Quantile factor models (QFM) represent a new class of factor models for high‐dimensional panel data. Unlike approximate factor models (AFM), which only extract mean factors, QFM also allow unobserved factors to shift other relevant parts of the distributions of observables. We propose a quantile regression approach, labeled Quantile Factor Analysis (QFA), to consistently estimate all the quantile‐dependent factors and loadings. Their asymptotic distributions are established using a kernel‐smoothed version of the QFA estimators. Two consistent model selection criteria, based on information criteria and rank minimization, are developed to determine the number of factors at each quantile. QFA estimation remains valid even when the idiosyncratic errors exhibit heavy‐tailed distributions. An empirical application illustrates the usefulness of QFA by highlighting the role of extra factors in the forecasts of U.S. GDP growth and inflation rates using a large set of predictors.

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well–known Dickey–Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d0) versus FI(d1), with d1<d0. When d0=1, the proposed test statistics are based on the OLS estimator, or its t–ratio, of the coefficient on Δd1yt−1 in a regression of Δyt on Δd1yt−1 and, possibly, some lags of Δyt. When d1 is not taken to be known a priori, a pre–estimation of d1 is needed to implement the test. We show that the choice of any T1/2–consistent estimator of d1∈[0 ,1) suffices to make the test feasible, while achieving asymptotic normality. Monte–Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.