Jackknife estimation with a unit root

Jackknife Bias Reduction in the Presence of a Near-Unit Root

“…This paper builds on two of the strands of Peter's research, namely jackknife bias reduction and the analysis of nonstationary time series. Indeed, our own work on the jackknife (Chambers 2013(Chambers , 2015Chambers and Kyriacou 2013) was inspired by Peter's work on this topic with Jun Yu, published as Phillips and Yu (2005), and the current contribution also extends the results on moment generating functions (MGFs) contained in Phillips (1987a).…”

“…Within the class of stationary autoregressive time series models, Phillips and Yu (2005) show that the jackknife can effectively reduce bias in the pricing of bond options in finance, while Chambers (2013) analyses the performance of jackknife methods based on a variety of sub-sampling procedures. In subsequent work, Chambers and Kyriacou (2013) demonstrate that the usual jackknife construction in the time series case has to be amended when a unit root is present, while Chen and Yu (2015) show that a variance-minimising jackknife can be constructed in a unit root setting that also retains its bias reduction properties. In addition, Kruse and Kaufmann (2015) compare bootstrap, jackknife and indirect inference estimators in mildly explosive autoregressions, finding that the indirect inference estimator dominates in terms of root mean squared error, but that the jackknife excels for bias reduction in stationary and unit root situations.…”

“…Its construction proceeds by finding a set of weights that, when applied to a full-sample estimator and a set of sub-sample estimators, is able to eliminate fully the first-order term in the resulting jackknife estimator's bias expansion. In stationary time series settings, the bias expansions are common to both the full-sample and sub-sample estimators, but Chambers and Kyriacou (2013) pointed out that this property no longer holds in the case of a unit root. This is because the initial values in the sub-samples are no longer negligible in the asymptotics and have a resulting effect on the bias expansions, thereby affecting the optimal weights.…”

A Jackknife Correction to a Test for Cointegration Rank

“…, m), all share the same expansion in Equation (8) as the full-sample statistic S r . It has been shown by Chambers and Kyriacou [19,20] that, in a univariate setting with a unit root, the sub-sample statistics (for j = 1) have different properties to the full sample statistic in the limit due to the stochastic order of magnitude of the pre-sub-sample value, and the same phenomenon also arises here in case 1. The implication is that the limiting distributions of the sub-sample statistics S rj (at least for j = 1) will differ from that of S r ; the expansions of E(S rj ) will differ from that of E(S r ); and, hence, the jackknife statistic (as defined above) will not fully eliminate the O(T −1 ) term in the bias.…”

“…In the case of stationary autoregressive time series Chambers [18] has shown that jackknife methods are capable of producing substantial bias reductions as well as reductions in root mean square errors compared to other methods in the estimation of model parameters; the jackknife results were also shown to be robust to departures from normality and conditional heteroskedasticity as well as other types of misspecification. However, some care has to be taken when applying these techniques with non-stationary data, as pointed out by Chambers and Kyriacou [19,20], who propose methods that can be used to ensure that the jackknife procedure achieves the bias reduction as intended in the case of non-stationarity.…”