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A.  The ICAPM

Multifactor models can be theoretically motivated by the ICAPM of Merton (1973), in which the state‐variable‐mimicking portfolios arise as additional factors (see also, e.g., Breeden 1979; Campbell 1993, 1996; Fama 1996). Merton’s ICAPM is a generalization of the CAPM, and multifactor efficiency generalizes mean‐variance efficiency. The assumptions imply that ICAPM investors do not only dislike wealth uncertainty but also want to hedge more specific uncertainties about future consumption‐investment opportunities for which they use the state‐variable‐mimicking portfolios. These hedging demands typically lead to optimal ME portfolios that are not mean‐variance efficient (MVE).

The investment universe consists of a risk‐free asset (with risk‐free rate r_f), N test portfolios (with excess returns over the risk‐free rate denoted by the matrix R_N), a value‐weighted market portfolio (with excess returns denoted by the vector R_p), plus a set of K state‐variable‐mimicking portfolios (with excess returns over the risk‐free rate denoted by the matrix R_K).

A sufficient condition to characterize ME portfolios is that the joint distribution of the returns of all portfolios in the investment universe is multivariate normal (see Merton [1973], Long [1974], and Fama [1996]). With this distributional assumption, we can write where such that μ_N, μ_p, and μ_K are the mean excess returns of R_N, R_p, and R_K, respectively; V_N is the variance‐covariance matrix of R_N; V_p is the variance of R_p; V_N,p is the covariance matrix between R_N and R_p; is the covariance matrix between R_N and R_K; and finally is the covariance matrix between R_p and R_K. The mean and variance‐covariance matrix of are denoted by and , respectively.

Given this normality assumption, ICAPM investors choose an ME portfolio such that the portfolio weights

Assuming a risk‐free asset exists, the returns on all MVE portfolios are spanned by the returns on any one MVE portfolio in combination with the risk‐free rate (see Black 1972). Equivalently, the returns on any linearly independent ME portfolios span the returns on all ME portfolios in combination with the risk‐free asset. Finally, any portfolio of ME portfolios is itself ME (see, e.g., Jobson and Korkie 1985; Huberman, Kandel, and Stambaugh 1987).

These spanning portfolios can be characterized by K state‐variable‐mimicking portfolios R_K plus any MVE portfolio (see Fama 1996). In this article, we consider three popular state‐variable‐mimicking portfolios. First, we use the two Fama and French (1992, 1993) factors, namely, the SMB (small‐minus‐big) and HML (high‐minus‐low) portfolios, which capture the difference between small‐ versus large‐market‐capitalization firms and between high versus low BM firms, respectively. Second, we use the UMD (up‐minus‐down) momentum factor, which captures the difference between high‐ and low‐prior‐return portfolios.

The portfolio needed for the spanning set of all ME portfolios is the MVE‐tangency portfolio R_MVE, a weighted average of all portfolios in (see, e.g., Merton 1972), where the vector is the weight of the tangency portfolio on the portfolios in in equation (1). Intuitively, ICAPM investors use the MVE portfolio for an optimal trade‐off between expected return and non‐state‐variable‐return variance and combine the tangency portfolio with the K state‐variable‐mimicking portfolios to hedge uncertainty about future consumption‐investment opportunities. As a result, each ME portfolio can be written as the weighted average of the tangency portfolio and the state‐variable‐mimicking portfolios. The complete set of excess returns of the ME spanning portfolios—the MVE tangency portfolio plus the K mimicking portfolios—is denoted by .

The testable implication of the ICAPM is that the market portfolio is ME, thus, an exact linear combination of the spanning portfolios. Consider the regression of the market portfolio R_p on the ME spanning portfolios in , where denotes the loading of the market portfolio on the spanning portfolios in . Equation (5) decomposes the excess market return into two parts: first, the part of the excess market return R_p that loads on and second, the regression residuals ε_p, which capture the “idiosyncratic” component of the excess market return and have variance . The first component in equation (5), denoted by , is a weighted average of the ME spanning portfolios in using as weights. As a result, is also ME.5 The second component, ε_p, will be independent from the first component, such that , where denotes the variance of .

The ICAPM holds if the market portfolio is ME, which implies that regression equation (5) has a perfect fit, such that or . In that case, the correlation between R_p and , denoted by ρ_p, equals 100% (see Fama [1996] for the ICAPM and Fama [1976] and Roll [1977] for the CAPM).

If the ICAPM holds and the market portfolio is indeed ME, the market portfolio can replace as the spanning portfolio. Therefore, another pricing restriction is that the regression constant equals zero in the regression of the test portfolios on the market portfolio and the mimicking portfolios (see Huberman and Kandel 1987): where ι_T denotes a vector of ones, α_N are the regression constants, the regression residuals ε_N have variance , and . We denote the proportion of explained variation of R_N in regression equation (6) by .

B.  Evaluation of Asset‐Pricing Model Performance: A New Inefficiency Measure

In this subsection, we introduce a new inefficiency measure that can be used to test the ICAPM implication that the market portfolio is ME. We also briefly describe three other well‐known inefficiency measures. Each measure provides a different metric of how close the market portfolio is to being ME and thus allows a different evaluation of the pricing performance. Also, even if the asset‐pricing model as specified in equation (6) holds exactly, nonzero pricing errors could result from the fact that the mimicking portfolios contain estimation risk. Furthermore, the market portfolio R_p may be an imperfect proxy; that is, the empirical counterpart to the theoretical stochastic discount factor is error ridden (see, e.g., Roll 1977). If the asset‐pricing model is viewed as an approximation and does not correctly price all portfolios, we still want to evaluate its performance and are interested in how good an approximation it is. As a result, we will discuss the interpretation of each inefficiency measure under the alternative where the market portfolio is not exactly ME.

The new inefficiency measure, ρ_p, is the maximum correlation between the market portfolio and any ME portfolio. This measure generalizes for multifactor models the notion of how close the market portfolio is to the efficient frontier. The maximum correlation of the market portfolio and any MVE portfolio is discussed in, for example, Kandel and Stambaugh (1987), Shanken (1987b), Gibbons et al. (1989), and Kandel et al. (1995) and used to test the CAPM or the mean‐variance efficiency of the market portfolio. The ICAPM implication is that . The next proposition shows how to express ρ_p in terms of and .

Proposition 1. Investors choose ME portfolios under the ICAPM, thus, according to (i) and (ii) as defined above, maximizing the expected return given the variance and minimizing the variance given the expected return, both conditional on the loadings on the mimicking portfolios. Given the market portfolio R_p, the maximum correlation ρ_p between R_p and any ME portfolio is given by the correlation between R_p and as defined in equation (5), where

Proof. The ICAPM holds, thus gives a complete set of ME spanning portfolios. Consider the regression in equation (5), where is the variance of the residuals ε_p. These residuals are independent of and thus also independent of , the component of R_p in equation (5) that loads on and whose variance is denoted by . The regression coefficients in equation (5) are such that they minimize the residual variance . As equation (7) shows, minimizing implies maximizing the correlation between R_p and , which is the weighted average of the portfolios in using the weights . Each weighted average of is ME, such that is ME.6 The correlation is maximized over all possible ME portfolios, because all ME portfolios could be attained by varying . In turn, this is because all ME portfolios can be uniquely described by a linear combination of spanning ME portfolios.

