The Pricing Error Function

27 Pages Posted: 27 Mar 2008

See all articles by Huntley Schaller

Huntley Schaller

Carleton University - Department of Economics

Lynda Khalaf

Carleton University

Date Written: March 21, 2008


Measurement error can lead to biased parameter estimates and misleading test results. In cross-sectional asset pricing models (e.g., CAPM, Fama-French, good beta/bad beta), measurement error can easily arise in calculating portfolio betas. A common approach to the errors-in-variables problem is instrumental variables. Econometrically, conditional asset pricing models effectively serve as instrumental variables (IV) estimators of asset pricing models.

A large body of econometric research has shown that IV estimation performs poorly with weak instruments. Weak instruments can lead to highly misleading parameter estimates and standard errors, so that inference is severely distorted. Biases and size distortions due to weak instruments often do not vanish in large samples. Previous work (based on econometric theory and Monte Carlo simulations) has shown that the confidence intervals can be so misleading that they do not include the true values of the model parameters. Moreover, under plausible circumstances, the usual techniques to correct these problems (including bootstrap techniques) also fail.

The fundamental reason for the measurement error and weak instrument problems is weak identification. When parameters are not identifiable, it is important to allow for unbounded confidence intervals. Usual confidence intervals of the form {estimate ± asymptotic cut-off point × asymptotic standard error} are bounded by construction. When a parameter is hard to identify from the available data, the fact that any value in its parameter space is equally acceptable should be reflected in the confidence interval. Weak identification should lead to diffuse confidence sets that alert the researcher to the problem. Unfortunately, if usual confidence intervals are constructed when estimating weakly-identified parameters, the researcher may obtain very tight confidence intervals that are focused on "wrong" values.

In traditional estimation methodology, a point estimate is found first, and confidence intervals are then constructed. Our approach proceeds in reverse: first, we build a confidence set; then we obtain a point estimate from the confidence set. The confidence set is obtained by "inverting" an artificial regression-based test designed to achieve type-I error control (at the desired level, say 5%) whether the instruments are weak or strong. The pricing error function given the data Y is denoted e(Y; à3B3), where à3B3 represents the parameters of the asset pricing model. Under the null hypothesis, if the pricing errors are regressed on the instruments X, the coefficients should be zero. This condition reflects the same orthogonality principle that underlies GMM. Testing this condition can be interpreted as assessing whether additional explanatory variables hold further information on returns that is not explained by the betas.

This paper examines the importance of measurement error and weak instruments for existing tests of cross-sectional asset pricing models. Our paper is constructive, in the sense that: 1) we develop empirical techniques that allow researchers to determine whether measurement error or weak instruments problems are important; 2) we show that some existing asset pricing models are consistent with the data; and 3) we develop empirical techniques that are robust to measurement error and weak instruments.

Keywords: asset pricing, measurement error, weak instruments, testing

JEL Classification: G12

Suggested Citation

Schaller, Huntley and Khalaf, Lynda, The Pricing Error Function (March 21, 2008). Available at SSRN: or

Huntley Schaller

Carleton University - Department of Economics ( email )

1125 Colonel By Drive
Ottawa, Ontario K1S 5B6
613-520-3751 (Phone)
613-520-3906 (Fax)

Lynda Khalaf (Contact Author)

Carleton University ( email )

1125 colonel By Drive
Ottawa, Ontario K1S 5B6

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