Factor Models: Testing and Forecasting

Abstract

This dissertation focuses on two aspects of factor models, testing and forecasting. For testing, we investigate a more general high-dimensional testing problem, with an emphasis on panel data models. Specifically, we propose a novel technique to boost the power of testing a high-dimensional vector against sparse alternatives. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers, whereas more powerful tests such as thresholding and extreme-value tests require either stringent conditions or bootstrap to derive the null distribution, and often suffer from size distortions. Based on a screening technique, we introduce a ``power enhancement component", which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. As a byproduct, the power enhancement component also consistently identifies the elements that violate the null hypothesis.

Next, we consider forecasting a single time series using many predictors when nonliearity is present. We develop a new methodology called sufficient forecasting, by connecting sliced inverse regression with factor models. The sufficient forecasting correctly estimates projections of the underlying factors and provides multiple predictive indices for further investigation. We derive asymptotic results for the estimate of the central space spanned by these projection directions. Our method allows the number of predictors larger than the sample size, and therefore extends the applicability of inverse regression. Numerical experiments demonstrate that the proposed method improves upon a linear forecasting model. Our results are further illustrated in an empirical study of macroeconomic variables, where sufficient forecasting is found to deliver additional predictive power over conventional methods.