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Intermediary and downside-risk asset-pricing theories lay the foundations for spanning the multi-asset return space by a small number of risk factors. Recent studies document strong empirical\ud support for such factors across major asset classes. We revisit these results and show that robust evidence for common factor pricing remains elusive. Importantly, the proposed risk factors do not\ud seem to provide incremental information to the traditional market factor. We argue that most of the economic and statistical challenges are not specific to these analyses and, with the aid of a placebo test, offer general recommendations for improving empirical tests, thus adding to the prescriptions in Lewellen, Nagel, and Shanken (2010).

We propose a methodology for estimating and testing beta-pricing models when a large number of assets is available for investment but the number of time-series observations is fixed. We first consider the case of correctly specified models with constant risk premia, and then extend our framework to deal with time-varying risk premia, potentially misspecified models, firm characteristics, and unbalanced panels. We show that our large cross-sectional framework poses a serious challenge to common empirical findings regarding the validity of beta-pricing models. In the context of pricing models with Fama-French factors, firm characteristics are found to explain a much larger proportion of variation in estimated expected returns than betas. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Recurrence relations for integrals that involve the density of multivariate normal distributions are developed. These recursions allow fast computation of the moments of folded and truncated multivariate normal distributions. Besides being numerically efficient, the proposed recursions also allow us to obtain explicit expressions of low-order moments of folded and truncated multivariate normal distributions. Supplementary material for this article is available online.

This paper is concerned with statistical inference and model evaluation in possibly misspecified and unidentified linear asset-pricing models estimated by maximum likelihood. Strikingly, when spurious factors (that is, factors that are uncorrelated with the returns on the test assets) are present, the models exhibit perfect fit, as measured by the squared correlation between the model's fitted expected returns and the average realized returns. Furthermore, factors that are spurious are selected with high probability, while factors that are useful are driven out of the model. While ignoring potential misspecification and lack of identification can be very problematic for models with macroeconomic factors, empirical specifications with traded factors (e.g., Fama and French, 1993, and Hou, Xue, and Zhang, 2015) do not suffer of the identification problems documented in this study.

We provide an in-depth analysis of the theoretical properties of the Hansen–Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. Under a multivariate elliptical distribution assumption, we present explicit expressions for the HJ-distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the stochastic discount factor (SDF) parameters in the case of linear SDFs. This allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to evaluate and rank models. We also study the asymptotic and finite-sample properties of the sample constrained HJ-distance. Finally, we demonstrate the practical relevance of our theoretical findings in an empirical illustration of some popular asset-pricing models.

We show how to conduct asymptotically valid tests of model comparison when the extent of model mispricing is gauged by the squared Sharpe ratio improvement measure. This is equivalent to ranking models on their maximum Sharpe ratios, effectively extending the GRS test to accommodate comparison of non-nested models. Mimicking portfolios can be substituted for any nontraded model factors and estimation error in the portfolio weights is taken into account in the statistical inference. A variant of the Fama and French (2018) six-factor model, with a monthly-updated version of the usual value spread, emerges as the dominant model.

Since Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), the two-pass cross- sectional regression (CSR) methodology has become the most popular approach for estimating and testing asset pricing models. Statistical inference with this method is typically conducted under the assumption that the models are correctly specified, that is, expected returns are exactly linear in asset betas. This assumption can be a problem in practice since all models are, at best, approximations of reality and are likely to be subject to a certain degree of misspecification. We propose a general methodology for computing misspecification-robust asymptotic standard errors of the risk premia estimates. We also derive the asymptotic distribution of the sample CSR R 2 and develop a test of whether two competing linear beta pricing models have the same population R 2 . This test provides a formal alternative to the common heuristic of simply comparing the R 2 estimates in evaluating relative model performance. Finally, we provide an empirical application, which demonstrates the importance of our new results when applied to a variety of asset pricing models. JEL classification: G12

In this article, we examine the limiting behavior of generalized method of moments (GMM) sample moment conditions and point out an important discontinuity that arises in their asymptotic distribution. We show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the appropriate asymptotic (weighted chi-squared) distribution when this degeneracy occurs and show how to conduct asymptotically valid statistical inference. We also propose a new rank test that provides guidance on which (standard or nonstandard) asymptotic framework should be used for inference. The finite-sample properties of the proposed asymptotic approximation are demonstrated using simulated data from some popular asset pricing models.

