Lower Risk Bounds and Properties of Confidence Sets for Ill-Posed Estimation Problems with Applications to Spectral Density and Persistence Estimation, Unit Roots, and Estimation of Long Memory Parameters

Important estimation problems in econometrics like estimating the value of a spectral density at frequency zero, which appears in the econometrics literature in the guises of heteroskedasticity and autocorrelation consistent variance estimation and long run variance estimation, are shown to be "ill-posed" estimation problems. A prototypical result obtained in the paper is that the minimax risk for estimating the value of the spectral density at frequency zero is infinite regardless of sample size, and that confidence sets are close to being uninformative. In this result the maximum risk is over commonly used specifications for the set of feasible data generating processes. The consequences for inference on unit roots and cointegration are discussed. Similar results for persistence estimation and estimation of the long memory parameter are given. All these results are obtained as special cases of a more general theory developed for abstract estimation problems, which readily also allows for the treatment of other ill-posed estimation problems such as, e.g., nonparametric regression or density estimation