledge of the coefficient’s sign

is necessary. We derive the asymptotic distribution of the estimator and show that

for a wide range of parameter values, asymptotic mean-squared error is superior to

bagging a standard t-test. The improvements can be substantial and are not sensitive

to the choice of the critical value, which is a remaining tuning parameter. We obtain

forecast improvements if the data-generating parameter is small but non-zero. If the

data-generating parameter is indeed zero, however, our estimator has a large bias

and is therefore imprecise.

Bootstrap aggregation, bagging, was proposed by Breiman (1996) as a method

to improve forecast accuracy by smoothing instabilities from modeling strategies

that involve hard-thresholding and pre-testing. With bagging, the modeling strategy

is applied repeatedly to bootstrap samples of the data, and the final prediction is

obtained by averaging over the predictions from the bootstrap samples. Bühlmann

1.1. INTRODUCTION 3

and Yu (2002) show theoretically how bagging reduces variance of predictions and

can thus lead to improved accuracy. Stock and Watson (2012) derive a shrinkage

representation for bagging a hard-threshold variable selection based on the t-statistic.

This representation shows that standard t-test bagging is asymptotically equivalent

to shrinking the unconstrained coefficient estimate to zero. The degree of shrinkage

depends on the value of the t-statistic.

Bagging is becoming a standard forecasting technique for economic and financial

variables. Inoue and Kilian (2008) consider different bagging strategies for forecasting

US inflation with many predictors, including bagging a factor model where factors are

included if they are significant in a preliminary regression. They find that forecasting

performance is similar to other forecasting methods such as shrinkage methods

and forecast combination. Rapach and Strauss (2010) use bagging to forecast US

unemployment changes with 30 predictors. They apply bagging to a pre-test strategy

that uses individual t-statistics to select variables, and find that this delivers very

competitive forecasts compared to forecast combinations of univariate benchmarks.

Hillebrand and Medeiros (2010) apply bagging to lag selection for heterogeneous

autoregressive models of realized volatility, and they find that this method leads to

improvements in forecast accuracy.

Our method requires a sign restriction in order to impose the null. We focus on

a single predictor variable, because in this case, intuition and economic theory can

be used to derive sign restrictions. For models with multiple correlated predictors,

sign restrictions are harder to justify. In the literature, bagging has been applied for

reducing variance from imposing sign restrictions on parameters. A hard-threshold es-

timator with sign restriction sets the estimate to zero if the sign restriction is violated.

Gordon and Hall (2009) consider bagging the hard-threshold estimator and show

analytically that bagging can reduce variance. Sign restrictions arise naturally in pre-

dicting the equity premium, see Campbell and Thompson (2008) for a hard-threshold,

and Pettenuzzo, Timmermann, and Valkanov (2013) for a Bayesian approach. Hille-

brand, Lee, and Medeiros (2013) analyze the bias-variance trade-off from bagging

positive constraints on coefficients and the equity premium forecast itself, and they

find empirically that bagging helps improving the forecasting performance.

The remainder of the chapter is organized as follows. In Section 1.2, the bagging

estimator for weak predictors is presented and asymptotic properties are analyzed.

Monte Carlo results for small samples are presented in Section 1.3. In Section 1.4, the

estimator is applied to CPI inflation forecasting using the unemployment rate and

industrial production as predictors. Concluding remarks are given in Section 1.5.


1.2 Bagging Predictors

Let y be the target variable we wish to forecast h-steps ahead, for example consumer

price inflation. The variables x is a potential predictor variable that can be used to

forecast the target variable y . Let T be the sample size. At time t , we forecast yt+h,T

using the scalar variable xt as predictor and a model estimated on the available data.

In our framework we consider the simple regression relation

yt+h,T =µ+βT xt +ut+h , (1.1)

that is used to obtain h-steps ahead forecasts of the variable y , and where βT is a

coefficient that depends on the sample size to reflect a weak predictive relation.

The focus of our analysis is estimation of the coefficient βT of the predictor

variable x. We start with the following assumptions regarding the unrestricted least-

squares estimate of the coefficient, βT , and the estimator of its asymptotic variance,

σ2∞,T . To reduce notational clutter we suppress the dependence of the asymptotic

variance on the fixed forecast horizon h.

Assumption 1.1

T 1/2(βT −βT )d−→N (0,σ2

∞), (1.2)

and let σ2∞,T > 0 be a consistent estimator of σ2∞ <∞, i.e., σ2

∞,T −σ2∞p−→ 0.

Given the asymptotic variance from Assumption 1.1, we analyze weak predictors

by considering the following parameterization,

βT = T −1/2bσ∞, (1.3)

where we assume that the sign of b is known. Without loss of generality, we assume

that b is strictly positive, i.e., si g n(b) = 1.

For a given sample of length T and a given forecast horizon h, we start with

considering two forecasting models, the unrestricted model (UR) that includes the

predictor variable xt and the restricted model (RE) that contains only an intercept.

Let µRET and (µU R

T , βT )′ be the OLS parameter estimates from the restricted model and

the unrestricted model, respectively. The forecasts for yt+h,T from the unrestricted

and restricted models are denoted

yU Rt+h,T = µU R

T + βT xt , (1.4)

and

yREt+h,T = µRE

T , (1.5)

respectively.

In practice, we are often not certain whether to include the weak predictor xt in

the forecast model or not, i.e., whether RE or UR yields more accurate forecasts. In

1.2. BAGGING PREDICTORS 5

such a situation, it is common to use a pre-test estimator. Typically, the t-statistic

τT = T 1/2βT σ−1∞,T is used to decide whether or not to include the predictor variable.

Let I(.) denote the indicator function that takes value 1 if the argument is true and 0

otherwise. The one-sided pre-test estimator is

βPTT = βT I(τT > c), (1.6)

for some critical value c, for example 1.64 for a one-sided test at the 5% level. We

focus on one-sided testing because we assumed that the sign of β is known.

The hard-threshold indicator function involved in the pre-test estimator intro-

duces estimation uncertainty, and it is not well designed to improve forecasting per-

formance. Bootstrap aggregation (bagging) can be used to smooth the hard-threshold

and thereby improve forecasting performance (see Bühlmann and Yu, 2002; Breiman,

1996). The bagging version of the pre-test estimator is defined as

βBGT = 1

B

B∑b=1

β∗b I(τ∗b > c), (1.7)

where β∗b and τ∗b are calculated from bootstrap samples, and B is the number of

bootstrap replications.

The bagging estimator and the underlying t-statistic pre-test estimator are based

on a test for β = 0. We use the estimated value of the coefficient, βT , if this null

hypothesis can be rejected at some pre-specified significance level, e.g., 5%. However,

this test does not directly address the actual question of the model selection decision,

i.e., whether or not the coefficient can be estimated accurately enough to be useful

for forecasting for the given sample size. Rather, it is a test for whether the coefficient

is zero or not.

Clark and McCracken (2012) (CM henceforth) propose an asymptotic in-sample

test for predictive ability for weak predictors to test whether estimation uncertainty

outweighs the predictive power of a predictor in terms of mean-squared error. The

null hypothesis equates asymptotic estimation variance and squared bias. In terms of

squared bias and variance of the estimator of the coefficient βT , this null hypothesis

becomes

H0,C M : limT→∞

TE[(βT )2] = limT→∞

TE[(βT −βT )2]. (1.8)

Under Assumption 1.1 and the parameterization (1.3), we have that the null hypothe-

sis H0,C M is true for b2σ2∞ =σ2∞. Thus the null hypothesis is true for b = 1 as we have

assumed that b is positive. Looking at the distribution of the t-test statistic under the

null hypothesis H0,C M , Assumption 1.1, and using Equation (1.3), we get

T 1/2βT σ−1∞,T = T 1/2(βT −βT )σ−1

∞,T +T 1/2T −1/2bσ∞σ−1∞,T

d−→ N (0,1)+1. (1.9)

The distribution under the null is a non-central distribution. This non-central asymp-

totic distribution is used to obtain critical values c for the t-statistic under the hy-

pothesis H0,C M following Clark and McCracken (2012) .


The asymptotic distribution is non-central, because under the null hypothesis

the coefficient is not zero. The critical values c are different than for the standard

significance test and depend on the sign of b (see Clark and McCracken, 2012, for de-

tails). More importantly, imposing the null hypothesis of the CM-test is not achieved

by setting β= 0. Therefore we cannot set β= 0 if the CM-test does not reject the null

hypothesis. Instead, we impose this null hypothesis, which can be achieved by setting

the coefficient to an estimate of the asymptotic variance,

β0,C M =√

var[βT ] =√

T −1σ2∞,T = T −1/2σ∞,T . (1.10)

Note that we utilized the sign restriction on b to identify the sign of β0,C M under the

null.

This results in the following pre-test estimator based on the CM-test, which we

call CMPT (Clark-McCracken Pre-Test).

βC MPTT = βT I(τT > c)+T −1/2σ∞,T I(τT ≤ c), (1.11)

where, for the same confidence level, the critical value c is different from the crit-

ical value c used in the standard pre-test estimator (1.6), because the asymptotic

distributions of the test statistics differ.

The bagging version of the CMPT estimator (1.11), henceforth called CMBG, is

defined as

βC MBGT = 1

B

B∑b=1

[β∗

b I(τ∗b > c)+T −1/2σ∞,T I(τ∗b ≤ c)]

. (1.12)

The first term in the sum is exactly the standard bagging estimator, except for the

different critical values. The critical values for C MBG come from the normal distri-

bution N (1,1), while critical values for standard bagging come from the standard

normal distribution. The second term in the sum of Equation (1.12) stems from the

cases where the null is not rejected for bootstrap replication b. Note that we do not

re-estimate the variance under the null, σ2∞,T , for every bootstrap sample. The main

reason to apply bagging are hard-thresholds, which are not involved in the estimation

of σ2∞,T , such that there is no obvious reason for bagging the variance estimator.

1.2.1 Asymptotic Distribution and Mean-Squared Error

We have proposed an estimator that is based on the CM-test and better reflects our

goal of improving forecast accuracy rather than testing statistical significance. In this

section, we derive the asymptotic properties of this estimator to see if, and for which

parameter configurations, this estimator indeed improves the asymptotic mean-

squared error (AMSE). The asymptotic distribution for bagging estimators has been

analyzed for bagging t-tests by Bühlmann and Yu (2002), and for sign restrictions by

Gordon and Hall (2009). The following assumption on the bootstrapped least-squares

estimator β∗T is needed for the analysis of the bagging estimators.


Assumption 1.2 (Bootstrap consistency)

supv∈R

|P∗[T 1/2(β∗T − βT ) ≤ v]−Φ(v/σ∞)| = op (1), (1.13)

where P∗ is the bootstrap probability measure.

In Assumption 1.2 we assume that the bootstrap distribution converges to the

asymptotic distribution of the CLT in Assumption 1.1. Under Assumption 1.2, with

a local-to-zero coefficient given by model (1.3), Bühlmann and Yu (2002) derive

the asymptotic distribution for two-sided versions of the pre-test and the bagging

estimators. The one-sided versions considered in this chapter follow immediately as

special cases. Let φ(.) denote the pdf andΦ(.) the cdf of a standard normal variable.

Proposition 1.1 (Special case of Bühlmann and Yu (2002), Proposition 2.2)

Under Assumption 1.1, and model (1.3)

T 1/2σ−1∞,T β

PTT

d−→ (Z +b)I(Z +b > c), (1.14)

and, with additionally Assumption 1.2,

T 1/2σ−1∞,T β

BGT

d−→ (Z +b)Φ(Z +b − c)+φ(Z +b − c), (1.15)

where Z is a standard normal random variable.

The proposition follows immediately from Bühlmann and Yu (2002). The asymp-

totic distributions depend on the predictor strength b and the critical value c. For

the pre-test estimator, the indicator function enters the asymptotic distribution. The

distribution of the bagging estimator, on the other hand, contains smooth functions

of b and c. Bühlmann and Yu (2002) show how this can reduce the variance of the

estimator substantially for certain values of b and c. We adapt this proposition to

derive the asymptotic distributions of the estimators CMPT, given by Equation (1.11),

and CMBG, given by Equation (1.12).

Proposition 1.2Under Assumption 1.1, and model (1.3)

T 1/2σ−1∞,T β

C MPTT

d−→ (Z +b)I(Z +b > c)+ I(Z +b ≤ c), (1.16)

and, with additionally Assumption 1.2,

T 1/2σ−1∞,T β

C MBGT

d−→ (Z +b)Φ(Z +b − c)+φ(Z +b − c)+1−Φ(Z +b − c), (1.17)

where Z is a standard normal variable.

The proof of the proposition is given in the appendix. The asymptotic distribu-

tions are similar to those of the pre-test and bagging estimators (BG and PT), but


involve extra terms due to the different null hypothesis. For CMPT, the extra term is

simply an indicator function, and for CMBG it involves the standard normal cdfΦ(·).

Figures 1.1 and 1.2 show asymptotic mean-squared error, asymptotic bias, asymp-

totic squared bias, and asymptotic variance of the pre-test and bagging estimators

for test levels 5% and 1%, respectively. Note that the t-test and the CM-test use dif-

ferent critical values, c and c. The results for the two different significance levels,

5% and 1%, are qualitatively identical. The effect of choosing a lower significance

level is that the critical values increase, and the effects from pre-testing become

more pronounced. For the asymptotic mean-squared error (AMSE), we get the usual

picture for PT and PTBG (see Bühlmann and Yu, 2002). Bagging improves the AMSE

compared to pre-testing for a wide range of values of b, except at the extremes. CMBG

compares similarly to CMPT, but shifted towards the right compared to BG and PT.

When looking at any given value b, there are striking differences between the es-

timators based on the CM-test and the ones based on the t-test. Both CMPT and

CMBG do not perform well for b close to zero, but the AMSE decreases as b increases,

before starting to slightly increase again. For values of b from around 0.5 to 3, CMBG

performs better than BG. For values larger than 3 the estimators PT, BG, and CMBG

perform similarly and get closer as b increases. Thus, the region where CMBG does

not perform well are values of b below 0.5.

The asymptotic biases for CMPT and CMBG are largest at b = 0. For all estima-

tors, the bias can be both positive or negative, depending on b. Bagging can reduce

bias compared to the corresponding pre-test estimation, in particular in the region

where the pre-test estimator has the largest bias. CMPT and CMBG have very low

variance for b close to zero, because the CM-test almost never rejects for these param-

eters. However, as the null hypothesis is not close to the true b in this region, CMPT

and CMBG are therefore very biased. As b increases slightly, CMBG has the lowest

asymptotic variance for b up to around 3.

The asymptotic results show that imposing a different null hypothesis dramat-

ically changes the characteristics of the estimators. The estimator based on the

CM-test is not intended to work for b very close to zero. In this case, the standard

pre-test estimator has much better properties. For larger b, the CM-based estimators

give substantially better forecasting results. These results highlight that the CM-based

estimators will be useful for relations where the coefficient is expected to be strictly

positive or strictly negative, but too small to exploit with an unrestricted coefficient

estimator.

1.2.2 Asymptotic Shrinkage Representation

Stock and Watson (2012) provide an asymptotic shrinkage representation of the BG

estimator. This representation, henceforth called BGA , is given by

βBG AT = βT

[1−Φ(c − τT )+ τ−1

T φ(c − τT )]

(1.18)


0 1 2 3 4 5

01

23

4

AMSE

b

0 1 2 3 4 5

−1.

0−

0.5

0.0

0.5

1.0

Abias

b

PTBGCMPTCMBG

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Squared Abias

b

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Avar

b

Figure 1.1. Comparison of asymptotic mean-squared error (AMSE), asymptotic bias (Abias),asymptotic square bias (Abias square), and asymptotic variance (Avar) as a function of b for5% significance level.


0 1 2 3 4 5

01

23

4

AMSE

b

0 1 2 3 4 5

−1.

0−

0.5

0.0

0.5

1.0

Abias

b

PTBGCMPTCMBG

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Squared Abias

b

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Avar

b

Figure 1.2. Comparison of asymptotic mean-squared error (AMSE), asymptotic bias (Abias),asymptotic square bias (Abias square), and asymptotic variance (Avar) as a function of b for1% significance level.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

β

BGACMBGA

Figure 1.3. Shrinkage estimators BGA and CMBGA (y-axis) for a given value of the unrestrictedparameter estimate β (x-axis) for σ∞ = 0.2 and 5% level. Dotted line is 45◦ line.

and Stock and Watson (2012, Theorem 2) show under general conditions that βBGT =

βBG AT +oP (1). This allows computation without bootstrap simulation. While boot-

strapping can improve test properties, bagging can improve forecasts even without

actual resampling. There is no reason to suspect that the estimator based on the

asymptotic distribution will be inferior to the standard bagging estimator. Therefore,

we consider a version of the bagging estimators that samples from the asymptotic,

rather than the empirical, distribution of βT . We can find closed-form solutions

for estimators that do not require bootstrap simulations. The asymptotic version of

CMBG is henceforth referred to as CMBGA.

Proposition 1.3 (Asymptotic Shrinkage representation)

Apply CMBG with the asymptotic distribution of βT under Assumption 1.2, then

βC MBG AT = βT

[1−Φ(c − τT )+ τ−1

T φ(c − τT )+ τ−1T Φ(c − τT )

]. (1.19)

The proof of the proposition is given in the appendix. The representation is very

similar to BGA in Equation (1.18), with an extra term for the contribution for the null

of the CM-test. Note that we can express βC MBG AT as the OLS estimator βT multiplied

by a function that depends on the data only through the t-statistic τT , just like βBG AT .

Figure 1.3 plots BGA and CMBGA against the OLS estimate βT . The vertical

deviation from the 45◦ line indicates the degree and direction of shrinkage applied

by the estimator to the OLS estimate βT . This reveals the main difference between

BGA and CMBGA. Rather than shrinking towards zero, CMBGA shrinks towards σ∞,T ,


which makes a substantial difference for b close to 0. For larger βT , the CMBGA, and

thus CMBG, shrink more heavily downwards than BGA.

1.3 Monte Carlo Simulations

The asymptotic analysis suggests that our modified bagging estimator can yield

significant improvements in MSE for the estimation of β. This section uses Monte

Carlo simulations to investigate the performance for the prediction of yt+h,T in small

samples using the estimators presented above. In our linear model (1.1), lower MSE

for estimation of β can be expected to translate directly into lower MSE for prediction

of yt+h,T .

For the Monte Carlo simulations, we generate data from the following model that

is designed to resemble the empirical application of inflation forecasting:

yt+h,T =µ+βT xt +ut+h

ut+h = εt+h +θ1εt+h−1 +·· ·+θh−1εt+1

xt =φxt−1 + vt

εt ∼N (0,σ2ε )

vt ∼N (0,σ2v ). (1.20)

We allow for serially correlated errors in the form of an MA(h-1) model. The choice

of AR(1) for xt is guided by the model for the monthly unemployment change series

selected by AIC.

As we vary the sample size, the predictor variable xt modeled as a weak predictor

with coefficient βT = T −1/2bσ∞. We consider values b ∈ {0,0.5,1,2,4}. For b = 1, we

are indifferent between estimating β unrestrictedly and no using the predictor vari-

able. For higher (lower) values of b, including the predictor variable should improve

(deteriorate) the forecasting performance. Table 1.1 presents an overview of all these

methods.

We are interested in the small-sample properties and consider sample sizes T ∈{25,50,200}. Furthermore, we set µ= 0.1 and φ= 0.66, which we take from the our

empirical example, i.e., monthly changes in unemployment. Additionally, we consider

φ= 0.9 to investigate the behavior for more persistent processes. Finally, we consider

the forecast horizons h = 1 and h = 6. The MA coefficients are set to θi = 0.4i for

1 ≤ i ≤ h − 1, and 0 otherwise. The critical values are taken from the respective

asymptotic distribution of both tests for significance levels 5% and 1%. We run 10,000

Monte Carlo simulations and use 299 bootstrap replications for bagging.

Columns 2 through 9 of Tables 1.2-1.5 show the MSE for the different estimators

listed in Table 1.1. The last two columns show the rejection frequencies for the t-test

and CM-test. The MSE is reported in excess of var[ut+h], which does not depend on

the forecasting model, such that the true model with known parameters will have

MSE of zero.

1.3. MONTE CARLO SIMULATIONS 13

Table 1.1. Forecasting methods for Monte Carlo and empirical application

Name Method Formula for forecast yt+h,T

RE Restricted Model µRET

UR Unestricted Model µU RT + βT xt

PT Pre-Test t-test µU RT + I(τT > c)βT xt

BG Bagging t-test µU RT + 1

B∑B

b=1 β∗b I(τ∗b > c)xt

BGA Asymptotic BG µU RT + βT

[1−Φ(c − τT )+ τ−1

T φ(c − τT )]

xt

CMPT Pre-Test CM-test µU RT +

(βT I(τT > c)+T−1/2σ∞,T I(τT ≤ c)

)xt

CMBG Bagging CM-test µU RT +

(1B

∑Bb=1 β

∗b I(τ∗b > c)+T−1/2σ∞,T I(τ∗b ≤ c)

)xt

CMBGA Asymptotic CMBG µU RT +

(βT

[1−Φ(c − τT )+ τ−1

T φ(c − τT )+ τ−1T Φ(c − τT )

])xt

Note: µT and βT are the OLS estimates that depend on the forecast horizon.

For different values of b, we get the overall patterns expected from the asymptotic

results for all parameter configurations, sample sizes T , persistence parameters φ,

and forecast horizons h. For b = 0 the restricted model is correct. Forecast errors

of the restricted model stem only from mean estimation. The CM-based methods

perform worst, as the null hypothesis b = 1 is incorrect, and the CM-test rejects very

infrequently. The null of the t-test-based pre-test estimator is correct and is imposed

whenever the test fails to reject, which happens frequently under all parameter

configurations. This allows PT and its bagging version to achieve a lower MSE than

the unrestricted model.

