Sufficient Forecasting Using Factor Models

32 Pages Posted: 20 May 2015

See all articles by Jianqing Fan

Jianqing Fan

Princeton University - Bendheim Center for Finance

Lingzhou Xue

Pennsylvania State University - Department of Statistics

Jiawei Yao

Princeton University

Date Written: December 10, 2014


We consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect. The dimensionality was first reduced via a high-dimensional factor model implemented by the principal component analysis. Using the extracted factors, we develop a link-free forecasting method, called the sufficient forecasting, which provides several sufficient predictive indices, inferred from high-dimensional predictors, to deliver additional predictive power. Our method is also applicable to cross-sectional sufficient regression using extracted factors. {The connection between the sufficient forecasting and the deep learning architecture is explicitly stated.} The sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function. The proposed method extends the sufficient dimension reduction to high-dimensional regimes by condensing the cross-sectional information through factor models. We derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices. We also show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace. Our method and theory allow the number of predictors to be larger than the number of observations. We finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables.

Keywords: Regression, forecast, deep learning, approximate factor model, principal components, learning indices, sliced inverse regression, dimension reduction.

JEL Classification: C13, C30, C33

Suggested Citation

Fan, Jianqing and Xue, Lingzhou and Yao, Jiawei, Sufficient Forecasting Using Factor Models (December 10, 2014). Available at SSRN: or

Jianqing Fan

Princeton University - Bendheim Center for Finance ( email )

26 Prospect Avenue
Princeton, NJ 08540
United States
609-258-7924 (Phone)
609-258-8551 (Fax)


Lingzhou Xue (Contact Author)

Pennsylvania State University - Department of Statistics ( email )

326 Thomas Building
University Park, PA 16802
United States

Jiawei Yao

Princeton University ( email )

22 Chambers Street
Princeton, NJ 08544-0708
United States

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