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Financial Markets and Portfolio Management

22.02.2024

A simple test of misspecification for linear asset pricing models

verfasst von: Antoine Giannetti

Erschienen in: Financial Markets and Portfolio Management

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Abstract

A fundamental implication of asset pricing theory is that investors must earn risk-premiums for bearing exposure to systematic risk. The two-pass cross-sectional regression is a popular approach for risk-premium estimation. The empirical literature has found that this approach often delivers estimates that significantly differ from their time-series counterparts. The paper explores a test of model misspecification that exploits the difference between cross-sectional and time-series risk-premium estimates. The suggested approach complements traditional misspecification tests and may be applied as an alternative to the deployment of misspecification-robust standard errors to test risk-premium significance.

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1 Joint significance of the risk-premiums may be tested with a standard F test (see Goyal (2012)).

For instance, while testing the CAPM, Black, Jensen and Scholes (1972) report a zero-beta rate significantly larger than the average risk-free rate and a cross-sectional slope significantly smaller than the average market risk premium.

When some or all the risk factors included in the model are not traded assets (macroeconomic variables like inflation, GDP growth, etc. fill the bill), the researcher may elect to substitute mimicking portfolios of traded assets in lieu of the non-traded factors. See Breeden (1979).

Lewellen, Shanken and Nagel (2010) argue that one may reasonably expect that “the risk premium associated with a factor portfolio should be the factor’s expected excess return.”.

A closely related but distinct issue is the useless factor problem. Useless factors are factors that are uncorrelated with test-assets’ returns. The useless factor problem refers to the Kan and Zhang (1999) finding that low correlation between risk factors and test-assets results in the singularity of the factor loadings matrix and faulty inference on risk premiums. The problem is most acute for non-traded factors like macroeconomic variables.

Incidentally, Kleibergen and Zhan (2020) pp. 511 also acknowledge the importance of the factor model being correctly specified.

In their “Prescription 4,” Lewellen, Nagel and Shaken (2010) argue that “a GLS cross-sectional regression, when a traded factor is included as a test-asset, is similar to the time-series approach of Black, Jensen and Scholes (1972) and Gibbons, Ross and Shanken (1989).”

For instance, in the case of the Fama–French 3 factor model, the small growth portfolio is substantially affecting the outcome of the asset pricing test.

If the i.i.d. assumption for returns is violated, the GLS estimator is not fully efficient (see Cochrane (2005)). In this case, one may proceed by computing a robust Hausman test (see Cameron and Trivedi (2005).

Roll and Ross (1994) is an early reference. Recently, in dealing with the omitted variable problem in large cross sections, Giglio and Xiu (2021) emphasize the relevance of identifying a cross-sectional market risk premium close to its time-series average.

Because the risk factors are traded they must price themselves. Hence, for the \(K\) factors represented by assets \(i=N+1,\dots ,N+K\), one must have \({a}_{i}=0\) and \({\beta }_{i,k}=1.0\) for \(i=k\), \({\beta }_{i,k}=0\) for \(i\ne k\).

This setting is referred to as conditional homoscedasticity since the asset returns are independent and identically distributed conditional on the time series of the risk factors (Shanken (1992) Assumption 1). Violations of conditional homoscedasticity are theoretically explored in Jagannathan and Wang (1998) and Kan, Robotti and Shanken (2013a, b).

The \(\left(N+K\right)\) square covariance matrix \(\Sigma\) of the \(N+K\) augmented test-assets as well as \({\Sigma }_{f}\) the \(K\times K\) covariance matrix of the K factors are assumed full-rank.

When spread-portfolios like SMB and HML are used as factors, Ferson, Sarkissian and Simin (1999) argue that the market factor Rm-Rf is essentially included to capture the grand mean of the asset returns.

More specifically, the authors point out that when the asset pricing in (2) holds, risk-premiums estimators are independent of the weight matrix used to perform second-pass estimation.

See Lewellen, Nagel and Shanken (2010) Proposition 4: “If a proposed factor is a traded portfolio, include it as one of the test-assets on the left-hand side of the cross-sectional regression.”

This only holds true if factor loadings are nonzero, i.e., risk factors are not useless. In the context of this paper, a misspecified model is one such that risk factors are useful (i.e., risk premiums are statistically nonzero) but they also are not equal to expected factors’ realizations.