For b = 0.5, the predictor is still so weak that the unrestricted model always

performs best. The difference between using t-tests and CM-tests is not as large as

it is for b = 0. Setting b = 1 imposes that the unrestricted and restricted methods

asymptotically have the same MSE for estimation of β. For T = 25, however, the

restricted model has substantially lower MSE than the unrestricted model for the

prediction of yt+h,T . The difference disappears as the sample size grows. The rejection

frequency for the CM-tests is fairly close to the nominal size for h = 1. For h = 6 the

test is over-sized in small samples. Despite these small sample issues of the test, the

CM-based estimators work well when b = 1 even for T = 25 with φ= 0.66 in Tables 1.2

and 1.4. For φ= 0.9, shown in Tables 1.3 and 1.5, CM-test and t-test-based estimators

perform very similarly for T = 25.

For b = 2, the CM-based method is able to improve the MSE, even though the null

hypothesis is not precisely true. The magnitude of the improvement depends on the

persistence parameter φ, critical value, and sample size. For b = 4 the coefficient is

large enough such that the unrestricted model dominates. All other models except

RE provide very similar performance. Both the CM test and the standard significance


test reject very frequently, such that the different values of the coefficient under the

two null hypotheses are less important.

Our Monte Carlo simulations confirm that the asymptotic properties of the co-

efficient estimators carry over to the small sample behavior of the estimators and

the resulting forecasting performance for the target variable. The bagging version

of the CM-test can be expected to perform well when bias is not too small relative

to the estimation uncertainty, i.e., b is not close to zero. If bias is much smaller than

estimation uncertainty, then methods that shrink towards zero dominate. Our esti-

mators will work well if the predictor is weak but the coefficient is large enough that

excluding the predictor induces a substantial bias.


Table 1.2. Monte Carlo Results for φ= 0.66 and c0.95

MSE Rejection %

RE UR PT PTBG PTBGA CMPT CMBG CMBGA t-test CM-testPanel 1: b = 0

h=1T = 25 6.84 21.74 11.55 12.63 12.54 21.60 23.39 23.19 5.90 0.60T = 50 4.69 9.41 6.19 6.42 6.41 10.07 10.70 10.69 4.95 0.70

T = 200 0.74 1.78 0.97 1.03 1.01 1.73 1.88 1.87 4.60 0.55h=6

T = 25 19.43 42.22 28.92 30.66 29.81 39.33 42.17 41.16 14.00 6.10T = 50 13.07 22.34 15.97 16.60 16.20 21.15 22.65 22.17 9.80 2.55

T = 200 2.60 4.60 3.10 3.34 3.22 4.84 5.18 5.07 5.85 1.00

Panel 2: b = 0.5h=1

T = 25 9.36 20.96 15.72 13.95 13.73 15.64 17.50 17.28 13.20 2.70T = 50 6.63 10.69 8.18 7.50 7.48 7.60 8.24 8.23 11.95 2.15

T = 200 0.45 1.35 0.89 0.68 0.67 0.63 0.81 0.80 13.85 1.90h=6

T = 25 27.22 46.59 38.00 35.50 35.39 35.07 37.03 36.43 25.15 11.90T = 50 13.82 21.23 17.03 16.26 15.82 16.12 17.59 16.93 20.00 6.75

T = 200 3.87 5.62 4.60 4.41 4.22 4.16 4.66 4.47 14.70 2.45

Panel 3: b = 1h=1

T = 25 17.67 23.81 22.89 18.13 18.00 14.43 15.48 15.50 23.80 6.10T = 50 9.30 10.07 10.71 8.48 8.42 6.80 7.18 7.17 24.40 4.85

T = 200 2.06 2.26 2.43 1.89 1.87 1.35 1.44 1.43 27.05 5.25h=6

T = 25 44.20 46.14 45.87 39.96 40.41 36.61 36.49 36.24 42.60 23.40T = 50 17.97 20.42 20.02 17.08 16.81 13.84 14.60 13.91 34.90 14.75

T = 200 4.85 4.80 5.28 4.36 4.22 3.47 3.77 3.55 26.40 7.10

Panel 4: b = 2h=1

T = 25 48.44 23.89 32.83 24.03 23.81 20.79 17.56 17.57 53.75 21.35T = 50 21.81 9.94 14.16 10.49 10.42 9.58 7.76 7.73 58.50 22.75

T = 200 4.87 1.83 2.76 2.03 2.01 1.97 1.47 1.47 63.30 25.80h=6

T = 25 96.54 46.97 56.01 46.86 47.80 44.48 40.44 40.79 70.95 49.05T = 50 46.09 21.85 27.92 22.41 22.79 21.54 19.01 18.89 66.25 38.45

T = 200 11.64 5.16 7.14 5.48 5.57 5.48 4.62 4.51 64.60 29.75

Panel 5: b = 4h=1

T = 25 149.03 21.10 26.21 24.11 23.82 29.24 22.90 22.78 93.65 74.20T = 50 74.98 9.74 11.08 10.76 10.65 12.84 10.92 10.83 96.55 83.70

T = 200 18.29 2.74 2.84 2.90 2.90 3.30 3.05 3.04 98.80 90.15h=6

T = 25 302.49 40.50 44.53 42.16 42.87 51.46 42.16 45.52 96.95 87.95T = 50 147.60 21.55 22.86 22.23 22.68 25.83 22.42 23.12 98.30 89.15

T = 200 34.15 4.31 4.57 4.45 4.64 5.59 4.49 4.83 98.40 87.50

Notes: MSE calculated in excess of var[ut+h ], and multiplied by 100.



MSE Rejection %


h=1T = 25 6.82 39.64 22.50 23.47 23.49 40.09 43.48 43.14 6.98 0.80T = 50 3.60 13.89 7.72 8.07 8.03 13.36 14.46 14.38 5.64 0.74

T = 200 0.91 2.20 1.31 1.36 1.36 2.26 2.43 2.43 5.46 0.44h=6

T = 25 21.89 84.13 55.36 57.93 56.25 78.06 83.07 81.65 15.20 6.45T = 50 13.49 35.24 23.79 25.76 24.68 34.48 37.33 36.31 9.50 3.00

T = 200 3.24 6.39 4.09 4.50 4.27 6.46 7.06 6.85 5.90 0.62

Panel 2: b = 0.5h=1

T = 25 12.17 41.66 27.36 26.12 25.94 32.45 36.03 35.62 10.64 1.62T = 50 5.35 13.74 9.70 8.84 8.77 9.95 11.07 11.03 10.86 1.58

T = 200 1.40 2.47 1.87 1.69 1.68 1.73 1.90 1.90 11.98 1.84h=6

T = 25 24.19 77.86 56.40 53.55 52.63 58.47 62.08 61.08 23.90 11.25T = 50 15.02 34.25 25.23 24.92 23.84 25.05 28.46 27.17 18.80 6.65

T = 200 3.66 5.99 4.77 4.60 4.28 4.48 5.20 4.87 14.02 2.50

Panel 3: b = 1h=1

T = 25 17.49 37.97 30.36 25.42 25.43 23.88 26.93 26.92 17.24 3.36T = 50 8.18 12.51 11.21 8.77 8.68 7.18 8.13 8.07 20.36 3.76

T = 200 2.07 2.37 2.36 1.83 1.82 1.31 1.42 1.41 23.86 4.24h=6

T = 25 43.83 84.33 70.23 63.89 63.15 68.45 70.06 70.60 35.15 18.00T = 50 24.15 30.77 28.57 24.50 23.59 21.09 23.42 22.05 31.65 12.90

T = 200 5.71 6.25 6.52 5.46 5.17 4.22 4.78 4.36 27.00 7.38

Panel 4: b = 2h=1

T = 25 52.56 39.39 46.45 34.30 34.30 26.47 26.01 26.06 38.94 12.62T = 50 24.42 14.11 18.03 13.28 13.16 10.53 9.51 9.45 47.92 16.88

T = 200 5.57 2.47 3.51 2.60 2.58 2.42 1.94 1.93 59.18 23.00h=6

T = 25 128.12 82.76 91.29 73.93 74.94 65.95 63.16 62.30 58.20 37.95T = 50 59.63 30.62 38.18 29.77 29.92 26.82 24.87 24.02 58.40 32.85

T = 200 13.22 5.93 8.41 6.17 6.22 5.74 4.98 4.70 59.84 27.30

Panel 5: b = 4h=1

T = 25 175.35 39.32 56.76 43.18 42.70 44.32 33.49 33.57 76.24 47.18T = 50 83.22 13.60 18.01 15.59 15.38 18.25 13.91 13.79 88.14 64.32

T = 200 17.05 2.08 2.43 2.37 2.35 2.93 2.39 2.37 96.22 81.52h=6

T = 25 404.67 72.41 89.23 74.60 76.49 79.20 68.59 69.56 88.25 72.60T = 50 205.15 33.91 41.26 35.47 36.85 41.92 33.88 35.09 92.05 75.40

T = 200 44.01 6.15 6.91 6.34 6.74 8.38 6.28 6.85 96.78 81.46




MSE Rejection %


h=1T = 25 7.26 21.02 10.54 11.06 10.91 21.38 22.26 21.98 1.42 0.16T = 50 2.93 8.09 3.67 3.88 3.86 8.12 8.48 8.42 1.20 0.04

T = 200 0.63 1.67 0.74 0.78 0.78 1.71 1.76 1.76 1.04 0.10h=6

T = 25 18.45 42.04 26.89 27.70 26.98 37.10 39.17 37.99 8.15 3.45T = 50 10.35 21.45 12.85 13.90 13.24 19.93 20.95 20.23 3.95 1.45

T = 200 3.39 5.39 3.62 3.83 3.69 5.52 5.73 5.62 1.56 0.30

Panel 2: b = 0.5h=1

T = 25 11.02 21.62 14.24 13.30 13.21 15.37 16.26 16.03 3.74 0.72T = 50 5.62 10.08 6.73 6.29 6.25 6.66 7.06 7.02 3.62 0.36

T = 200 1.42 2.26 1.62 1.48 1.48 1.49 1.57 1.57 3.06 0.26h=6

T = 25 21.72 42.23 32.86 30.26 30.34 30.34 31.96 31.05 16.95 7.65T = 50 10.73 18.75 13.89 13.21 12.76 13.12 14.29 13.48 8.90 3.10

T = 200 2.84 4.46 3.22 3.13 2.94 3.01 3.35 3.15 4.42 0.60

Panel 3: b = 1h=1

T = 25 17.39 22.84 20.74 16.60 16.57 12.09 12.92 12.71 9.08 1.54T = 50 8.18 9.41 9.42 7.33 7.34 4.94 5.24 5.22 9.96 1.34

T = 200 1.75 1.89 2.04 1.51 1.49 0.88 0.95 0.94 9.46 0.94h=6

T = 25 41.53 45.77 45.05 38.09 39.21 33.44 33.88 33.13 28.55 14.70T = 50 20.87 22.21 22.59 19.34 19.22 15.60 16.46 15.50 17.80 7.60

T = 200 4.52 4.81 4.98 4.17 4.02 2.97 3.29 3.03 9.88 1.98

Panel 4: b = 2h=1

T = 25 47.77 23.19 38.34 26.24 26.28 19.64 16.40 16.56 33.86 10.12T = 50 20.74 9.30 16.73 11.02 10.98 8.37 6.61 6.65 33.70 9.18

T = 200 5.27 2.15 4.22 2.79 2.76 2.36 1.81 1.80 37.70 10.20h=6

T = 25 84.60 40.83 57.31 42.06 44.63 39.57 34.09 34.60 52.35 32.35T = 50 48.41 22.29 34.13 24.36 25.91 23.47 19.73 20.04 50.10 26.25

T = 200 11.17 4.88 8.70 5.77 6.05 5.30 4.36 4.25 41.00 13.84

Panel 5: b = 4h=1

T = 25 148.82 22.75 40.96 31.12 30.87 40.47 27.96 28.23 80.40 53.04T = 50 71.20 9.13 14.79 12.52 12.37 18.16 12.44 12.46 88.04 62.20

T = 200 17.51 2.31 3.05 2.92 2.90 4.40 3.20 3.18 93.68 70.42h=6

T = 25 345.55 49.56 64.55 54.44 57.74 67.39 54.12 57.22 93.55 81.45T = 50 154.77 20.70 29.62 23.07 25.63 32.96 23.24 26.19 92.10 75.20

T = 200 37.91 4.95 6.22 5.36 5.95 8.45 5.59 6.52 95.06 76.14




MSE Rejection %


h=1T = 25 7.70 40.13 20.26 21.18 20.98 40.48 42.39 41.94 1.28 0.08T = 50 3.76 13.34 6.46 6.74 6.69 13.41 13.90 13.83 1.10 0.08

T = 200 0.93 2.30 1.15 1.19 1.19 2.23 2.31 2.30 0.96 0.04h=6

T = 25 21.81 81.10 48.04 50.87 48.61 106.76 110.36 108.59 5.62 2.00T = 50 10.76 32.52 18.37 19.82 18.55 32.27 34.25 33.05 3.78 1.02

T = 200 2.04 5.17 2.56 2.92 2.64 5.09 5.46 5.25 1.66 0.18

Panel 2: b = 0.5h=1

T = 25 12.23 40.64 22.78 21.74 21.49 30.68 32.76 32.24 3.00 0.26T = 50 6.25 14.07 9.09 8.60 8.58 10.67 11.27 11.19 3.46 0.46

T = 200 1.22 2.26 1.48 1.34 1.34 1.39 1.49 1.48 3.48 0.24h=6

T = 25 32.00 80.32 54.94 53.58 51.93 61.73 67.38 63.71 14.35 7.50T = 50 16.48 35.75 25.92 25.07 23.81 27.35 29.73 28.22 11.10 3.35

T = 200 3.49 5.95 4.23 4.21 3.86 4.04 4.58 4.22 4.58 0.64

Panel 3: b = 1h=1

T = 25 17.16 39.57 27.37 24.15 24.13 24.46 26.54 26.08 5.86 0.84T = 50 9.25 13.95 11.92 9.85 9.84 7.70 8.30 8.24 7.88 0.94

T = 200 1.86 2.14 2.19 1.67 1.66 1.07 1.14 1.13 7.90 0.88h=6

T = 25 54.06 82.65 69.21 61.55 60.74 72.97 73.95 73.73 18.10 8.26T = 50 23.57 32.24 28.70 24.88 23.95 19.93 22.05 20.35 14.76 4.86

T = 200 5.52 6.48 6.27 5.49 5.15 3.80 4.43 3.92 10.82 1.98

Panel 4: b = 2h=1

T = 25 46.31 36.34 43.88 31.21 31.50 21.13 20.94 20.74 17.50 4.28T = 50 23.56 13.75 20.90 14.05 14.03 8.79 7.97 7.96 24.50 6.28

T = 200 5.31 2.26 4.28 2.79 2.76 2.16 1.72 1.71 33.78 8.60h=6

T = 25 127.54 81.29 100.74 75.29 77.74 64.03 58.88 59.03 39.90 21.04T = 50 59.66 33.76 47.63 34.44 35.36 28.31 25.72 25.16 34.58 15.44

T = 200 14.60 6.58 11.28 7.47 7.87 6.66 5.71 5.48 37.94 12.38

Panel 5: b = 4h=1

T = 25 173.15 36.79 73.40 48.12 48.62 49.19 33.34 34.30 57.28 29.28T = 50 80.02 13.95 27.18 19.35 19.18 22.39 15.46 15.48 71.10 42.02

T = 200 17.42 2.14 3.57 2.98 2.96 4.42 3.03 3.00 87.96 62.18h=6

T = 25 420.58 82.77 133.75 94.18 102.51 105.78 82.30 87.08 76.10 58.90T = 50 206.64 31.99 56.46 36.95 42.37 48.94 34.13 37.89 80.70 60.90

T = 200 43.69 5.67 9.06 6.31 7.53 10.86 6.23 7.65 88.52 64.06


1.4. APPLICATION TO CPI INFLATION FORECASTING 19

1.4 Application to CPI Inflation Forecasting

Inflation is a key macroeconomic variable, measuring changes in consumer price

levels. Clearly, these price levels depend on the demand and supply for production

and consumer goods. Thus, one would expect them to be linked negatively to unem-

ployment and positively to industrial production. While economists and the media

pay attention to such variables to assess inflationary pressure, the variables do not

help to forecast inflation more accurately than univariate models. Cecchetti, Chu,

and Steindel (2000) find that using popular candidate variables as predictors fails to

provide more accurate forecasts for US inflation, and that the relationship between

inflation and some of the predictors is of the opposite sign as one would expect. Thus,

they conclude that single predictor variables provide unreliable inflation forecasts.

Atkeson and Ohanian (2001) consider more complex autoregressive distributed-

lags models for inflation forecasting and conclude that none of the models out-

perform a random walk model. Stock and Watson (2007) argue that the relative

performance of inflation forecasting methods depends crucially on the time period

considered. Not only does the relative performance of forecasting methods change

over time, but coefficients in the models are also likely to be time-varying. Stock and

Watson (2009) go so far as to call it the consensus that including macroeconomic

variables in models does not improve inflation forecasts over univariate benchmarks

that do not utilize information other than past inflation.

We denote inflation by

πht = ln(Pt+h/Pt ), (1.21)

where Pt is the level of the US consumer price index (CPI, All Urban Consumers: All

Items). We specify our models in terms of changes in inflation and aim to forecast

these changes for different forecast horizons h. We define the change in inflation

as ∆πht = h−1πh

t −π1t−1, i.e, the change of average inflation over the next h month

compared to the most recent inflation rate. The forecast models are the specified as

∆πht =µ+βxt +εt+h , (1.22)

where xt is some predictor variable. For example, with a forecast horizon of 6 months

(h = 6), we forecast the change in average inflation over the next 6 months compared

to the current month’s inflation. Figure 1.4 shows the target variable ∆πht for different

forecast horizons h. Even at the longest forecast horizon of 12 months, where we are

forecasting annual inflation, the series is not very persistent. The estimation methods

used to determine the parameters are the same as the ones used for the Monte Carlo

simulations and are summarized in Table 1.1.

As predictor variables xt , we use unemployment changes (UNEMP) and growth

in industrial production (INDPRO). Both variables are seasonally adjusted. We use

the latest data vintage available from St. Louis Fed’s FRED1 on August 21, 2013 for

1URL: http://research.stlouisfed.org/fred2

http://research.stlouisfed.org/fred2


monthly data over the period 1:1948–7:2013. Considering changes in unemployment

and growth in industrial production rather than the levels of the two series ensures

that the predictor variables are stationary.

For multiple-step ahead forecasts, we choose a direct forecasting approach. Thus,

the test statistics and parameter estimates depend on the forecasting horizon and

can differ. For all forecast horizons, we use a short estimation window to allow for

parameter instability. We use estimation window lengths of 24 and 60 months, which

are reasonable sample sizes as we use only one predictor variable.

Bagging is conducted using a block bootstrap with block-length optimally chosen

by the method of Politis and White (2004), applying the correction of Patton, Politis,

and White (2009). For multiple-month forecasts (h > 1), we calculate standard errors

using the method of Newey and West (1987) to account for serial correlation.

In Table 1.6, we show the MSE results for the pseudo out-of-sample forecasting

exercise. The maximal out-of-sample period depends on the estimation window

length m and the forecast horizon h. For example, for m = 24 and h = 6 we forecast

inflation over 3:1953–7:2013 (725 observations) and for m = 60 and h = 6 over 3:1956–

7:2013 (689 observations).

The first observation, in line with the existing literature on inflation forecasting, is

that the restricted model is very hard to beat. The unrestricted model never performs

better than the restricted model. The relative performance of the forecasting methods

depends on the forecast horizon h. We apply the model confidence set of Hansen,

Lunde, and Nason (2011) to the resulting loss series in order to determine whether

the out-of-sample results are statistically significant. The MulCom package version

3.002 for the Ox programming language (see Doornik, 2007) is used to construct the

model confidence sets3.

The forecasting results show that the performance of the models is hard to distin-

guish statistically. The model confidence set contains many models in most cases. In

particular, CMBGA and CMBG are never excluded from the 95% model confidence

set, such that there is no statistical evidence against these two forecasting methods. In

terms of mean-squared error, CMBGA and CMBG perform well compared to standard

bagging, BG and BGA, and the unrestricted model. The different critical values, for

the significance levels 5% and 1%, have only a minor effect on the performance of

the predictors.

The performance differences between the bootstrap and the asymptotic versions

of the bagging estimators are small. Thus, the asymptotic versions BGA and CMBGA

offer computationally attractive alternatives to the bootstrap-based predictors BG

and CMBG.

Figures 1.5 and 1.6 display the time series of coefficients from unrestricted esti-

2Available from the homepage http://mit.econ.au.dk/vip_htm/alunde/MULCOM/MULCOM.HTM.3 We use the following settings for the model confidence set construction in MulCom: 9999 boostrap

replication with block bootstrapping, block size equal to forecasting horizon, the range test for equalpredictive ability δR,M , and the range elimination rule eR,M , see Hansen et al. (2011) for details.

http://mit.econ.au.dk/vip_htm/alunde/MULCOM/MULCOM.HTM

1.4. APPLICATION TO CPI INFLATION FORECASTING 21

Table 1.6. MSE relative to restricted model for out-of-sample inflation forecasting.

