Small sample properties of the GMM specification test based on the Hansen–Jagannathan distance

Following Hansen and Jagannathan (J. Finance 52 (1997) 557), Jagannathan and Wang (J. Finance 51 (1996) 3) propose a distance measure that estimates the maximum pricing error generated by a linear asset model. Jagannathan and Wang propose a test of this HJ-distance using an empirical p-value as an alternative generalized method of moments (GMM) measure to Hansen's (Econometrica 50 (1982) 1029) GMM specification test. Using Monte Carlo analysis, we examine the finite sample properties of these specification tests. While the Hansen test mildly overrejects correct models in commonly used sample size, the empirical p-value of the HJ-distance rejects correct models too severely in such samples to provide a valid test of such models.     

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.     

Improvement in finite sample properties of the Hansen–Jagannathan distance test

Jagannathan and Wang [Jagannthan, R., and Wang, Z., “The conditional CAPM and the cross-section of expected returns.” Journal of Finance, 51 (1996), 3–53] derive the asymptotic distribution of the Hansen–Jagannathan distance (HJ-distance) proposed by Hansen and Jagannathan [Hansen, L.P., and Jagannathan, R., Assessing specific errors in stochastic discount factor models." Journal of Finance, 52 (1997), 557–590], and develop a specification test of asset pricing models based on the HJ-distance. While the HJ-distance has several desirable properties, Ahn and Gadarowski [Ahn, S.C., and Gadarowski, C., “Small sample properties of the GMM specification test based on the Hansen–Jagannathan distance.” Journal of Empirical Finance, 11 (2004), 109–132] find that the specification test based on the HJ-distance overrejects correct models too severely in commonly used sample size to provide a valid test. This paper proposes to improve the finite sample properties of the HJ-distance test by applying the shrinkage method [Ledoit, O., and Wolf, M., “Improved estimation of the covariance matrix of stock returns with an application to portfolio selection.” Journal of Empirical Finance, 10 (2003), 603–621] to compute its weighting matrix. The proposed method improves the finite sample performance of the HJ-distance test significantly.     

Nonparametric estimation and testing of stochastic discount factor

This paper attempts to estimate stochastic discount factor (SDF) proxies nonparametrically using the conditional Hansen–Jagannathan distance. Nonparametric estimation can not only avoid misspecification when dealing with nonlinearity in the model but also provide more precise information about the local properties of the estimators. Empirical studies show that our method performs better than the alternative parametric polynomial models, and furthermore, we find that the return on aggregate wealth can sufficiently explain the SDF proxies when one deals with nonlinearity appropriately.     

Evaluating asset pricing models in the Korean stock market

This paper evaluates and compares asset pricing models in the Korean stock market. The asset pricing models considered are the CAPM, APT-motivated models, the Consumption-based CAPM, Intertemporal CAPM-motivated models, and the Jagannathan and Wang conditional CAPM model. By using various test portfolios as well as individual stocks, we conduct time-series tests and cross-sectional regression tests based on individual t-tests, the joint F-tests, the Hansen and Jagannathan (1997) distance, and R-squares. Overall, the Fama and French (1993) five-factor model performs most satisfactorily among the asset pricing models considered in explaining the intertemporal and cross-sectional behavior of stock returns in Korea. The Fama and French (1993) three-factor model, the Chen et al. (2010) three-factor model, and the Campbell (1996) model are the next. The results indicate that the two bond portfolios, term spread and default spread, play an important role in explaining stock returns in Korea.     

Return Distributions and Improved Tests of Asset Pricing Models

We compare and contrast some existing ordinary least squares(OLS)- and generalized method of moments (GMM)-based tests ofasset pricing models with a new more general test. This newtest is valid under the assumption that returns are ellipticallydistributed, a necessary and sufficient assumption of the linearcapital asset pricing model (CAPM). This new test fails to rejectthe CAPM on a dataset of stocks sorted by market valuations,whereas similar tests constructed from OLS and GMM estimationmethods reject the linear CAPM. We also find that outliers reducethe OLS-estimated mispricing of the linear CAPM on monthly returnssorted by previous performance, that is, momentum. Monte Carloevidence supports superior size and power properties of thenew test relative to OLS- and GMM-based tests.     

A best choice among asset pricing models? The Conditional Capital Asset Pricing Model in Australia

We use Australian data to test the Conditional Capital Asset Pricing Model (Jagannathan and Wang, 1996). Our results are generally supportive: the model performs well compared with a number of competing asset pricing models. In contrast to the study by Jagannathan and Wang, however, we find that the inclusion of the market for human capital does not save the concept of the time‐independent market beta (it remains insignificant). We find support for the role of a small‐minus‐big factor in pricing the cross‐section of returns and find grounds to disagree with Jagannathan and Wang's argument that this factor proxies for misspecified market risk.     

