A Nonlinear Unit Root Test in the Presence of an Unknown Break
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(2) Ruhr Economic Papers Published by Ruhr-Universität Bochum (RUB), Department of Economics Universitätsstraße 150, 44801 Bochum, Germany Technische Universität Dortmund, Department of Economic and Social Sciences Vogelpothsweg 87, 44227 Dortmund, Germany Universität Duisburg-Essen, Department of Economics Universitätsstraße 12, 45117 Essen, Germany Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI Essen) Hohenzollernstrasse 1/3, 45128 Essen, Germany Editors: Prof. Dr. Thomas K. Bauer RUB, Department of Economics Empirical Economics Phone: +49 (0) 234/3 22 83 41, e-mail: [email protected] Prof. Dr. Wolfgang Leininger Technische Universität Dortmund, Department of Economic and Social Sciences Economics – Microeconomics Phone: +49 (0) 231 /7 55-32 97, email: [email protected] Prof. Dr. Volker Clausen University of Duisburg-Essen, Department of Economics International Economics Phone: +49 (0) 201/1 83-36 55, e-mail: [email protected] Prof. Dr. Christoph M. Schmidt RWI Essen Phone: +49 (0) 201/81 49-227, e-mail: [email protected] Editorial Office: Joachim Schmidt RWI Essen, Phone: +49 (0) 201/81 49-292, e-mail: [email protected]. Ruhr Economic Papers #45 Responsible Editor: Volker Clausen All rights reserved. Bochum, Dortmund, Duisburg, Essen, Germany, 2007 ISSN 1864-4872 (online) – ISBN 978-3-86788-046-6 The working papers published in the Series constitute work in progress circulated to stimulate discussion and critical comments. Views expressed represent exclusively the authors’ own opinions and do not necessarily reflect those of the editors..
(3) Ruhr Economic Papers #45. Stephan Popp. UN I V ERS I T Ä T. D U I S B U R G E S S E N.
(4) Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar.. ISSN 1864-4872 (online) ISBN 978-3-86788-046-6.
(5) Stephan Popp*. A Nonlinear Unit Root Test in the Presence of an Unknown Break Abstract The Perron test is the most commonly applied procedure to test for a unit root in the presence of a structural break of unknown timing in the trend function. Deriving the Perron-type test regression from an unobserved component model, it is shown that the test regression in fact is nonlinear in coefficient. Taking account of the nonlinearity leads to a test with properties that are exclusively assigned to Schmidt-Phillips LM-type unit root tests. JEL Classification: C12, C22 Keywords: Unit root tests, nonlinear regression, structural breaks, innovational outliers May 2008. * University of Duisburg-Essen, Faculty of Economics, Department of Statistics and Econometrics, 45117 Essen, Germany, Phone: Tel.: +49 201 183-3516, e-mail: [email protected],.
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(367)
(368) - 3 < 2 = - 3 F =. .
(369) . . . . . .
(370) .
(371) - 3 < 2 = - 3 F =. . . . . . . .
(372). - . .
(373).
(374).
(375).
(376)
(377)
(378) . . . . - . . . . .
(379)
(380)
(381) . . . - . . . . .
(382)
(383)
(384) .
(385). . . - . . .
(386). .
(387)
(388)
(389) . . . . - .
(390). . . .
(391)
(392)
(393) . . . . - . . . .
(394)
(395)
(396) . . . .
(397)
(398) .
(399) . .
(400) . .
(401) .
(402). .
(403).
(404)
(405) . . . . .
(406) . . . .
(407) . .
(408) .
(409) .
(410)
(411) . . . . .
(412)
(413) . . . . .
(414)
(415) . .
(416) . . .
(417)
(418) . . . .
(419)
(420) . . .
(421) . .
(422)
(423) . . . .
(424)
(425) .
(426) . . . . .
(427). . .
(428). . .
(429). . .
(430). . .
(431). . .
(432).
(433)
(434)
(435)
(436) - 3 < 2 = - 3 F = -
(437)
(438)
(439) . .
(440). . . . .
(441). . .
(442). . . . . .
(443)
(444) - 3 < 2 = - 3 F = -
(445)
(446)
(447) . .
(448) . . . .
(449) .
(450) . . . .
(451) . . .
(452). .
(453) .
(454) - 3 < 2 = - 3 F = -
(455)
(456)
(457) . . . . . . . . . . . . . . .
(458)
(459)
(460)
(461) - 3 < 2 = - 3 F = -
(462)
(463)
(464) . . . . .
(465) . . .
(466). . . .
(467) . . .
(468)
(469) - 3 < 2 = - 3 F = -
(470)
(471)
(472) .
(473).
(474) .
(475)
(476).
(477). . . . . . .
(478) . . . .
(479) .
(480) - 3 < 2 = - 3 F = -
(481)
(482)
(483) .
(484). . . . . .
(485). .
(486). . . . .
(487) . .
(488) . . . .
(489). .
(490) . .
(491) . . . .
(492) . . . . . .
(493) .
(494) . . . . .
(495) .
(496) .
(497) . .
(498). .