Although it is of interest to test whether or not a particular asset pricing model is literally true, a more useful task for empirical researchers is to determine how wrong a model is and to compare the performance of competing asset pricing models. In this paper, we propose a new methodology to test whether or not two competing linear asset pricing models have the same Hansen-Jagannathan distance. We show that the asymptotic distribution of the test statistic depends on whether the competing models are correctly specified or misspecified, and on whether the competing models are nested or non-nested. In addition, given the increasing interest in misspecified models, we propose a simple methodology for computing the standard errors of the estimated stochastic discount factor parameters that are robust to model misspecification. Using monthly data on 25 size and book-to-market ranked portfolios and the one-month T-bill, we show that the commonly used returns and factors are, for the most part, too noisy for us to conclude that one model is superior to the other models in terms of Hansen-Jagannathan distance. Specifically, there is little evidence that conditional and intertemporal capital asset pricing model (CAPM)-type specifications outperform the simple unconditional CAPM. In addition, we show that many of the macroeconomic factors commonly used in the literature are no longer priced once potential model misspecification is taken into account. The Author 2008. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

This paper studies some seemingly anomalous results that arise in possibly misspecified and unidentified linear asset-pricing models estimated by maximum likelihood and one-step generalized method of moments (GMM). Strikingly, when useless factors (that is, factors that are independent of the returns on the test assets) are present, the models exhibit perfect fit, as measured by the squared correlation between the model's fitted expected returns and the average realized returns, and the tests for correct model specification have asymptotic power that is equal to the nominal size. In other words, applied researchers will erroneously conclude that the model is correctly specified even when the degree of misspecification is arbitrarily large. We also derive the highly nonstandard limiting behavior of these invariant estimators and their t-tests in the presence of identification failure. These results reveal the spurious nature of inference as useless factors are selected with high probability, while useful factors are driven out from the model. The practical relevance of our findings is demonstrated using simulations and an empirical application.

This paper shows that in misspecified models with risk factors that are uncorrelated with the test asset returns, the conventional inference methods tend to erroneously conclude, with high probability, that these factors are priced. Our proposed model selection procedure, which is robust to identification failure and potential model misspecification, restores the standard inference and proves to be effective in eliminating factors that do not improve the model's pricing ability. Applying our methodology to several popular asset-pricing models suggests that only the market and book-to-market factors appear to be priced, while the statistical evidence on the pricing ability of many macroeconomic factors is rather weak.

We derive new results on the asymptotic behavior of the estimated parameters of a linear asset pricing model and their associated t-statistics in the presence of a factor that is independent of the returns. The inclusion of this "useless" factor in the model leads to a violation of the full rank (identification) condition and renders the inference nonstandard. We show that the estimated parameter associated with the useless factor diverges with the sample size but the misspecification-robust t-statistic is still well-behaved and has a standard normal limiting distribution. The asymptotic distributions of the estimates of the remaining parameters and the model specification test are also affected by the presence of a useless factor and are nonstandard. We propose a robust and easy-to-implement model selection procedure that restores the standard inference on the parameters of interest by identifying and removing the factors that do not contribute to improved pricing. The finite-sample properties of our asymptotic approximations and the practical relevance of our results are illustrated using simulations and an empirical application.

We provide an in-depth analysis of the theoretical properties of the Hansen-Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. Under a multivariate elliptical distribution assumption, we present explicit expressions for the HJ-distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the SDF parameters in the case of linear SDFs. This approach allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to rank asset pricing models.

Over the years, many asset pricing studies have employed the sample cross-sectional regression (CSR) R2 as a measure of model performance. We derive the asymptotic distribution of this statistic and develop associated model comparison tests, taking into account the inevitable impact of model misspecification on the variability of the two-pass CSR estimates. We encounter several examples of large R2 differences that are not statistically significant. A version of the intertemporal CAPM exhibits the best overall performance, followed by the "three-factor model" of Fama and French (1993). Interestingly, the performance of prominent consumption CAPMs proves to be sensitive to variations in experimental design.

Using data for the Philippines, I develop and estimate a heterogeneous agent model to analyze the role of monetary policy in a small open economy subject to sizable remittance fluctuations. I include rule-of-thumb households with no access to financial markets and test whether remittances are countercyclical and serve as an insurance mechanism against macroeconomic shocks. When evaluating the welfare implications of alternative monetary rules, I consider both an anticipated large secular increase in the trend growth of remittances and random cyclical fluctuations around this trend. In a purely deterministic framework, a nominal fixed exchange rate regime avoids a rapid real appreciation and performs better for recipient households facing an increasing trend for remittances. A flexible floating regime is preferred when unanticipated shocks driving the business cycle are also part of the picture.

