Nonparametric Estimation and Conformal Inference of the Sufficient Forecasting With a Diverging Number of Factors

The sufficient forecasting (SF) provides a nonparametric procedure to estimate forecasting indices from high-dimensional predictors to forecast a single time series, allowing for the possibly nonlinear forecasting function. This article studies the asymptotic theory of the SF with a diverging number of factors and develops its predictive inference. First, we revisit the SF and explore its connections to Fama–MacBeth regression and partial least squares. Second, with a diverging number of factors, we derive the rate of convergence of the estimated factors and loadings and characterize the asymptotic behavior of the estimated SF directions. Third, we use the local linear regression to estimate the possibly nonlinear forecasting function and obtain the rate of convergence. Fourth, we construct the distribution-free conformal prediction set for the SF that accounts for the serial dependence. Moreover, we demonstrate the finite-sample performance of the proposed nonparametric estimation and conformal inference in simulation studies and a real application to forecast financial time series.