Forecasting Asset Returns in State Space Models Dominik Boos Prof. Dr. Alex Keel Prof. Paul

149 Pages·2011·0.69 MB·English

Forecasting Asset Returns in State Space Models
D I S S E R T A T I O N
of the University St. Gallen,
Graduate School of Business Administration,
Economics, Law and Social Sciences (HSG)
to obtain the title of
Doctor of Economics
submitted by
Dominik Boos
from
Amden (St. Gallen)
Approved on the application of
Prof. Dr. Alex Keel
and
Prof. Paul S¨oderlind, PhD
Dissertation Nr. 3812
ADAG Copy AG, St. Gallen 2011
The University of St. Gallen, Graduate School of Business Administration,
Economics,LawandSocialSciences(HSG)herebyconsentstotheprintingof
the present dissertation, without hereby expressing any opinion on the views
herein expressed.
St. Gallen, October 26, 2010
The President:
Prof. Ernst Mohr, PhD
Acknowledgements
FirstandforemostIamverygratefultothemembersofmythesiscommittee,
Prof. Dr. Alex Keel and Prof. Paul S¨oderlind, PhD for their willingness to
supportmychosentopicandtoundertakethe tasksofadvisingme inwriting
this thesis.
I also owe many thanks to Lukas W¨ager for numerous discussions on the
Kalman filter and interest rate models as well as for the proofreading of my
dissertation. Additionalthanks gotoJackCorrigallfor hiscarefulproofread-
ing and language quality control.
Finally,IoweverymuchtoBirgitwhohasbeenparticularlyunderstanding
and supportive during all stages of my dissertation.
Winterthur, January 2011 Dominik Boos
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Summary
Expectations about the future evolution of the economy are of immense im-
portance for taking the right decisions in a stochastic environment. Econo-
metricians have long been studying forecasting techniques to this end. Most
of this work is based on regressiontechniques such as OLS or GMM.
I propose a different approach: state-space models and their estimation
by means of maximum likelihood using the Kalman filter. While the two
techniques are often identical in an environment with clean data; state-space
models are clearly superior if the observed data is affected by measurement
error or displays a seasonal pattern. In this case, state-space models allow
the separation of the true underlying signal from the measurement noise. As
only the signal is relevant for prediction, this can considerably improve the
quality of the forecast.
In particular, I use the state space framework to estimate affine yield
curve models and find that the implied return forecasts for long bonds is
much more reliable than that implied by a linear regression, although the
implied insample R2 is lower. Moreover,I detect substantial predictability of
long/short portfolios not properly revealed by a linear regression.
I then generalize the affine yield curve models such that they can include
persistent shocks or state variables not spanned by yields. Firstly, these un-
spanned factor models are used to further improve the yield-curve forecast
by including expected inflationasanadditionalstate variable. Inthis model,
the R2 of the annual term premium forecast is above 30 percent. Secondly, I
buildajointstock-bondmodelthatmergestheyieldcurvemodelwithastock
marketmodelusingtheprice-dividendratioasanadditionalvariable. Thisis
achievedby linearizationusing the Campbell-Shiller approximation. Thirdly,
the cross-section of assets is enlarged by including size and book-to-market
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sorted portfolios. This model provides evidence for substantial variation in
the dividend growth rate. Once the model captures this feature, it is able to
explain a large fraction of the value premium by a higher exposure of value
stocks to the single persistent shock of the system.
Finally, this thesis uses rank-reduction techniques to explore the return
predictability pattern. This analysis provides strong evidence for at least
two independent predictability factors: the term premium and the equity
premium.
