Asset Pricing Model Uncertainty: A Tradeoff between Bias and Variance

• DOI: http://doi.org/10.1111/irfi.12112
• Date: 01 June 2017
• Published date: 01 June 2017
• Author: Qing Zhou

Asset Pricing Model Uncertainty: A

Tradeoff between Bias and Variance

QING ZHOU

†,‡

†

UQ Business School, The University of Queensland, Queensland, Australia and

‡

School of Management, Xi’an Jiaotong University, China

ABSTRACT

This paper draws a parallel between model combination and the mean–

variance tradeoff in Modern Portfolio Theory (Markowitz 1952) and proposes

a bias–variance tradeoff framework. Building on the bias–variance tradeoff

framework, the paper proposes a Model Portfolio Approach (MPA) and a

Global Minimum Variance(GMV) weighting scheme to mitigate asset pricing

model uncertainty. Using a well-conditioned pricing covariance estimator,

the proposed approach improves out-of-sample pricing performance over six

widely used asset pricing models, a model selection method and two most

popular benchmarks in existing model combination studies, that is, the

simple arithmetic average (“1/N”) and Ordinary Least Square (OLS) weighting

methods.

JEL Codes: G11; G12; D81; E37

I. INTRODUCTION

Out-of-sample asset pricing model uncertainty arises primarily from economic

uncertainty (asset return volatility) and modeling uncertainty (model specifica-

tion uncertainty and parameter uncertainty). While the economic uncertainty

affecting the modeling process is irreducible, and thus appears as a lower bound

on out-of-sample pricing uncertainty, model and parameter uncertainty can be

reduced. Hence, reducing modeling uncertainty is the major task in out of-

sample asset pricing modeling. Out-of-sample performance relies largely on the

tradeoff between bias and variance. As asset pricing models are relatively

unbiased and the large out of-sample pricing error is due primarily to the variance

(Simin 2008), the diversification of asset pricing variability is critical to the

improvement of equilibrium asset pricing model out of-sample pricing

performance. Inspired by the mean–variance tradeoff framework in Modern

Portfolio Theory (Markowitz 1952), which states that a properly weighted

portfolio of assets diversifies asset return uncertainty, I propose a bias–variance

tradeoff framework for constructing an optimal “model portfolio”. Under this

framework, I derive a Model Portfolio Approach (MPA) and a Global Minimum

© 2016 International Review of Finance Ltd. 2016

International Review of Finance, 17:2, 2017: pp. 289–324

DOI: 10.1111/irfi.12112

Variance (GMV) weighting scheme for pooling a set of individual asset pricing

models to diversify asset pricing model uncertainty.

Model selection has long been widely used to mitigate asset pricing model

uncertainty. However, the choice of one asset pricing model to the exclusion of

another is an inherently misguided strategy (O’Doherty et al. 2012). The

underlying assumption of both non-Bayesian and Bayesian model selection is

that the model space is complete and, hence, the true model is in the model

space. However, the finance literature is far from consensus on whether a

particular asset pricing model is a true model. Worse, omitting useful informa-

tion in other, abandoned models is detrimental to accurate asset pricing.

Moreover, sampling error is also a concern. The best performing model for one

sample may prove to be the worst for another sample. Despite the winner’s curse

problem (see Hansen 2009), unobservable changes in the economic structure

may also increase the risk of excluding the seemingly worse model under a given

regime. This situation is akin to the six blind monks who encountered an

elephant for the first time–each monk grasping a different part of the beast and

coming to a wholly different conclusion as to what an elephant is but no one

giving a true picture of the elephant. Disciples of different pricing models have

captured different features of the same financial asset price, but none of them

has a completely true description. A collection of all opinions provides a closer

illustration of the truth.