Note that the systematic risk characteristics of R_p and in equation (5) are identical, as they have the same expected return and the same loadings on the state variables. As a result, under the alternative that the market portfolio is not ME, the maximum correlation ρ_p can be interpreted as the proportion of the volatility of R_p, denoted by V_p, that can be related to its loadings on the risk factors. That is, is that part of the total variance of the market portfolio that is not compensation for any of the risk factors. This part of the total variance V_p could have been avoided by instead choosing the ME portfolio while keeping the same expected return and the same loadings on the risk factors. Similarly, indicates the percentage of the volatility of R_p due to “missing factors” under the ICAPM assumptions.

A modified version of this correlation measure can also be used to help rank the performance of inefficient portfolios. In a CAPM world, portfolios can be ranked by their Sharpe ratio or equivalently by their correlation with the tangency portfolio. In particular, in that setting any such ranking is independent of investor preferences. Similarly, in a multifactor world, mean‐variance investors have preferences that do not include hedging demands or the covariance with the state variables. Therefore, for such investors all portfolios can still be ranked by ρ_MVE, but for investors with more general preferences that include hedging demands, this is not the case.

The following proposition establishes a multifactor generalization of this well‐known result for the CAPM. We generalize the investor’s investment‐opportunity set and assume that it consists of the K mimicking portfolios and the risk‐free asset (or more generally any set of K ME spanning portfolios, one short of a complete spanning set), in addition to some set of inefficient portfolios. If investors want to hold one of these inefficient portfolios in addition to the set of K ME portfolios, then proposition 2 shows that all investors, regardless of their preferences, would achieve an identical ranking of all the inefficient portfolios.

Proposition 2. Investors choose ME portfolios under the ICAPM, according to (i) and (ii) as defined above, and have access to the risk‐free asset and the following J portfolios R_j: the K mimicking portfolios ( ) and J‐K inefficient portfolios ( ). The maximum correlation of each portfolio R_j, ρ_j, with any ME portfolio is attained with , where , where the variance of ε_j equals .

If investors want to choose one of the J‐K inefficient portfolios to combine with the K mimicking portfolios, they would assign identical ranks (i.e., regardless of investor preferences) to the inefficient portfolios according to the modified multifactor correlation measure , where

Proof. See appendix B.

As an example, consider the case of the performance evaluation of a mutual fund manager who commits to a certain “style,” which can be defined by the set of state variables that are assumed to be of hedging concern (see Sharpe 1992; Brown and Goetzmann 1997; Pástor and Stambaugh 2002a). The mutual fund’s investors will typically have diverse preferences that may lead to greatly different loading constraints. Therefore, using proposition 2, a fair way to evaluate the manager is to take into account the exposure of the managed portfolio to the priced risk factors according to the modified ρ* of the mutual fund. A second application for ρ_p and the modified is in optimal portfolio decisions in a multifactor world (see, e.g., Pástor 2000; Pástor and Stambaugh 2000). Instead of choosing optimal portfolio weights to maximize the Sharpe ratio, ICAPM investors would choose weights to maximize ρ_p or the modified .

For comparison purposes, next to the maximum correlation of the market portfolio with any ME portfolio, ρ_p, we use the following three inefficiency measures as well:

The average, absolute alpha, , measures how well the market portfolio and the mimicking portfolios explain the expected returns of the test portfolios in R_N. However, as Roll and Ross (1994) and Kandel and Stambaugh (1995) show, if the market portfolio is not exactly efficient, then perturbations of the reference assets could generate almost any pricing error α_N and almost any of regression equation (6). Therefore, the finding of low, average, absolute pricing errors for the test portfolios does not guarantee that other assets will have a similar small, absolute pricing error, even if the reference assets plus the market portfolio span their returns. However, this robustness for any perturbation of the reference assets is offered by the other three inefficiency measures considered.

The maximum pricing error, denoted by δ, was developed by Hansen and Jagannathan (1997). It measures the pricing error of the most severely mispriced portfolio or the upper bound for the pricing error of any portfolio constructed from . The inefficiency measure δ is normalized such that the payoffs have a second moment equal to unity. Hansen and Jagannathan (1997) show that if the stochastic discount proxy is linear, the maximum pricing error is given by the minimum least‐squares distance between the stochastic discount proxy and the stochastic discount factor that is admissible or correctly prices all payoffs.

The admissible stochastic discount factor (see Hansen and Richard 1987) where R* equals the excess returns of the MVE portfolio with a minimum second (uncentered) moment, and r_f denotes the return on the risk‐free asset. Because R* is on the efficient frontier of excess returns, we know (see, e.g., Jobson and Korkie 1985) that its weights on the excess returns in must be proportional to . Writing , then the weights w* that produce the MVE portfolio that has the minimum second uncentered moment are such that and

The least‐squares distance between p* and the proxy stochastic discount factor can be found by regressing p* on a constant c*, the market portfolio, and the mimicking portfolios, where denotes the variance of the residuals ε*, such that the Hansen‐Jangannathan distance δ can be defined as8

The statistic ρ_MVE is the maximum correlation between the tangency portfolio and any combination of the market portfolio and the mimicking portfolios. Given and , the inefficiency measure ρ_MVE can be expressed as (see, e.g., Kandel and Stambaugh 1987) where denotes the variance of the tangency portfolio R_MVE in equation (4), and is the variance of the residuals ε_MVE resulting from regressing the tangency portfolio on the market portfolio and the mimicking portfolios as in

In particular, ρ_MVE can be directly linked to α_N and Σ_N in regression equation (6). Gibbons et al. (1989) show that and where and denote the expectation and variance of in equation (6), θ_MVE is the maximum Sharpe ratio from all portfolios in and thus equals the Sharpe ratio of the tangency portfolio or any MVE portfolio, and is the maximum Sharpe ratio from just using the market portfolio and the mimicking portfolios. Therefore, ρ_MVE can be seen as a particular transformation of the Gibbons et al. (1989) measure , which measures the difference between the maximum squared Sharpe ratio from using all portfolios and the maximum Sharpe ratio from using just the spanning portfolios (under the model) in . The inefficiency measure ρ_MVE provides a more intuitive comparison of these two Sharpe ratios. Specifically, ρ_MVE equals the maximum Sharpe ratio from investing in the portfolios in , only over the Sharpe ratio of the tangency portfolio (see Kandel and Stambaugh 1987; Shanken 1987a).

