Semiparametric inference in multivariate fractionally cointegrated systems

Abstract

a b s t r a c tA semiparametric multivariate fractionally cointegrated system is considered, integration orders possibly being unknown and I(0) unobservable inputs having nonparametric spectral density. Two estimates of the vector of cointegrating parameters ν are considered. One involves inverse spectral weighting and the other is unweighted but uses a spectral estimate at frequency zero. Both corresponding Wald statistics for testing linear restrictions on ν are shown to have a standard null χ 2 limit distribution under quite general conditions. Notably, this outcome is irrespective of whether cointegrating relations are ''strong'' (when the difference between integration orders of observables and cointegrating errors exceeds 1/2), or ''weak'' (when that difference is less than 1/2), or when both cases are involved. Finite-sample properties are examined in a Monte Carlo study and an empirical example is presented.

Nonparametric predictive regression

“…Section 5 of the paper provides simulation results for bandwidths of the formĥ n = n −δ m − 1 2  wherem is an estimator for m. Memory estimators are common in inferential methods for fractional systems (e.g. Robinson and Hualde (2003), Marmol and Velasco (2004) and Hualde and Robinson (2010). Preliminary findings of the authors indicate that under certain conditions, Theorems 1 and 2 also hold when a stochastic bandwidth of the 3 We leave detailed theoretical explorations of this matter to future work.…”

Persistence of airline accidents

“…In particular, some new developed methods for fractional cointegration have been proposed by Hualde and Robinson (2006), Chen andHurvich (2006), Johansen (2006) and others. The implementation of some of these methods in our data will be examined in future papers.…”

Small‐band Fixed‐bAsymptotics for Weighted Covariance Estimation in Fractional Cointegration

“…Then, noting that B 2 ( r , δ 2 − δ 1 ) and B 1.2 ( r ) are independent processes, the fixed‐ b limiting distribution of can be easily shown to be mixed Gaussian. This is a crucial result that as in, for example Hualde and Robinson (), can be exploited to construct a Wald test statistic to test for values of ν with a χ 2 fixed‐ b null limiting distribution. Additionally, in the present setting, the fixed‐ b theory enjoys other attractive features.…”

“…However, evidence showing that the fixed‐ b approximation is more accurate than the traditional one might support the empirical relevance of appropriate modifications of the WC estimator, which would lead to Wald test statistics with pivotal fixed‐ b limits. This appears to be especially relevant in view of the size problems displayed by the Wald test statistics based on second‐stage estimators (like those of Robinson and Hualde, (), or Hualde and Robinson, (), whose standard limiting distribution is pivotal. We address briefly this issue in Remark .…”