Panel 1: m = 24

c0.99 c0.95

h = 1 3 6 12 1 3 6 12RE 1* 1* 1* 1* 1* 1* 1* 1*

INDPROUR 1.095* 1.116 1.072 1.142 1.095* 1.116 1.072 1.142PT 1.050* 1.050* 0.994* 0.995* 1.046* 1.076* 1.005* 1.020*BGA 1.032* 1.060* 1.004* 1.018 1.048* 1.067* 1.018* 1.049BG 1.036* 1.057* 1.009* 1.015 1.053* 1.067* 1.025* 1.045CM 1.079 1.075* 1.038* 1.009* 1.117 1.079* 1.042* 1.020*CMBGA 1.018* 1.041* 0.993* 0.986* 1.033* 1.052* 1.000* 1.005*CMBG 1.023* 1.038* 0.999* 0.991* 1.037* 1.050* 1.004* 1.008*

UNEMPUR 1.059 1.092 1.080* 1.066 1.059 1.092 1.080 1.066*PT 1.004* 0.994* 1.007* 0.973* 1.010* 1.042* 1.020* 1.020*BGA 1.006* 1.015* 1.008* 0.980* 1.019* 1.035* 1.026* 0.991*BG 1.006* 1.025* 1.012* 0.980* 1.020* 1.043* 1.027* 0.989*CM 1.025* 1.043* 1.007* 0.981* 1.044* 1.044* 1.015* 0.983*CMBGA 0.991* 0.999* 0.988* 0.965* 1.001* 1.010* 0.999* 0.973*CMBG 0.991* 1.011* 0.992* 0.967* 1.000* 1.021* 1.003* 0.973*

Panel 2: m = 60

c0.99 c0.95

h = 1 3 6 12 1 3 6 12RE 1* 1* 1* 1 1* 1* 1* 1*

INDPROUR 1.019* 1.057* 1.024* 1.021 1.019* 1.057* 1.024* 1.021PT 1.001* 1.003* 1.002* 1.000* 0.998* 1.005* 1.005* 0.999*BGA 0.999* 1.014* 1.000* 0.995* 1.002* 1.026* 1.004* 0.997*BG 0.999* 1.021* 1.000* 0.995* 1.003* 1.032* 1.004* 0.998*CM 1.025* 1.026* 1.025* 1.011* 1.025* 1.030* 1.025* 1.012*CMBGA 0.999* 1.003* 0.999* 0.995* 1.000* 1.009* 1.000* 0.995*CMBG 1.000* 1.010* 0.999* 0.993* 1.000* 1.016* 0.999* 0.994*

UNEMPUR 1.022* 1.042* 1.044* 1.018* 1.022* 1.042* 1.044* 1.018*PT 0.998* 1.004* 1.012* 1.006* 1.000* 1.022* 1.043* 1.003*BGA 1.000* 1.009* 1.019* 0.997* 1.004* 1.021* 1.031* 1.002*BG 1.001* 1.014* 1.029* 0.995* 1.005* 1.025* 1.037* 1.000CM 1.005* 1.018* 1.003* 0.997* 1.005* 1.022* 1.005* 1.001*CMBGA 0.996* 1.000* 1.002* 0.991* 0.998* 1.006* 1.011* 0.994*CMBG 0.997* 1.003* 1.009* 0.988* 0.999* 1.009* 1.021* 0.992*

Notes: An asterisk (*) indicates that the model is included in 95% model confidence set (MCS). The MCSare computed for all methods with same m, c , and h, i.e., for every column in each panel. Thus, each MCS

is computed for 15 models.


mation and CMBGA for m = 24 and m = 60, respectively. For m = 24, the coefficients

from unrestricted estimation are very volatile and frequently change sign for both pre-

dictor variables. CMBGA imposes the sign restriction by construction and shrinks the

coefficients heavily towards the null hypothesis, which results in much less volatile

coefficients. For m = 60, the coefficients from unrestricted estimation are more sta-

ble, and sign changes of the coefficients are less frequent. CMBGA again shrinks the

coefficients substantially and imposes the sign restriction.

Overall, the proposed methods CMBG and CMGA provide competitive forecasting

results and are never excluded from the model confidence set. We find, however, that

no method is significantly better than the random walk benchmark, i.e., the forecasts

from the restricted model. Inflation is a difficult time series to forecast and using other

economic variables as predictors is of limited value in the framework considered in

this paper.

1.5 Conclusion

Bootstrap aggregation (bagging) is typically applied to t-tests of whether coefficients

are significantly different from zero. In finite samples, a significantly non-zero coef-

ficient is not sufficient to guarantee that including the predictor improves forecast

accuracy. Instead, estimation variance has to be taken into account and weighed

against bias from excluding the predictor.

We propose a novel bagging estimator that is based on the in-sample test for

predictive ability of Clark and McCracken (2012), which directly addresses the bias-

variance trade-off. We show that this estimator performs well when bias and variance

are of similar magnitude. This is achieved by shrinking the coefficient towards an

estimate of the estimation variance rather than shrinking towards zero. In order

to find this shrinkage target, the sign of the coefficient has to be known. Thus, the

method is appropriate for predictor variables for which theory postulates the sign of

the relation, as is often the case for economic variables.

The new bagging estimator is shown to have good asymptotic properties, domi-

nating the standard bagging estimator if bias and estimation variance are of similar

magnitude. If, however, the data-generating coefficient is very close to zero, such

that the forecasting power of the predictor is completely dominated by estimation

uncertainty, the new estimator is very biased and thus performs poorly.

In this chapter, we have been concerned with improving accuracy of a single

predictor variable when predictive power is diluted by estimation variance. Using

single predictors for forecasting is important, as many inflation predictors, for ex-

ample, are considered individually to assess their predictive power (cf. Cecchetti

et al., 2000). Econometric forecasting models, however, typically include multiple

correlated predictor variables. In this context, our estimator could be applied to the

individual predictor variables, just as standard bagging is applied in this context by,

1.5. CONCLUSION 23

e.g., Inoue and Kilian (2008). The drawbacks of applying our estimator in this context

to each predictor is that, first, it is harder to motivate sign restrictions on coefficients

and, second, covariances are ignored when assessing the estimation uncertainty.

The second issue can be fixed by using orthogonal factors instead of the original

predictors, which makes it potentially even harder to find credible sign restrictions.

The extension to multivariate specifications is left to future research.


1960 1980 2000

−5

05

h= 1

1960 1980 2000

−6

−4

−2

02

46

h= 3

1960 1980 2000

−6

−4

−2

02

46

h= 6

1960 1980 2000

−6

−4

−2

02

46

8

h= 12

Figure 1.4. Time series of the target variables ∆πht at the different forecasting horizons h.

1.5. CONCLUSION 25

0

0OLS CMBGA

1950 1960 1970 1980 1990 2000 2010

−4

−2

02

4

URCMBGA

(a) Coefficients for unemployment changes (UNEMP).

1950 1960 1970 1980 1990 2000 2010

−1.

00.

01.

0

(b) Coefficients for industrial production growth (INDPRO).

Figure 1.5. Recursive coefficients for UR and CMBGA in forecast regressions of inflationchanges on (a) unemployment changes and (b) industrial production growth. Forecast horizonh = 12 and significance level 1%. Estimation window length m = 24.


00

OLS CMBGA

1960 1970 1980 1990 2000 2010

−1

01

23

4

(a) Coefficients for unemployment changes (UNEMP).

1960 1970 1980 1990 2000 2010

−0.

50.

00.

51.

0

(b) Coefficients for industrial production growth (INDPRO).

Figure 1.6. Recursive coefficients for UR and CMBGA in forecast regressions of inflationchanges on (a) unemployment changes and (b) industrial production growth. Forecast horizonh = 12 and significance level 1%. Estimation window length m = 60.

1.6. REFERENCES 27

1.6 References

Atkeson, A., Ohanian, L. E., 2001. Are phillips curves useful for forecasting inflation?

Federal Reserve Bank of Minneapolis Quarterly Review 25 (1), 2–11.

Breiman, L., 1996. Bagging predictors. Machine Learning 24, 123–140.

Bühlmann, P., Yu, B., 2002. Analyzing bagging. The Annals of Statistics 30 (4), 927–961.

Campbell, J. Y., Thompson, S. B., 2008. Predicting excess stock returns out of sample:

Can anything beat the historical average? Review of Financial Studies 21 (4), 1509–

1531.

Cecchetti, S. G., Chu, R. S., Steindel, C., 2000. The unreliability of inflation indicators.

Federal Reserve Bank of New York: Current Issues in Economics and Finance. 4 (6).

Cheung, Y.-W., Chinn, M. D., Pascual, A. G., 2005. Empirical exchange rate models

of the nineties: Are any fit to survive? Journal of International Money and Finance

24 (7), 1150–1175.

Clark, T. E., McCracken, M. W., 2012. In-sample tests of predictive ability: A new

approach. Journal of Econometrics 170 (1), 1–14.

Doornik, J. A., 2007. Object-Oriented Matrix Programming Using Ox, 3rd ed. Timber-

lake Consultants Press and Oxford: www.doornik.com., London.

Gordon, I. R., Hall, P., 2009. Estimating a parameter when it is known that the param-

eter exceeds a given value. Australian & New Zealand Journal of Statistics 51 (4),

449–460.

Hansen, P. R., Lunde, A., Nason, J. M., 3 2011. The model confidence set. Econometrica

79 (2), 453–497.

Hillebrand, E., Lee, T.-H., Medeiros, M. C., 2013. Bagging constrained equity premium

predictors. In: Haldrup, N., Meitz, M., Saikkonen, P. (Eds.), Essays in Nonlinear

Time Series Econometrics (Festschrift for Timo Teräsvirta). Oxford University Press

(forthcoming).

Hillebrand, E., Medeiros, M. C., 2010. The benefits of bagging for forecast models of

realized volatility. Econometric Reviews 29 (5-6), 571–593.

Inoue, A., Kilian, L., 2008. How useful is bagging in forecasting economic time series?

a case study of us consumer price inflation. Journal of the American Statistical

Association 103 (482), 511–522.

Meese, R. A., Rogoff, K., 1983. Empirical exchange rate models of the seventies: do

they fit out of sample? Journal of International Economics 14 (1), 3–24.


Newey, W. K., West, K. D., 1987. A simple, positive semi-definite, heteroskedasticity

and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–708.

Patton, A., Politis, D. N., White, H., 2009. Correction to "automatic block-length

selection for the dependent bootstrap" by d. politis and h. white. Econometric

Reviews 28 (4), 372–375.

Pettenuzzo, D., Timmermann, A., Valkanov, R., 2013. Forecasting stock returns under

economic constraints. CEPR Discussion Papers No. 9377.

Politis, D. N., White, H., 2004. Automatic block-length selection for the dependent

bootstrap. Econometric Reviews 23 (1), 53–70.

Rapach, D. E., Strauss, J. K., 2010. Bagging or combining (or both)? an analysis based

on forecasting us employment growth. Econometric Reviews 29 (5-6), 511–533.

Stock, J. H., Watson, M. W., 2007. Why has us inflation become harder to forecast?

Journal of Money, Credit and Banking 39 (s1), 3–33.

Stock, J. H., Watson, M. W., 2009. Phillips curve inflation forecasts. In: Fuhrer, J., Ko-

drzycki, Y., Little, J., Olivei, G. (Eds.), Understanding Inflation and the Implications

for Monetary Policy: A Phillips Curve Retrospective. MIT Press, pp. 99–186.

Stock, J. H., Watson, M. W., 2012. Generalized shrinkage methods for forecasting using

many predictors. Journal of Business & Economic Statistics 30 (4), 481–493.

1.7. APPENDIX 29

1.7 Appendix

1.7.1 Proof of Proposition 1.2

The proof follows Bühlmann and Yu (2002), Proposition 2.2. From Assumption 1.1

and βT = T −1/2bσ∞, we get

T 1/2σ−1∞,T βT = (σ∞,T /σ∞)×T 1/2σ−1

∞ βTd−→ Z +b.

For CMPT we have

T 1/2σ−1∞,T β

C MPTT = T 1/2σ−1

∞,T βT 1(T 1/2σ−1∞,T βT ≥ c)

+T 1/2σ−1∞,T T −1/2σ∞,T 1(T 1/2σ−1

∞,T βT < c)

= T 1/2σ−1∞,T βT 1(T 1/2σ−1

∞,T βT ≥ c)+1(T 1/2σ−1∞,T βT < c).

The expression on the last line is a function of T 1/2σ−1∞,T βT that is continuous except

for points of measure 0, therefore the continuous mapping theorem applies and we

get

T 1/2σ−1∞,T βT 1(T 1/2σ−1

∞,T βT ≥ c)+1(T 1/2σ−1∞,T βT < c)

d−→ (Z+b)1(Z+b ≥ c)+1(Z+b < c),

where Z is a standard normal random variable.

Next consider the bagged version,

T 1/2σ−1∞,T β

C MBGT = 1

B

B∑b=1

[T 1/2σ−1∞ β∗

b I(T 1/2σ−1∞,T β

∗b > c)

+T 1/2σ−1∞,T β

∗b T −1/2σ∞,T β

∗b I(T 1/2σ−1

∞,T β∗b ≤ c)],

= 1

B

B∑b=1

[T 1/2σ−1

∞,T β∗b I(T 1/2σ−1

∞,T β∗b > c)+ β∗

b I(T 1/2σ−1∞,T β

∗b ≤ c)

].

From Assumption 1.2, we get

T 1/2(β∗T − βT )

d∗−−→N (0,σ2

∞),

and thus

T 1/2σ−1∞,T (β∗

T − βT )d∗−−→N (0,1).

This can be expressed as

T 1/2σ−1∞,T βT

d−→ Z +b, Z ∼N (0,1),

T 1/2σ−1∞,T β

∗T

d∗−−→ W ∼ |Z N (Z +b,1),


where W ∼ |Z denotes the distribution of W conditional on Z . Then, again using

continuity almost everywhere of the estimator, we get

1

B

B∑b=1

[T 1/2σ−1

∞,T β∗b I(T 1/2σ−1

∞,T β∗b > c)+ I(T 1/2σ−1

∞,T β∗b ≤ c)

],

d∗−−→ EW

[W I(W > c)+ I(W ≤ c)|Z ]

,

= EW [W |Z ]−EW[W I(W ≤ c)|Z ]+EW

[I(W ≤ c)|Z ]

,

= Z +b −EW[W I(W ≤ c)|Z ]+Φ(c −Z −b).

Next, we use that for x ∼ N (m,1) we have (see Eqn. (6.3) in Bühlmann and Yu,

2002),

E[xI(x ≤ k)] = mΦ(k −m)−φ(k −m),

and thus

Z +b −EW[W I(W ≤ c)|Z ]+Φ(c −Z −b),

= Z +b − (Z +b)Φ(c −Z −b)+φ(c −Z −b)+1−Φ(Z +b − c),

= Z +b − (Z +b)(1−Φ(Z +b − c))+φ(c −Z −b)+1−Φ(Z +b − c),

= (Z +b)Φ(Z +b − c)+φ(Z +b − c)+1−Φ(Z +b − c),

which completes the proof.

1.7.2 Proof of Proposition 1.3

Let βA ∼N (βT ,T −1σ2∞,T ), the random variable sampled from the asymptotic distri-

bution of the OLS estimator for given βT and σ∞,T . First, we consider the asymptotic

version of the standard bagging estimator, BGA. By the same arguments as used in

the proof of Proposition 1.2 we get

βBG AT = E[βAI(T 1/2σ−1

∞,TβA > c)]

= β−E[βAI(T 1/2σ−1∞,TβA ≤ c)]

= βT −T −1/2σ∞,T E[T 1/2σ−1∞,TβAI(T 1/2σ−1

∞,TβA ≤ c)]

= βT − βTΦ(c −T 1/2σ−1∞,T βT )+T −1/2σ∞,Tφ(c −T 1/2σ−1

∞,T βT ).

With τT = T 1/2σ−1∞,T βT we get

βBG AT = β

[1−Φ(c − τT )

]+T −1/2σ∞,Tφ(c − τT ),

= β[

1−Φ(c − τT )+ τ−1T φ(τT − c)

].

1.7. APPENDIX 31

We proceed along the same lines for βC MBG AT :

βC MBG AT = E[βAI(T 1/2σ−1

∞,TβA > c)+T −1/2σ∞,T I(T 1/2σ−1∞,TβA ≤ c)]

= E[βAI(T 1/2σ−1∞,Tβ> c)]+E[T −1/2σ∞,T I(T 1/2σ−1

∞,TβA ≤ c)]

= βBG AT +T −1/2σ∞,T E[I(T 1/2σ−1

∞,TβA ≤ c)]

= βBG AT +T −1/2σ∞,TΦ(c − τT ),

which gives the desired result:

βC MBG AT = β

[1−Φ(c − τT )+ τ−1

T φ(c − τT )]+T −1/2σ∞,TΦ(c − τT ),

= β[

1−Φ(c − τT )+ τ−1T φ(c − τT )+ τ−1

T Φ(c − τT )]

.

CH

AP

TE

R

2RETURN PREDICTABILITY, MODEL

UNCERTAINTY, AND ROBUST INVESTMENT

Manuel Lukas


Abstract

Under stock return predictability, investment decisions are based on conditional

expectations of stock returns. The choice of appropriate predictor variables is, how-

ever, subject to great uncertainty. In this chapter, we use the model confidence set

approach to quantify uncertainty about expected utility from stock market invest-

ment, accounting for potential return predictability, over the sample period 1966:01–

2002:12. We find that confidence sets imply economically large and time-varying

uncertainty about expected utility from investment. We propose investment strate-

gies aimed at reducing the impact of model uncertainty. Reducing model uncertainty

requires lower investment in stocks, but the return predictability still leads to eco-

nomic gains for investors.

Keywords: Return predictability, Model uncertainty, Model confidence set, Portfolio

choice, Loss function.

33

34 CHAPTER 2. RETURN PREDICTABILITY, MODEL UNCERTAINTY, AND ROBUST INVESTMENT

2.1 Introduction

There is substantial disagreement regarding the relevant conditioning variables,

model specification, and economic significance of stock return predictability. The

large literature on return predictability documents that certain variables, for example

valuation ratios, help predicting stock market excess returns (see, e.g., Fama and

French, 1988; Barberis, 2000; Lewellen, 2004; Ang and Bekaert, 2007; Lettau and Lud-

vigson, 2010, among many other studies). Strongly supportive evidence of predictive

power mostly stems from in-sample analysis. Robustness, stability, and economic

significance of predictability is still disputed, as out-of-sample result are much less

conclusive (Timmermann, 2008). For example, Welch and Goyal (2008) find that

forecasts based on the historical average (HA) are not consistently outperformed by

a wide range of predictor variables in univariate predictive regression, and that the

performance of predictive regressions changes over time. In some periods certain

variables seem to predict excess returns, while, in other periods, return prediction

models perform poorly.

The evidence on return predictability is not only sensitive to whether we look

at in-sample or out-of-sample performance, but also to the measure by which re-

turn forecasts are evaluated (see, e.g., Pesaran and Timmermann, 1995). Kandel and

Stambaugh (1996) use an economic measure based on the real-time performance of

an investor, which provides a more relevant performance measure than statistical

criteria. Cenesizoglu and Timmermann (2012) document that statistical measures

are not very informative about the performance with economic measures.

Several empirical studies accounted for model uncertainty, rather than investigat-

ing return predictability for single model specifications. Cremers (2002) documents

that even when taking model uncertainty into account by Bayesian model averaging,

return prediction models are superior to unconditional forecasts. Using Bayesian

model averaging followed by optimal investment within the average model, Avramov

(2002) finds that the Bayesian investor successfully uses return prediction models

for portfolio choice. Wachter and Warusawitharana (2009) consider a Bayesian in-

vestor who puts low prior probability on return predictability. Even though this

investor is skeptical about return predictability, the predictive content in the data is

strong enough to influence investment decisions. Dangl and Halling (2012) consider

a Bayesian investor who averages over time-varying coefficients models and find

robust economic gains from return predictability both during recessions and expan-

sions. Aiolfi and Favero (2005) document that asset allocation based on multiple

models, rather than a single model, can increase investors’ utility. Using forecast

combination, Rapach, Strauss, and Zhou (2010) find that the historical average can

be significantly outperformed, even when the individual forecasts perform poorly.

Overall, there is evidence that return prediction models can benefit investors,

even when the investment decision takes model uncertainty into account. Investment

strategies based on multiple models are able to increase the unconditional expected

2.1. INTRODUCTION 35

utility, as measured by the average over many sequential investment decisions. Model

uncertainty induces uncertainty about the conditional expectation of utility. Previous

approaches do not investigate and measure this uncertainty. In particular, it is ignored

how an investment strategy would perform under other reasonable return models.

In this chapter, we use the model confidence set approach of Hansen et al. (2011)

to quantify the uncertainty stemming from potential return predictability. In particu-

lar, we construct confidence sets for the expected utility from investment based on

the models in the model confidence set. For this, we consider a small investor with

CRRA utility, who allocates wealth to stocks and the risk-free asset. The confidence

sets contain expected utility under the return models that are not rejected by the data

for a given confidence level. Return predictability implies that expected utilities, and

thus the confidence sets, depend on the predictor variables of the return prediction

models. First, we construct such confidence sets for a standard investor who does

not use a return prediction model, but relies on the historical average (HA) of returns

to estimate expected returns. Second, we consider investment strategies that are

designed to reduce uncertainty about expected utility for a given set of models. A

robust strategy is proposed for which the investor chooses stock investment such that

the minimal element of the confidence set is maximized. This corresponds to maxi-

mizing the lowest expected utility over all models in the confidence set. Additionally,

we consider two less conservative investment strategies; one based on averaging and

one based on the majority forecasts along the lines of Aiolfi and Favero (2005) that

also take into account the model uncertainty as measured by the model confidence

set.

The methodology described above is applied to monthly returns on the US stock

market for 1945:12–2002:12. The potential predictors are 14 variables from the popu-

lar data set of Welch and Goyal (2008). Each of the variables is used individually in a

simple regression model. Additionally we consider multivariate prediction strategies

based on principal components and on the complete subset regression of Elliott,

Gargano, and Timmermann (2013).

