A theory of robust long-run variance estimation

Abstract

The paper studies the robustness of long-run variance estimators employed for conducting Wald-type tests in standard time series models. It is shown that all long-run variance estimators that are consistent for the variance of Gaussian White Noise lack robustness in the sense that they yield arbitrary results for some underlying process that satisfies a Functional Central Limit Theorem. An analytical measure of robustness of long-run variance estimators is suggested that captures the degree of this fragility.A family of inconsistent long-run variance estimators is derived that optimally trade off this measure of robustness against efficiency. A minor modification of these optimal estimators lead to asymptotically F-distributed test statistics under the null hypothesis, so that robust large sample inference can be conducted very similarly to well-understood small sample Gaussian inference.JEL Classification: C13, C12, C32

AsymptoticFandttests in an efficient GMM setting

“…where f j (r)g are orthonormal basis functions on L 2 [0; 1] satisfying R 1 0 j (r) dr = 0: In the econometric literature, the series LRV estimator has been recently used, for example, in Phillips (2005), Müller (2007), and Sun (2011Sun ( , 2013.…”

Testing Models of Low-Frequency Variability

“…As shown by Müller (2007a), these tests, when applied to X T (i.e., v T ), are optimal in the sense that they maximize (weighted average) power among all scale invariant tests whose asymptotic rejection probability is smaller than or equal to the nominal level for all data generating processes that satisfy X T ⇒ X ∼ N (0 Σ 0 ). In other words, if the convergence X T ⇒ X of (4) completely summarizes the implications for data y t generated by a given low-frequency model, then the test statistics derived in this section applied to v T are asymptotically most powerful (in a weighted average sense) among all scale invariant asymptotically valid tests.…”

Powerful Unit Root Tests Free of Nuisance Parameters

“…In practice, the bandwidth choice for PP‐type tests is as delicate an issue as the lag length selection for ADF‐type tests. Moreover, works by Kiefer et al () or Müller () show that the estimation of the long‐run variance can have adverse effects on the testing procedures in general. In particular, Müller () reinforced the earlier arguments on ‘near observational equivalence’ of I(1) and I(0) processes by Blough () and Faust ().…”