Empirical evaluation of asset pricing models: Arbitrage and pricing errors in contingent claims

Hansen and Jagannathan (1997) have developed two measures of pricing errors for asset-pricing models: the maximum pricing error in all static portfolios of the test assets and the maximum pricing error in all contingent claims of the assets. In this paper, we develop simulation-based Bayesian inference for these measures. While the literature reports that the time-varying extensions substantially reduce pricing errors of classic models on the standard test assets, our analysis shows that the reduction is much smaller based on the second measure. Those time-varying models have large pricing errors on the contingent claims of the test assets because their stochastic discount factors are often negative and admit arbitrage opportunities.     

Empirical Evaluation of Asset–Pricing Models: A Comparison of the SDF and Beta Methods

The stochastic discount factor (SDF) method provides a unified general framework for econometric analysis of asset–pricing models. There have been concerns that, compared to the classical beta method, the generality of the SDF method comes at the cost of efficiency in parameter estimation and power in specification tests. We establish the correct framework for comparing the two methods and show that the SDF method is as efficient as the beta method for estimating risk premiums. Also, the specification test based on the SDF method is as powerful as the one based on the beta method.     

Evaluation of linear asset pricing models by implied portfolio performance

We present a theoretical perspective that motivates the use of the Generalized Least Squares R-Square, prominently advocated by Lewellen et al. [Lewellen, J., Nagel, S., Shanken, J., forthcoming. A skeptical appraisal of asset-pricing tests. Journal of Financial Economics], as an evaluation measure for multivariate linear asset pricing models. Adapting results from Shanken [Shanken, J., 1985. Multivariate tests of the zero-beta CAPM. Journal of Financial Economics 14, 327–348] and Kandel and Stambaugh [Kandel, S., Stambaugh, R.F., 1995. Portfolio inefficiency and the cross-section of expected returns. Journal of Finance 50, 157–184], we provide various interpretations and a graphical account in mean-variance space of this measure, facilitating a better understanding of its properties. We furthermore relate it to another leading evaluation metric, the HJ-distance of Hansen and Jagannathan [Hansen, L.P., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 557–590]. Additionally, we present a comparison between these evaluation measures using mean-variance mathematics in risk-return space, and we provide a simple formula for calculating both model evaluation measures that involves only the parameters of the mean-variance asset and factor frontiers.     

Market proxies as factors in linear asset pricing models: Still living with the roll critique

A new model misspecification measure for linear asset pricing models is proposed for the case where misspecification maps to latency of one of the pricing factors; in this case, the market return. This measure is suited both for testing models that include the market return as a pricing factor in a traditional sense (i.e., whether the chosen model does or does not price a collection of risky assets) and ranking those models (i.e., determining which model performs best). The proposed measure is used in pricing portfolios reflecting the size, value, and momentum premia. The conditional CAPM of Jagannathan and Wang (1996) is found to best the performance of both the simple CAPM and the ICAPM of Petkova (2006). Moreover, it is discovered that winner stocks in a momentum portfolio may have higher market betas than loser stocks.     

Estimating and testing beta pricing models: Alternative methods and their performance in simulations

We conduct a simulation analysis of the Fama and MacBeth[1973. Risk, returns and equilibrium: empirical tests. Journal of Political Economy 71, 607–636.] two-pass procedure, as well as maximum likelihood (ML) and generalized method of moments estimators of cross-sectional expected return models. We also provide some new analytical results on computational issues, the relations between estimators, and asymptotic distributions under model misspecification. The generalized least squares estimator is often much more precise than the usual ordinary least squares (OLS) estimator, but it displays more bias as well. A “truncated” form of ML performs quite well overall in terms of bias and precision, but produces less reliable inferences than the OLS estimator.     

Evaluating a non-linear asset pricing model on international data

The paper analyses the ability of a non-linear asset pricing model suggested by Dittmar [Dittmar, R.F., 2002. Non-linear pricing kernels, kurtosis preference, and the cross-section of equity returns. Journal of Finance 57, 369-403] to explain the returns on international value and growth portfolios. For comparison we use competing pricing models such as the ICAPM, the exchange rate risk augmented ICAPM and the international two-factor model proposed by Fama and French [Fama, E.F., French, K. R., 1998. Value versus growth: The international evidence. Journal of Finance 53, 1975-1999]. All models are evaluated both unconditionally and conditionally. The models are evaluated by applying the Hansen and Jagannathan distance measure, and we also employ several alternative measures to ensure a robust comparison of the models. We find support for the model of Dittmar [Dittmar, R.F., 2002. Non-linear pricing kernels, kurtosis preference, and the cross-section of equity returns. Journal of Finance 57, 369-403]. Evaluated conditionally, this model successfully passes all the different diagnostic tests performed in the analysis.     

