Interval Estimation with Model Uncertainty : A Method and Applications

Classical interval estimation ignores misspecification uncertainty that is almost inevitable in practice. This paper proposes an approach to construct an uncertainty interval that incorporates misspecification based on an $f$-divergence. We construct the uncertainty interval estimators using various members in the $f$-divergence class, and find that the estimator with a Pearson's \chi^2-divergence produces an analytical solution while others do not. The analytical solution also implies a maximal level of the amount of uncertainty, instead of being no restriction. We further demonstrate the approach to construct tractable interval estimators for parameters of first and second moments, and particularly a simple linear regression model. At last, we empirically examine the approach based on a CAPM for the factor portfolios of momentum, SMB, and HML, respectively. It shows that the out-of-sample uncertainty prediction intervals include observations about twice as many as the classical ones. Moreover, the results suggest that portfolio anomalies can be explained to a larger extent