For this model universe of 23 models, the model uncertainty is substantial: In the

beginning of the out-of-sample period 1966–2002, no model can be excluded from

the model confidence set in real-time for common confidence levels. The large model

uncertainty translates into a large economic uncertainty regarding expected utility.

As we move further on in the sample, models start getting excluded from the model

confidence set and the uncertainty regarding the expected utility is reduced. The mag-

nitude of uncertainty, measured by width of confidence sets, changes significantly

with the predictor variables. The robust investment strategy leads to investments that

are much lower than for the HA model, in particular in the first half of our sample. All

of the proposed investment strategies lead to economic out-of-sample gains from

return prediction. There are gain from return prediction both during recessions and

expansions, but during recession the gains are substantially higher.


Our findings add to the literature on model uncertainty in stock return prediction.

The economic significance of uncertainty about conditional expected utility under

different return predictions models has not been documented before. Our approach

reveals that model uncertainty translates into substantial uncertainty about expected

utility from investing in stocks, which varies over time and becomes less pronounced

later in the sample. We show that, for the universe of models considered in this

chapter, it is possible for investors to benefit from return predictability while reducing

the model uncertainty.

The remainder of the chapter is structured as follows: In section 2.2 we present

the investment problem, the econometric approach for constructing confidence sets,

and the confidence set based investment strategies. Section 2.3 discusses data and

models used in the empirical analysis. Section 2.4 presents the empirical results.

Concluding remarks are given in section 2.5.

2.2 Investment and Confidence Sets

This section sets up the investment problem, presents the econometric methodology

for model confidence set construction, and presents the investment strategies based

on the model confidence set.

2.2.1 The Investment Problem

We study the real-time investment decisions of an small investor in the spirit of Kandel

and Stambaugh (1996). The investor faces a one-period portfolio selection problem

with a monthly horizon. The return on the risk-free asset is r ft+1 and the excess return

on stocks, the risky asset, is rt+1. Returns are continuously compounded. At time t

the risk-free rate r ft+1 is known, while the excess stock return rt+1 is uncertain with

rt+1|t ∼ N (µt+1,σ2t+1). The investor has initial wealth of 1 to invest at every time t .

At time t the investor has to decide what share of wealth θt to invest in stocks. The

remaining wealth 1−θt is held in the risk-free asset. The investor’s final wealth at

time t +1 is

Wt+1 = θt exp(r ft+1 + rt+1)+ (1−θt )exp(r f

t+1). (2.1)

The investor’s utility for wealth level W is given by constant relative risk aversion

(CRRA) utility,

Uγ(W ) = W 1−γ

1−γ , (2.2)

with constant relative risk aversion coefficient γ> 1. As a function of investment and

excess return, the utility is

Uγ(θt ,rt+1) = 1

1−γ (θt exp(r ft+1 + rt+1)+ (1−θt )exp(r f

t+1))1−γ. (2.3)

2.2. INVESTMENT AND CONFIDENCE SETS 37

In order to calculate expected utilities, the investor needs a model for rt+1. Given a

model of conditional returns, the investor can maximize expected utility. The expected

utility from investing θt in stocks is

Et

[Uγ(θt ,rt+1)

]= 1

1−γEt

[(θt exp(r f

t+1 + rt+1)+ (1−θt )exp(r ft+1))1−γ

]. (2.4)

The investor maximizes his expected utility in period t by investing

θt = argmaxθ∈[0,1.5]

Et

[Uγ(θ,rt+1)

], (2.5)

where we impose the standard restriction θt ∈ [0,1.5] of Campbell and Thompson

(2008) throughout our analysis.

The optimal portfolio in (2.5) requires a model for the conditional distribution

of returns. Given the high uncertainty regarding model specification of return pre-

dictability, the investor might be unable to specify a unique conditional model for

returns. The major challenge is to specify a model for the conditional mean. For some

conditioning set Di ,t , this conditional mean is given by

µi ,t+1 = Ei ,t [rt+1] = E[rt+1|Di ,t ], (2.6)

where i = 1, . . . ,I is one of I possible conditioning sets. The uncertainty regarding the

choice of Di ,t entails uncertainty regarding the expected utility (2.4) and thus the

optimal investment in (2.5).

Given the large uncertainty about the conditioning variables Di ,t , we want to

address the question whether return predictability can be exploited when the investor

is not willing to choose a particular set of conditioning variables, but rather maintains

multiple reasonable conditioning sets. We study investment strategies that work well

on different conditioning sets in the manner of Aiolfi and Favero (2005). The model

confidence set is used to identify reasonable conditioning sets, e.g., return models

that are not rejected by the data. Thus, we are interested in the conditional expected

utility for different conditioning variables, i.e.,

E[Uγ(θ,rt+1)|Di ,t ] for i = 1, . . . ,I . (2.7)

In the following, we measure the variation of expected utility over different reasonable

conditioning sets, and investigate the performance of strategies that are based on the

conditional expected utility for return models.

2.2.2 Expected Utility Confidence Sets

Assume there is a set Mt = {1, . . . ,m} of potential return prediction models, including

the unconditional HA model. Every model specifies a conditional density for return

rt+1, and thus a conditional expectation. Let Et ,i be the conditional expectation under


model i ∈Mt . For such a set of models Mt , we construct the model confidence set

(MCS) at every time t . We denote the MCS by M∗t , and by m∗

t = #M∗t the number of

models in the MCS at time t .

Loosely speaking, the MCS of Hansen et al. (2011) is a subset of the models,

M∗t ⊆Mt , which contains the best model with 1−α confidence. The best model is the

one with highest expected utility in our setting, or equivalently the lowest expected

loss, where loss is defined as the negative of the utility. The MCS is constructed

using past observation on outcomes (returns) and past predictions (in our case,

optimal investments) from all models in Mt . The confidence level 1−α controls

how strong the statistical evidence against a model needs to be in order to exclude it

from the MCS. The MCS approach captures statistical model uncertainty. The harder

it is to identify the best model, the more models are included in the MCS. If one

model performs significantly better than all its competitors, then it becomes the only

element of M∗t . Details on the implementation of the MCS approach are given in

section 2.3.3.

Based on model confidence set M∗t , we can construct a confidence set for ex-

pected utility from investment θt as

Ct (θt ,M∗t ) = {Et ,i [Uγ(θt ,rt+1)] : i ∈M∗

t }. (2.8)

The confidence set Ct (θt ,M∗t ) is a measure of uncertainty about expected utility for

investment θt . It contains the expected utility for all models that cannot be excluded

from the model confidence set. As the expected utility depends on investment θt , so

does the confidence set.

For easier interpretation, we transform expected utilities to the corresponding

certainty equivalent returns. The certainty equivalent return (CER) under model i at

time t for investment θt is

C ERi ,t (θt ) =((1−γ)Et ,i [Uγ(θt ,rt+1)]

)1/(1−γ) −1. (2.9)

Calculating the CER for all elements of Ct (θt ,M∗t ), we get a time t confidence set for

the CER of investment θt . In the following we focus on the CER confidence range,

defined as the highest and the lowest CER of all models in the model confidence set.

The confidence set and the confidence range presented above are tools to quantify

uncertainty regarding expected utility associated with a certain investment strategy. It

allows us to quantify uncertainty for a standard investor who uses the historical mean

to guide his investment decision. Beyond this, we are interested in characterizing

investment for which the uncertainty is lower, in a way that we shall discuss in the

next section.

2.2.3 Robust Investment

The confidence sets for expected utility are a function of investment θt . We use this

to explore how the investor needs to set investment in order to reduce uncertainty

2.2. INVESTMENT AND CONFIDENCE SETS 39

about expected utility. Specifically, we construct a robust investment strategy that can

provide non-negative certainty equivalent returns for all conditional expectations

that are not rejected by the data, i.e., for all return prediction models in the model

confidence set. This robust investment is constructed by maximizing the lowest

expected utility over all models in the confidence set. In terms of CER confidence set,

the robust investment θRt ∈ [0,1.5] is given by

θRt = argmax

θ∈[0,1.5]

(minCt (θ,M∗

t ))

. (2.10)

The robust investment leads to non-negative certainty equivalent returns under all

modes in the MCS.

The robust investment in equation (2.10) is a special version of maxmin invest-

ment. Maxmin investment rules have drawn some attention in the portfolio choice

literature (see, e.g., Epstein and Wang, 1994; Maenhout, 2004; Garlappi, Uppal, and

Wang, 2007). Maxmin rules reflect an extreme attitude toward model uncertainty,

i.e., they reflect model uncertainty aversion (see, e.g., Gilboa and Schmeidler, 1989;

Hansen and Sargent, 2001). Our robust investment strategy applies the maxmin rule

over the model confidence set, such that it can be interpreted as an investor who is

averse to uncertainty over the set of models that are not rejected by the data.

The robust investment (2.10) is very conservative. As a less conservative alterna-

tive that takes into account all models in the confidence set, we consider maximizing

average utility over the elements of the confidence set, i.e.,

θAv gt = argmax

θ∈[0,1.5]

1

m∗t

∑i∈M∗

t

Et ,i [Uγ(θt ,rt+1)]

. (2.11)

The averaging (avg) investment θAv gt does not ensure any properties of the expected

utility for the individual model in the confidence set. The number of models over

which the average is taken depends on the model confidence set.

As a third investment strategy based on the model confidence set, we use a major-

ity strategy similar to the investment strategies considered in Aiolfi and Favero (2005).

Let nh,t be the number of models in the MCS for which the time t optimal investment

θi ,t is higher than θH At , and nh,t the number of models with lower investment than

for the HA model. The majority investment is then given by

θMt =

1

nh,t

∑i∈M∗

tθi ,t 1(θi ,t > θH A

t ) for nh,t > nl ,t ,1

nl ,t

∑i∈M∗

tθi ,t 1(θi ,t < θH A

t ) for nh,t < nl ,t ,

θH At for nh,t = nl ,t ,

(2.12)

where 1(.) is the indicator function. This investment rule only invests below or above

the HA investment, if the majority of models in the MCS imply such an investment

decision. In contrast to the robust and averaging investment strategies, the majority

investment is a function of the investment decision of the models in the confidence

set rather than their conditional distributions.


2.3 Models and Data

This section discusses data, forecasting methods, estimation, and model confidence

set construction.

2.3.1 Variables and Data

A major contribution to the uncertainty in return predictability stems from uncer-

tainty regarding which variables should be used as predictors. The benchmark model,

over which the investor wishes to improve expected utility, is the unconditional HA

model:

• Using no predictor variables gives the historical average (HA) model, which

specifies expected excess returns as a constant.

We consider predictors from the popular data set1 of Welch and Goyal (2008). Stock

returns are calculated from Center for Research in Security Prices (CRSP) data on the

S&P 500 index. We follow Welch and Goyal (2008) in the construction of the variables

from this data set. The 14 variables can be roughly grouped in four categories.

Predictor variables describing the state of the financial market are:

• long-term rate of return (ltr),

• the variance of stock returns computed from daily returns (vars),

• and the cross-section beta premium (csp) of Polk, Thompson, and Vuolteenaho

(2006).

The bond market and macroeconomic conditions are captured by the predictors:

• default yield spreads (dfy) measured by yield difference between AAA and

BAA-rated corporate bonds,

• term spread between long-term bond and Treasury bill yields (tms),

• default return spread (dfr) between long-term corporate bonds and long term

government bonds,

• long-term yields (ltr),

• and inflation (inf ).

The valuations ratios considered are:

• dividend-price ratio (dp),

• dividend yield (dy),

1The data set is available from Amit Goyal’s homepage http://www.hec.unil.ch/agoyal/.

http://www.hec.unil.ch/agoyal/

2.3. MODELS AND DATA 41

• 10-year moving average of earnings-price ratio (ep10),

• and the book-to-market ratio (bm).

Finally, we consider:

• dividend-earnings ratio (d/e) and

• ratio of 12-month net equity issues over end-of-year market capitalization

(ntis),

as predictors related to corporate finance decisions.

2.3.2 Return Prediction Models and Forecasts

In this next section we discuss the estimation and forecasting approach taken to

model the conditional distribution of returns.

For the conditional mean, linear models are considered. For each of the 14 vari-

ables, xv , v = 1, . . . ,14, we estimate a univariate regression model,

rt+1 = cv +β′v xv,t +εv,t+1, (2.13)

where cv is the intercept, βv is the slope parameter, and εv,t+1 are zero mean error

terms. For the HA model, equation (2.13) only features a constant and no predictors.

Using data up to time t , we get the least-squares estimates cv,t and βv,t using an

expanding estimation window. From this estimated model, we get the conditional

mean forecast,

µv,t+1 = cv,t + βRv,t xv,t , (2.14)

for the univariate predictive regression with variable xv . For the individual predic-

tors, we use the sign restrictions of Campbell and Thompson (2008) on the slope

coefficients, such that forecasts are obtained for βRv,t = max(0,βv,t ), where βv,t is the

unrestricted estimate.

In addition to the individual predictors, we consider return models that use the

information in all the predictors. In the context of return prediction, multivariate

least-squares regression is known to produce noisy estimates, and very poor out-of-

sample performance (see, e.g., the kitchen sink model in Welch and Goyal, 2008). To

make such models produce accurate forecasts, we need to reduce the dimensionality

or the estimation variance.

We apply two multivariate approaches. First, we use a linear principal compo-

nents model. The first 1 to 3 principal components, extracted from all 14 predictors,

are used as variables in a regression model. The resulting models are called PC1, PC2,

and PC3, and are estimated by least-squares. Second, we use the complete subset

regression of Elliott et al. (2013), which is a forecast combination approach that has

been shown to be successful in stock return prediction. The complete subset regres-

sion combines all possible models that contain k of the 14 predictors. We consider


k = 1, . . . ,5 with corresponding models labelled CSR1, . . . , CSR5. For k = 1, the com-

plete subset regression corresponds to the combination of univariate regressions

applied in Rapach et al. (2010).

The univariate regressions, together with the principal components and complete

subset regression models, give us a total of 23 candidate models. For all models, we

impose a non-negative equity premium forecast, such that the final forecast for the

conditional mean is max(0,µi ,t+1) where µi ,t+1 is the forecast from model i .

The conditional variances σ2t+1 are computed from model-based residuals using

a ten year rolling window of monthly returns. We are mainly concerned with model

uncertainty, i.e., which variables to condition on, but it is also relevant to account for

parameter estimation uncertainty. We account for estimation uncertainty by adjust-

ing the conditional variance for uncertainty about the coefficients of the conditional

mean model. The adjusted conditional variance is given by

Var(rt+1|Di ,t ) = E[Var{rt+1|ci ,βi ,Di ,t }|Di ,t ]+Var{E[rt+1|ci ,βi ,Di ,t ]|Di ,t }, (2.15)

where the first term is estimated from the model-based residual, and the second term

captures the estimation variance, see Pástor and Stambaugh (2012) for a detailed

discussion. An estimate of this estimation variance is obtained from the asymptotic

covariance matrix of the parameters in each of the considered conditional mean

models.

2.3.3 Model Confidence Set Construction

Next, we discuss the exact implementation of the model confidence set (MCS) pro-

cedure of Hansen et al. (2011) used in this chapter. The MCS is a subset of Mt that

contains the best model with 1−α confidence level. The best model is the one with

lowest expected loss for a given loss function. The MCS at time t is denoted by M∗t ,

suppressing the dependence on the confidence level 1−α. To construct M∗t , a sample

of E losses up to time t for each model in Mt is needed. The MCS algorithm uses

sequential testing of equal predictive ability. At every step of the sequential testing,

critical values are obtained using a moving-block bootstrap, and one model is elim-

inated from the confidence set until the null hypothesis of equal predictive ability

(EPA) is not rejected. The tests for EPA are based on the max statistic and the max

elimination rule is used in each elimination step. This statistic and elimination rule

are based on the maximum average t-statistic, where for each model the average is

taken over the t-statistics from all pairwise loss differentials, see Hansen et al. (2011)

for details. The critical values are obtained from 1999 bootstrap replications using

a block bootstrap.2 Based on this sequential testing, a p-value for each model is

2The np package (see Hayfield and Racine, 2008) for R (see R Core Team, 2013) is used to find theoptimal block length for the block bootstrap using the methods of Politis and White (2004) and Patton et al.(2009).

2.4. EMPIRICAL RESULTS 43

obtained. These p-values tell us whether a certain model is member of the MCS for a

given confidence level.

The investor’s relevant loss function, here taken as the negative of his realized

CRRA utility, is used to obtain the sample of E losses for each model. Results from

forecast comparison for models of financial returns and volatility depend on the

loss function and can, e.g., differ between utility-based and statistical loss functions

(see, e.g., West, Edison, and Cho, 1993; Gonzàlez-Rivera, Lee, and Mishra, 2004;

Skouras, 2007; Cenesizoglu and Timmermann, 2012). The investor’s realized losses

are based on forecasts, and thus cannot be computed from the beginning of the

available sample. We therefore reserve the first M observations for initial parameter

estimation, such that when we have a sample of N observations at time t , the MCS is

based on E = N −M losses:

M︷︸︸︷t −N +1, . . . ,t −E ,

E︷︸︸︷t −E +1, . . . ,t −1,t︸︷︷︸

N

.

Later in the sample, more data are available to construct the MCS. A larger sample

will give the MCS more power to exclude models. If, however, the performance of

models varies over time, having a longer history of past losses is not necessarily more

informative regarding expected performance.

2.4 Empirical Results

Our sample spans over the period 1946:1 to 2002:12 (N = 684). All variables are at

monthly frequency. The first 120 observations are reserved for initial estimation (M =120). Another 120 observations are used for construction of the first model confidence

set. The model confidence sets are constructed using an expanding window. Thus,

the out-of-sample period, for which we observe investments and confidence sets, is

1966:01–2002:12 (444 observations). All results in this section are for an investor with

risk aversion parameter γ= 5.

Return predictability appears to interact strongly which the business cycle. There

is evidence that expected excess returns are higher during recessions (see, e.g., Fama

and French, 1989; Henkel, Martin, and Nardari, 2011). We therefore identify NBER

recessions in the results.

Before looking at the confidence sets, we evaluate the performance of the 23 re-

turn prediction models in our model universe. For this purpose, the out-of-sample R2

( OOS-R2), Sharpe ratio (SR), and certainty equivalent return relative to the historical

average investment (∆C ERi ) are computed for each return prediction model. For


model i , the OOS-R2 is defined as:

R2i = 1−

1S

S∑t=1

(µi ,t − rt )2

1S

S∑t=1

(µH At − rt )2

, (2.16)

where µH At is the historical average and S = 564 is the number of out-of-sample

observations. The OOS-R2 measures the statistical accuracy of the conditional mean

forecast. The Sharpe ratios are calculated from the realized excess return for the

optimal investment for each model. The ∆C ERi is given by:

∆C ERi =(

(1−γ)1

S

S∑t=1

Uγ(θi ,t ,rt+1)

)1/(1−γ)

−(

(1−γ)1

S

S∑t=1

Uγ(θH At ,rt+1)

)1/(1−γ)

,

(2.17)

where θH At is the time t investment based on the historical average and θi ,t is the

investment for model i . Return prediction models that lead to economic gains for

investors have a positive ∆C ERi .

Table 2.1 summarizes the forecasting performance for the individual predictors

and the multivariate forecasting strategies. We see that many individual predictors

perform poorly out-of-sample in statistical terms, but most have higher certainty

equivalent returns than the HA model. In terms of certainty equivalent returns, the in-

vestments based on the complete subset regressions perform best. These models also

achieve the highest out-of-sample R2. The HA investment is rejected by the model

confidence, such that we conclude that return predictability leads to improvements

in the unconditional expected utility for investors. The variables csp, tms, and infl

are the only univariate models that are in the model confidence set for the 90% con-

fidence level. Five models have a negative out-of-sample R2, but a higher certainty

equivalent return than the historical average model. Such a result is not unusual, as

Cenesizoglu and Timmermann (2012) find that statistical and economic measures of

returns predictability are only very weakly correlated.

2.4.1 Model Confidence Sets and Investment

First we look at the evidence of real-time model uncertainty by computing series

of model confidence sets for different confidence levels 1−α. For every month in

the out-of-sample period, Figures 2.1 to 2.3 show which models are included in

the MCS. For α= 0.05, and thus a confidence level of 0.95, we find that the model

confidence sets change substantially over the sample 1966:01–2002:12. Up to the

mid-1970s, all 23 models are included in the model confidence set every month.