We provide an in-depth analysis of the theoretical and statistical properties of the Hansen- Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. We show that for stochastic discount factors (SDF) that are spanned by the returns on the test assets, testing the equality of HJ distances with no-arbitrage constraints is the same as testing the equality of HJ distances without no- arbitrage constraints. A discrepancy can exist only when at least one SDF is a function of factors that are poorly mimicked by the returns on the test assets. Under a joint normality assumption on the SDF and the returns, we derive explicit solutions for the HJ distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the SDF parameters in the case of linear SDFs. This solution allows us to show that nontrivial differences between HJ distances with and without no-arbitrage constraints can arise only when the volatility of the unspanned component of an SDF is large and the Sharpe ratio of the tangency portfolio of the test assets is very high. Finally, we present the appropriate limiting theory for estimation, testing, and comparison of SDFs using the HJ distance with a no-arbitrage constraint.

The risk premia of linear factor models on economic (non-traded) risk factors can be decomposed into: i) the premium on maximum-correlation portfolios mimicking the factors; ii) (minus) the covariance between the non-traded components of the pricing kernel and the factors; and iii) (minus) the mispricing of the maximum-correlation portfolios. For a given set of assets available for investment, the first component is the same across models and is typically estimated with little bias and high precision. We conclude that the premia on maximum-correlation portfolios are appealing alternatives to the risk premia of linear factor models, with the dividend yield being the only economic factor significantly priced.

We derive asymptotic standard errors of risk premia estimates based on the popular two-pass cross-sectional regression methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) when univariate betas are used as regressors. Our standard errors are robust to model misspecification and allow for general distributional assumptions. In testing whether the beta risk of a given factor is priced, our misspecification robust standard error can lead to economically different conclusions from those based on the Jagannathan and Wang (1998) standard error which is derived under the correctly specified model.

Working Paper 2009-12 March 2009 Abstract: Since Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), the two-pass cross-sectional regression (CSR) methodology has become the most popular tool for estimating and testing beta asset pricing models. In this paper, we focus on the case in which simple regression betas are used as regressors in the second-pass CSR. Under general distributional assumptions, we derive asymptotic standard errors of the risk premia estimates that are robust to model misspecification. When testing whether the beta risk of a given factor is priced, our misspecification robust standard error and the Jagannathan and Wang (1998) standard error (which is derived under the correctly specified model) can lead to different conclusions. JEL classification: G12 Key words: two-pass cross-sectional regressions, risk premia, model misspecification, simple regression betas, multivariate betas Introduction In the empirical asset pricing literature, the popular two-pass cross-sectional regression (CSR) methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) is often used for estimating risk premia and testing pricing models that relate expected security returns to security betas on economic factors (beta pricing models). Although there are many variations of this two-pass methodology, its basic approach always involves two steps. In the first pass, the betas of the test assets are estimated using the usual ordinary least squares (OLS) time series regression of returns on some common factors. In the second pass, the returns on test assets are regressed on the estimated betas obtained from the first pass. By running this second-pass CSR on a period-by-period basis, we obtain time series of the intercept and the slope coefficients. The average values of the intercept and the slope coefficients are then used as estimates of the zero-beta rate and the risk premia. Usually, asset betas are defined as the OLS slope coefficients in the multiple regression of asset returns on factors and are referred to as multiple regression or multivariate betas. However, there is a potential issue with the use of multiple regression betas: unless the factors are uncorrelated, the beta of an asset with respect to a particular factor in general depends on what other factors are included in the first-pass time series OLS regression. As a result, a factor can possess additional explanatory power for the cross-sectional differences in expected returns but yet have a zero risk premium in a model with multiple factors. This makes it problematic to use the risk premium of a factor for the purpose of model selection. To overcome this problem, Chen, Roll, and Ross (1986) and Jagannathan and Wang (1996, 1998) define the beta of an asset with respect to a given factor as the OLS slope coefficient in a simple regression of its return on the factor. These betas are normally referred to as simple regression or univariate betas. In models with simple regression betas, adding or deleting a factor in a model will not change the values of the betas corresponding to the other factors and selecting models based on risk premia becomes more meaningful. (1) Jagannathan and Wang (1998) present an asymptotic theory for models with simple regression betas. (2) However, their asymptotic results rest on the assumption that expected returns are exactly linear in the betas, i.e., the beta pricing model is correctly specified. It is difficult to justify this assumption when estimating the zero-beta rate and risk premia parameters for many different models because some (if not all) of the models are bound to be misspecified. Since asset pricing models are, at best, approximations of reality, it is inevitable that we will often, knowingly or unknowingly (since asset pricing tests have limited power), estimate an expected return relation that departs from exact linearity in the betas. …

