On the Irrelevance of Impossibility Theorems: The Case of the Long-run Variance

It has been argued that estimating the spectral density function of a stationary stochastic process at the zero frequency (or the so-called long-run variance) is an illposed problem so that any estimate will have an infinite minimax risk (e.g., Pötscher 2002). Most often it is a nuisance parameter that is present in the limit distribution of some statistic and one then needs an estimate of it to obtain test statistics that have a pivotal distribution. In this context, we argue that such an impossibility result is irrelevant. We show that, in the presence of the discontinuities that cause the illposedness of the estimation problem for the long-run variance, using the true value of the spectral density function at frequency zero leads to tests that have either 0 or 100% size and, hence, lead to confidence intervals that are completely uninformative. On the other hand, tests based on standard estimates of the long-run variance will have well defined limit distributions and, accordingly, be more informative.