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Confidence Intervals for Diffusion Index Forecasts and Inference for Factor-Augmented Regressions

Jushan Bai
Serena Ng

Abstract

We consider the situation when there is a large number of series, N, each with T observations, and each series has some predictive ability for some variable of interest. A methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors. In this paper, we show that the least squares estimates obtained from these factor-augmented regressions are $\sqrt{T}$T consistent and asymptotically normal if $\sqrt{T}/N\rightarrow0$T/N→0. The conditional mean predicted by the estimated factors is $\min[\sqrt {T},\sqrt{N}\,]$min[T,N] consistent and asymptotically normal. Except when T/N goes to zero, inference should take into account the effect of "estimated regressors" on the estimated conditional mean. We present analytical formulas for prediction intervals that are valid regardless of the magnitude of N/T and that can also be used when the factors are nonstationary.