High Dimensional Estimation and Multi-Factor Models

The purpose of this paper is to test a multi-factor model for realized returns implied by the generalized arbitrage pricing theory (APT) recently developed by Jarrow and Protter (2016) and Jarrow (2016). This model relaxes the convention that the number of risk-factors is small. We estimate this model using a new approach for identifying risk-factors. We first obtain the collection of all possible risk-factors and then provide a simultaneous test, security by security, of which risk-factors are significant for which securities. Since the collection of risk-factors is large and highly correlated, high-dimension methods (including the LASSO and prototype clustering) are used. The multi-factor model is shown to have a significantly better fit than the Fama-French 5-factor model. Robustness tests are also provided.