Alternative beta risk estimators and asset pricing tests in emerging markets: The case of Pakistan

This paper tests and compares the applicability of two asset pricing models specifically, the CAPM and the Fama–French three factor models for an emerging stock market namely, Pakistan. The paper analyses a number of beta risk estimators, including OLS, the Dimson thin trading estimator, a trade-to-trade estimator and a sample selectivity estimator. To uncover any possible influence of the return interval and the type of the market index, the analysis is carried out on three data frequencies namely daily, weekly and monthly as well as for a value and an equally weighted market index. The alternative beta estimators appear to correct thin trading bias but their effects on asset pricing tests are not visible. Moreover contrary to the expectations the test results for monthly and weekly frequencies are not promising. Instead for daily data the cross-section of returns are explained by a number of risk factors and trading volume.     

Cross‐sectional Tests of Conditional Asset Pricing Models: Evidence from the German Stock Market

We study the performance of conditional asset pricing models and multifactor models in explaining the German cross‐section of stock returns. We focus on several variables, which (according to previous research) are associated with market expectations on future market excess returns or business cycle conditions. Our results suggest that the empirical performance of the Capital Asset Pricing Model (CAPM) can be improved when allowing for time‐varying parameters of the stochastic discount factor. A conditional CAPM using the term spread explains the returns on our size and book‐to‐market sorted portfolios about as well as the Fama‐French three‐factor model and performs best in terms of the Hansen‐Jagannathan distance. Structural break tests do not necessarily indicate parameter instability of conditional model specifications. Another major finding of the paper is that the Fama‐French model – despite its generally good cross‐sectional performance – is subject to model instability. Unconditional models, however, do a better job than conditional ones at capturing time‐series predictability of the test portfolio returns.     

Financial Intermediaries and the Cross‐Section of Asset Returns

Financial intermediaries trade frequently in many markets using sophisticated models. Their marginal value of wealth should therefore provide a more informative stochastic discount factor (SDF) than that of a representative consumer. Guided by theory, we use shocks to the leverage of securities broker‐dealers to construct an intermediary SDF. Intuitively, deteriorating funding conditions are associated with deleveraging and high marginal value of wealth. Our single‐factor model prices size, book‐to‐market, momentum, and bond portfolios with an R² of 77% and an average annual pricing error of 1%—performing as well as standard multifactor benchmarks designed to price these assets.     

Modeling daily price limits

This paper characterizes the behavior of observed asset prices under price limits and proposes the use of two-limit truncated and Tobit regression models to analyze regression models whose dependent variable is subject to price limits. Through a proper arrangement of the sample, these two models, the estimation of which is easy to implement, are applied only to subsets of the sample under study, rather than the full sample. Using the estimation of simple linear regression model as an example, several Monte Carlo experiments are conducted to compare the performance of the maximum likelihood estimators (MLEs) based on these two models and a generalized method of moments (GMM) estimator developed by K. C. John Wei and R. Chiang. The results show that under different price limits and various distributional assumptions for the error terms, the MLEs based on the two-limit Tobit and truncated regression models and the GMM estimator perform reasonably well, while the naive OLS estimator is downward biased. Overall, the MLE based on the two-limit Tobit model outperforms the other estimators.     

Credibility ratemaking using collateral information

Credibility ratemaking is a technique used in pricing health care, property and casualty, workers’ compensation, and group life coverages. It has been a part of actuarial practice since the time of Mowbray's (1914) contribution. In earlier work, we showed how many types of credibility models could be expressed as special cases of mixed linear models. This article extends this approach to credibility by formally introducing collateral information through the use of Bayesian methods.

Specifically, we derive credibility estimators and mean square errors for normal hierarchical linear models. We provide intuition for the credibility estimators by establishing the link between these estimators and homogeneous and inhomogeneous estimators that appear in non-Bayesian credibility theory.     

Temporal Aggregation and the Continuous-Time Capital Asset Pricing Model

We examine how the empirical implications of the Capital Asset Pricing Model (CAPM) are affected by the length of the period over which returns are measured. We show that the continuous-time CAPM becomes a multifactor model when the asset pricing relation is aggregated temporally. We use Hansen's Generalized Method of Moments (GMM) approach to test the continuous-time CAPM at an unconditional level using size portfolio returns. The results indicate that the continuous-time CAPM cannot be rejected. In contrast, the discrete-time CAPM is easily rejected by the tests. These results have a number of important implications for the interpretation of tests of the CAPM which have appeared in the literature.     

An examination of alternative CAPM-based models in UK stock returns

We evaluate the performance of unconditional and conditional versions of seven stochastic discount factor models in UK stock returns between January 1975 and December 2001. We find that the conditional four-moment capital asset pricing model (CAPM) has the best performance among the models we consider in terms of the lowest [Hansen, L.P., Jagannathan, R., 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 591–607] distance measure and explaining the time-series predictability of industry portfolio excess returns. Conditional models also do a better job than unconditional models. However we find that the superior performance of the conditional four-moment CAPM, and conditional models in general, arises in part due to overfitting the data.