SDF-based estimation of linear factor models with alternative loss functions

Hansen and Jagannathan (1997) introduce a measure of model misspecification which is based on the L2-norm and which has been wildly used in recent years in order to estimate the parameters of linear factor models. Given the observed asymmetry and excess kurtosis of financial returns, this paper introduces two alternative estimation methods which follow the same approach but replace its loss function. The first one is based on the absolute value of the corresponding deviations while the second one uses a gain-loss ratio-based loss function. We show how these two estimation methods can be implemented by means of simple linear programming and Monte Carlo simulations are undertaken to assess the relative performance of all three methods under varying distributional assumptions. Our results show a promising behaviour of the gain-loss ratio-based estimates and they also emphasise the important gains that may be accomplished by using positivity constraints on the model's associated stochastic discount factor.