Limit theory for moderate deviations from a unit root

Abstract

An asymptotic theory is given for autoregressive time series with a root of the form r n ¼ 1 þ c=k n , which represents moderate deviations from unity when ðk n Þ n2N is a deterministic sequence increasing to infinity at a rate slower than n, so that k n ¼ oðnÞ as n ! 1. For co0, the results provide a ffiffiffiffiffiffiffi nk n p rate of convergence and asymptotic normality for the first order serial correlation, partially bridging the ffiffi ffi n p and n convergence rates for the stationary (k n ¼ 1) and conventional local to unity (k n ¼ n) cases. For c40; the serial correlation coefficient is shown to have a k n r n n convergence rate and a Cauchy limit distribution without assuming Gaussian errors, so an invariance principle applies when r n 41. This result links moderate deviation asymptotics to earlier results on the explosive autoregression proved under Gaussian errors for k n ¼ 1, where the convergence rate of the serial correlation coefficient is ð1 þ cÞ n and no invariance principle applies. r 2005 Elsevier B.V. All rights reserved.JEL classification: C22

Testing for a Housing Bubble at the National and Regional Level: The Case of Israel

“…The asymptotic theory of mildly explosive processes is developed in Phillips and Magdalinos (2007). Large sample properties of the bubble date-stamping procedure are developed in Phillips and Yu (2009).…”

Portfolio diversification under local and moderate deviations from power laws

“…Reminiscent of the terminology in the time series unit root literature (see Phillips 1988, Phillips & Magdalinos 2004, it is natural to refer to distributions whose tails satisfy one of relation (3) as exhibiting "local" or "moderate" deviations from power laws. Corollaries 1 and 2 provide results for when diversification will and will not be preferred for such deviations.…”

Detection of multiple bubbles in South African electricity prices

“…Phillips, Wu and Yu (2011) suggested the following equation as the main econometric model, representing the middly-integrated root as specified in Phillips and Magdalinos (2007) and denoting a right-sided test:…”