Regression and SDF approaches with centered or uncentered moments and symmetric or asymmetric normalizations are commonly used to empirically evaluate linear factor pricing models. We show that unlike two-step or iterated GMM procedures, single-step estimators such as continuously updated GMM yield numerically identical risk prices, pricing errors, and overidentifying restrictions tests irrespective of the model validity and regardless of the factors being traded, or the use of excess or gross returns. We illustrate our results with Lustig and Verdelhan’s (2007) currency returns, propose tests to detect some problematic cases, and provide Monte Carlo evidence on the reliability of asymptotic approximations.

We derive Lagrange multiplier and likelihood ratio specification tests for the null hypotheses of multivariate normal and Student-t innovations using the generalized hyperbolic distribution as our alternative hypothesis. We decompose the corresponding Lagrange multiplier-type tests into skewness and kurtosis components. We also obtain more powerful one-sided Kuhn-Tucker versions that are equivalent to the likelihood ratio test, whose asymptotic distribution we provide. Finally, we conduct detailed Monte Carlo exercises to study the size and power properties of our proposed tests in finite samples.