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The Fama-French three factor models are commonly used in the description of asset returns in finance. Statistically speaking, the Fama-French three factor models imply that the return of an asset can be accounted for directly by the Fama-French three factors, i.e. market, size and value factor, through a linear function. A natural question is: would some kind of transformed Fama-French three factors work better than the three factors? If so, what kind of transformation should be imposed on each factor in order to make the transformed three factors better account for asset returns? In this paper, we are going to address these questions through nonparametric modelling. We propose a data driven approach to construct the transformation for each factor concerned. A generalised maximum likelihood ratio based hypothesis test is also proposed to test whether transformations on the Fama-French three factors are needed for a given data set. Asymptotic properties are established to justify the proposed methods. Intensive simulation studies are conducted to show how the proposed methods work when sample size is finite. Finally, we apply the proposed methods to a real data set, which leads to some interesting findings.

Sufficient dimension reduction [J. Amer. Statist. Assoc. 86 (1991) 316-342] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an adaptive composite quantile approach. Compared to existing methods, (1) it requires minimal assumptions and is capable of revealing all dimension reduction directions; (2) it is robust against outliers and (3) it is structure-adaptive, thus more efficient. Asymptotic results are proved and numerical examples are provided, including a real data analysis., Comment: Published in at http://dx.doi.org/10.1214/14-AOS1242 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The single-index model is one of the most popular semiparametric models in Econometrics. In this paper, we define a quantile regression single-index model, which includes the single-index structure for conditional mean and for conditional variance., Comment: 32pages

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes $\{(Y_{i},\underline{X}_{i})\}$. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging in such estimators into other functionals where some control over higher order terms are required. We apply our results to the estimation of an additive M-regression model., Comment: 40 pages