(499) . . . . .
(500)
(501)
(502) . . . .
(503)
(504)
(505) . . . .
(506)
(507)
(508) . . . .
(509) .
(510)
(511)
(512) . .
(513). . .
(514)
(515)
(516) . .
(517) .
(518). .
(519)
(520)
(521) . . . . . +/0:3 @3831B7=< 4@3?C3<1G /<2 >@=0/07:7BG =4 23B31B7<5 B63 B@C3 0@3/9 2/B3 4=@ % .
(522)
(523)
(524). . . . . . . .
(525) .
(526) .
(527) . .
(528) .
(529) .
(530) . .
(531) .
(532) .
(533) .
(534) . . . . . . . . . . . .
(535) . . . .
(536). . . .
(537). . . .
(538) . . .
(539) . . .
(540). . . . . . .
(541)
(542) . .
(543). . . . . . . . .
(544)
(545) . . . . .
(546) . . .
(547) . .
(548)
(549) .
(550) .
(551) . . . . . . . .
(552)
(553)
(554)
(555) - 3 < 2 = - 3 F = -
(556)
(557)
(558) . . . . . .
(559) . .
(560). . . . . . . . . .
(561). .
(562). .
(563).
(564). . . . . . . .
(565). . .
(566). . . . . . .
(567).
(568) .
(569)
(570) - 3 < 2 = - 3 F = -
(571)
(572)
(573) .
(574)
(575). . . . .
(576). . . . . . . . . .
(577). . . . .
(578).
(579). . . . . . . . . . . . . . . . . . . . . .
(580) .
(581) - 3 < 2 = - 3 F = -
(582)
(583)
(584) . . . .
(585). .
(586)
(587).
(588) . .
(589) . . .
(590) .
(591). . . . . . . . . . . . . . .
(592). . . . . . .
(593). .
(594). . . . . . . .
(595) . .
(596) .
(597) .
(598) . . .
(599).
(600) .
(601).
(602). . . .
(603) . . . . . . .
(604) .
(605) .
(606).
(607).
(608) . . .
(609) . .
(610) .
(611) . .
(612). . . .
(613). . . .
(614)
(615)
(616) . .
(617).
(618)
(619). . . .
(620)
(621). . . . .
(622)
(623)
(624) . . .
(625) .
(626) .
(627) . . . . .
(628)
(629)
(630) . . . .
(631) . .
(632) . . .
(633) .
(634). +/0:3 @3831B7=< 4@3?C3<1G /<2 >@=0/07:7BG =4 23B31B7<5 B63 B@C3 0@3/9 2/B3 4=@ %
(635) .
(636)
(637) . . . . . . . .
(638) .
(639) .
(640) . .
(641) .
(642) .
(643) . .
(644) .
(645) .
(646) .
(647) . . . .
(648). . . . . . . .
(649) . . .
(650) . . . . . . . . .
(651)
(652) . .
(653). . . . . . . .
(654). . .
(655)
(656) . . . . . . . . . . .
(657). .
(658)
(659) .
(660) . . .
(661). . .
(662). . .
(663)
(664) . . . . . . .
(665) . . .
(666)
(667)
(668)
(669) - 3 < 2 = - 3 F = -
(670)
(671)
(672) . . . . . .
(673) .
(674) . .
(675) . . .
(676). . .
(677).
(678). . . . . .
(679). . . . . . . . . . . . . . . . . . . . .
(680)
(681) - 3 < 2 = - 3 F = -
(682)
(683)
(684) . .
(685). . . .
(686). . .
(687)
(688) .
(689) .
(690) .
(691) . . . . . . .
(692). . .
(693). . . . . . . . . . . . . .
(694). . . . . . .
(695). .
(696) .
(697) - 3 < 2 = - 3 F = -
(698)
(699)
(700) . . . . . . . . . . . .
(701). . . .
(702) . . . . . . . .
(703). . . . .
(704). . . . . . . . . .
(705). . . . . .
(706) . . . . . . . . . . . . . .
(707) .
(708).
(709)
(710) .
(711) .
(712) .
(713) . . . .
(714) . . . . .
(715) . . . . . . . . . . . .
(716) .
(717)
(718)
(719) . .
(720) . . . . . .
(721) .
(722). . .
(723)
(724)
(725) . . .
(726) . .
(727) .
(728) . .
(729). .
(730) .
(731)
(732)
(733) . .
(734) . . . . .
(735). .
(736). . +/0:3 @3831B7=< 4@3?C3<1G /<2 >@=0/07:7BG =4 23B31B7<5 B63 B@C3 0@3/9 2/B3 4=@ %
(737) .
(738) +/0:3 )3AC:BA =4 C<7B @==B B3ABA 4=@ /<<C/: 7<J/B7=< @/B3A. *3@73A /</2/ @/<13 !B/:G "/>/< ,# ,*. */;>:3
(739).
(740).
(741).
(742).
(743).
(744). .
(745) .
(746)
(747)
(748) .
(749) . . %.
(750) .
(751)
(752) . . . . .
(753) .
(754) . . . . .
(755)
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