The measure ρ_MVE plays a central role in Roll (1977). Roll’s critique concerning the unobservability of the market portfolio underscores the importance of estimating the level of inefficiency using various metrics and viewing the model as an approximation. Also, Roll argues that high values of ρ_MVE should give one reservations about rejecting the CAPM simply because one rejects the exact MVE of R_p. Furthermore, as Roll observes: “most reasonable proxies will be highly correlated with each other and with the true market” (130), which is verified by Stambaugh (1982). Therefore, while our choice of the market portfolio is necessarily a proxy, as long as it is highly correlated with the true market, our inference should still hold. At the same time, the use of a proxy of the market portfolio indicates that exact pricing (i.e., all pricing errors are exactly zero) may not be found even if the ICAPM would hold. This motivates the estimation and pricing performance evaluation of models that are a priori only expected to hold approximately, using informative priors that allow for mispricing (see, e.g., Pástor 2000; Pástor and Stambaugh 2000). Other Bayesian studies focusing on ρ_MVE in the context of the CAPM are Shanken (1987a), Harvey and Zhou (1990), and Kandel et al. (1995). This article is the first that applies the measure ρ_MVE to multifactor models in a Bayesian framework or to show how to estimate the Hansen‐Jagannathan distance using noninformative or informative priors.

A.  Noninformative versus Informative Priors

An important methodological choice is between noninformative versus informative priors for and . Using noninformative priors, the posteriors will be very similar to the classical maximum‐likelihood results. However, informative priors allow additional inference of the information in the data. Generally, one can compare how the data change informative prior views of the four different statistics into posterior distributions over a wide range of prior views. If the data shift the values of any inefficiency measure closer to its value under exact efficiency, this is evidence that the data support the aspect of the model that is evaluated by that particular statistic. In addition, prior views on the basic parameters of the models can be chosen such that the marginal, unconditional priors of the inefficiency measures can be kept basically identical across different models. One can then conduct a horse race by comparing the posterior results of these inefficiency measures across models where any difference can be attributed by the data.

There are three prime motivations for using informative priors: to avoid some problematic properties of noninformative priors, to explicitly allow for prior model mispricing or for models to hold only approximately, and finally to adjust the estimates of the inefficiency measures for model size when comparing across models. In this section, we present the estimation results from using noninformative priors of and to motivate the use of informative priors in the remainder of the article.

First, in the context of the CAPM, Kandel et al. (1995) show that for data samples of typical size noninformative priors can have highly undesirable properties that make posterior inference about the evidence in the data problematic (which we will shortly confirm for multifactor models and multiple inefficiency measures). In short, the problem is that data samples commonly used are too short to have different priors converge to a common posterior opinion. In particular, noninformative priors for the basic model parameters, such as α_N, , and Σ_N in equation (6), are shown to lead to highly informative priors of nonlinear transformations of these parameters, such as ρ_p in equation (7), δ in equation (13), and ρ_MVE in equation (14). Such implicitly highly informative priors on these inefficiency measures make it very hard to use posteriors to distinguish the evidence in the data from the content of the imposed priors.

Instead, Kandel et al. (1995) advocate conducting careful prior elicitation, comparing posteriors to priors in order to investigate the information in the data, and comparing posteriors across a wide range of priors in order to investigate the sensitivity of the results to the priors. New in this article is the comparison of pricing ability across models by choosing basically identical priors for the inefficiency measures in each of the models under comparison.

Second, the “exact efficiency” paradigm holds that all pricing errors are zero, and the model restrictions hold exactly. However, it is not unreasonable to a priori expect a model to hold only approximately—leading to lower pricing errors than using any other model but still allowing nonzero pricing errors. One reason models might hold only approximately is the existence of measurement problems for both the market portfolio (see Roll 1977) and the state‐variable‐mimicking portfolios. Further, the fact that when using noninformative priors or classical analysis the literature is typically able to reject that any of the considered models holds exactly is another impetus to investigate whether or not these models may hold approximately. For example, Davis et al. (2000) can reject the FF model for portfolios formed from independent sorts of stocks on size and BM using the Gibbons et al. (1989) test, writing that “this result shows that the three‐factor model is just a model and thus an incomplete description of expected returns” (405).

Informative priors enable us to explicitly deal with the case where models are expected to hold only approximately. The prior view on model mispricing is incorporated through the prior of α_N in equation (6). The priors of α_N used in this article are the same as in Pástor (2000) and Pástor and Stambaugh (2000), which investigate portfolio decisions if the investor a priori expects the models to hold only approximately. Choosing a prior of α_N that is very tightly centered around zero reflects the view of an investor who strongly believes in the pricing ability of the models. The choice of a prior of α_N with a prior expectation of zero but a large standard deviation reflects large prior uncertainty about the model’s pricing ability. However, neither Pástor (2000) nor Pástor and Stambaugh (2000) use such priors to directly evaluate pricing performance in the context of models holding approximately. Also, neither seeks to answer the question of whether adding factors to the CAPM decreases pricing errors relative to those of the CAPM.9

Third, another aspect of the small‐sample problem is that if factors are added to the model, this can only decrease the posterior mean of δ and increase the posterior mean of ρ_MVE and ρ_p. This again motivates the use of informative priors, because the specification of informative priors can directly account for model size in the priors. As a solution, we compare posteriors across models by using informative priors that are basically similar for the different models. This directly addresses any model size differences in the comparison.

A natural question arises as to whether the resulting posteriors are not largely driven more by the priors and much less by the data. While the priors are an important determinant of the posteriors, any shift from prior to posterior is obviously due to the data, so the extent to which the data shift the prior over a wide range of priors gives some indication on how much information there is in the prior relative to the data. Further and most important, all empirical conclusions reached in this article were ensured to be robust to significant changes to the priors and hold over a wide range of reasonable prior views.

B.  Data and Factors

The sample period used is from January 1954 to December 2001, for a total of 576 monthly observations. The data from January 1927 to December 1953, for a total of 324 monthly observations, are used as a training sample to assist in the choice of the informative priors (see the next section for the details).

We choose two different sets of test portfolios. The first set consists of the 25 BM/size‐sorted value‐weighted portfolios that result from the sorting according to Davis et al. (2000). These BM/size‐sorted portfolios are formed by first sorting all New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and NASDAQ stocks in the intersection of the Center for Research in Security Prices (CRSP) and Compustat files into five groups according to market capitalization (size) and then sorting each into five more groups according to the ratio of book value of equity to market capitalization (book to market). The second set of test portfolios consists of 30 industry‐sorted portfolios. The market proxy R_p is the value‐weighted portfolio of all stocks.

We consider three factors for the mimicking portfolios in R_K. The first two factors are from the FF model: the SMB size factor, capturing the difference between small‐ and large‐capitalization stocks, and the HML BM factor, capturing the difference between stocks with a high and low book value of equity relative to market capitalization. Fama and French (1992, 1993, 1996) show that these two additional factors explain much variation in the cross section of stock returns and claim that they also significantly improve pricing, that is, reduce pricing errors of the test portfolios used.

The third factor considered is the UMD momentum factor (see, e.g., Jegadeesh and Titman 1993; Carhart 1997). This momentum factor is constructed from six value‐weighted portfolios formed on size and the prior 12‐month return, which are the intersections of two portfolios formed on size and three portfolios formed on the prior 12‐month return. The UMD‐mimicking portfolio is the average return on the two high prior‐return portfolios minus the average return on the two low prior‐return portfolios (see Fama and French 1996; Grundy and Martin 2001; Jegadeesh and Titman 2001).