Subsequently, some models, notably the HA model, are excluded from the MCS. In

the early 1980s, more than half of all models are excluded. After 1995, a number of

models make a reappearance in the MCS. The time variation in the MCS can either be


Table 2.1. Out-of-sample performance of the historical average (HA) model, the 14 predictorvariables, the complete subset regresssions (CSR1, . . . , CSR5), and the principal componentmodels (PC1, PC2, and PC3). Columns 2 to 5 show mean, minimum, maximum and varianceof the conditional mean forecasts. Out-of-sample R2 (OOS-R2) and certainty equivalent returndifference to HA investment (∆CER) are reported in percentage points for an investor with riskaversion γ= 5. Sharpe ratios (SR) are calculated based on excess returns. Model confidenceset p-values (MCS-p) are based on the CRRA loss function in (2.2) with γ= 5. Sample period1956:01–2002:12 (564 observations).

mean max min var OOS-R2 ∆CER SR MCS-pHA 0.655 1.193 0.416 0.037 0.000 - 0.087 0.006dp 0.279 1.215 0.000 0.121 0.700 0.069 0.077 0.030dy 0.278 1.325 0.000 0.126 0.739 0.103 0.084 0.030ep10 0.568 2.400 0.014 0.202 -0.824 0.072 0.087 0.030bm 0.412 1.544 0.141 0.053 0.258 0.108 0.085 0.030ltr 0.788 5.717 0.000 0.525 -0.724 0.167 0.131 0.064svar 0.693 1.510 0.424 0.027 -0.139 0.031 0.099 0.030csp 0.290 1.162 0.000 0.079 0.716 0.168 0.100 0.182ntis 0.827 2.223 0.000 0.221 -0.062 -0.003 0.113 0.030de 1.062 2.143 0.464 0.107 -2.048 -0.254 0.098 0.030dfy 0.796 2.420 0.293 0.138 -0.628 -0.022 0.097 0.030tms 0.854 2.751 0.000 0.391 -0.112 0.127 0.141 0.275infl 0.652 1.837 0.000 0.160 1.261 0.239 0.146 0.275lty 0.283 1.006 0.000 0.070 0.695 0.162 0.100 0.030dfr 0.651 2.683 0.000 0.138 -0.721 0.000 0.086 0.030CSR1 0.527 1.277 0.018 0.034 1.121 0.208 0.127 0.030CSR2 0.456 1.550 0.000 0.059 1.847 0.313 0.155 0.363CSR3 0.421 1.797 0.000 0.102 2.223 0.362 0.168 0.804CSR4 0.412 2.022 0.000 0.157 2.439 0.373 0.171 1.000CSR5 0.418 2.455 0.000 0.224 2.494 0.370 0.169 0.834PC1 0.270 0.759 0.000 0.032 0.700 0.167 0.100 0.030PC2 0.304 1.442 0.000 0.099 0.284 0.124 0.082 0.030PC3 0.513 4.170 0.000 0.446 -1.152 0.142 0.106 0.030

due to increased power as the sample size grows, or be caused by changes in relative

performance of the models. The fact that models, that have been previously excluded,

later reenter the MCS is an indication of time-variation in predictive content of the

variables. Alternatively, higher variance in the loss series might cause the MCS to

retain more models.

For the lower confidence level of 0.90 in Figure 2.2, the MCS contains fewer

models by construction. The variation over time is very similar to the 0.95 confidence

level, and the MCS contains a similar number of models most of the time. For a 0.99

confidence level in Figure 2.3, the MCS retains a much larger number of models in all


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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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HAdpdy

ep10bm

ltrsvarcspntisdedfytmsinflltydfr

CSR1CSR2CSR3CSR4CSR5

PC1PC2PC3

1970 1980 1990 2000

Figure 2.1. Inclusion in model confidence set (MCS) forα= 0.05. A dot indicates that the modelis included in the real-time MCS for this month. Loss is based on risk aversion of γ= 5. Dashedred lines indicate start- and end-dates of NBER recessions. Sample period is 1966:1–2002:12.

months. Only the HA model is consistently excluded after the early 1980s.

The model confidence sets suggest that real-time model uncertainty is high, and

that investors cannot identify the single best model based on past performance, as

the MCS always contains more than one model. However, the HA model is excluded

from the MCS in that latter part of the sample for all three α. Thus, while there is

statistical uncertainty about the best model, the evidence for return predictability is

rather strong.


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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

HAdpdy

ep10bm



PC1PC2PC3

1970 1980 1990 2000



Figure 2.4 shows the series of investments in stocks for the HA model and the

three investment strategies based on the model confidence set. The HA investment

series is most stable. In the beginning of the sample, the upper limit on investment

is binding. The robust investment is substantially lower than the HA investment

and flat throughout most of the first half of the sample. The averaging and majority

investment always allocate a significant share of wealth to stocks, and both series

follow on similar paths.

Figure 2.5 shows confidence ranges for certainty equivalent returns that are con-

structed from model confidence sets and investment series, i.e., the CER confidence

range is spanned by the lowest and highest certainty equivalent returns from models

in the model confidence sets. For HA investment, the width of the confidence range

varies over time. In the beginning of the sample, the lower bound is below the risk-

free rate. In the second half of the sample, the confidence ranges become narrower,

and the lowest CER occasionally lies above the risk-free rate. The CER confidence

ranges for the robust investment look quite differently, reflecting the behavior of the

robust investment series. In the beginning of the sample, the robust investment rule

allocates a very small share of wealth to stocks, such that the CER confidence ranges

are narrow. When the model uncertainty is reduced and the robust investment rule

leads to higher stock holdings, the width of CER confidence range increases, but

by construction the lowest element is never below the risk-free rate. The evolution

of the CER confidence ranges for the averaging and majority investment rules are

qualitatively similar to the one for HA investment, and both become narrower in the

second half of the sample.

Table 2.2 presents of the out-of-sample performance, as measured by Sharpe ratio

and certainty equivalent return, for the investment strategies based on the model

confidence sets and the multivariate models. Over the full sample, all investment

strategies outperform the historical average model both in terms of Sharpe ration and

CER. Thus, we find strong evidence that return predictability benefits investors even

after taking model uncertainty into account. Among our MCS-based strategies, the

majority investment performs best for all three choices of α. The CER improvements

in investment performance are present both in recessions and outside of recessions,

but are substantially larger in magnitude during recessions.

To test whether the investment strategies significantly outperform the HA model,

we perform pair-wise Diebold and Mariano (1995) tests using the CRRA utility as

loss function. The Bonferroni correction is applied to conservatively account for

distortions from multiple testing. The averaging and majority investment rules sig-

nificantly outperform the HA model, while the gains for the robust investment rule

are not significant at the 10% level. For the two subsamples, we only find statistically

significant outperformance of the averaging and majority investment rules during

recessions.

The empirical findings can be summarized as follows. The model confidence sets


(a) HA investment:

0.50

0.75

1.00

1.25

1.50

1970 1980 1990 2000

(b) MCS robust investment:

0.25

0.50

0.75

1.00

1970 1980 1990 2000

(c) MCS averaging investment:

0.4

0.8

1.2

1970 1980 1990 2000

(d) MCS majority investment:

0.4

0.8

1.2

1970 1980 1990 2000

Figure 2.4. Optimal investments in stocks, θt , for model historical average (HA) model, ro-bust investment, averaging investment, and majority investment. Risk aversion γ = 5 andmodel confidence sets with α= 0.05. Dashed red lines indicate start- and end-dates of NBERrecessions.


Table 2.2. Sharpe ratios (SR) and certainty equivalent return relative to the historical averageinvestment (∆C ER). p-value of Diebold-Mariano test (p) for expected utility equal to HAinvestment. Rejection of the null hypothesis after Bonferroni correction are indicates as: * for10% level, ** for 5% level, and *** for 1% level. Sample period 1966:01–2002:12 with total of 444observations, of which 65 are during recessions.

Full Sample Recession No Recession

α SR ∆C ER p SR ∆C ER p SR ∆C ER pHA - 0.048 - - −0.097 - - 0.080 - -Robust 0.01 0.083 0.212 0.216 −0.024 0.917 0.085 0.107 0.089 0.617Robust 0.05 0.076 0.232 0.156 0.072 0.988 0.057 0.077 0.100 0.554Robust 0.10 0.060 0.213 0.187 0.036 0.946 0.064 0.065 0.086 0.609Avg 0.01 0.115 0.291 0.000*** 0.064 0.804 0.002** 0.128 0.202 0.011Avg 0.05 0.122 0.298 0.000*** 0.101 0.939 0.001** 0.127 0.186 0.031Avg 0.10 0.119 0.291 0.001** 0.121 1.031 0.001** 0.120 0.162 0.075Majority 0.01 0.133 0.364 0.002** 0.155 1.205 0.002** 0.129 0.219 0.063Majority 0.05 0.135 0.369 0.002** 0.188 1.340 0.001** 0.124 0.202 0.094Majority 0.10 0.136 0.371 0.002** 0.214 1.446 0.000*** 0.119 0.187 0.129PC1 - 0.073 0.228 0.047 −0.013 0.846 0.043 0.087 0.120 0.288PC2 - 0.058 0.193 0.155 −0.053 0.749 0.076 0.076 0.096 0.498PC3 - 0.087 0.190 0.185 0.061 0.827 0.068 0.092 0.079 0.596CSR1 - 0.108 0.271 0.000*** 0.057 0.854 0.005* 0.118 0.169 0.013CSR2 - 0.147 0.402 0.000*** 0.190 1.328 0.002** 0.139 0.242 0.023CSR3 - 0.165 0.463 0.001*** 0.244 1.553 0.002** 0.148 0.275 0.042CSR4 - 0.168 0.478 0.002** 0.252 1.609 0.002** 0.150 0.284 0.064CSR5 - 0.168 0.482 0.002** 0.254 1.645 0.002** 0.149 0.283 0.080

vary substantially over time and always contain multiple models. The uncertainty,

measured by width of the confidence range, is substantial in economic terms and

varies over time. Confidence ranges for expected utility for HA investment frequently

contain certainty equivalent returns below the risk-free rate. It is, however, possible to

devise investment strategies that reduce model uncertainty and still lead to economic

gains from return predictability.


(a) CER confidence range for HA investment:

0

1

2

3

4

1970 1980 1990 2000

(b) CER confidence range for robust investment:

0.5

1.0

1.5

2.0

1970 1980 1990 2000

(c) CER confidence range for averaging investment:

0

2

4

6

8

1970 1980 1990 2000

(d) CER confidence range for majority investment:

0.0

2.5

5.0

7.5

1970 1980 1990 2000

Lowest CER Highest CER Risk free Rate

Figure 2.5. Certainty equivalent returns (CER) confidence ranges forα= 0.05. CER and risk-freerate in percentage returns.


2.4.2 Impact of Risk Aversion

To assess the sensitivity to changes in the specification of the investor, we repeat the

empirical analysis for different values of the risk aversion parameter γ. Changing γ

affects the optimal investment, realized excess returns, and thus the realized utilities.

As a consequence, the MCS and CER confidence ranges can be affected trough

changes in the loss series.

In Figure 2.6, the results for a less risk averse investor withγ= 2 are presented. This

change of risk aversion causes dramatic changes in the results. As we see from Panel

(a), the MCS includes all the models in every month in our sample. Panel (b) gives

insight as to why this happens. The lower risk aversion increases optimal investments

to the extent that the upper bound of holding 150% of wealth in stocks is binding

throughout large parts of the sample. This is true also for the other models, and

therefore the investment decision is identical for many models for most observations.

Identical investment leads to identical loss, such that the model confidence sets

cannot distinguish between the models. The increased statistical model uncertainty

leads to CER confidence range that always include a CER below the risk-free rate in

Panel (c).

For a higher risk aversion of γ= 10, shown in Figure 2.7, the model confidence

sets remain similar to the ones for γ= 5. In Panel (b) we see that the constraints of

the investment are never binding. The dynamics of the CER confidence ranges for

HA investment do not change qualitatively.

Table 2.3 summarizes investment performance under the two alternative risk

aversion coefficients γ = 2 and γ = 10. The increased uncertainty for γ = 2 has a

strong effect on the performance of the robust investment, while the remaining

investment strategies are not effected so dramatically. Because of the high model

uncertainty for γ= 2, the robust investment strategy is not able to produce significant

economic gains from return prediction. For γ= 10, where the model confidence sets

remain similar to γ= 5, economic gains are observed for all the investment strategies

based on the MCS. Thus, being able to narrow down the set of models using the

model confidence set appears to be a crucial ingredient for the success of the robust

investment strategy.

From changing the risk aversion, we have learned that the results change if con-

straints are binding frequently, because it makes it impossible to distinguish different

return prediction models in terms of economic investment performance. When no

such effects are present, which is the case when we increase risk aversion to 10, the

findings remain qualitatively unchanged.


(a) Model inclusion:

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HAdpdy

ep10bm



PC1PC2PC3

1970 1980 1990 2000

(b) HA investments:

1.2

1.3

1.4

1.5

1970 1980 1990 2000

(c) CER confidence ranges:

0.0

2.5

5.0

7.5

1970 1980 1990 2000


Figure 2.6. Results for lower risk aversion, γ = 2. The panels show (a) inclusion in modelconfidence set for α= 0.05, (b) investment based on historical average (HA) , and (c) lowestand highest element of CER confidence set (CER confidence ranges) along with risk-free ratein percentage returns. Dashed red lines indicate start- and end-dates of NBER recessions.


(a) Model inclusion:

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●

HAdpdy

ep10bm



PC1PC2PC3

1970 1980 1990 2000

(b) HA investments:

0.4

0.6

0.8

1970 1980 1990 2000

(c) CER confidence ranges:

0

1

2

1970 1980 1990 2000


Figure 2.7. Results for higher risk aversion, γ= 10. The panels show (a) inclusion in modelconfidence set for α= 0.05, (b) investment based on historical average (HA), and (c) lowestand highest element of CER confidence set (CER confidence ranges) along with risk-free ratein percentage returns. Dashed red lines indicate start- and end-dates of NBER recessions.


Table 2.3. Sharpe ratios (SR) and certainty equivalent return relative to the historical averageinvestment ( ∆C ER) for different investment strategies for γ = 2 and γ = 10. Robust, Avg,and Majority investment rules are based on MCS with α= 0.05. p-value of Diebold-Marianotest (p) for null hypothesis that expected loss is equal to expected loss from HA investment.Rejection of the null hypothesis after Bonferroni correction are indicates as: * for 10% level, **for 5% level, and *** for 1% level. The Bonferroni correction is based on the number of testsconducted in each panel for the same subsample. Sample period 1966:01–2002:12. Based on444 observations, of which 65 are during recessions.

Full Sample Recession No Recession

SR ∆C ER p SR ∆C ER p SR ∆C ER p(a) γ= 2HA 0.080 - - −0.046 - - 0.111 - -Robust 0.085 0.006 0.981 −0.024 1.057 0.238 0.110 −0.176 0.530Avg 0.090 0.079 0.004* −0.009 0.323 0.004** 0.116 0.037 0.158Majority 0.120 0.272 0.045 0.086 1.295 0.014 0.129 0.094 0.467CSR1 0.096 0.117 0.076 0.016 0.685 0.085 0.113 0.017 0.608CSR2 0.121 0.283 0.019 0.120 1.498 0.014 0.121 0.073 0.442CSR3 0.137 0.358 0.031 0.159 1.730 0.011 0.132 0.121 0.434CSR4 0.145 0.386 0.047 0.163 1.754 0.013 0.141 0.149 0.439CSR5 0.148 0.394 0.056 0.174 1.816 0.012 0.143 0.149 0.474PC1 0.080 0.075 0.648 −0.013 0.881 0.188 0.096 −0.065 0.676PC2 0.068 0.036 0.863 −0.074 0.576 0.390 0.098 −0.059 0.782PC3 0.099 0.167 0.400 −0.011 0.793 0.246 0.122 0.058 0.773(b) γ= 10HA 0.045 - - −0.097 - - 0.077 - -Robust 0.065 0.111 0.184 0.017 0.466 0.073 0.074 0.049 0.569Avg 0.114 0.146 0.001*** 0.078 0.429 0.002** 0.123 0.097 0.026Majority 0.143 0.202 0.002** 0.197 0.692 0.001** 0.132 0.118 0.071CSR1 0.108 0.140 0.000*** 0.057 0.425 0.005* 0.118 0.091 0.009CSR2 0.148 0.206 0.000*** 0.189 0.659 0.003** 0.139 0.128 0.018CSR3 0.166 0.239 0.001*** 0.235 0.769 0.002** 0.151 0.148 0.032CSR4 0.172 0.255 0.001** 0.247 0.826 0.003** 0.156 0.158 0.052CSR5 0.174 0.261 0.003** 0.258 0.883 0.003** 0.155 0.156 0.084PC1 0.073 0.119 0.040 −0.013 0.420 0.044 0.087 0.067 0.246PC2 0.063 0.106 0.119 −0.053 0.373 0.078 0.081 0.060 0.400PC3 0.076 -0.011 0.924 0.073 0.401 0.108 0.077 −0.082 0.533


2.5 Conclusion

We have used the model confidence set (MCS) approach of Hansen et al. (2011) to

measure and describe the model uncertainty in return predictability over the sample

period 1966:01–2002:12. For the universe of 23 models considered in this chapter,

the model uncertainty is substantial both in statistical and economic terms. Model

confidence sets change substantially over time and contain less models in the second

half of our sample. Investors are exposed to large model uncertainty in the sense that

for different return models, that all cannot be rejected by the data, the conditional

expected utility from investment is very different.

We have proposed three investment strategies based on model confidence sets

that account for, and reduce, the model uncertainty. All three investment strategies

lead to economic gains from using the predictor variables in the data set of Welch and

Goyal (2008). In particular, we show that a robust investment rule, that is designed

to perform well under all models in the MCS, produces economic gains from return

predictability. Reducing the model uncertainty with this robust investment strategy,

requires lower investments in stocks compared to investments based on expected

return forecasts from historical averages. In the first half of the sample, the stock

investment for the robust strategy is very low, but it increases with lower model

uncertainty in the second half of the sample. For the robust investment, it is crucial to

narrow down the set of candidate models using the MCS. When this is not possible, as

it is the case for investors with low risk aversion, for which the investment constraints

are binding for many models, the robust strategies perform poorly.

2.6. REFERENCES 57

2.6 References

Aiolfi, M., Favero, C., 2005. Model uncertainty, thick modelling and the predictability

of stock returns. Journal of Forecasting 24 (4), 233–254.

Ang, A., Bekaert, G., 2007. Stock return predictability: Is it there? Review of Financial

Studies 20 (3), 651–707.

Avramov, D., 2002. Stock return predictability and model uncertainty. Journal of

Financial Economics 64 (3), 423–458.

Barberis, N., 2000. Investing for the long run when returns are predictable. The Journal

of Finance 55 (1), 225–264.

Campbell, J., Thompson, S., 2008. Predicting excess stock returns out of sample: Can

anything beat the historical average? Review of Financial Studies 21 (4), 1509–1531.

Cenesizoglu, T., Timmermann, A., 2012. Do return prediction models add economic

value? Journal of Banking & Finance.

Cremers, K., 2002. Stock return predictability: A bayesian model selection perspective.

Review of Financial Studies 15 (4), 1223–1249.

Dangl, T., Halling, M., 2012. Predictive regressions with time-varying coefficients.

Journal of Financial Economics 106 (1), 157–181.

Diebold, F. X., Mariano, R. S., July 1995. Comparing predictive accuracy. Journal of

Business & Economic Statistics 13 (3), 253–63.

Elliott, G., Gargano, A., Timmermann, A., 2013. Complete subset regressions. Journal

of Econometrics.

Epstein, L. G., Wang, T., March 1994. Intertemporal asset pricing under knightian

uncertainty. Econometrica 62 (2), 283–322.

Fama, E., French, K., 1988. Dividend yields and expected stock returns. Journal of

Financial Economics 22 (1), 3–25.

Fama, E., French, K., 1989. Business conditions and expected returns on stocks and

bonds. Journal of Financial Economics 25 (1), 23–49.

Garlappi, L., Uppal, R., Wang, T., 2007. Portfolio selection with parameter and model

uncertainty: A multi-prior approach. Review of Financial Studies 20 (1), 41–81.

Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior.

Journal of Mathematical Economics 18 (2), 141–153.


Gonzàlez-Rivera, G., Lee, T.-H., Mishra, S., 2004. Forecasting volatility: A reality check

based on option pricing, utility function, value-at-risk, and predictive likelihood.

International Journal of Forecasting 20 (4), 629–645.

Hansen, L. P., Sargent, T. J., 2001. Robust control and model uncertainty. The American

Economic Review 91 (2), 60–66.

Hansen, P. R., Lunde, A., Nason, J. M., 2011. The model confidence set. Econometrica

79 (2), 453–497.

Hayfield, T., Racine, J. S., 2008. Nonparametric econometrics: The np package. Journal

of Statistical Software 27 (5).

URL http://www.jstatsoft.org/v27/i05/

Henkel, S., Martin, J., Nardari, F., 2011. Time-varying short-horizon predictability.


Kandel, S., Stambaugh, R., 1996. On the predictability of stock returns: An asset-

allocation perspective. The Journal of Finance 51 (2), 385–424.

Lettau, M., Ludvigson, S., 2010. Measuring and Modeling Variation in the Risk- Return

Tradeoff, Handbook of Financial Econometrics. Vol. 1. Elsevier Science B.V., North

Holland, Amsterdam, pp. 617–690.

Lewellen, J., 2004. Predicting returns with financial ratios. Journal of Financial Eco-

nomics 74 (2), 209–235.

Maenhout, P. J., 2004. Robust portfolio rules and asset pricing. The Review of Financial

Studies 17 (4), 951–983.

Pástor, L., Stambaugh, R. F., 2012. Are stocks really less volatile in the long run? The

Journal of Finance 67 (2), 431–478.

Patton, A., Politis, D. N., White, H., 2009. Correction to "automatic block-length

selection for the dependent bootstrap" by d. politis and h. white. Econometric

Reviews 28 (4), 372–375.

Pesaran, M. H., Timmermann, A., 1995. Predictability of stock returns: Robustness

and economic significance. The Journal of Finance 50 (4), pp. 1201–1228.

Politis, D. N., White, H., 2004. Automatic block-length selection for the dependent

bootstrap. Econometric Reviews 23 (1), 53–70.

Polk, C., Thompson, S., Vuolteenaho, T., 2006. Cross-sectional forecasts of the equity

premium. Journal of Financial Economics 81 (1), 101–141.

http://www.jstatsoft.org/v27/i05/

2.6. REFERENCES 59

R Core Team, 2013. R: A Language and Environment for Statistical Computing. R

Foundation for Statistical Computing, Vienna, Austria.

URL http://www.R-project.org/

Rapach, D., Strauss, J., Zhou, G., 2010. Out-of-sample equity premium prediction:

Combination forecasts and links to the real economy. Review of Financial Studies

23 (2), 821–862.

Skouras, S., April 2007. Decisionmetrics: A decision-based approach to econometric

modelling. Journal of Econometrics 137 (2), 414–440.

