Cesare Robotti

Specification Tests of Asset Pricing Models Using Excess Returns

In this paper, we discuss the impact of different formulations of asset pricing models on the outcome of specification tests that are performed using excess returns. We point out that the popular way of specifying the stochastic discount factor (SDF) as a linear function of the factors is problematic because (1) the specification test statistic is not invariant to an affine transformation of the factors, and (2) the SDFs of competing models can have very different means. In contrast, an alternative specification that defines the SDF as a linear function of the de-meaned factors is free from these two problems and is more appropriate for model comparison. In addition, we suggest that a modification of the traditional Hansen-Jagannathan distance (HJ-distance) is needed when we use the de-meaned factors. The modified HJ-distance uses the inverse of the covariance matrix (instead of the second moment matrix) of excess returns as the weighting matrix to aggregate pricing errors. Asymptotic distributions of the modified HJ-distance and of the traditional HJ-distance based on the de-meaned SDF under correctly specified and misspecified models are provided. Finally, we propose a simple methodology for computing the standard errors of the estimated SDF parameters that are robust to model misspecification. We show that failure to take model misspecification into account is likely to understate the standard errors of the estimates of the SDF parameters and lead us to erroneously conclude that certain factors are priced.

Mimicking Portfolios, Economic Risk Premia, and Tests of Multi-Beta Models

We consider two formulations of the linear factor model (LFM) with nontraded factors. In the first formulation, LFM, risk premia and alphas are estimated by a cross-sectional regression of average returns on betas. In the second formulation, LFM**, the factors are replaced by their projections on the span of excess returns, and risk premia and alphas are estimated by time series regressions. We compare the two formulations and study the small-sample properties of estimates and test statistics. We conclude that the LFM** formulation should be considered in addition to, or even instead of, the more traditional LFM formulation. For corrected versions of Tables 2, 6, and 7, please see the supplemental files posted with this article.

Misspecification-Robust Inference in Linear Asset-Pricing Models with Irrelevant Risk Factors

We show that in misspecified models with useless factors (for example, factors that are independent of the returns on the test assets), the standard inference procedures tend to erroneously conclude, with high probability, that these irrelevant factors are priced and the restrictions of the model hold. Our proposed model selection procedure, which is robust to useless factors and potential model misspecification, restores the standard inference and proves to be effective in eliminating factors that do not improve the models pricing ability. The practical relevance of our analysis is illustrated using simulations and empirical applications.

Asset-pricing Models and Economic Risk Premia: A Decomposition

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.

On the Hansen-Jagannathan distance with a no-arbitrage constraint

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 Exact Distribution of the Hansen–Jagannathan Bound

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.

Evaluation of Asset Pricing Models Using Two-Pass Cross-Sectional Regressions

This chapter provides a review of the two-pass cross-sectional regression methodology, which over the years has become the most popular approach for estimating and testing linear asset pricing models. We focus on some of the recent developments of this methodology and highlight the importance of accounting for model misspecification in estimating risk premia and in comparing the performance of competing asset pricing models.

Further Results on the Limiting Distribution of GMM Sample Moment Conditions

In this paper, we extend the results in Hansen (1982) regarding the asymptotic distribution of generalized method of moments (GMM) sample moment conditions. In particular, 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 asymptotic distribution for a given linear combination of the sample moment conditions and show how to conduct statistical inference. We demonstrate the finite-sample properties of the proposed asymptotic approximation using simulation.

Playing the Field: Geomagnetic Storms and the Stock Market

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 peoples moods, and, in turn, peoples 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 weeks 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.

Minimum-Variance Kernels, Economic Risk Premia, and Tests of Multi-Beta Models

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).