Under the assumption of multivariate normality of asset returns, this paper presents a geometric interpretation and the finite-sample distributions of the sample Hansen–Jagannathan bounds on the variance of admissible stochastic discount factors, with and without the nonnegativity constraint on the stochastic discount factors. In addition, since the sample Hansen–Jagannathan bounds can be very volatile, we propose a simple method to construct confidence intervals for the population Hansen–Jagannathan bounds. Finally, we show that the analytical results in the paper are robust to departures from the normality assumption. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2222 . This paper was accepted by Jerome Detemple, operations management.

The risk premia assigned to economic (non-traded) risk factors can be decomposed into three parts: i) the risk premia on maximum-correlation portfolios mimicking the factors; ii) (minus) the covariance between the non-traded components of the candidate pricing kernel of a given model and the factors; and iii) (minus) the mis-pricing assigned by the candidate pricing kernel to the maximum-correlation mimicking portfolios. The first component is the same across asset-pricing models, and is typically estimated with little (absolute) bias and high precision. The second component, on the other hand, is essentially arbitrary, and can be estimated with large (absolute) biases and low precisions by multi-beta models with non-traded factors. This second component is also sensitive to the criterion minimized in estimation. The third component is estimated reasonably well, both for models with traded and non-traded factors. We conclude that the economic risk premia assigned by multi-beta models with non-traded factors can be very unreliable. Conversely, the risk premia on maximum-correlation portfolios provide more reliable indications of whether a non-traded risk factor is priced. These results hold for both the constant and the time-varying components of the factor risk premia.

This paper considers two alternative formulations of the linear factor model (LFM) with nontraded factors. The first formulation is the traditional LFM, where the estimation of risk premia and alphas is performed by means of a cross-sectional regression of average returns on betas. The second formulation (LFM*) replaces the factors with their projections on the span of excess returns. This formulation requires only time-series regressions for the estimation of risk premia and alphas. We compare the theoretical properties of the two approaches and study the small-sample properties of estimates and test statistics. Our results show that when estimating risk premia and testing multi-beta models, the LFM* formulation should be considered in addition to, or even instead of, the more traditional LFM formulation.

Explaining movements in daily stock prices is one of the most difficult tasks in modern finance. This paper contributes to the existing literature by documenting the impact of geomagnetic storms on daily stock market returns. A large body of psychological research has shown that geomagnetic storms have a profound effect on people's moods, and, in turn, people's moods have been found to be related to human behavior, judgments and decisions about risk. An important finding of this literature is that people often attribute their feelings and emotions to the wrong source, leading to incorrect judgments. Specifically, people affected by geomagnetic storms may be more inclined to sell stocks on stormy days because they incorrectly attribute their bad mood to negative economic prospects rather than bad environmental conditions. Misattribution of mood and pessimistic choices can translate into a relatively higher demand for riskless assets, causing the price of risky assets to fall or to rise less quickly than otherwise. The authors find strong empirical support in favor of a geomagnetic-storm effect in stock returns after controlling for market seasonals and other environmental and behavioral factors. Unusually high levels of geomagnetic activity have a negative, statistically and economically significant effect on the following week's stock returns for all U.S. stock market indices. Finally, this paper provides evidence of substantially higher returns around the world during periods of quiet geomagnetic activity.

This paper documents the impact of geomagnetic storms (GMS) on world and country-specific stock market returns. For the world index and for most of the international indices in our sample, we find that the previous week's unusually high levels of geomagnetic activity have a negative, statistically and economically significant impact on today's stock returns. Our results are consistent with psychological theories of "misattribution of mood," since GMS have been found to negatively affect people's judgment and behavior.