Timmermann, A., 2008. Elusive return predictability. International Journal of Fore-

casting 24 (1), 1–18.

Wachter, J., Warusawitharana, M., 2009. Predictable returns and asset allocation:

Should a skeptical investor time the market? Journal of Econometrics 148 (2),

162–178.



West, K. D., Edison, H. J., Cho, D., 1993. A utility-based comparison of some models

of exchange rate volatility. Journal of International Economics 35 (1-2), 23–45.

http://www.R-project.org/

CH

AP

TE

R

3FREQUENCY DEPENDENCE IN THE

RISK-RETURN RELATION

Bent Jesper Christensen and Manuel Lukas


Abstract

The risk-return trade-off is typically specified as a linear relation between the condi-

tional mean and conditional variance of returns on financial assets. In this chapter

we analyze frequency dependence in the risk-return relation using a band spectral

regression approach that is robust to contemporaneous leverage and feedback effects.

For daily returns and realized variances from high-frequency data on the S&P 500

from 1995 to 2012 we strongly reject the null of no frequency dependence. Although

the risk-return relation is positive on average over all frequencies, we find a large and

statistically significant negative coefficient for periods of around one week. Subsam-

ple analysis reveals that the negative effect at the higher frequency is not statistically

significant before the financial crisis, but very strong after July 2007. Accounting for

the frequency dependence in the risk-return relation improves the forecasting of

stock returns after 2007.

Keywords: Risk-Return Relation, Band Spectral Regression, Realized Variance, Leverage

Effect.

61

62 CHAPTER 3. FREQUENCY DEPENDENCE IN THE RISK-RETURN RELATION

3.1 Introduction

Financial theory predicts that investors need to be compensated for taking on greater

risks through higher expected returns, such that the conditional mean and condi-

tional variance of stock market returns are positively related. Empirical estimates

of the risk-return relation are abundant in the literature (see Lettau and Ludvigson,

2010, for an extensive survey). The risk-return relation is typically specified as a lin-

ear relation between stock returns and some measure of the conditional variance,

motivated by the Merton (1973) intertemporal capital asset pricing model. Although

the linear specification is predominant in the literature, Rossi and Timmermann

(2011) find that the relation between conditional mean and conditional variance is

distinctly non-linear and even non-monotonic. The risk-return relation is also found

to be non-linear by Christensen, Dahl, and Iglesias (2012) who use a semi-parametric

estimation approach. The non-linearity also shows up in a frequency domain analysis.

Bollerslev, Osterrieder, Sizova, and Tauchen (2013) get different slope coefficients

in the risk-return relation when re-estimating it using different frequencies, which

would not have been seen in case of a linear relation between conditional mean and

conditional variance.

In this chapter, we estimate the risk-return relation by band spectral regression in

order to explicitly allow for a frequency-dependent relation between the conditional

mean and conditional variance of stock market returns. If regression coefficients

depend on the frequencies used, the band spectral regression model is a natural

means of allowing the simultaneous presence of different frequency-dependent

slopes within the same risk-return model. Contrary to linear models of the risk-

return relation, the frequency-dependent model allows the impact on the conditional

mean return from movements in conditional variance to depend on whether these

movements are short-lived or persistent

An empirical measure of conditional variance is needed for estimation of the risk-

return relation. Already Merton (1980) regressed returns on sample variances of intra-

period returns covering the same interval. This early work used daily returns for the

intra-period calculation and monthly variances and returns for the regression. With

the advent of high-frequency data, it has become possible to calculate daily variance

measures from intra-daily returns, and consider daily level regressions. Although

much recent research (e.g., Bollerslev and Zhou, 2006; Bollerslev et al., 2013) follows

Merton (1980) in regressing returns on variances covering the same period, it is well

motivated based on asset pricing theory to consider conditional variance measures

that are in investor’s information set at the start of the period (see, e.g., Ghysels,

Santa-Clara, and Valkanov, 2005). Following this idea, we base our work on realized

variances calculated from high-frequency returns, and use these to construct proxies

for conditional variance using different versions of the heterogeneous auto-regressive

(HAR) model of Corsi (2009). In particular, there is a need to accommodate the

leverage effect, i.e., a possible negative contemporaneous relation between variance

3.1. INTRODUCTION 63

and return. Following Black (1976) and Christie (1982), a drop in stock price increases

the debt-equity ratios and the expected risk. Empirical research on realized variances

documents strong leverage effects, see, e.g., Christensen and Nielsen (2007) and Yu

(2005). Thus, following Corsi and Renò (2012), we use a version of the HAR model

extended with lagged leverage effects, the LHAR. With this, we study the risk-return

relation at a daily horizon using band spectral regression, thus allowing the regression

coefficients to differ across the chosen frequency bands. The presence of the leverage

effect in the risk-return relation would render the classical band spectral regression

(Engle, 1974; Harvey, 1978) inconsistent. The problem is that the error term (the

expectation error in returns) is in the information set in subsequent conditional

variances. To achieve consistency, we use the one-sided filtering approach to band

spectral regression suggested by Ashley and Verbrugge (2008).

Our empirical study uses realized variances from high-frequency data for the S&P

500 from 1995 to 2012. We implement band spectral regression using the one-sided fil-

ter, and use the estimated regression to assess the presence of frequency dependence

in the risk-return relation by testing for equal parameters across frequency bands.

Indeed, we find that the linear relation (common parameters across frequency bands)

is rejected consistently across different model specifications. These results strongly

suggest that the relation between risk and return depends on the frequency (period

length) considered. We specifically find that conditional variance fluctuations with

periods of around one month and one week have significantly positive, respectively

negative, effects on expected returns, indicating that the risk compensation effect is

at work at the lower frequency. When the sample is split at the start of the financial

crisis in 2007, we find that the negative relation at the weekly frequency becomes

much stronger following the onset of the crisis and is not statistically significant

before the crisis.

To further assess the importance of the frequency dependence in the risk-return

relation, we compare forecasting performance with and without this extension. Be-

cause we use a one-sided filter, it is possible to obtain real-time forecasts of stock

returns from the band spectral regression on conditional variances. However, due

to estimation uncertainty, a good in-sample fit of multivariate regressions for stock

returns often fails to translate into accurate real-time forecasts. To mitigate the effect

of estimation uncertainty in the construction of our return forecasts, we combine the

Ashley and Verbrugge (2008) one-sided filter with the complete subset regression ap-

proach of Elliott et al. (2013). We show that allowing for frequency dependence in the

regression relation using this combined approach helps improve return forecasting

performance after July 2007.

While our results therefore strongly support frequency dependence in the risk-

return relation, and the significantly positive relation around the one-month period

is consistent with the need for risk compensation from asset pricing theory, the

significantly negative relation between return and conditional variance around the


one-week period calls for some attention. In the GARCH literature, the risk-return

relation is studied using GARCH-in-mean (or GARCH-M) models, following Engle,

Lilien, and Robins (1987). Studies in this literature frequently find a negative relation,

e.g., Nelson (1991) and Glosten, Jagannathan, and Runkle (1993). There are economic

forces that move risk and return in opposite directions over the same time period.

In case of the leverage effect, a negative realized return increases risk, thus leading

to a negative contemporaneous relation. Volatility feedback has a similar effect.

Following French, Schwert, and Stambaugh (1987) and Campbell and Hentschel

(1992), if the conditional variance increases and the risk-return relation is positive,

then discount rates increase, and therefore the stock price drops. If this effect plays

out instantaneously, then this again leads to a negative contemporaneous relation

of realized returns and risk. If this effect from conditional variance to price does not

transmit instantaneously, then it could lead to a negative relation at, for example, the

weekly period. Recent asset pricing research further examines the possibility that

stocks that move with variance pay off in bad states and therefore should command

lower risk premia. Ang, Hodrick, Xing, and Zhang (2006) document negative cross-

sectional premia based on this idea, i.e., a negative price of volatility risk.

Thus, leverage, feedback, or a negative volatility risk price may confound evi-

dence on the risk compensation effect. There are only subtle measurable differences

between the confounding effects. Causality runs from return to risk in case of the

leverage effect, and from risk to return for the feedback and negative risk price effects.

Furthermore, both the leverage effect and the cross-sectional risk price effect are

likely to be strongest at the firm level, whereas the volatility feedback effect is opera-

tional at the market level. As a negative volatility feedback effect, firstly, is consistent

with a positive risk compensation effect, secondly, should be at work at the market

level, and thirdly, should lead to an effect of changes in conditional variance on sub-

sequent returns, we may cautiously interpret our findings of a negative coefficient

at one week and a simultaneous positive coefficient at one month as evidence of

volatility feedback, respectively risk compensation. However, as a caveat, the effect

through discount rates should also have some long-lived features, in addition to the

immediate price drop following an increase in risk, so the interpretation remains

delicate. At any rate, whatever the reason for the negative risk-return relation around

the weekly frequency, our work establishes its empirical importance. The negative

relation at the weekly frequency co-exists with the positive risk-return tradeoff or risk

compensation effect at the monthly frequency, and accommodating both improves

the forecasting of future stock returns.

Our findings shed some light on the empirical risk-return relation, which has

been of interest in the finance literature for a long time (see, e.g., Merton, 1973,

1980). Advances in variance estimation have provided new and more precise ways of

estimating the risk-return relation. Using high-frequency volatility measurements to

study the risk-return relation at a daily horizon, Bali and Peng (2006) find a positive

3.2. THE EMPIRICAL RISK-RETURN RELATION 65

and significant relation that is robust to different model specifications and methods

of volatility measurement. Harrison and Zhang (1999) find that a positive risk-return

relation is only present for longer holding periods but not, for example, at the monthly

horizon. Ghysels et al. (2005) use a mixed-frequency approach to construct monthly

conditional variance forecasts and find a positive and significant risk-return relation.

Our findings confirm that the risk-return relation is positive on average, i.e., when we

do not allow for frequency dependence. When we allow for frequency dependence,

the relation is strongly negative at certain frequencies during the financial crisis. In

line with Rossi and Timmermann (2011) and Christensen et al. (2012), this suggests

that the risk-return relation is non-linear and that the estimated coefficient in a linear

regression model thus depends on sample period and sampling frequency.

The remainder of the chapter is organized as follows. In Section 3.2 we discuss the

specification of the risk-return relation, conditional variance modeling, and the data.

Section 3.3 presents the band spectral regression results for the risk-return relation.

Section 3.4 uses band spectral regression for real-time forecasting of stock market

returns. Concluding remarks are given in Section 3.5.

3.2 The Empirical Risk-Return Relation

Theory suggests that investors are compensated for taking on greater risks by higher

expected returns, such that the conditional mean is positively related to the con-

ditional variance of returns. Let rt+1 denote the return on day t + 1. The time t

conditional expectation of the variance of rt+1 is denoted Et [σ2t+1]. The risk-return

relation is empirically often specified as the linear relation

rt+1 =µ+γEt [σ2t+1]+ut+1, (3.1)

where ut+1 are zero-mean innovations. Equation (3.1) is in the tradition of the inter-

temporal capital asset pricing model (ICAPM, see Merton, 1973). Ghysels et al. (2005)

estimate a positive γ in model (3.1) with a mixed-frequency approach using monthly

returns while calculating conditional variances from daily returns. Bali and Peng

(2006) consider model (3.1) with daily returns and conditional variances measured

from intra-day data, and consistently find a positive risk-return relation across dif-

ferent volatility measurements and model specifications. Bollerslev and Zhou (2006)

also use high-frequency data, but specify a contemporaneous regression that is in-

fluenced by leverage and feedback effects. In a closely related paper, Corsi and Renò

(2012) use conditional variance constructed from heterogeneous autoregressive (HAR

) models in the linear risk-return model (3.1) and find a positive and significant γ for

daily returns.

A major challenge in estimating the risk compensation is to separate it from

leverage and volatility-feedback effects that can produce negative contemporaneous

correlation between risk and return (see, e.g., Black, 1976; Christie, 1982; Campbell


and Hentschel, 1992; Wu, 2001). Contemporaneous risk-return regressions will give a

biased estimate of the risk compensation coefficient because of this negative con-

temporaneous correlation from leverage and volatility-feedback. Using conditional

expectations of the variance that are only based on lagged information, as in model

(3.1), has the advantage that the estimate is not influenced by contemporaneous

leverage and volatility-feedback effects, see French et al. (1987). Lagged leverage

effects are, however, allowed in the conditional variance as we discuss in Section

3.2.2.

3.2.1 Data and Volatility Measurement

Previous literature, for example Bollerslev, Litvinova, and Tauchen (2006), has shown

that high-frequency data is crucial for precision in the estimation of the risk-return re-

lation. We use intraday high-frequency returns on the SPDR S&P 500 exchange-traded

fund (ticker SPY) that tracks the S&P 500 index.1 Our sample runs from 1995/01/03

to 2012/07/31, with a total of 4426 trading days.

Daily realized variances are computed with the standard approach from intra-day

returns. The intraday realized variance on day t , RVt , is calculated by

RVt =I∑

i=1y2

i , (3.2)

where yi are the intra-day returns over some short time intervals. We choose 5-minute

intra-day returns, of which there are 79 for most trading days (I = 79). The 5-minute

sampled realized variances avoid market microstructure noise by sparse sampling,

but this discarding of data leads to an inefficient estimator if returns are observed

at a higher frequency than 5-minute intervals. We also consider realized kernels

(RK) as an alternative, more efficient, estimator of the variance that remains robust

to market microstruture effects (see Hansen and Lunde, 2006; Barndorff-Nielsen,

Hansen, Lunde, and Shephard, 2008). Instead of sampling a fixed time interval, we

use all available returns, after appropriate data cleaning (see Barndorff-Nielsen,

Hansen, Lunde, and Shephard, 2009). Let zi be the i th intra-day return, of which we

assume there are J . The time interval for these intra-day returns is not fixed, as price

observations are not equally spaced. The realized kernels are computed, as described

in Barndorff-Nielsen et al. (2009), by

RKt =H∑

h=1k

(h −1

H

)(ρh +ρ−h) (3.3)

ρh =I∑

j=|h|+1z j z j−|h|, (3.4)

1We are grateful to Asger Lunde for providing us with the data used in this chapter.


1995 2000 2005 2010

−0.

100.

000.

05

Returns

1995 2000 2005 2010

010

3050

RV

1995 2000 2005 2010

−12

−10

−8

−6

log RV

Figure 3.1. Time series plots of daily returns on S&P 500, intraday realized variance (RV), andlog realized variance (log RV). Sample period is 1995/01/03 to 2012/07/31.

where k(.) is a kernel function and H the bandwidth parameter. The Parzen kernel,

given by

k(x) =

1−6x2 +6x3 for 0 ≤ x < 1/2,

2(1−x)3 for 1/2 ≤ x ≤ 1,

0 for x > 1,

(3.5)

is used as kernel function for the construction of RKt .

Figure 3.1 shows the time series of daily returns, realized variances, and natural

logarithm of realized variances over the full sample period. The realized variances

series shows very pronounced spikes. After the log transformation, the series is much

more stable and no outliers are apparent. Modeling realized variances after the log

transformation has been advocated by Andersen, Bollerslev, Diebold, and Labys

(2003) and has since become the standard approach.


3.2.2 Modeling the Conditional Variance

Before we can estimate the risk-return regression (3.1), we have to construct a proxy

for conditional variance Et [σ2t+1]. We consider different model specifications for the

conditional variance. As a starting point we use the contemporaneous realized vari-

ance spanning the same period as the left-hand side returns in Equation (3.1). The

problem with using contemporaneous realized variance is that γ cannot be inter-

preted as the risk-return trade-off parameters, because contemporaneous leverage

and volatility-feedback effects will be captured by the estimate. Both effects are likely

to induce a negative relation between contemporaneous returns and changes in

variance. This problem can be mitigated by using lagged realized variance as a proxy.

Using lagged realized variance amounts to a random walk model for realized

variance. While not as persistent as a random walk, realized variance series are still

highly persistent. The heterogeneous autoregressive (HAR) model of Corsi (2009)

provides a parsimonious approximation of the variance dynamics that works well in

forecasting, and thus provides good conditional variance proxies. This conditional

variance proxy has been used by Corsi and Renò (2012) to estimate the risk-return

relation for daily returns and intra-day realized variances. They use the standard

HAR model, a HAR model with jumps, and a HAR model with jumps and lagged

leverage effects to obtain the conditional variance forecasts. The risk-return relation

is found to be positive and statistically significant for all three conditional variance

proxies. While Corsi and Renò (2012) find that incorporating jumps has almost no

effect on the estimate, they find that allowing for lagged leverage effects matters for

the risk-return relation and makes it more significant.

Let r vt = ln(RVt ) denote the natural logarithm of the realized variance. The HAR

model is given by

r vt = c +φ1r vt−1 +φ2r v5,t−1 +φ3r v22,t−1 +εt , (3.6)

where r v l ,t−1 = 1l

∑lj=1 r vt− j is the average of r vt over the past l days up to day

t −1. Thus, realized variance is predicted by simple moving averages of past realized

variances. The HAR model is remarkably successful in capturing the time series

properties of realized variance in a parsimonious fashion. The inclusion of lags

5 and 22 corresponds to averages over one trading week and one trading month,

respectively. With estimated coefficients (c,φ1,φ2,φ3) the variance forecast is

r v t+1 = c + φ1r vt + φ2r v5,t + φ3r v22,t , (3.7)

which is known at time t .

The simple structure of the HAR model allows for additional regressors. We add

lagged returns, and absolute values of lagged returns, to account for lagged leverage

effects in the manner of Corsi and Renò (2012). The HAR model with leverage (LHAR)

is given by

r vt = c +φ1r v t−1 +φ2r v5,t−1 +φ3r v22,t−1 +λ1rt−1 +λ2|rt−1|+εt , (3.8)


from which we construct the variance forecasts as

r v t+1 = c + φ1r v t + φ2r v5,t + φ3r v22,t + λ1rt + λ2|rt |. (3.9)

Table 3.1 shows the in-sample estimation results of the HAR and LHAR models.

Additionally, we include a model that only includes one lagged variance. Allowing for

leverage does not substantially increase the adjusted R2, but both lagged returns and

lagged absolute returns are significant, showing that leverage effects are important.

Consistent with the leverage argument, we find that the coefficient on lagged absolute

returns have the opposite sign than the coefficient on lagged returns. This implies

that negative returns have a positive effect while positive returns has a much smaller

effect on variance. Indeed, the hypothesis H0 :λ1 =−λ2 is not rejected, leading us to

the conclusion that only negative returns matter for the conditional variance. The

results for using realized kernel are very similar to the results for 5-minute realized

variance.

To construct the conditional variances, we do not use the full sample parameter

estimates for the HAR and LHAR models, but recursively estimate the coefficients

only using information prior the period for which the conditional variance is con-

structed. The parameters are estimated using a rolling estimation window of length

440, roughly one tenth of the sample. We have specified the variance models in terms

of the logarithm of realized variance. The risk-return trade-off is, however, specified

for the untransformed conditional variance. Thus, we transform the forecast back by

σ2t+1 = exp(r v t+1 +0.5σ2

ε) for the analysis of the risk-return relation, where σ2ε is an

estimate of the error variance in the model of r vt+1. For realized kernels we obtain

forecasts from the HAR and LHAR models in the exact same way as we have just

described for the 5-minute realized variances.

3.2.3 Linear Risk-Return Regression

In the linear model returns are regressed on conditional variances, i.e.,

rt+1 =µ+γσ2t+1 +ut+1. (3.10)

Let T be the sample size. It is convenient for the further presentation to write the

regression relation in vector notation as

R = Zµ+ V γ+U , (3.11)

where R = (r1, . . . ,rT )′ , V = (σ21, . . . ,σ2

T )′, and U = (u1, . . . ,uT )′. Here Z is a T ×1 vector

of ones. In general it is also possible to include regressors the are not frequency-

dependent.

Table 3.2 shows the estimation results for regression model (3.11) using differ-

ent conditional variance models. The coefficients are estimated by least-squares

with asymptotic standard errors based on the Newey and West (1987) covariance


Table

3.1.E

stimated

mo

dels

for

log

realizedvarian

cean

dlo

grealized

kernelfo

rfu

llsamp

le.

Realized

Varian

ceR

ealizedK

ernel

LagR

VH

AR

LHA

RLag

RK

HA

RLH

AR

c−

1.533 ∗∗∗−

0.417 ∗∗∗−

0.875 ∗∗∗−

1.671 ∗∗∗−

0.441 ∗∗∗−

0.942 ∗∗∗(0.098)

(0.088)(0.096)

(0.094)(0.091)

(0.101)φ

10.839 ∗∗∗

0.407 ∗∗∗0.323 ∗∗∗

0.827 ∗∗∗0.401 ∗∗∗

0.317 ∗∗∗(0.010)

(0.021)(0.020)

(0.010)(0.024)

(0.024)φ

20.364 ∗∗∗

0.394 ∗∗∗0.367 ∗∗∗

0.396 ∗∗∗(0.033)

(0.030)(0.034)

(0.033)φ

30.185 ∗∗∗

0.198 ∗∗∗0.186 ∗∗∗

0.197 ∗∗∗(0.026)

(0.025)(0.026)

(0.025)λ

1−

9.931 ∗∗∗−

9.675 ∗∗∗(0.734)

(0.725)λ

29.498 ∗∗∗

10.147 ∗∗∗(1.109)

(1.094)A

dj.R

20.704

0.7530.754

0.6840.743

0.744N

um

.ob

s.4425

44044404

44254404

4404H

0:λ

1 =−λ

20.734

0.6975

No

te:Signifi

cance

levelsw

ithN

ewey-W

eststand

arderro

rs:***:1%,**:5%

,and

*:10%.Lastrow

show

sp

-values

for

testing

the

hyp

oth

esisH

0:λ

1 =−λ

2 .


Tab

le3.

