Recursive Right-Tailed Unit Root Tests for an Explosive Asset Price Bubble

D I Harvey,R Sollis,S J Leybourne

doi:10.1093/jjfinec/nbt025

Abstract

In this article, we compare the local asymptotic and finite sample power of two recently proposed recursive right-tailed Dickey–Fuller-type tests for an explosive rational bubble in asset prices. It is shown that the power of the two tests can differ substantially depending on the location of the explosive regime, and whether such a regime ends in collapse. Since this information is typically unknown to the practitioner, we propose a union of rejections strategy that combines inference from the two individual tests. We find that, for a given specification of the explosive regime, the union of rejections strategy always attains power close to the better of the individual tests considered. An empirical illustration using the Nasdaq composite price index is also provided.

Highlights

A substantial body of theoretical and empirical research exists on statistically testing for explosive asset price bubbles
When applied to an asset price series and the associated fundamentals series, tests of this type proposed by PWY and HB o er methods of detecting explosive rational bubbles
This paper has focused on the relative local asymptotic and nite sample power of the forward recursive PWY and backward recursive HB tests when the series contains a single explosive period, possibly with some form of collapse

Summary

Introduction

A substantial body of theoretical and empirical research exists on statistically testing for explosive asset price bubbles. These results raise the interesting possibility that when the timing of the bubble is unknown, as it would be in practice, a composite test based on a union of rejections strategy applied to the PWY and HB test statistics could yield bene ts to practitioners relative to either of these tests being used individually This type of strategy, based on rejecting the null hypothesis if any of a number of individual tests indicate rejection, has previously been employed in the literature on testing for a unit root against a stationary alternative, for example when uncertainty exists regarding the presence of a trend in the data, or when uncertainty surrounds the nature of the initial value of the series (see Harvey et al, 2009, 2012). The following notation is used: `b c' denotes the integer part, `!d ' denotes weak convergence, `!p ' denotes convergence in probability, and I(:) denotes the indicator function

Recursive right-tailed unit root tests

Model and asymptotic results

Finite sample power comparison

A union of rejections strategy

Empirical illustration

Conclusion