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

Abstract

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.

A theory of robust long-run variance estimation

“…Sims (1971), Sims (1972), Faust (1999) and Pötscher (2002). These papers show the impossibility of obtaining correct confidence intervals for the spectral density at a given point for any sample size when the underlying parametric structure of a time series model is too rich in some sense.…”

Some properties of tests for parameters that can be arbitrarily close to being unidentified

“…This suggests that the use of confidence intervals to measure the precision of an estimator may be problematic even when the sample size is very large (cf. Bahadur and Savage (1956), Singh (1963) andPötscher (2002)). The results of LeCam and Schwartz (1960) and Pötscher (2002) suggest that no uniformly consistent estimator exists for parameters that can be arbitrarily close to being unidentified.…”

“…Bahadur and Savage (1956), Singh (1963) andPötscher (2002)). The results of LeCam and Schwartz (1960) and Pötscher (2002) suggest that no uniformly consistent estimator exists for parameters that can be arbitrarily close to being unidentified. When identification fails, standard estimators have non-standard asymptotic distributions.…”

“…In this paper we extend the impossibility results of Dufour (1997), Pfanzagl (1998) and Pötscher (2002) to tests of hypotheses, to help understand what optimal properties a test can have in situations where the parameters can be arbitrarily close to being unidentified. Our results are as follows:…”

“…The source of the inferential problems about parameters that can be arbitrarily close to being unidentified is the discontinuity of the functional defining the parameter of interest at a point in the space of probability measures, as in Dufour (1997), Pfanzagl (1998 and Pötscher (2002). We characterise the problem by showing that the closures of the sets of probability measures under the null and alternative hypotheses have a nonempty intersection, even if the closures of the sets of the values that the parameter of interest can take on under the null and alternative hypotheses do have empty intersection.…”