2.E

stim

atio

nre

sult

sfo

rlin

ear

risk

-ret

urn

regr

essi

on.L

agR

Van

dla

gR

Ku

seth

efi

rstl

agof

the

real

ized

vari

ance

and

real

ized

kern

el,r

esp

ecti

vely

.Fo

rH

AR

and

LHA

Rth

eco

nd

itio

nal

vari

ance

sar

eo

bta

ined

fro

mre

curs

ive

esti

mat

ion

wit

hro

llin

gw

ind

owo

flen

gth

M=

440.

Rea

lize

dV

aria

nce

Rea

lized

Ker

nel

RV

Lag

RV

HA

RLH

AR

RK

Lag

RK

HA

RLH

AR

µ0.

000

−0.0

01∗

−0.0

01∗

−0.0

01∗∗

∗0.

000

−0.0

01∗∗

−0.0

01∗∗

∗−0

.001

∗∗∗

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

γ−0

.182

2.70

8∗∗∗

2.89

3∗∗

4.02

4∗∗∗

−0.7

273.

014∗

∗∗3.

318∗

∗∗4.

999∗

∗∗(0

.763

)(0

.762

)(0

.991

)(0

.784

)(1

.427

)(0

.591

)(0

.862

)(1

.919

)R

20.

001%

0.31

6%0.

214%

0.65

6%0.

020%

0.34

9%0.

269%

0.85

4%N

um

.ob

s.39

8539

8539

8539

8539

8539

8539

8539

85

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ign

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ests

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rors

:***

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5%,a

nd

*:10

%.


estimator with lag length 44, corresponding to roughly two months of trading days.

For contemporaneous variance (RVt and RKt ) we get a negative but insignificant

coefficient γ. When we use lagged realized variance, the coefficient becomes positive

and statistically significant. For conditional variances from both HAR and LHAR the

coefficient γ increases further and remains statistically significant. The explanatory

power, as measured by R2, is highest for the LHAR model. The estimates are close to

the results in Corsi and Renò (2012), who analyze a different, yet overlapping, sample

period from 1982 to 2009 for the S&P 500.

The standard errors in Table 3.2 do not account for the additional uncertainty

from the use of generated regressors, i.e., the fact that the conditional variances are

obtained from estimated HAR models (see, e.g., Pagan (1984) and Pagan and Ullah

(1988)). Murphy and Topel (2002) suggest a covariance estimator that corrects for

generated regressors. When the regressors are generated by a linear model and errors

are homoskedastic, the standard errors are inflated by a factor 1+γ2 var(ut )/var(εt ).

Using an AR model selected by AIC on the original (without log transformation)

realized variance series, we get an estimate var(ut )/var(εt ) = 1.58×10−4, such that

the correction factor is negligible for all reasonable values of γ. For example, for γ= 5

the correction factor is 1.0048. Even though our data is clearly heteroskedastic, the

minuscule correction factors indicate that correcting for generated regressors is not

important for our data. The error variance in the conditional variance models is much

smaller than in the second step, i.e., the risk-return regression. This is in line with

French et al. (1987), who also find that adjusting the standard errors in model (3.10)

for generated regressors leads to negligible adjustments.

3.3 Frequency Dependence in the Risk-Return Relation

To allow for frequency dependence in the risk-return relation we apply the band

spectral regression (see Engle (1974) and Harvey (1978)). Bands spectral regression

has been applied to detect frequency dependence in macroeconomic models by Tan

and Ashley (1999) and Ashley and Verbrugge (2008).

For the real-valued band spectral regression of Harvey (1978) we define the T ×T

discrete Fourier transform matrix AT with elements

ai j =

T −1/2, for i = 1;

(2T

)−1/2cos

(πi ( j−1)

T

), for i = 2,4,6, . . . ,(T −2) or (T −1);

(2T

)−1/2sin

(π(i−1)( j−1)

T

), for i = 3,5,7, . . . ,(T −1) or (T );

T −1/2(−1) j+1, for j = T if T is even.

(3.12)

3.3. FREQUENCY DEPENDENCE IN THE RISK-RETURN RELATION 73

Pre-multiplying the regression model (3.11) by AT we get

R∗ = Z∗µ+V ∗γ+U∗, (3.13)

where R∗ = AT R, Z∗ = AT Z , V ∗ = AT V , and U∗ = AT U . The elements of R∗ and

V ∗ correspond to different frequency components of returns and variances, respec-

tively. The first element corresponds to frequency 0 and the last element, element

T , corresponds to frequency π. In general, the element i corresponds to frequency

π(bi /2c/T ), where b.c rounds to the next lower integer. The constant µ and the regres-

sion coefficient γ in model (3.13) remain unaffected by this transformation.

If there is no frequency dependence in the risk-return relation, such that the

linear model is correctly specified, then the coefficient γ does not depend on which

frequencies we include in the regression. To detect frequency dependence for the

conditional variances V , we allow the coefficients to vary for different frequency

bands. Let B be the chosen number of frequency bands. Define the T ×1 vectors

D∗b , b = 1, . . . ,B , as the dummy variables with the observations of V ∗ belonging to

frequency band b, and zeros for all other frequencies. The frequency-dependent

model in the frequency domain then becomes

R∗ = Z∗µ+B∑

b=1D∗

bβb +U∗, (3.14)

where βb are the coefficients that can be different for the frequency bands. By pre-

multiplying by A′T we can transform the regression (3.14) back to the time domain,

R = Zµ+B∑

b=1Dbβb +U , (3.15)

where Db = A′T D∗

b is now a time series of the frequency component corresponding to

frequency band b (see Ashley and Verbrugge, 2008, for details). The null hypothesis

of no frequency dependence corresponds to testing

H0 :β1 =β2 = ·· · =βB (3.16)

in time domain regression (3.15). The important difference of the band spectral

approach to other frequency domain approaches is that we keep all frequencies and

do not focus exclusively on certain frequencies. This allows us to get a complete

picture of the frequency dependence in the risk-return relation.

The standard band spectral regression is based on a two-sided filter, i.e., the

time domain dummies Db in (3.15) are constructed from past and future observa-

tions. Contemporaneous correlation and feedback effects play an important role

in the risk-return regression, as we have seen in the regression results in Table 3.2.

Parameter estimates based on a two-sided filter will therefore be influenced by the

contemporaneous correlation and likely be downwards biased. We therefore adapt


the one-sided filtering approach to the band spectral regression proposed by Ashley

and Verbrugge (2008). In this approach we extract the frequency components by

recursively applying a one-sided filter with a moving window of size w , such that

for every time t we construct the frequency bands Db,t , for b = 1, . . . ,B using only

information up to, and including, time period t . Because we only apply the filter on

w observation, the number of frequencies is reduced compared to two-sided band

spectral regression. To avoid endpoint problems associated with one-sided filtering

we use the approach of Stock and Watson (1999), in which we use an auto-regressive

(AR) model to forward pad (in the terminology of Stock and Watson (1999)) in each

step of the recursive filtering. We choose a moving window of w = 600 observations

and pad 300 observations forward.

The number of bands B in the band spectral regression can be selected by in-

formation criteria. Table 3.3 shows AIC and BIC for 1 to 16 bands for the different

regressors in the band spectral regression. The number of bands selected by BIC is

very unstable across regressors, ranging from 1 band for lagged RV to 12 bands for

LHAR. Using AIC we select 14 bands for RV and 12 bands for the three other regressors.

This leads to the use of 12 bands (B = 12) in the following analysis, because we are

most interested in the results for lagged RV, HAR, and LHAR.

In Table 3.4 we show the periods that correspond to the 12 frequency bands. These

values are a function of our choice of number of bands B and length of window w for

the one-sided filter. The lowest frequency band contains all fluctuations with periods

higher than roughly 2 months. We will not be able to analyze frequency dependence

at lower frequencies such as business cycle frequencies. Thus, the following analysis

will detect frequency dependence at higher frequencies, such as weekly and monthly

periods.

Figure 3.2 shows four of the extracted frequency band components Db in the

time domain. The frequency components are for 12 bands and use the LHAR condi-

tional variances estimated with realized variances. The lowest frequency component

captures the persistent component of realized variance. All frequency components

capture some of the erratic behavior of variance at the beginning of the financial

crisis around July 2007.

In contrast to the standard band spectral regression, the one-sided filtering ap-

proach does not guarantee orthogonal frequency components. Table 3.5 shows the

correlation between the extracted components in the time domain. Most entries in

the correlation matrix are low, but for neighboring frequency bands the correlation

can be substantial. The correlation of neighboring frequency bands is particularly

strong for the high frequencies. For example, the correlation of the two highest

frequency components is 0.48. The correlations between high and low frequency

components are close to zero. This means that the association of the components Db

with a certain frequency is very rough, in particular at the high frequencies.

Figure 3.3 shows the coefficients of the band spectral regression (3.15) with 12


Tab

le3.

3.In

form

atio

ncr

iter

iafo

rd

iffe

ren

tn

um

ber

so

ffr

equ

ency

ban

ds

Ban

dco

nd

itio

nal

vari

ance

pro

xies

asre

gres

sors

.Co

nd

itio

nal

vari

ance

sco

nst

ruct

edu

sin

gre

aliz

edva

rian

ce.T

he

low

estv

alu

eso

fAIC

and

BIC

are

show

nin

bo

ld.

Reg

ress

or:

RV

LRV

HA

RLH

AR

BA

ICB

ICA

ICB

ICA

ICB

ICA

ICB

IC1

−207

49.1

4−2

0730

.76

−207

59.4

5−2

0741

.07

−207

56.1

9−2

0737

.8−2

0760

.05

−207

41.6

72

−207

92.9

5−2

0768

.44

−207

65.1

9−2

0740

.68

−207

54.9

5−2

0730

.44

−207

67.0

1−2

0742

.00

3−2

0780

.87

−207

50.2

3−2

0767

.49

−207

36.8

6−2

0760

.45

−207

29.8

1−2

0774

.46

−207

43.8

34

−208

05.3

2−2

0768

.55

−207

73.0

2−2

0736

.25

−207

70.8

1−2

0734

.05

−208

28.7

4−2

0791

.98

5−2

0797

.88

−207

54.9

9−2

0767

.34

−207

24.4

5−2

0763

.98

−207

21.0

9−2

0802

.66

−207

59.7

76

−207

98.3

6−2

0749

.33

−207

67.6

8−2

0718

.66

−207

60.2

2−2

0711

.20

−208

25.7

0−2

0776

.68

7−2

0789

.51

−207

34.3

6−2

0774

.92

−207

19.7

7−2

0767

.87

−207

12.7

2−2

0828

.40

−207

73.2

58

−208

05.7

4−2

0744

.46

−207

69.8

4−2

0708

.57

−207

66.1

8−2

0704

.90

−208

80.9

8−2

0819

.71

9−2

0796

.69

−207

29.2

8−2

0789

.25

−207

21.8

5−2

0776

.97

−207

09.5

7−2

0862

.95

−207

95.5

510

−208

16.8

1−2

0743

.27

−207

61.3

0−2

0687

.77

−207

62.0

8−2

0688

.55

−208

66.6

8−2

0793

.16

11−2

0791

.45

−207

11.7

9−2

0770

.76

−206

91.1

0−2

0772

.98

−206

93.3

2−2

0884

.46

−208

04.8

012

−208

16.7

1−2

0730

.92

−208

07.2

0−2

0721

.41

−208

12.3

4−2

0726

.56

−209

32.6

6−2

0846

.88

13−2

0810

.84

−207

18.9

2−2

0795

.25

−207

03.3

4−2

0796

.03

−207

04.1

2−2

0884

.67

−207

92.7

514

−208

33.4

0−2

0735

.35

−207

81.5

3−2

0683

.49

−207

87.3

9−2

0689

.36

−208

93.3

6−2

0795

.33

15−2

0830

.95

−207

26.7

8−2

0793

.04

−206

88.8

7−2

0787

.44

−206

83.2

7−2

0899

.70

−207

95.5

316

−208

23.0

9−2

0712

.79

−207

84.8

2−2

0674

.52

−207

85.1

6−2

0674

.86

−209

06.7

5−2

0796

.46


2000 2005 2010

0.00

000.

0015

D1

2000 2005 2010

−6e

−04

0e+

004e

−04

D2

2000 2005 2010

−2e

−04

1e−

04

D11

2000 2005 2010

−6e

−04

0e+

006e

−04

D12

Figure 3.2. Time series of frequency components Db in time domain from band spectralregression with 12 frequency bands obtained from one-sided filter applied to LHAR withrealized variances. From top to bottom the plot shows the lowest, second lowest, secondhighest, and highest frequency band.


Table 3.4. Periods (in trading days) included in each of the 12 frequency bands.

Band Highest Lowest1 ∞ 48.652 47.37 24.003 23.68 15.934 15.79 11.925 11.84 9.526 9.47 7.937 7.89 6.828 6.79 5.989 5.96 5.33

10 5.31 4.8011 4.79 4.3712 4.36 2.00

Table 3.5. Correlations for time series of frequency components from one-sided filter appliedto LHAR with realized variance. The frequency components are ordered from lowest to highestfrequency, i.e., D1 corresponds to the band with the lowest, and D12 to the band with thehighest frequencies. Correlations are calculated from 3386 observations.

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12

D1 1.00 0.13 0.06 0.05 0.02 0.01 -0.02 0.00 0.02 -0.00 -0.00 0.01D2 0.13 1.00 -0.03 -0.03 -0.02 -0.01 0.01 0.01 0.01 -0.01 -0.00 0.01D3 0.06 -0.03 1.00 0.04 0.02 0.03 0.03 0.04 0.04 0.02 0.03 0.03D4 0.05 -0.03 0.04 1.00 0.04 0.03 0.02 0.08 0.09 0.06 0.09 0.09D5 0.02 -0.02 0.02 0.04 1.00 0.14 0.11 0.15 0.11 0.05 0.07 0.08D6 0.01 -0.01 0.03 0.03 0.14 1.00 0.16 0.17 0.11 0.03 0.06 0.07D7 -0.02 0.01 0.03 0.02 0.11 0.16 1.00 0.28 0.09 0.03 0.09 0.07D8 0.00 0.01 0.04 0.08 0.15 0.17 0.28 1.00 -0.06 -0.06 -0.01 -0.00D9 0.02 0.01 0.04 0.09 0.11 0.11 0.09 -0.06 1.00 0.06 0.14 0.11

D10 -0.00 -0.01 0.02 0.06 0.05 0.03 0.03 -0.06 0.06 1.00 0.40 0.31D11 -0.00 -0.00 0.03 0.09 0.07 0.06 0.09 -0.01 0.14 0.40 1.00 0.38D12 0.01 0.01 0.03 0.09 0.08 0.07 0.07 -0.00 0.11 0.31 0.38 1.00

bands for the different regressors. For RV, where contemporaneous leverage and

volatility feedback effects affect the coefficients, the coefficients become negative for

periods lower than 8 days. For the other three regressor,s the majority of coefficients

are positive. Coefficients for lagged RV and HAR are very similar, with a pronounced

negative coefficient at frequency band 10. For LHAR conditional variance we see

the strongest negative coefficients around the weekly period. The results for realized

kernels in Figure 3.4 show no qualitative differences.

Based on the band spectral regression we test for frequency dependence by testing


(a) RV−

80−

60−

40−

200

20

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(b) Lagged RV

−50

050

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(c) HAR

−15

0−

100

−50

050

100

150

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(d) LHAR

−15

0−

100

−50

050

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

Figure 3.3. Coefficients from band spectral regression with 12 bands. Regressors are RK, laggedRK, HAR, and LHAR. Red dotted lines show 95% confidence intervals.

for equal parameters. The test is implemented as a robust Wald test with Newey-

West covariance matrix. The coefficient β1 associated with the lowest frequency

component is excluded from the test, such that we test H 20 : β2 = β3 = ·· · = β12. We

excludeβ1 for robustness, because the lowest frequency component is very persistent

(see Figure 3.2). Clearlyl, rejection of H 20 implies rejection of the more restrictive null

hypothesis that all frequency bands have equal coefficients. However, by testing H 20

instead of H0, we sacrifice power.

Table 3.6 shows adjusted R2 for the band spectral regressions, Wald test statistics,


(a) RK

−10

0−

500

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(b) Lagged RK

−50

050

100

Period (days)10

0

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(c) HAR

−15

0−

100

−50

050

100

150

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(d) LHAR

−15

0−

100

−50

050

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

Figure 3.4. Coefficients from band spectral regression with 12 bands for realized kernels.Regressors are RK, lagged RK, HAR, and LHAR. Red dotted lines show 95% confidence intervals.

and associated p-values. The null hypothesis of equal parameters (H 20 ) is strongly

rejected for all regressors. Thus, the linear regression model is not a good approx-

imation for the risk-return relation over the full sample. Adjusted R2 is highest for

LHAR compared to the other regressors. This is in line with Corsi and Renò (2012),

who also find that lagged leverage effects are very important in the construction of

the conditional variance when used in the risk-return relation.


Table 3.6. Test for no frequency dependence. The table shows adjusted R2 from band spectralregression with one-sided filter, and the Wald statistic W , and p-value from test for equalparameters across frequency bands 2 to 12, H2

0 : β2 = β3 = ·· · = β12. The test is based onNewey-West covariance matrix with 44 lags. Sample size is 3386.

Realized Variance Realized Kernel

adj. R2 W p-value adj. R2 W p-valueRV/RK 1.913% 4.101 0.000 2.993% 9.947 0.000

Lag RV/RK 1.794% 3.174 0.000 2.383% 4.871 0.000HAR 2.147% 4.725 0.000 3.550% 6.894 0.000

LHAR 5.563% 19.759 0.000 5.079% 7.405 0.000

3.3.1 Frequency Dependence during the Financial Crisis

Stock markets have experienced a sharp rise in variance after the beginning of the

financial crisis in mid-2007. The effects are very pronounced in the frequency com-

ponents of the realized variance series in Figure 3.2. Both low and high frequency

components show more erratic behavior after the start of the financial crisis. To ana-

lyze the frequency dependence during the period with increased variance we split the

sample into the two subsamples 1995/01/03–2007/07/31 and 2007/08/01–2012/07/3,

such that the first subsample stops before the financial crisis starts. The subsamples

are labeled pre-crisis and crisis, respectively.

Table 3.7 shows estimation results for the linear risk-return model on the two

subsamples. With contemporaneous variance the estimated coefficient is positive

in the first, and negative in the second subsample. This suggests that leverage and

feedback effects are more important during the crisis. For lagged RV, HAR, and LHAR,

the coefficient estimate is always positive, but much higher for the pre-crisis sample.

During the crisis the risk-return relation is still positive and statistically significant

for lagged RV, HAR, and LHAR.

The coefficient estimates of the band spectral regression for HAR and LHAR

variances for the pre-crisis and crisis sample are shown in Figure 3.5. In the pre-crisis

sample there is no statistically significant evidence of negative dependence at any

frequency. In the crisis sample, however, we get strong negative dependence around

the one week period. Two frequency bands for HAR and three frequency bands for

LHAR have statistically significant negative coefficients.

Tests for no frequency dependence are shown in Table 3.8. The null is rejected

for all regressors in both subsamples. The test statistic is, however, much larger in

the crisis than pre-crisis, i.e., the evidence against the null hypothesis is stronger in

the second subsample. The band spectral regression has a higher adjusted R2 for the

crisis sample than for the pre-crisis sample for all regressors.

These findings suggest that during the financial crisis and its aftermath the short

lived volatility fluctuation, with period of around one week and below, had a distinctly


Tab

le3.

7.Su

bsa

mp

lees

tim

atio

nre

sult

sfo

rli

nea

rri

sk-r

etu

rnre

gres

sio

n.

1995

/01/

03–2

007/

07/3

120

07/0

8/01

–201

2/07

/31

RV

Lag

RV

HA

RLH

AR

RV

Lag

RV

HA

RLH

AR

µ0.

000

0.00

0∗−0

.001

∗∗∗

−0.0

01∗∗

∗0.

000

0.00

0∗∗∗

0.00

00.

000

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

γ0.

793

4.41

0∗∗∗

6.16

3∗∗

10.5

86∗∗

∗−0

.712

1.89

0∗∗∗

1.86

0∗∗∗

2.00

7∗∗∗

(2.5

89)

(1.3

33)

(2.8

28)

(3.2

97)

(1.4

86)

(0.4

89)

(0.1

41)

(0.3

25)

Ad

j.R

2−0

.002

%0.

380%

0.31

7%1.

722%

-0.0

19%

0.19

2%0.

104%

0.23

0%N

um

.ob

s.22

8322

8322

8322

8314

1814

1814

1814

18

(b)

RK

1995

/01/

03–2

007/

07/3

120

07/0

8/01

–201

2/07

/31

RK

Lag

RK

HA

RLH

AR

RK

Lag

RK

HA

RLH

AR

µ0.

000

0.00

0∗−0

.001

∗∗∗

−0.0

01∗∗

∗0.

000

0.00

0∗∗∗

0.00

00.

000

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

γ1.

390

4.45

3∗∗

7.08

6∗∗

12.4

04∗∗

∗−1

.574

2.38

2∗∗∗

2.37

8∗∗∗

2.53

8∗∗∗

(2.7

93)

(1.7

54)

(3.2

31)

(3.7

65)

(1.5

92)

(0.5

17)

(0.2

33)

(0.4

21)

Ad

j.R

2−0

.004

%0.

296%

0.33

3%2.

120%

0.10

5%0.

309%

0.20

5%0.

321%

Nu

m.o

bs.

2283

2283

2283

2283

1418

1418

1418

1418

No

te:S

ign

ifica

nce

leve

lsw

ith

New

ey-W

ests

tan

dar

der

rors

:***

:1%

,**:

5%,a

nd

*:10

%.