We consider four different asset‐pricing models: (1) the CAPM ( ), (2) the FF model ( ), (3) the two‐factor model including the market portfolio plus UMD ( ), and (4) the four‐factor model including the three factors of the FF model plus UMD ( ). All three factors were proposed in the literature to address specific shortcomings of the CAPM. All returns are in excess of the risk‐free asset, which is the 1‐month Treasury bill rate.10

C.  Results Using Noninformative Priors

In order to compare to the previous literature, illustrate the small‐sample problem, and empirically motivate the use of informative priors, we first use noninformative priors for and . The posterior results for the CAPM and the FF model are computed using 10,000 draws from the posterior distributions of and . In table 1, we report the posterior mean and standard deviation of the four inefficiency measures using noninformative priors.

Table 1 Posterior Results Using Noninformative Priors for and

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At first glance, the results in table 1 seem to show dramatically bad performance for all models under consideration, for all four inefficiency measures. For example, and again for all four asset‐pricing models considered, the noninformative priors result in dramatically low posterior means of both correlation measures ρ_p and ρ_MVE and dramatically high posterior means of the Hansen‐Jagannathan distance δ. However, a simulation study shows that such posterior means could be obtained with relatively large probabilities using a sample size of 576 months, even if the respective models would hold exactly.

In figure 1a, we plot the posterior distribution of ρ_p for the CAPM using the improper priors (using the actual data and the 25 FF portfolios), together with two histograms of the posterior means of ρ_p for the CAPM using the improper priors from two simulation exercises assuming normal distributions. In the first exercise, we conduct 1,000 simulations where returns are simulated such that the model under evaluation—in this case the CAPM—holds exactly, using the and maximum‐likelihood estimates and constraining all pricing errors to be exactly zero. In the second exercise, we conduct 1,000 simulations using the unconstrained and maximum‐likelihood estimates. We find that the posterior mean of ρ_p for the CAPM in the actual data is greater than 23.6% of the posterior means of ρ_p for the CAPM estimated from the simulations, such that the CAPM holds exactly. Moreover, comparing the two histograms of the posterior means of ρ_p, we find that the probability that the posterior mean in the unconstrained simulations is greater than in the constrained simulations equals 22.1%.11

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Fig. 1.— Posteriors of ρ and for the CAPM, with histograms of their means. Using improper priors, we plot the resulting posteriors of ρ_p (a), which equals ρ_MVE, and (b) in conjunction with two histograms of the respective posterior means from 1,000 simulations, using the CAPM. In the first simulation the data are such that the CAPM holds exactly, using the and maximum‐likelihood estimates of the actual data constrained such that all CAPM pricing errors are exactly zero. In the second simulation the unconstrained maximum‐likelihood estimates of the actual data are used. Both simulations assume normally distributed returns and use the 25 FF BM/size‐sorted test portfolios.

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As a second illustration and to show that this problem is not limited to ρ_p (which equals ρ_MVE for the CAPM), figure 1b plots the corresponding results for the posterior means of the average, absolute pricing error, , under the CAPM using noninformative priors and the actual data using the 30 industry‐sorted portfolios in conjunction with, again, two histograms of the posterior means of for the CAPM using the improper priors, under the CAPM holding exactly and using the unconstrained and maximum‐likelihood estimates. In this case, the posterior mean of for the CAPM in the actual data is smaller than 26.2% of the posterior means of for the CAPM estimated from the simulations, such that the CAPM holds exactly. In addition, comparing the two histograms of the posterior means of , we find that the probability that the posterior mean in the unconstrained simulations is greater than in the constrained simulations equals 4.2%.

These results confirm the large estimation uncertainty and hence the severe small‐sample problems in estimating the asset‐pricing models using noninformative priors as also reported in Kandel et al. (1995). They interpret this as a case where noninformative priors about and imply very sharp priors about statistics such as ρ_MVE. In a large‐enough sample, the posteriors would be such that even sharp priors about a particular statistic are dominated by the data, which the posterior p‐values show is not the case here. Therefore, we refer to this issue as a small‐sample problem rather than just a problem associated with the use of noninformative priors. As we will show, even if informative priors are used, the data are not always able to converge different priors to a common posterior opinion.

These small‐sample problems suggest that classical studies using a fixed uncertainty level may reject models too often (see, e.g., Shanken 1987a) and may have low power against alternatives. This lack of power is illustrated by the posterior p‐values. It is also made clear by our finding that these p‐values hardly change if the simulations used to calculate the p‐values are done using the unconstrained maximum‐likelihood estimates of and , such that pricing errors are not constrained to be zero. In addition, the severity of the small‐sample problem may depend on model size, which complicates the comparison of, for example, the CAPM to the FF model.

Finally, the corresponding results for the other asset‐pricing models are much less dramatic. However, to the extent that the CAPM is our benchmark model for which we want to investigate the impact of added factors, our focus on the CAPM seems warranted. Further, as the results for proper or informative priors in the following sections will show, for the other asset‐pricing models the data are also clearly unable to have different priors converge to a common posterior distribution. Further discussion of the pricing comparison of these models will be done using the informative priors.

A.  Methodology of Choosing Informative Priors

We choose a similar framework for choosing priors of and , as in Kandel et al. (1995). We use the regression framework in equation (8) for specifying priors of and . Therefore, the specification of the priors for and in combination with priors for and Σ_N results in priors for and using and where for the CAPM, , and β_K is set to zero.

For the prior distributions for and , we choose the normal inverse‐Wishart conjugate priors. This choice greatly improves the clarity and tractability of the posterior analysis, because the prior and the likelihood combine such that the prior and the posterior have the same distributional forms. Specifically, the prior can be directly interpreted as implied from the data in the training sample: where N(·) denotes the multivariate normal distribution, and IW(·) denotes the inverse‐Wishart distribution such that and are the prior expectations of and . Furthermore, t₀ is the degree of freedom in the priors. Combined with a normal likelihood of [ ] given and , the posteriors are and where and are the maximum‐likelihood estimates of and . The posterior expectation of is a weighted average of its prior expectation and maximum‐likelihood estimate, where the weights are determined by the degrees of freedom in the prior, t₀, and the number of observations, T.

Similar to Kandel et al. (1995), we use the 324 monthly returns in the period 1927–53 as a training sample for prior elicitation. The prior means of and , denoted by and , are set equal to their maximum‐likelihood estimates in the training sample. Furthermore, we choose , equal to the number of observations used in the training sample. We find that with , the posterior results are robust to large changes in and .

For the regression parameters α_N, , and Σ_N in equation (8), we choose the following normal inverse‐Wishart conjugate priors, and where , , and are the prior expectations of α_N, , and Σ_N, respectively.

Similar to Kandel et al. (1995), Pástor (2000), and Pástor and Stambaugh (2000, 2002a, 2002b), we choose priors that center around the pricing restriction. As a result, we set the prior expectation of α_N equal to zero, such that , and choose a nonzero prior variance of α_N that allows for mispricing. Furthermore, the prior expectations of and Σ_N are set at their respective maximum‐likelihood estimates in the training sample.

For the prior variance‐covariance matrix of α_N and , Λ₀, we choose a setting that resembles the g‐prior specification, a popular choice in Bayesian statistics (see Zellner 1986), such that where the scalars λ_α and determine the prior variance of α_N and , respectively.

