HEGY test under seasonal heterogeneity

HEGY test under seasonal heterogeneity Nan Zou ∗ and Dimitris Politis arXiv:1608.04039v1 [stat.ME] 14 Aug 2016 Department of Mathematics, University of California-San Diego, La Jolla, CA 92093 Abstract Both seasonal unit roots and seasonal heterogeneity are common in seasonal data. When testing seasonal unit roots under seasonal heterogeneity, it is unclear if we can apply tests designed for seasonal homogeneous settings, for example the non-periodic HEGY test (Hylleberg, Engle, Granger, and Yoo, 1990). In this paper, the validity of both augmented HEGY test and unaugmented HEGY test is analyzed. The asymptotic null distributions of the statistics testing the single roots at 1 or −1 turn out standard and pivotal, but the asymptotic null distributions of the statistics testing any coexistence of roots at 1, −1, i, or −i are non-standard, non- pivotal, and not directly pivotable. Therefore, the HEGY tests are not directly applicable to the joint tests for the concurrence of the roots. As a remedy, we bootstrap augmented HEGY with seasonal independent and identically distributed (iid) bootstrap, and unaugmented HEGY with seasonal block bootstrap. The consistency of both bootstrap procedures is established. Simulations indicate that for roots at 1 and −1 seasonal iid bootstrap augmented HEGY test prevails, but for roots at ±i seasonal block bootstrap unaugmented HEGY test enjoys better performance. Keywords: Seasonality, Unit root, AR sieve bootstrap, Block bootstrap, Functional central limit theorem. Introduction Seasonal unit roots and seasonal heterogeneity often coexist in seasonal data, hence the importance to design seasonal unit root tests that allow for seasonal heterogeneity. In particular, given the following heterogeneous quarterly data {Y 4t+s : t = 1, ..., T , s = −3, ..., 0} (see also Ghysels and Osborn, 2001, and Franses and Paap, 2004), generated by α s (L)Y 4t+s = V 4t+s . Suppose V t = (V 4t−3 , ..., V 4t ) 0 is a weakly stationary vector-valued process. Suppose for all s = −3, ..., 0, the roots of α s (L) are on or outside the unit circle. If for some s, the roots of α s (L) are all outside the unit circle, suppose the data are a stretch of a process {Y 4t+s , t = 1, 2, ..., s = −3, ..., 0}; otherwise, suppose Y −3 = Y −2 = Y −1 = Y 0 = 0, all α s (L) share the same set of roots on the unit circle, and this set of roots on the unit circle is a subset of {1, −1, ±i}. We aim to test if all α s (L) share roots at 1, −1, or ±i. To address this task, Franses (1994) and Boswijk, Franses, and Haldrup (1997) limit their scope to finite order seasonal AutoRegressive (AR) data, and apply Johansen’s method (1988) to seasonal unit root tests in seasonal heterogeneous setting. However, ∗ Corresponding author. Email address: nzou@ucsd.edu.