(a) HAR Pre-Crisis−

200

−10

00

100

200

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(b) LHAR Pre-Crisis

−10

0−

500

5010

0Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(c) HAR Crisis

−15

0−

100

−50

050

100

150

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

(d) LHAR Crisis

−20

0−

150

−10

0−

500

50

Period (days)

100

32.1

4

19.1

5

13.6

4

10.5

9

8.65

7.32

6.36

5.62

5.04

4.57

4.18

Figure 3.5. Coefficients from band spectral regression with 12 bands for the two subsamples1995/01/03–2007/07/31 (pre-crisis) and 2007/08/01–2012/07/31 (crisis). Regressor is HAR orLHAR conditional variance from realized variances. Red dotted lines show 95% confidenceintervals.

negative effect on returns. The linear regression model fails to capture this feature of

the risk-return relation that is clearly present after July 2007.

3.4. FREQUENCY-DEPENDENT REAL-TIME FORECASTS 83

Table 3.8. Test for no frequency dependence for subsamples. The table shows adjusted R2

from band spectral regression with one-sided filter, and the Wald statistic W , and p-value fromtest for equal parameters across frequency bands 2 to 12, H2

0 :β2 =β3 = ·· · =β12. The test isbased on Newey-West covariance matrix with 44 lags.

Sample: 1995/01/03–2007/07/31 2007/08/01–2012/07/31

adj. R2 W p-value adj. R2 W p-valueRV 1.367% 6.486 0.000 3.090% 9.044 0.000

Lag RV 1.445% 5.607 0.000 4.084% 4.488 0.000HAR 1.625% 3.636 0.000 4.515% 7.364 0.000

LHAR 1.326% 2.839 0.001 11.978% 52.687 0.000RK 1.172% 6.278 0.000 5.982% 18.878 0.000

Lag RK 1.255% 5.095 0.001 5.943% 5.146 0.000HAR 1.596% 3.701 0.000 8.143% 10.484 0.000

LHAR 1.261% 2.671 0.002 11.278% 18.450 0.000

3.4 Frequency-Dependent Real-Time Forecasts

We have documented strong in-sample evidence of frequency dependence in the

risk-return relation. In this section we make an out-of-sample forecasting experiment

to investigate whether real-time forecast of returns can be improved by allowing for

frequency dependence. The one-sided filter described in Section 3.3 makes it possible

to extract the frequency components in real-time and thus allows us to construct the

forecasts from the band spectral regression.

As a benchmark, we obtain conditional mean forecasts from the linear regression

model by

rt+1 = µ+ γσ2t+1, (3.17)

where the parameter estimates and the conditional variance σ2t+1 are based on data

up to time t . A rolling window of length R is used for parameter estimation. Forecasts

from the band spectral regression with B bands are calculated as

r Ft+1 = µ+

B∑b=1

βbDb,t , (3.18)

where Db,t , the B frequency components of σt+1, are constructed using the one-sided

filter based on observations up to time t as described in Section 3.3.

The forecasts from the band spectral regression in Equation (3.18) are based

on B +1 estimated parameters. Regression models with multiple regressors do not

work well for return prediction, even with a modest number of regressors (see, e.g.,

Welch and Goyal, 2008). Rapach et al. (2010) have shown that forecast combination

can be used to improve forecasting accuracy compared to a multivariate regression

approach for stock returns.


3.4.1 Complete Subset Regression

Elliott et al. (2013) propose the complete subset regression (CSR) as a flexible fore-

cast combination approach. Complete subset regression performs equal-weighted

forecast combination over all possible regression models that contain k out of the B

available regressors. Varying k allows the econometrician to change properties of the

forecast by trading off bias and variance. For small k the coefficients contain a strong

omitted variable bias, while for large k the predictor will suffer from high estimation

variance.

The parameter estimates from CSR are computed as

βk,B = 1

nk,B

nk,B∑i=1

βSi , (3.19)

where nk,B = B !/((B − k)!k !). The parameter vectors βSi (i = 1. . . ,nk,B ) contain all

possible parameter estimates based on k of the B variables, i.e., for each i a different

combination of k variables is included in the model. For the variables that are not

included in model i , the entries in βSi are zero. The coefficients for CSR are obtained

as the average over all nk,B possible combinations of k out of B variables. The total

number of models, nk,B , can be quite large. In this application we have B is 12,

because each frequency component is a potential predictor. The number of models

is highest for k = 6, which results in n6,12 = 924 models.

The forecasting performance of the CSR depends crucially on the choice of k. For

k = B , the complete subset regression is equivalent to multiple regression estimated

by least squares including all variables at once. For k = 1, CSR corresponds to forecast

combination of univariate regression for each predictor variable, equivalent to the

approach of Rapach et al. (2010).

We follow Elliott et al. (2013) in choosing k by minimizing an estimate of the

asymptotic mean squared error (AMSE). The AMSE can be derived under an IID

assumption and the local model β= T −1/2bσu , where σu is the standard deviation

of innovations. The parameter b controls the strength of the predictor and thus

determines which choice of k is optimal. Let ΣX be the covariance matrix of the

predictor variables. Elliott et al. (2013) show (Theorem 2) that the AMSE, scaled by

σ−2u , can be expressed as

σ−2u MSE(k) ≈

B∑j=1

η j +b′(Λk,B − IB )′ΣX (Λk,B − IK )b (3.20)

where η j is the j th eigenvalue ofΛ′B ,BΣXΛk,BΣ

−1X , and

Λk,B = 1

nk,B

nk,B∑i=1

(S′iΣX Si )−1(S′

iΣX ), (3.21)

where Si is selection matrix for all combinations, with ones on the diagonal for

included variables and zeros everywhere else. The AMSE, as a function of k, still

depends on the parameter b.


(a) AMSE for HAR Realized Variance

0 1 2 3 4 5 6

0.2

0.6

1.0

k

AM

SE

●

●

●

(b) AMSE for LHAR Realized Variance

0 1 2 3 4 5 6

0.2

0.6

1.0

k

AM

SE

●

●

●

(c) AMSE for HAR Realized Kernel

0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

k

AM

SE

●

●

●

(d) AMSE for LHAR Realized Kernel

0 1 2 3 4 5 6

0.2

0.6

1.0

1.4

k

AM

SE

●

●

●

Figure 3.6. Asymptotic mean squared error (AMSE) curves for the complete subset regressionwith B = 12 based on the first 440 observations. The curves correspond to b′b = 1,2,3, in thisorder from lowest to highest. In each plot the lowest asymptotic MSE for each γ is marked witha circle.


Figure 3.6 plots the AMSE as a function of k for different values of b. The first

R = 440 observations are used to estimate ΣX and σu in (3.20), i.e., we base the

selection of k on first estimation window and not on observations in the out-of-

sample period. For the different b, the optimal k varies from k = 0 to k = 4 for all

regressors, with the optima most frequently being at k = 1, k = 2, and k = 3. The

curves are all very flat around their minima, such that the forecasting performance

should not be very sensitive to the choice of k. This leads us to consider k = 1, k = 2,

and k = 3, for the forecasting.

Besides providing guidance for choosing k, some further insights into the ex-

pected forecasting performance can be gained from Figure 3.6. For k larger than 4

the MSE monotonically increases in all cases. Using the unrestricted band spectral re-

gression corresponds to k = 12, which is far away from the optimum for all regressors.

Therefore, we expected the MSE for the unrestricted band spectral regression to be

much higher than for smaller k.

3.4.2 Results

We evaluate the out-of-sample performance by root mean squared error (RMSE), out-

of-sample R2, and model confidence set p-values. The out-of-sample R2 (OOS-R2)

for model i is given by

OOS-R2i = 1−

∑Pt=1(rt ,i − rt )2∑Pt=1(at − rt )2

, (3.22)

where rt ,i is the forecast form model i , and at is the forecast from the constant model,

i.e., the historical average (see Campbell and Thompson, 2008). To test whether

differences in RMSE among the models are statistically significant we report p-values

obtained by the model confidence set approach of Hansen et al. (2011). We apply the

model confidence set to each group of models that use the same regressors and are

based on the same sample period.2

The forecasting results in Table 3.9 show that not all in-sample results are con-

firmed by the out-of-sample results. The linear model has negative OOS-R2 for all

regressors and subsamples. The unrestricted band spectral regression leads to dismal

forecasting performance, particularly in the crisis sample. The poor performance

was, however, already expected from the AMSE estimates in Figure 3.6 and can be

explained by the overwhelming estimation variance. Additionally, our subsample

analysis indicates that the band spectral coefficients change over time, which further

deteriorates the forecasting performance.

2The MulCom package version 3.00 for the Ox programming language (see Doornik, 2007) is usedto construct the model confidence sets. The MulCom package is available from http://mit.econ.au.dk/vip_htm/alunde/MULCOM/MULCOM.HTM. The following settings are used for the model confidenceset construction in MulCom: 9999 boostrap replication with block bootstrapping, block length 44, therange test for equal predictive ability δR,M , and the range elimination rule eR,M .




Table 3.9. Results from out-of-sample forecasting for realized variances (RV) and realizedkernels (RK). All models are estimated using a rolling window with R = 440 observations.Out-of-sample period 2000/11/09–2012/07/31 (2947 observations). We consider the linearmodel (Linear), band spectral regression with 12 bands (BSR), and complete subset regression(CSR) with k = 1,2,3. Root mean squared errors (RMSE) are multiplied by 100. OOS-R2 is theout-of-sample R2. MCS-p are p-values from the model confidence set. The model confidenceset is calculated for each group of 5 models that use the same regressors and sample.

(a) Full Sample (b) Pre-Crisis (b) Crisis

RMSE OOS-R2 MCS-p RMSE OOS-R2 MCS-p RMSE OOS-R2 MCS-p

(a) HAR RV

Linear 1.154 -4.852% 0.307 0.969 -0.100% 0.951 1.363 -8.328% 0.314BSR 1.272 -27.322% 0.090 0.984 -3.281% 0.021 1.576 -44.908% 0.109

CSR k = 1 1.131 -0.786% 1.000 0.968 0.009% 1.000 1.319 -1.368% 1.000CSR k = 2 1.136 -1.693% 0.183 0.968 -0.031% 0.951 1.328 -2.908% 0.191CSR k = 3 1.142 -2.701% 0.111 0.791 -0.123% 0.765 1.339 -4.587% 0.121

(b) LHAR RV

Linear 1.140 -2.304% 0.063 0.973 -0.946% 0.471 1.331 -3.297% 0.095BSR 1.222 -17.508% 0.033 0.986 -3.802% 0.001 1.479 -27.534% 0.060

CSR k = 1 1.125 0.288% 0.835 0.968 -0.030% 1.000 1.306 0.520% 0.740CSR k = 2 1.125 0.390% 1.000 0.969 -0.122% 0.471 1.305 0.764% 1.000CSR k = 3 1.125 0.307% 0.835 0.970 -0.273% 0.350 1.305 0.731% 0.954

(c) HAR RK

Linear 1.155 -5.031% 0.316 0.969 -0.149% 0.961 1.365 -8.601% 0.328BSR 1.390 -52.094% 0.061 0.982 -2.764% 0.016 1.796 -88.177% 0.065

CSR k = 1 1.132 -0.985% 1.000 0.968 0.034% 1.000 1.321 -1.731% 1.000CSR k = 2 1.138 -2.053% 0.187 0.968 0.019% 0.961 1.333 -3.568% 0.185CSR k = 3 1.146 -3.444% 0.095 0.968 -0.048% 0.833 1.348 -5.927% 0.099

(d) LHAR RK

Linear 1.140 -2.280% 0.046 0.972 -0.860% 0.481 1.331 -3.319% 0.075BSR 1.197 -12.858% 0.010 0.982 -2.961% 0.003 1.435 -20.098% 0.019

CSR k = 1 1.126 0.211% 0.887 0.968 0.023% 1.000 1.307 0.347% 0.891CSR k = 2 1.126 0.258% 1.000 0.968 -0.016% 0.716 1.307 0.457% 1.000CSR k = 3 1.126 0.142% 0.797 0.969 -0.114% 0.480 1.307 0.328% 0.891


The forecasting performance is substantially improved when complete subset re-

gression is used to produce forecasts from the band spectral regression. Even though

the performance is much better than for the unrestricted BSR, for the full sample CSR

only gives a positive OOS-R2 for the LHAR conditional variance. Complete subset

regression with the HAR model, that does not include leverage effects in the condi-

tional variance, does not produce a positive OOS-R2 over the full sample. The LHAR

model provides similar performance using realized variance or realized kernels. For

LHAR using realized kernels, CSR significantly outperforms the linear model and BSR

according to the model confidence set p-values, as both the linear model and the

band spectral regression are excluded from the 5% model confidence set for the full

sample.

The performance differences before and after July 2007 are striking. While there is

little evidence of forecast improvements from frequency dependence in the pre-crisis

subsample, the CSR forecasts based on the LHAR model work very well in the crisis

subsample. Thus, taking into account the different effects of variance movement

at different frequencies does improve forecasting during the crisis. These findings

confirm our in-sample results that the linear risk-return model is not well-specified

to describe the risk-return relation in the second subsample.

3.5 Conclusion

In this chapter we document strong evidence of frequency dependence in the relation

of conditional mean and conditional variance for daily returns on the S&P 500. Our

analysis is based on a band spectral regression approach with one-sided filtering,

which is robust to contemporaneous leverage and feedback effects and allows us to

obtain real-time forecast. The findings provide further evidence against a linear risk-

return relation. After July 2007 there is a distinct negative relation at high frequencies

with periods of around one week and less, which is not statistically significant before

the financial crisis. Taking into account this frequency dependence can improve

forecasting performance. Our results suggest that estimates of the risk-return relation

from linear models are both sensitive to the sampling frequency of the data and to

the state of the financial market.

As a consequence of the data sample used in this chapter, our analysis focuses

on fluctuations with monthly and weekly periods. In order to analyze frequency

dependence at lower frequencies, such as business cycle frequencies, different data

must be used, for example, monthly returns with variances from daily returns, for

which time series with much longer time span are available. Due to the focus on high

frequencies, our results are complementary to the literature on asset pricing with

different risk components, such as Adrian and Rosenberg (2008), that uses monthly

returns.

We have largely refrained from structural interpretations of the negative risk-

3.5. CONCLUSION 89

return relation found at certain frequencies. Volatility feedback effects are typically

assumed to have instantaneous impact, i.e., when expected variance increases the

prices drops immediately. If this is not true, but instead such adjustments take time

in the market, then the volatility feedback effect can explain the negative risk-return

relation that we find. Our findings are also consistent with the empirical evidence

from the literature on bear and bull markets, e.g., Maheu, McCurdy, and Song (2012),

where high volatility is typically associated with bear markets, i.e., periods with

declining prices. However, such bull and bear markets are typically identified as

market regimes with long duration, lasting several months or years, while we have

documented non-linear effects with much shorter duration.


3.6 References

Adrian, T., Rosenberg, J., 2008. Stock returns and volatility: Pricing the short-run and

long-run components of market risk. The Journal of Finance 63 (6), 2997–3030.

Andersen, T. G., Bollerslev, T., Diebold, F. X., Labys, P., 2003. Modeling and forecasting

realized volatility. Econometrica 71 (2), 579–625.

Ang, A., Hodrick, R. J., Xing, Y., Zhang, X., 2006. The cross-section of volatility and

expected returns. The Journal of Finance 61 (1), 259–299.

Ashley, R., Verbrugge, R. J., 2008. Frequency dependence in regression model coeffi-

cients: An alternative approach for modeling nonlinear dynamic relationships in

time series. Econometric Reviews 28 (1-3), 4–20.

Bali, T. G., Peng, L., 2006. Is there a risk–return trade-off? evidence from high-

frequency data. Journal of Applied Econometrics 21 (8), 1169–1198.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2008. Designing

realized kernels to measure the ex post variation of equity prices in the presence of

noise. Econometrica 76 (6), 1481–1536.

Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., Shephard, N., 2009. Realized kernels

in practice: Trades and quotes. The Econometrics Journal 12 (3), C1–C32.

Black, F., 1976. Studies of stock price volatility changes. In: Proceedings of the 1976

Meetings of the American Statistical Association, Business and Economics Statistics

Section. pp. 177–181.

Bollerslev, T., Litvinova, J., Tauchen, G., 2006. Leverage and volatility feedback effects

in high-frequency data. Journal of Financial Econometrics 4 (3), 353–384.

Bollerslev, T., Osterrieder, D., Sizova, N., Tauchen, G., 2013. Risk and return: Long-run

relations, fractional cointegration, and return predictability. Journal of Financial

Economics 108 (2), 409 – 424.

Bollerslev, T., Zhou, H., 2006. Volatility puzzles: a simple framework for gauging

return-volatility regressions. Journal of Econometrics 131 (1), 123–150.

Campbell, J. Y., Hentschel, L., 1992. No news is good news: An asymmetric model of

changing volatility in stock returns. Journal of Financial Economics 31 (3), 281–318.

Campbell, J. Y., Thompson, S. B., 2008. Predicting excess stock returns out of sample:

Can anything beat the historical average? Review of Financial Studies 21 (4), 1509–

1531.

Christensen, B. J., Dahl, C. M., Iglesias, E. M., 2012. Semiparametric inference in a

garch-in-mean model. Journal of Econometrics 167 (2), 458–472.

3.6. REFERENCES 91

Christensen, B. J., Nielsen, M. Ø., 2007. The effect of long memory in volatility on

stock market fluctuations. The Review of Economics and Statistics 89 (4), 684–700.

Christie, A. A., 1982. The stochastic behavior of common stock variances: Value,

leverage and interest rate effects. Journal of Financial Economics 10 (4), 407–432.

Corsi, F., 2009. A simple approximate long-memory model of realized volatility. Jour-

nal of Financial Econometrics 7 (2), 174–196.

Corsi, F., Renò, R., 2012. Discrete-time volatility forecasting with persistent leverage

effect and the link with continuous-time volatility modeling. Journal of Business &

Economic Statistics 30 (3), 368–380.

Doornik, J. A., 2007. Object-Oriented Matrix Programming Using Ox, 3rd ed. Timber-

lake Consultants Press and Oxford: www.doornik.com., London.

Elliott, G., Gargano, A., Timmermann, A., 2013. Complete subset regressions. Journal

of Econometrics 177 (2), 357–373.

Engle, R. F., 1974. Band spectrum regression. International Economic Review 15 (1),

1–11.

Engle, R. F., Lilien, D. M., Robins, R. P., March 1987. Estimating time varying risk

premia in the term structure: The arch-m model. Econometrica 55 (2), 391–407.

French, K. R., Schwert, G. W., Stambaugh, R. F., 1987. Expected stock returns and

volatility. Journal of Financial Economics 19 (1), 3–29.

Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. There is a risk-return trade-off after all.


Glosten, L. R., Jagannathan, R., Runkle, D. E., 1993. On the relation between the

expected value and the volatility of the nominal excess return on stocks. The

journal of finance 48 (5), 1779–1801.

Hansen, P. R., Lunde, A., 2006. Realized variance and market microstructure noise.

Journal of Business & Economic Statistics 24 (2), 127–161.

Hansen, P. R., Lunde, A., Nason, J. M., 03 2011. The model confidence set. Economet-

rica 79 (2), 453–497.

Harrison, P., Zhang, H. H., 1999. An investigation of the risk and return relation at

long horizons. Review of Economics and Statistics 81 (3), 399–408.

Harvey, A. C., 1978. Linear regression in the frequency domain. International Eco-

nomic Review 19 (2), 507–512.


Lettau, M., Ludvigson, S., 2010. Measuring and modeling variation in the risk- return

tradeoff. In: Ait-Sahalia, Y., Hansen, L.-P. (Eds.), Handbook of Financial Economet-

rics. Vol. 1. Elsevier Science B.V., North Holland, Amsterdam, pp. 617–690.

Maheu, J. M., McCurdy, T. H., Song, Y., 2012. Components of bull and bear markets:

bull corrections and bear rallies. Journal of Business & Economic Statistics 30 (3),

391–403.

Merton, R. C., 1973. An intertemporal capital asset pricing model. Econometrica,

867–887.

Merton, R. C., 1980. On estimating the expected return on the market: An exploratory

investigation. Journal of Financial Economics 8 (4), 323–361.

Murphy, K. M., Topel, R. H., 2002. Estimation and inference in two-step econometric

models. Journal of Business & Economic Statistics 20 (1), 88–97.

Nelson, D. B., 1991. Conditional heteroskedasticity in asset returns: A new approach.

Econometrica, 347–370.

Newey, W. K., West, K. D., 1987. A simple, positive semi-definite, heteroskedasticity

and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–708.

Pagan, A., 1984. Econometric issues in the analysis of regressions with generated

regressors. International Economic Review, 221–247.

Pagan, A., Ullah, A., 1988. The econometric analysis of models with risk terms. Journal

of Applied Econometrics 3 (2), 87–105.

Rapach, D. E., Strauss, J. K., Zhou, G., 2010. Out-of-sample equity premium prediction:

Combination forecasts and links to the real economy. Review of Financial Studies

23 (2), 821–862.

Rossi, A., Timmermann, A., 2011. What is the shape of the risk-return relation? Work-

ing paper, UCSD.

Stock, J. H., Watson, M. W., 1999. Business cycle fluctuations in us macroeconomic

time series. In: Taylor, J., Woodford, M. (Eds.), Handbook of Macroeconomics. Vol. 1.

Amsterdam: Elsevier, pp. 3–64.

Tan, H. B., Ashley, R., 1999. Detection and modeling of regression parameter variation

across frequencies. Macroeconomic Dynamics 3 (01), 69–83.



Wu, G., 2001. The determinants of asymmetric volatility. Review of Financial Studies

14 (3), 837–859.

3.6. REFERENCES 93

Yu, J., 2005. On leverage in a stochastic volatility model. Journal of Econometrics

127 (2), 165–178.

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