With a prior expectation of α_N of zero, the choice of a larger prior variance of α_N or a large value of λ_α reflects less prior confidence in the pricing ability of the model, which increases the prior means of and δ and lowers the prior mean of ρ_MVE and ρ_p. Finally, we choose very large prior variances of β_p and β_K by setting , such that their prior means become relatively unimportant.

Our choice of the prior of α_N dependent on the variance matrix of the regression residuals Σ_N is motivated by MacKinlay (1995). He shows that if α_N is not linked to Σ_N under a risk‐based model, very large Sharpe ratios could potentially be obtained by combining the assets in R_N with those in . Therefore, he argues that the risk‐based nature of the model should make large values of mispricing less likely than under non‐risk‐based alternatives. The prior of α_N reflects this notion: for a given λ_α, large values of mispricing receive lower probabilities relative to the case where α_N would be a priori independently distributed from all other parameters and in particular from Σ_N.

Moreover, MacKinlay (1995) argues that any mispricing (nonzero α_N) will be accompanied by a higher residual variance, which is reflected in the prior by the link between α_N and Σ_N. These same motivations underlie the prior choices in Pástor and Stambaugh (1999, 2000, 2002a) and Pástor (2000). In the context of the CAPM, Kandel et al. (1995) choose the prior variance of α_N dependent on μ_p but independent of Σ_N. However, they suggest our alternative approach as a possible improvement but leave it for further research.

Combined with the normal likelihood of R_N given α_N, , and Σ_N, the posteriors are (see Poirier 1995, 540–43, 597–99) and where , see equation (6), and , , and are the posterior expectations of α_N, , and Σ_N, respectively, which can be written as and where , , , and are the maximum‐likelihood estimates of α_N, β_p, β_K, and Σ_N, respectively.

B.  Posterior Sensitivity to the Prior

Extensive sensitivity analyses show that the posterior results are relatively robust to large changes in the priors of and Σ_N and in the priors of and (the means and variances of the factors). The only prior choice that has a profound impact on both the prior and posteriors of the four inefficiency measures is the prior variance of α_N, which is determined by λ_α.

In order to illustrate the sensitivity of the posteriors to large changes in those former prior choices, we report in panel A of table 2 the results for three different prior views of λ_K (which determines the size of the prior variance‐covariance matrix of ) and in panel B of table 2 the results for three different choices of t₀ (influencing the prior of Σ_N, , and ), whereas all the other prior choices (other than the one that is varied) are as described in the previous subsection, and . We report the posterior means and standard deviations of all four inefficiency measures for each of the respective prior choices but, in order to save space, only for the CAPM, the FF models, and the 25 FF test portfolios.

Table 2 Informative Priors and Resulting Posteriors: Varying λ_K and t₀

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A higher value of λ_K means a higher prior uncertainty about and less emphasis on the training sample prior estimates of in the posterior of ; see equations (29)–(35). First, notwithstanding the large range of values of λ_K considered (0.5–10), the prior means of the four inefficiency measures remain identical. Second, there is some effect of the choice of λ_K on the posterior mean of for the CAPM only, where larger values of λ_K lead to lower abnormal returns, but relative to the sensitivity of the posterior of for choices of λ_α (as we will show shortly), these differences are small. Moreover, we do not find this sensitivity with respect to λ_K for any of the other three inefficiency measures and the CAPM or for all four inefficiency measures and the FF model.

Next, a smaller value of t₀ means a higher prior uncertainty about and (the means and variances of the factors) as well as about Σ_N (the residual variance‐covariance matrix; see eqq. [21]–[30], [34], and [36]). For a broad range of values of t₀, from to (or from a quarter to twice the number of time‐series observations in the training sample, which is 324), we find both prior and posterior means of all four inefficiency measures and both the CAPM and the FF model to be quite stable.

Therefore, in the remainder of the article we only report results for four different prior views that differ only in their respective values of λ_α, ranging from a prior view that allows large mispricing ( ) to a prior view that reflects much confidence in exact efficiency ( ). All other prior choices remain fixed henceforth, such that , , the means of , , and are equal to the maximum‐likelihood estimates from the training sample, and pricing errors are a priori mean zero. A larger value of λ_α reflects the a priori expectation that mispricing under the model is larger. As a result, larger values of λ_α imply larger prior means of and δ and smaller prior means of ρ_MVE and ρ_p. Also, the results for are closest to the results that would arise from noninformative priors.

Furthermore, λ_α determines the prior variance of α_N conditional on Σ_N, while the unconditional variance of α_N equals . As a result, the average of the diagonal of times provides the average unconditional variance of α_N. For the BM/size‐sorted test portfolios, the (annualized) average of the diagonal of for the CAPM and FF model equals (40.44%)² and (26.73%)², respectively, and for the industry‐sorted portfolios, (31.66%)² and (28.90%)², respectively. For example, for the CAPM and the BM/size‐sorted portfolios, the average unconditional (annualized) variance of α_N equals (7.0%)² and (0.87%)² for and , respectively.

In table 3, we report results for the four inefficiency measures: the average, absolute pricing error, , in panel A; the Hansen‐Jagannathan distance, δ, in panel B; the maximum correlation of the market portfolio plus the included mimicking portfolios with the tangency portfolio, ρ_MVE, in panel C; and the maximum correlation of the market portfolio with any ME portfolio, ρ_p, in panel D. For each model and choice of test portfolios considered in the table, we report the prior and posterior mean and standard deviation of the particular inefficiency measure.12

Table 3 Informative Priors and Resulting Posteriors: Varying λ_α

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We start with addressing two questions that are critical for interpreting the results. The first question is how to choose priors to compare pricing performance across models. Because the only prior hyperparameter choice that is crucially important is the choice of λ_α, this question comes down to how to choose λ_α across models.

In general, the influence of λ_α on the prior and consequently on the posterior results is straightforward: decreasing λ_α for a given model means that less prior mass is given to larger pricing errors (i.e., more shrinkage), which lowers the prior and posterior means of and δ and increases the prior and posterior means of ρ_MVE and ρ_p. Similarly, if λ_α is kept constant but we add factors to a model, this means that for the larger model the prior residual variance is lowered. Crucially, because of the direct link between α_N and Σ_N in the prior distribution of α in equation (29), this lowers the prior standard deviation of α such that less prior mass is given to larger pricing errors in the larger model than in the smaller model. As a result, ceteris paribus adding a factor would lower the prior mean of and increase the prior mean of ρ_p for the larger relative to the smaller model.

We would argue that setting the marginal, unconditional priors of α_N equal across competing models is the most intuitive way to compare posteriors across models. We implement this by ensuring that the prior means of are equal across models. It turns out that this is generally sufficient in order to get basically identical marginal, unconditional priors for the other three inefficiency measures as well. As a result, we typically will choose a larger λ_α for the larger model with the smaller prior expected residual variances in , such that the unconditional variance of the alphas is still the same as those of the smaller model. Also note that in each model, the prior means of all alphas are always set equal to zero, as discussed above. For the remainder, the priors of the model parameters are determined as described in the previous subsection.