Model combination is another common approach for addressing model

uncertainty. The first use of model combination in econometric forecasting was

by Bates and Granger (1969), then extended by Granger and Ramanathan

(1984), which then spawned a large volume of literature. Some excellent reviews

include Granger (1989), Clemen (1989), Diebold and Lopez (1996), Clements

et al. (2002), Timmermann (2006) and Stock and Watson (2006). Recently,

forecast combinations have received renewed attention in the macroeconomic

forecasting literature with respect to forecasting inflation and real output growth

(e.g., Stock and Watson 2003). Des pite the increasing popularity of forecast

combination in economic forecasting, applications remain relatively scarce in

the finance forecasting literature. Only in recent several years has forecasting

combination been employed in asset pricing studies (e.g., Rapach et al. 2010;

O’Doherty et al. 2012; Durham and Geweke 2014). There is no consensus on

model combination weighting. A plethora of weighting schemes has been

developed in both non-Bayesian and Bayesian econometrics. However, it appears

that the simple arithmetic average weighting method (i.e., the “1/N”rule)

outperforms the existing, more complicated weights in most cases. Stock and

Watson (2004) find that among all the competing weights, the simple “1/N”rule

yields smallest mean squared forecasting error (MSFE). This “1/N”puzzle has

long haunted forecast combination practice. The common explanation for this

puzzle is that the weight estimation error is too large to be offset by the gains

from diversification because of the small effective sample size.

1

1 The number of models“N”is large relative to the sample size used to estimate the weights.

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© 2016 International Review of Finance Ltd. 2016290

I propose an MPA to diversify the out-of-sample mispricing uncertainty of

individual asset pricing models. My approach is in the same spirit as Modern

Portfolio Theory for asset allocation (Markowitz 1952). The core inspiration of

portfolio theory is that idiosyncratic risk can be diversified by optimally pooling

a set of assets and that the portfolio of assets will provide higher risk-adjusted

return than any individual asset. Mapping this approach to the construction of

an asset portfolio that is derived under the mean–variance framework using a

tradeoff between the return and the risk, I derive the optimal “model portfolio”

through a tradeoff between bias and variance. This bias–variance tradeoff is

critical to the success of out-of-sample prediction.

2

The optimal weighting in this

study is a by-product of the optimization of the objective function along the

bias–variance efficient frontier. Because the bias and variance of the out-

of-sample forecasting error are unobservable, the optimization is actually

performed along the estimated frontier rather than the true frontier. Considering

the frontier estimation error problem, I propose a GMV weighting scheme,

which uses the weights of the global minimum variance portfolio. The GMV

weighting in its basic form can unify the Granger–Bates–Ramanathan optimal

Ordinary Least Squares (OLS) weighting scheme. Moreover, by utilizing recent

developments in large-scale covariance matrix estimation techniques, GMV

weighting can be used to address the small effective sample size problem when

the number of models is large. The traditional Granger–Bates–Ramanathan OLS

weighting has theoretical optimality, but because of estimation error in small

samples, empirically, it usually under-performs other weighting schemes.

To justify the performance of the proposed MPA, I further provide simulation

and empirical analyses. The simulation shows that, when the true model is in the

model set, the true model performs best, and the MPA has the closest perfor-

mance to the true model and outperforms all other individual false models.

When the sample size increases, the MPA assigns a weight closer to 1 to the true

model. When the true model is not in the model set, the MPA outperforms all

individual asset pricing models. The MPA also performs better than the “1/N”

and OLS weightings. However, when I enlarge the sample size, the weights of

the individual asset pricing models become more equal and my approach obtains

closer performance to “1/N”weighting. The simulation also provides an explana-

tion for the improvement of combined asset pricing models in out-of-sample

pricing. In sum, when the true model is not the in the model set, the superior

performance of the MPA is most pronounced when the out-of-sample forecasting

errors are highly correlated and individual models perform differently. In my

empirical analyses, I include six popular asset pricing models: (i) the capital asset

pricing model (CAPM); (ii) a linear version of the consumption CAPM (CCAPM);

(iii) the Jagannathan and Wang (1996) unconditional version of conditional

CAPM (JW); (iv) a linear version of Campbell’s (1996) log-linear pricing model

(CAMP); (v) the Fama–French three-factor model (FF3); and (vi) the Fama–French

five-factor pricing model (FF5). I provide details of the six pricing models in

2 See Geman et al. (1992)

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