A unifying approach to the empirical evaluation of asset pricing models

Two main approaches are commonly used to empirically evaluate linear factor pricing models: regression and SDF methods, with centred and uncentred versions of the latter. We show that unlike standard two-step or iterated GMM procedures, single-step estimators such as continuously updated GMM yield numerically identical values for prices of risk, pricing errors, Jensen’s alphas and overidentifying restrictions tests irrespective of the model validity. Therefore, there is arguably a single approach regardless of the factors being traded or not, or the use of excess or gross returns. We illustrate our results by revisiting Lustig and Verdelhan’s (2007) empirical analysis of currency returns.CU-GMM, Factor pricing models, Forward premium puzzle, Generalised Empirical Likelihood, Stochastic discount factor.

On the efficiency and consistency of likelihood estimation in multivariate conditionally heteroskedastic dynamic regression models

We rank the efficiency of several likelihood-based parametric and semiparametric estimators of conditional mean and variance parameters in multivariate dynamic models with i.i.d. spherical innovations, and show that Gaussian pseudo maximum likelihood estimators are inefficient except under normality. We also provide conditions for partial adaptivity of semiparametric procedures, and relate them to the consistency of distributionally misspecified maximum likelihood estimators. We propose Hausman tests that compare Gaussian pseudo maximum likelihood estimators with more efficient but less robust competitors. We also study the efficiency of sequential estimators of the shape parameters. Finally, we provide finite sample results through Monte Carlo simulations.Adaptivity, ARCH, Elliptical Distributions, Financial Returns, Hausman tests, Semiparametric Estimators, Sequential Estimators.

Dynamic Specification Tests for Static Factor Models

We derive computationally simple score tests of serial correlation in the levels and squares of common and idiosyncratic factors in static factor models. The implicit orthogonality conditions resemble the orthogonality conditions of models with observed factors but the weighting matrices refl ect their unobservability. We derive more powerful tests for elliptically symmetric distributions, which can be either parametrically or semipametrically specified, and robustify the Gaussian tests against general non-normality. Our Monte Carlo exercises assess the finite sample reliability and power of our proposed tests, and compare them to other existing procedures. Finally, we apply our methods to monthly US stock returns.ARCH, Financial returns, Kalman filter, LM tests, Predictability

Spanning tests in return and stochastic discount factor mean-variance frontiers: A unifying approach

We propose new spanning tests that assess if the initial and additional assets share the economically meaningful cost and mean representing portfolios. We prove their asymptotic equivalence to existing tests under local alternatives. We also show that unlike two-step or iterated procedures, single-step methods such as continuously updated GMM yield numerically identical overidentifyng restrictions tests, so there is arguably a single spanning test. To prove these results, we extend optimal GMM inference to deal with singularities in the long run second moment matrix of the influence functions. Finally, we test for spanning using size and book-to-market sorted US stock portfolios.Asset Pricing, Continuously Updated GMM, Generalised Empirical Likelihood, Generalised Inverse, Representing Portfolios, Singular Covariance Matrix

Duality in mean-variance frontiers with conditioning information

Portfolio and stochastic discount factor (SDF) frontiers are usually regarded as dual objects, and researchers sometimes use one to answer questions about the other. However, the introduction of conditioning information and active portfolio strategies alters this relationship. For instance, the unconditional portfolio frontier in Hansen and Richard (1987) is not dual to the unconditional SDF frontier in Gallant, Hansen and Tauchen (1990). We characterise the dual objects to those frontiers, and relate them to the frontiers generated with managed portfolios, which are commonly used in empirical work. We also study the implications of a safe asset and other special cases.Asset Pricing, Dynamic Portfolio Strategies, Representing portfolios, Stochastic Discount Factors

Risk and return in the Spanish stock market

In this paper we use Spanish data to test the restrictions that a dynamic APT-type asset pricing model imposes on the risk-return relationship. For monthly returns on ten size-ranked portfolios and a value-weighted index, we find that those restrictions are rejected for different versions of the model over the period 1963-1992, as well as over two subsamples. The evidence for the conditional models suggests that the Spanish stock market is segmented, which probably reflects the fact that it is only deep for a few stocks

THE SCORE OF CONDITIONALLY HETEROSKEDASTIC DYNAMIC REGRESSION MODELS WITH STUDENT T INNOVATIONS, AN LM TEST FOR MULTIVARIATE NORMALITY

We provide numerically reliable analytical expressions for the score of conditionally heteroskedastic dynamic regression models when the conditional distribution is multivariate $t$. We also derive one-sided and 2-sided LM tests for multivariate normality versus multivariate $t$ based on the first two moments of the (squared) norm of the standardised innovations evaluated at the Gaussian quasi-ML estimators of the conditional mean and variance parameters. We reinterpret them as specification tests for multivariate excess kurtosis, and show that they have power against leptokurtic alternatives. Finally, we analyse UK stock returns, and confirm that their conditional distribution has fat tails.Kurtosis, Inequality Constraints, ARCH, Financial Returns.

CONSTRAINED EMM AND INDIRECT INFERENCE ESTIMATION

We develop generalised indirect inference procedures that handle equality and inequality constraints on the auxiliary model parameters. We obtain expressions for the optimal weighting matrices, and discuss as examples an MA(1) estimated as AR(1), an AR(1) estimated as MA(1), and a log-normal stochastic volatility process estimated as a GARCH(1,1) with Gaussian or t distributed errors. In the first example, the constraints have no effect, while in the second, they allow us to achieve full efficiency. As for the third, neither procedure systematically outperforms the other, but equality restricted estimators are better when the additional parameter is poorly estimated.

LIKELIHOOD-BASED ESTIMATION OF LATENT GENERALISED ARCH STRUCTURES

GARCH models are commonly used as latent processes in econometrics, financial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.Bayesian inference; Dynamic Heteroskedasticity; Factor models

Likelihood-based estimation of latent generalised ARCH structures

GARCH models are commonly used as latent processes in econometrics, financial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artificial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.Bayesian inference; Dynamic Heteroskedasticity; Factor models; Markov chain Monte Carlo; Simulated EM algorithm; Volatility.