The second question is whether the information in the prior drives our results. As is obvious from our results, the prior still matters: the posterior means of the inefficiency measures depend on the priors. However, we conduct extensive sensitivity analysis to ensure that the posterior results are robust to large changes in the priors. Still, in some cases the prior seems to completely dominate the posterior for the smaller values of λ_α. This is a particular problem for the Hansen‐Jagannathan distance δ, as evidence by panel B of table 3, which presents the prior and posterior mean of δ for different values of λ_α and the BM/size‐sorted test portfolios.13 For prior views with , the data change only the posterior of δ relative to the prior a few percentage points. For example, if one considers only , one might conclude that the data generally confirm the prior. However, the comparison to prior views with and 0.25 shows that the data seem to confirm these priors as well. This is illustrated in figure 2, which plots the prior and posterior distributions of δ for the FF model and the BM/size‐sorted test portfolios and , 0.5, and 0.25. For , the data clearly shift the posterior distribution of δ to lower values relative to the prior. However, for the other two prior views, the posterior and prior distributions of δ are virtually identical, and the data hardly influence the prior at all. Therefore, even using informative priors there is very limited information in the data about δ.

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Fig. 2.— Prior and posterior distributions of the Hansen‐Jagannathan distance, δ, for the FF model, the BM/size‐sorted test portfolios, and three different choices for the prior hyperparameter , 0.5, and 0.25.

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Critically, the limited ability of the data to change the priors is due to the small‐sample problem, that is, too few observations, and not to excessively strong priors per se. The simple reason is that this result remains when the amount of information in the prior relative to the data is lowered much, for example, by choosing much lower values of t₀ or for .14 Fortunately, this problem is clearly much less severe or nonexistent for the other three inefficiency measures with identical values of λ_α. This underlines the importance of comparing the posteriors for multiple prior views as well as multiple inefficiency measures. Finally, it is crucial to stress that even if priors play an important role in the individual posterior distributions, they do not at all drive any of the empirical conclusions in this article. We compare how the data change priors over a wide range and certify that reported findings hold over this wide range of priors.

C.  Posterior Results Using Informative Priors

In this section, we argue that the results from our methodology lead to the following three conclusions: that the market portfolio is useful for pricing, that none of the other factors robustly decrease pricing errors if added to the CAPM, and finally that the results for the FF model exhibit clear dependence on the choice of the test portfolios, while the CAPM results do not.

We investigate prior and posterior distributions while varying λ_α over a wide range and fixing all other prior choices as discussed above. First, we find that the market portfolio is clearly useful for pricing. It is “useful” in the sense that if one has a certain prior of expected returns, then applying the CAPM to the data should generally lead to posterior views of expected returns that are more in line with the evidence in the data. The evidence for this is given by the fact that the data generally move the priors of all four inefficiency statistics toward the values implied by the model. This reverses the findings of Kandel et al. (1995) as discussed in Section IV.E.

The relevant results are presented in table 3: panel A for , panel B for δ, and panel C for ρ_MVE (which equals ρ_p for the CAPM). For all prior choices and both sets of test portfolios, combining the data with the priors clearly leads to a lower posterior mean of and a higher posterior mean of ρ_MVE relative to their respective prior means. Such shifts toward the values implied by exact efficiency imply that the data support the efficiency of the model for the aspects that are evaluated by these statistics.

The support for the CAPM using the inefficiency measure δ is not as convincing as using either or ρ_MVE. This is because for the BM/size‐sorted test portfolios the data slightly decrease the posterior mean of δ for prior views with . However, as discussed, the posteriors for this particular inefficiency measure are influenced more by prior assumptions than by information from the data, such that this evidence against the CAPM is very weak. Furthermore, for the industry‐sorted test portfolios the data consistently decrease the posterior mean of δ for the CAPM for all prior views.

Second, there is little evidence that the three additional factors robustly decrease the pricing errors as measured by any of the four inefficiency measures if they are added to the CAPM. It is precisely in comparing pricing performance across models that informative priors are most useful. It provides a direct solution to the problem that, in a small sample, adding a factor to the CAPM necessarily decreases the posterior mean of δ and increases the posterior mean of ρ_MVE and ρ_p. The solution is to choose priors across models that are basically identical, such that any difference in posteriors can be attributed to the data. In particular, the differences in model size across models can be thus accounted for in the prior. This often involves choosing a smaller value for the prior hyperparameter λ_α for the CAPM than for the multifactor models. However, this in no way biases the procedure against or in favor of larger versus smaller models. As previously discussed, if we keep λ_α fixed then adding a factor will lower the prior residual variance (through the training sample) and thus decrease the overall prior variance of α_N through the prior link between α_N and Σ_N. As a result, choosing a lower value for λ_α for the CAPM than for a multifactor model “compensates” for the larger prior residual variance for the CAPM. This ensures that the unconditional prior variance of α_N (i.e., not dependent on Σ_N) is similar across models.

Because this is the most important empirical contribution of this article, we discuss the evidence for each of the four inefficiency measures separately. We start with the average, absolute pricing errors of the test portfolios, , in panel A of table 3. If we use the BM/size‐sorted test portfolios, then adding the FF factors to the CAPM reduces the average, absolute pricing errors relative to the market portfolio by itself. However, the reverse holds for the industry‐sorted test portfolios, for which the posterior mean of for the FF model is slightly larger than for the CAPM.

This result is best illustrated by figure 3, which plots a representative marginal prior and posterior distribution of for three different asset‐pricing models, namely, the CAPM, the FF model, and the two‐factor model including UMD.15 The results for the BM/size‐sorted test portfolios are given in figure 3a, and the results for the industry‐sorted test portfolios, in figure 3b. The plotted priors are such that the prior mean of equals approximately 0.23% in figure 3a and 0.20% in figure 3b. Therefore, in the prior for the FF model and when we use the BM/size‐sorted test portfolios, and for all other priors. Crucially, the marginal prior distributions of for the three different asset‐pricing models in each figure are highly comparable. As a result, the differences in the posteriors of the three models can be attributed to the data and not to the difference in the respective priors.

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Fig. 3.— Prior and posterior distributions of the average, absolute pricing errors, , for the CAPM, the FF model, and the two‐factor model ( , including the market portfolio plus the momentum factor UMD). We use the BM/size‐sorted test portfolios (a) and the industry‐sorted test portfolios (b). The prior hyperparameter for the FF model and the BM/size‐sorted test portfolios, and for all others.

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For the BM/size‐sorted test portfolios, figure 3a shows that the marginal posterior distribution of for the FF model lies to the left of the marginal posterior distribution for the CAPM and the two‐factor model including UMD. The marginal posterior distribution of in figure 3b using the industry‐sorted test portfolios gives an opposite result. In this case the posterior probability that for the CAPM is smaller than for the FF model equals 83%, with a corresponding prior probability of only 41%.16 This discrepancy of the results for the FF model with respect to the choice of test portfolios is particularly critical because the main evidence in favor of the model in Fama and French (1993, 1996) and Davis et al. (2000) is based on (average) alphas using the BM/size‐sorted test portfolios.

