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How well can we learn large factor models without assuming strong factors?

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Document pages: 45 pages

Abstract: In this paper, we consider the problem of learning models with a latentfactor structure. The focus is to find what is possible and what is impossibleif the usual strong factor condition is not imposed. We study the minimax rateand adaptivity issues in two problems: pure factor models and panel regressionwith interactive fixed effects. For pure factor models, if the number offactors is known, we develop adaptive estimation and inference procedures thatattain the minimax rate. However, when the number of factors is not specified apriori, we show that there is a tradeoff between validity and efficiency: anyconfidence interval that has uniform validity for arbitrary factor strength hasto be conservative; in particular its width is bounded away from zero even whenthe factors are strong. Conversely, any data-driven confidence interval thatdoes not require as an input the exact number of factors (including weak ones)and has shrinking width under strong factors does not have uniform coverage andthe worst-case coverage probability is at most 1 2. For panel regressions withinteractive fixed effects, the tradeoff is much better. We find that theminimax rate for learning the regression coefficient does not depend on thefactor strength and propose a simple estimator that achieves this rate.However, when weak factors are allowed, uncertainty in the number of factorscan cause a great loss of efficiency although the rate is not affected. In mostcases, we find that the strong factor condition (and or exact knowledge ofnumber of factors) improves efficiency, but this condition needs to be imposedby faith and cannot be verified in data for inference purposes.

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