In this paper, I study the behavior of an investor with unit risk aversion who maximizes a utility function defined over the mean and the variance of a portfolio's return. Conditioning information is accessible without cost and an unconditionally riskless asset is available in the market. ; The proposed approach makes it possible to compare the performance of a benchmark tangency portfolio (formed from the set of unrestricted estimates of portfolio weights) to the performance of a restricted tangency portfolio which uses single-index and multi-index asset pricing models to constrain the first moments of asset returns. ; The main findings of the paper are summarized as follows: i) The estimates of the constant and time-varying tangency portfolio weights are extremely volatile and imprecise. Using an asset pricing model to constrain mean asset returns eliminates extreme short positions in the underlying securities and improves the precision of the estimates of the weights. ii) Partially restricting mean asset returns according to single-index and multi-index asset pricing models improves the out-of-sample performance of the tangency portfolio. iii) Active investment strategies (i.e., strategies that incorporate the role played by conditioning information in investment decisions) strongly dominate passive investment strategies in-sample but do not provide any convincing pattern of improved out-of-sample performance.

The capital asset pricing model (CAPM), favored by financial researchers and practitioners fifteen years ago, holds that the extra return on a risky asset comes from bearing market risk only. But newer evidence supports the intertemporal CAPM (I-CAPM) theory (Merton 1973), which suggests that the premium on any risky asset is related not only to market risk but also to additional economic variables. ; This article reviews and interprets recent advances in the asset pricing literature. The study seeks to shed light on the sources of economic risk that investors should track and hedge against and the sign of the risk premia commanded by economic and financial risks. ; The author empirically measures the impact of prespecified financial and economic variables on the risk-return trade-off by looking at how they affect (or predict) the mean and the variance of asset returns. The analysis shows that variables such as the market portfolio, the term structure, the default premium, and the consumption-aggregate wealth ratio positively affect average asset returns and command positive risk premia while the inflation portfolio negatively affects returns and commands a negative premium. ; The article also provides extensive evidence of time variation in economic risk premia, showing that expected compensation for bearing different sorts of risk is larger at some times and smaller at others depending on economic conditions.

This paper uses minimum-variance (MV) admissible kernels to estimate risk premia associated with economic risk variables and to test multi-beta models. Estimating risk premia using MV kernels is appealing because it avoids the need to 1) identify all relevant sources of risk and 2) assume a linear factor model for asset returns. Testing multi-beta models in terms of restricted MV kernels has the advantage that 1) the candidate kernel has the smallest volatility and 2) test statistics are easy to interpret in terms of Sharpe ratios. The authors find that several economic variables command significant risk premia and that the signs of the premia mostly correspond to the effect that these variables have on the risk-return trade-off, consistent with the implications of the intertemporal capital asset pricing model (I-CAPM). They also find that the MV kernel implied by the I-CAPM, while formally rejected by the data, consistently outperforms a pricing kernel based on the size and book-to-market factors of Fama and French (1993).

This paper uses minimum-variance (MV) admissible kernels to estimate risk premia associated with inflation and foreign exchange risk and to test multi-beta models. Estimating risk premia using MV kernels is appealing because it avoids the need to: i) identify all relevant sources of risk; ii) assume a linear factor model for asset returns. Testing multi-beta models in terms of restricted minimum-variance kernels has the advantage that: i) the candidate kernel has the smallest volatility; ii) test statistics are easy to interpret in terms of Sharpe ratios. I find that only global market risk is significant both conditionally and unconditionally as indicated by the high positive and significant relative Sharpe ratio. When using the Hansen & Jagannathan (1991, 1997) variance bounds and distance measures as testing devices, I find that, while all international asset pricing models are formally rejected by the data, their pricing implications are significantly different. The international intertemporal CAPM in presence of deviations from Purchasing Power Parity (PPP) outperforms all the other asset pricing models.

This paper offers a novel way of testing whether prespecified risk variables command significant risk premia. Specifically, we construct portfolios of securities to mimick the variation in the chosen risk variables, and we estimate the conditional and unconditional expected returns on these portfolios in excess of the risk-free rate. We also test for the ability of these hedging portfolios to price the returns on a large collection of assets. Our approach has several advantages over more traditional approachs that model asset returns as linear functions of a given set of explicit factors. First, the risk premia that we estimate do not depend on the appropriate specification of either an asset-pricing model or a stochastic process for asset returns. Second, while we allow for time variation in the conditional risk premia associated with economic risks, our estimates of the unconditional premia do not require explicit modeling of such time variation. Third, we can introduce conditioning information effectively to expand the set of asset returns under scrutiny and improve the ability of the hedging portfolio returns to track the economic risks. Fourth, we are able to impose the no-arbitrage positivity restriction on the pricing kernel, a requirement missing from the standard formulation of multi-beta models.