Robust testing of time trend and mean with unknown integration order errors

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Accepted manuscript
Date
2022
Authors
Chang, Seong Yeon
Perron, Pierre
Xu, Jiawen
Version
Accepted manuscript
OA Version
Citation
S.Y. Chang, P. Perron, J. Xu. 2022. "Robust testing of time trend and mean with unknown integration order errors" Journal of Statistical Computation and Simulation, Volume 92, Issue 17, pp.3561-3582. https://doi.org/10.1080/00949655.2022.2074420
Abstract
We provide tests to perform inference on the coeโ€€cient of a linear trend assuming the noise to be a fractionally integrated process with memory parameter ๐‘‘ โˆˆ (โˆ’0.5; 1.5) excluding the boundary case 0.5 by applying a quasi-generalized least squares procedure using ๐‘‘-differences of the data. Doing so, the asymptotic distribution of the ordinary least squares estimators applied to quasi-differenced data and their t-statistics are unaffected by the value of d and have a normal limiting distribution. To have feasible tests, we use the exact local whittle estimator, valid for processes with a linear trend. The small sample properties of the tests are investigated via simulations and we provide comparisons with existing tests valid for a short-memory stationary, ๐ผ (0), or an autoregressive unit root, ๐ผ (1), noise. The results are encouraging in that our test is valid under more general conditions, yet has power approaching to the Perron and Yabu [Estimating deterministic trends with an integrated or stationary noise component. J. of Econometrics. 2009;151;56-69] tests that apply to the dichotomous cases with d either being 0 or 1. We also use our method of proof to show that the main result of Iacone, Leybourne and Taylor [Testing for a break in trend when the order of integration is unknown. J. of Econometrics. 2013;176:30-45] dealing with a test for a break in the slope of a trend function with a fractionally integrated noise is valid for ๐‘‘ โˆˆ (โ€”0.5) โˆช (0:5; 1:5).
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