For both sets of test portfolios and all prior views, we find that adding the momentum factor UMD to the CAPM increases the average, absolute pricing errors relative to the market portfolio by itself. For example, when we use the BM/size‐sorted test portfolios and (as plotted in fig. 3a), the posterior mean of equals 0.19% for the CAPM and 0.22% for the two‐factor model including UMD, while both prior means equal 0.23%. In this case the posterior probability that for the CAPM is smaller than for the two‐factor model including UMD equals 70%, with a corresponding prior probability of 48%.

Second, we consider the evidence for the Hansen‐Jagannathan distance using prior views for which the data clearly move the prior of δ, with , 1.75, and 1.5.17 In these cases, the results for the Hansen‐Jagannathan distance δ are similar to those for . We again find the discrepancy of the results depending on which of the two sets of test portfolios is used. For example, for the BM/size‐sorted test portfolios, the FF factors lead to lower posterior means of δ relative to the CAPM; for the industry‐sorted test portfolios, the FF factors lead to higher posterior means of δ.

This is again best illustrated graphically, in this case in figure 4, which plots the prior and posterior distribution of δ for the CAPM, the FF model, and the four‐factor model including UMD for the BM/size‐sorted test portfolios (fig. 4a) and the industry‐sorted test portfolios (fig. 4b) and . While the priors of δ for the three asset‐pricing models are almost identical, the posterior of δ for the CAPM clearly lies to the right of the posteriors of the other two models for the BM/size‐sorted test portfolios but to the left for the industry‐sorted test portfolios. For example, for the industry‐sorted test portfolios the posterior probability that δ for the CAPM is lower than for the FF model equals 79% with a corresponding prior probability of 48%.

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Fig. 4.— Prior and posterior distributions of δ for the CAPM, the FF model, and the four‐factor model ( , including the market portfolio, the two FF factors, plus the momentum factor UMD). We use the BM/size‐sorted test portfolios (a) and the industry‐sorted test portfolios (b), and the prior hyperparameter for all cases.

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Third, when we use the statistic ρ_MVE and the BM/size‐sorted test portfolios, there is some evidence that both the FF factors and the momentum factor UMD reduce pricing errors. For example, in figure 5a we plot the prior and posterior distributions of ρ_MVE for the CAPM, the FF model, and the two‐factor model including UMD for , 1, and 1.5, respectively, such that for all three models the prior expectation . Here, the priors of ρ_MVE are very similar, while the posteriors for both multifactor models are to the right of the posterior of the CAPM. For example, the prior probability that ρ_MVE for the FF model and the two‐factor model including UMD is larger than for the CAPM equals 49% and 50%, respectively. The respective posterior probabilities are 59% and 72%.

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Fig. 5.— Prior and posterior distributions of the maximum correlation measure ρ_MVE for the CAPM, the FF model, and the two‐factor model ( , including the market portfolio plus the momentum factor UMD). We use the BM/size‐sorted test portfolios (a), with , 1, and 1.5 for the CAPM, FF model, and the two‐factor model, respectively. We use the industry‐sorted test portfolios (b), with , 1.25, and 2 for the CAPM, FF model, and the two‐factor model, respectively.

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For the industry‐sorted test portfolios, we again find some improvement when we add the momentum factor UMD to the CAPM. On the contrary, there is no evidence in the data that adding the FF factors to the CAPM further improves ρ_MVE. These results are illustrated in figure 5b, which plots the marginal prior and posterior distributions of ρ_MVE for the CAPM, the FF model, and the two‐factor model including UMD for , 1.25, and 2, respectively, such that for all three models the prior expectation . The priors are again chosen such that they are very similar; for example, the prior probability that ρ_MVE for the FF model and the two‐factor model including UMD is larger than for the CAPM equals 55% and 51%, respectively.18 The corresponding posterior probabilities are 46% and 77%. Consequently, in figure 5b the posterior of ρ_MVE of the two‐factor model including UMD is clearly to the right of the posterior of the CAPM, while the posterior for the FF model is slightly to the left.

Fourth, the results for the final inefficiency measure ρ_p provide arguably the strongest evidence from the data against adding the FF factors or the momentum factor to the CAPM, as presented in panel D of table 3. For both sets of test portfolios, the data lower the posterior mean of ρ_p relative to the prior for all multifactor models considered for all but one prior view. The only exception occurs for the four‐factor model with , where the influence of the data on the posterior is minimal.

A typical example is given in figure 6, which plots the marginal prior and posterior distributions of ρ_p for the CAPM, FF model, and the two‐factor model including UMD using the BM/size‐sorted (fig. 6a) and the industry‐sorted (fig. 6b) test portfolios. We choose , 1.5, and 1.75, such that for all three models the prior expectation is . The posterior of ρ_p for the FF model is clearly to the left of its prior and similarly for the two‐factor model including UMD. For the CAPM, the data do not shift the posterior of ρ_p to lower values: for example, the posterior mean of ρ_p is 70.0% versus a prior mean of 63.4% for the industry‐sorted test portfolios. Also, and again for the industry portfolios, the median of the prior of ρ_p for the CAPM equals 53% (thus the prior probability that equals 50%), while the posterior probability that ρ_p is larger than the prior median equals 65%.

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Fig. 6.— Prior and posterior distributions of ρ_p for the CAPM, the FF model, and the two‐factor model ( , including the market portfolio plus the momentum factor UMD). We use the BM/size‐sorted test portfolios (a) and the industry‐sorted test portfolios (b), with , 1.5, and 1.75 for the CAPM, the FF model, and the two‐factor model, respectively.

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D.  Robustness

We perform two further robustness checks of our results, by considering very small values of λ_α and by comparing prior and posterior distributions from data simulated with and without each particular model holding. First, we consider the results for very small values of λ_α. In figure 7a, using the 25 BM/size‐sorted test portfolios, we plot the prior and posterior distributions of for both the CAPM and the FF model. We use very small values of λ_α equal to 0.1 and 0.2, respectively, such that the priors of are quite similar in both cases, each with a prior mean of 0.00018 (or 0.22% annualized). We find that in this case, the data still move the posterior of closer to zero. The resulting posterior distributions are quite similar across both models, with a posterior mean of of 0.00014 for the CAPM and 0.00013 for the FF model.

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Fig. 7.— a, prior and posterior distributions of for the CAPM and the FF model using and 0.2, respectively; b, prior and posterior distributions of ρ_p for the CAPM and the FF model using and 0.2, respectively, as well as ρ_MVE for the FF model using . Throughout, the BM/size‐sorted test portfolios are used.

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In figure 7b, again using the 25 BM/size‐sorted test portfolios, we plot the prior and posterior distributions of ρ_p and ρ_MVE for both the CAPM and the FF model. The very small values of λ_α used now are equal to 0.1 for the CAPM and either for the FF model and ρ_p or for the FF model and ρ_MVE. In all three cases, the posterior means of ρ_p or ρ_MVE are slightly larger than the respective prior means, even though the modes of the posterior distributions are slightly smaller (or to the left) relative to the modes of the respective prior distributions. Put differently, in all three cases the priors are more left skewed than the posteriors, such that the data still move the posteriors closer to 100%. For example, the prior and posterior means of ρ_p for the CAPM are 93.6% and 97.9%, respectively, while the prior and posterior modes are 99.4% and 99.1%, respectively.

