A misspecification test for the higher order co-moments of the factor model

The traditional estimation of higher order co-moments of non-normal random variables by the sample analog of the expectation faces a curse of dimensionality, as the number of parameters increases steeply when the dimension increases. Imposing a factor structure on the process solves this problem; however, it leads to the challenging task of selecting an appropriate factor model. This paper contributes by proposing a test that exploits the following feature: when the factor model is correctly specified, the higher order co-moments of the unexplained return variation are sparse. It recommends a general to specific approach for selecting the factor model by choosing the most parsimonious specification for which the sparsity assumption is satisfied. This approach uses a Wald or Gumbel test statistic for testing the joint statistical significance of the co-moments that are zero when the factor model is correctly specified. The asymptotic distribution of the test is derived. An extensive simulation study confirms the good finite sample properties of the approach. This paper illustrates the practical usefulness of factor selection on daily returns of random subsets of S&P 100 constituents.