Technically, from expression (8.34) in Cameron and Trivedi (2005), the Hausman test only requires that \(plim\left({\widehat{\uplambda }}_{OLS}-{\widehat{\uplambda }}_{GLS}\right)=0\) under the null hypothesis and \(plim\left({\widehat{\uplambda }}_{OLS}-{\widehat{\uplambda }}_{GLS}\right)\ne 0\) under the alternative. A stronger requirement imposed by Hausman (1978) is that \({\widehat{\uplambda }}_{GLS}\) should not only be consistent but also efficient under the null hypothesis. The latter is true under conditional homoscedasticity of the assets returns.

Harvey (1990) pp. 148 argues that “unlike a test of specification, therefore, a test of misspecification is constructed with no clear alternative in mind. As such, it is a procedure designed for assessing the goodness of fit of a model implied by a particular maintained hypothesis.”.

See Greene (2012) pp. 235 for details.

While the OLS estimation allows for a constant to account for some misspecification, the GLS estimation enforces the null hypothesis in (2). Hence, it is performed without a constant.

See Greene (2012) pp. 130 Theorem 5.1.

Details about the factors are found in Fama and French (1993), (1996) and (2015).

Also see Jagannathan and Wang (1998).

Explicitly, it does not rely on the main Hausman (1978) result that \(Var\left({\widehat{\uplambda }}_{OLS}-{\widehat{\uplambda }}_{GLS}\right)=Var\left({\widehat{\uplambda }}_{OLS}\right)-Var\left({\widehat{\uplambda }}_{GLS}\right)\) when the GLS estimator is fully efficient. See Cameron and Trivedi (2005) pp. 378.

Cochrane (2005) pp. 241–242 raises this point in the context of GMM estimation of linear asset pricing models under less restrictive assumptions about the return generating process. Shanken and Zhou (2007) have numerically verified that the GLS is the optimal GMM estimator in the sequential sense of Ogaki (1993).

From an earlier draft of the paper, simulations for data ending in December 2018 as opposed to December 2022 show power levels comparable to those of the FF3 factor model (0.722 for the 5 FF on the 32 P compared to 0.662 for the 3 FF on the 25 P with T = 960, no constant).

Observe that SMB displays an average mean return of 0.19 for the FF 3 factor model and 0.22 for the FF 5 factor model. The reason is SMB factor construction is slightly different across the two models. See Ken French webpage for further details.

Those risk premium point estimates are similar in magnitude to those reported by Kan et al. (2013b) Table IA.I.

See Appendix Eq. 10 for standard-error calculation formulas.

For a regression without a constant, the cross-sectional R² uses weighted sums of squared values of the dependent variable in the denominator and not squared deviations from the cross-sectional mean excess return (see Kan et al. (2013a, b) p. 2640).

Davidson and MacKinnon (1987) refers to those as "implicit alternative hypothesis".

See Kan, Robotti and Shanken (2013a, b) pp. 2633.

Incidentally, note that simply augmenting the test-assets by the traded factors is not sufficient for the CAPM OLS cross-sectional market risk-premium to match its time-series counterparts. Additionally, GLS must be performed on the factor-augmented data set to reach this outcome.

In their Table 3, Hou and Kimmel (2010) display similar results for the augmented FF 3 factor model estimated using both OLS and GLS.

Shanken and Zhou (2007) Table 13 fails to reject a test a difference that OLS and GLS risk-premium estimates are equal for the FF 3 factor model tested on the 25 P with a p value of 0.07. Note that their test is performed without augmentation of the test-assets by the risk factors.

A mimicking portfolio à la Breeden (1979)) could be used. Unfortunately, because it is a linear combination of the test-assets, such a portfolio would result in the singularity of the augmented test-assets covariance matrix, an undesirable outcome.

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Titel: A simple test of misspecification for linear asset pricing models
verfasst von: Antoine Giannetti
Publikationsdatum: 22.02.2024
Verlag: Springer US
Erschienen in: Financial Markets and Portfolio Management
Print ISSN: 1934-4554
Elektronische ISSN: 2373-8529
DOI: https://doi.org/10.1007/s11408-024-00445-6