The second robustness check is motivated by figure 1, which plots the histograms of the posterior means of and ρ_p for the CAPM simulated with and without the CAPM holding and using improper priors. We found there that the two histograms were quite similar, such that the data were not able to sufficiently distinguish between these two simulation schemes. Figure 8a repeats this exercise for . For both the CAPM and the FF model and using and the BM/size‐sorted test portfolios throughout, we plot both the histogram of the posterior mean of using data simulated with the model (either the CAPM or the FF model) holding and the posterior mean using data without the model holding. The results confirm the ability of the data to clearly decide between either simulation in combination with the use of informative prior information: for both models there is hardly any overlap between the two histograms of the posterior means simulated with and without the model holding. Further and as expected, the histograms of the posterior means for the simulations with the model holding are clearly to the left of the respective posterior histograms for the simulations without the model holding, and both are clearly to the left of the respective prior means of , which equal 0.280% and 0.232% for the CAPM and FF models, respectively.

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Fig. 8.— Using informative priors, , and the BM/size‐sorted portfolios throughout, both panels plot histograms of posterior means of the inefficiency measure from 1,000 simulations with and without the model holding (see fig. 1 for details). a, histograms of the posterior means of for the CAPM and the FF model; b, histograms of the posterior means of both ρ_p and ρ_MVE for the FF model. For the prior of ρ_p for the FF model, we use two different assumptions about the correlation structure of the factors (see the text for details).

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In figure 8b, we conduct an analogous procedure for ρ_p and the FF model, using the BM/size‐sorted portfolios and throughout, motivated by the potential concern that the failure of the FF model to increase ρ_p in the posterior could be caused by some hitherto unexplored prior assumptions. The results confirm that this indeed is a concern, because much of the histogram of the posterior mean of ρ_p under the FF model holding is to the left of the prior mean of ρ_p, which is equal to 66%. This can only be due to some important difference in the training versus the data samples. In particular, ρ_p is driven by the correlations of the market portfolio with the tangency portfolio (which is the same across models) as well as by the market portfolio’s correlation with the SMB and HML factors (in case of the FF model). However, the prior assumptions about the correlation structure of the market portfolio with these two FF model factors as formed by the training sample are quite different from the data sample. In the training sample (using stock returns from 1927–63), the correlation between the market portfolio with the SMB and HML factors equals 39% and 59%, respectively, while in the data sample (using returns from 1964–2001) these correlations equal 26% and −38%, respectively.

However, when we change the prior such that the correlation structure in the training sample is identical to that in the data sample while keeping the prior mean of ρ_p for the FF model at 66%, the resulting histogram of the posterior mean of ρ_p (also plotted in fig. 8b) is clearly to the right of the prior mean. Therefore, this feature of ρ_p for the FF model is indeed caused by this difference in correlations in the training versus the data samples.

It is critical to note that this happens exclusively for the single inefficiency measure ρ_p in combination with the FF model. For the CAPM and all four inefficiency measures as well as for the FF model and the other three inefficiency measures ( , δ, and ρ_MVE), simulating data under the model holding brings posteriors closer to efficiency relative to the relevant prior means (as evidenced, e.g., in fig. 8a). This again emphasizes the importance of considering multiple inefficiency measures.

Finally, the conclusion that the FF model does not perform well using the particular inefficiency measure ρ_p arguably remains valid for two reasons. First, as discussed before, ρ_p strongly depends on the correlation structure between the market portfolio and the included factors. In an unconditional model without regime switches, such a measure naturally performs badly when this correlation structure is highly unstable. In other words, our finding that the correlation between the market portfolio and the two FF factors is so different across time periods is in itself evidence against the FF model. Second, when the correlation structure between all factors in the prior is set equal to those in the data sample while keeping all other features the same, then for the FF model the data indeed generally increase ρ_p in the posterior relative to the prior but still less so than for the CAPM. So it remains the case that relative to the CAPM, adding the FF factors does not seem to improve multifactor efficiency as measured by the inefficiency measure ρ_p.

E.  Comparison to Kandel, McCulloch, and Stambaugh (1995)

The evidence in favor of the CAPM reported in this article is in sharp contrast to the results reported by Kandel et al. (1995) when they assume a risk‐free asset exists. In that case, using the market portfolio as the only risk factor, they find that the data consistently decrease the prior mean of ρ_p (or ρ_MVE), which is the sole inefficiency measure Kandel et al. (1995) consider. Although we follow Kandel et al. (1995) in their approach of using training samples to specify priors and in their interpretation of prior versus posterior distributions, there are some important differences in our methodology that explain the different results.

We conjecture that some unnecessary and highly restrictive prior assumptions in Kandel et al. (1995) are primarily responsible for this contrast in results. The most important indication for this is the very large discrepancy Kandel et al. (1995) report in their article for the cases where they assume that the risk‐free asset does and does not exist. If they do not assume the existence of a risk‐free asset, their posterior results are similar to our results and generally favorable to the unconditional mean‐variance efficiency of the market portfolio. In particular, these restrictions in their priors for the case where a risk‐free asset exists are missing in their prior specification for the case where a risk‐free asset does not exist. Moreover, our prior specification does not have any of these restrictions. Our prior design has the important additional advantage that it can be interpreted directly as arising from the posterior view from the inference on a training sample, which is not the case for the prior specification in Kandel et al. (1995).

The first important prior restriction in Kandel et al. (1995) for the case where they assume a risk‐free asset exists is their choice of the prior of α_N to be independent of Σ_N (see our earlier discussion of MacKinlay [1995]). In particular, this requires much smaller prior variances of α_N to reflect moderately high prior means of ρ_MVE relative to those required in our setting. For example, Kandel et al. (1995) need an annualized prior variance of α_N equal to (0.069%)² for a prior mean of ρ_MVE of 47%, while in our setup an annualized prior variance of α_N of, on average, (2.6%)² gives a prior mean of ρ_MVE equal to 53.50% (with ). Ceteris paribus, the larger the prior variance is, the more the data can dominate the posterior and the weaker the role of the prior in the posterior.19 This strongly indicates that the data have significantly more room to affect the posterior distribution of α_N in our prior specification than in Kandel et al. (1995). The link in the prior distribution between pricing errors and residual variance, although implicit, is not broken in their setting for the case where the risk‐free asset does not exist.

Second, several other assumptions in Kandel et al.’s (1995) prior settings are directly opposed to the findings in their or our posteriors. Even though Kandel et al. (1995) use a training sample to assist the prior elicitation, their choice of priors cannot be interpreted as arising directly from inference using the training sample, as is possible for our prior specification. For example, Kandel et al. (1995) assume that μ_p and V_p (the mean and variance of the return of the market portfolio) are a priori independent, while in the posteriors they will be clearly related (see eqq. [21] and [22] for our informative priors).

Third and finally, Kandel et al. (1995) consider a much more limited set of returns and data sample period: their test portfolios consist of 10 size‐sorted portfolios, and the data period is from 1963 to 1987, using a weekly frequency.