Contents
1 Introduction 1
2 State Space Models and Kalman Filter 7
2.1 State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Regression Forecasts of Bond Returns 15
3.1 Data and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.3 Measurement Error. . . . . . . . . . . . . . . . . . . . . 18
3.2 Bond Risk Premia . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Forecasting Long-Only Bond Returns . . . . . . . . . . 19
3.2.2 Graphical Analysis of the RegressionCoefficients . . . . 23
3.2.3 Is the Factor Spanned by Level, Slope and Curvature? . 25
3.3 A Second Risk Premium for Slope Risk . . . . . . . . . . . . . 27
3.3.1 Duration-Neutral Portfolios . . . . . . . . . . . . . . . . 27
3.3.2 Forecasting Slope Returns . . . . . . . . . . . . . . . . . 31
3.4 Curvature and other Risk Premia . . . . . . . . . . . . . . . . . 32
3.5 Summary of the Empirical Findings . . . . . . . . . . . . . . . 33
3.A Appendix: Robustness of Slope Return Forecasts . . . . . . . . 34
3.A.1 Measurement Error and Lagged Instruments . . . . . . 34
3.A.2 Different Data . . . . . . . . . . . . . . . . . . . . . . . 34
3.A.3 Data Range . . . . . . . . . . . . . . . . . . . . . . . . . 35
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4 Predictability in an Affine Model 37
4.1 Constructing Affine Yield Curve Models . . . . . . . . . . . . . 38
4.1.1 Prices and Yields . . . . . . . . . . . . . . . . . . . . . . 38
4.1.2 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.3 Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.4 Restricting the Number of Forecastable Factors . . . . 42
4.2 Parametrization. . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Rotating the Factors . . . . . . . . . . . . . . . . . . . . 43
4.2.2 An Almost General Parametrization . . . . . . . . . . . 44
4.3 Kalman Filter Estimation . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Is Kalman Filter Estimation more Efficient? . . . . . . 48
4.3.2 Model Selection: Testing for the Rank of λ . . . . . . . 49
1
4.3.3 Predicting the Excess Returns on Long Bonds. . . . . . 53
4.3.4 Predicting Slope Portfolios . . . . . . . . . . . . . . . . 57
4.3.5 Analysis of the Measurement Error . . . . . . . . . . . 59
4.4 Infinite Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Summary of the Empirical Findings . . . . . . . . . . . . . . . 64
4.A Appendix: Derivatives . . . . . . . . . . . . . . . . . . . . . . . 65
4.A.1 Derivatives of B . . . . . . . . . . . . . . . . . . . . . . 65
4.A.2 Derivatives of A . . . . . . . . . . . . . . . . . . . . . . 66
5 More about Affine Models 69
5.1 Expected Returns . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Unspanned Macro Factors . . . . . . . . . . . . . . . . . . . . 72
5.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Integrating Inflation Forecasts into an Affine Model . . 74
5.3 Shocks to Persistent Variables . . . . . . . . . . . . . . . . . . . 80
5.3.1 Incorporating Persistent Shocks . . . . . . . . . . . . . . 81
5.3.2 Example: A Simple Stock Bond Model . . . . . . . . . 82
5.4 Summary of the Empirical Findings . . . . . . . . . . . . . . . 85
6 Dividends and Returns 87
6.1 Campbell-Shiller Approximation . . . . . . . . . . . . . . . . . 88
6.2 State Space Framework with Noisy Dividends . . . . . . . . . . 90
6.2.1 Observable Variables . . . . . . . . . . . . . . . . . . . . 92
6.2.2 Specification of the Measurement Error . . . . . . . . . 93
6.2.3 Augmented State Equation . . . . . . . . . . . . . . . . 93
CONTENTS vii
6.2.4 Measurement Equation . . . . . . . . . . . . . . . . . . 94
6.3 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.1 Judging the Growth Rate Forecast . . . . . . . . . . . . 96
6.3.2 Empirical Results. . . . . . . . . . . . . . . . . . . . . . 97
7 Joint Stock-Bond Market Models 99
7.1 Unspanned Factors Again . . . . . . . . . . . . . . . . . . . . . 99
7.2 Campbell-Shiller Approximation and Affine Pricing Kernel . . 100
7.3 The Stock-Bond Framework . . . . . . . . . . . . . . . . . . . . 101
7.3.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . 102
7.3.2 Definition of the State Space Model . . . . . . . . . . . 102
7.3.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . 103
7.3.4 Enlarging the Cross-Section of Returns . . . . . . . . . 103
7.3.5 Overidentifying Macro Variables . . . . . . . . . . . . . 105
7.4 Increasing the Cross-Sectionof Prices . . . . . . . . . . . . . . 105
7.4.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4.2 The Predictability Pattern . . . . . . . . . . . . . . . . 106
7.5 Cross-Sectionof Returns . . . . . . . . . . . . . . . . . . . . . . 107
7.5.1 Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . 107
7.5.2 Loadings to the Priced Shocks . . . . . . . . . . . . . . 109
7.6 Statistical and other Problems . . . . . . . . . . . . . . . . . . 110
7.6.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.6.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.6.3 Analysis of the Measurement Error . . . . . . . . . . . . 112
7.7 Summary of the Empirical Findings . . . . . . . . . . . . . . . 113
8 Conclusion 115
A Description of the Data Sets 119
B Optimization 123
2.1 An Alternative Optimization Procedure . . . . . . . . . . . . . 123
2.2 Numerical Stability and Convergence . . . . . . . . . . . . . . . 124
C Dividends in Continuous Time 125
3.1 Definition of the Model . . . . . . . . . . . . . . . . . . . . . . 126
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