We study the asymptotic bias of the factor-augmented regression estimator and its correction, augmented by r factors extracted from N variables over T observations. We consider general weak latent factor models with r signal eigenvalues that may diverge at different rates N-alpha k, 0 < alpha(k) <= 1, k = 1, & mldr; ,r . In the literature, the bias has been derived using an approximation based on a specific data-dependent rotation matrix H for models with alpha(k) = 1 for all k . We instead derive the bias for general weak factor models without this restriction, and consider three rotation matrices: the data-dependent H and H-q, and their population counterpart H. We show that they induce distinct rotation-specific parameters, resulting in different asymptotic biases. This highlights the importance of explicitly specifying the rotation, as it determines the parameter being estimated and the associated asymptotic bias. Among these, the parameter associated with the population rotation matrix H is uniquely defined given signal components and particularly suitable for inference. Based on these results, we propose analytically bias-corrected estimators that do not require estimating alpha(k), and establish their asymptotic normality centered at zero. Finite sample experiments show that the proposed bias-corrected estimators perform very well, and an empirical application illustrates its practical usefulness.
This paper proposes a geometric model averaging (GMA) method in the framework of general likelihood estimation. Instead of using the classical weighted arithmetic mean, we consider a weighted geometric mean of the conditional probability density functions estimated from different candidate models, allowing both parameter uncertainty and model misspecification. The optimal weight is the minimizer of the Kullback-Leibler (KL) divergence of the true density to the weighted density. The geometric form of the weighted density makes it possible to find an approximately unbiased estimator of the KL divergence, and hence the optimal weight can be estimated consequently. Also, the geometric mean is naturally more applicable to density tail estimation than arithmetic mean due to its special form. The proposed GMA method provides a unified framework for a wide range of problems and is connected to many existing model averaging methods. When all the candidate models are misspecified, the weight of GMA is asymptotically optimal in the sense of achieving the lowest possible KL divergence. When correct candidate models are included, estimation consistency is established. Results of a comprehensive simulation study and two real data analyses show favorable performance of GMA relative to alternative model selection and model averaging methods.
