Unit root tests for panel data with AR(1) errors and small T

Abstract

We propose unit root tests for panel data with a small number of time periods, T , and increments that follow an AR(1) process under the null. The model is a fixed-effect panel version of the augmented Dickey-Fuller regression of order 1. Individual-specific linear trends may also be included. The test statistics are t-type statistics based on leastsquares estimates from which the Nickell bias is removed. Their limiting distributions (for an increasing number of independent cross-section units, N , and fixed T ) are standard normal. Our test generalizes the panel unit root test of Harris and Tzavalis, which is based on an unaugmented Dickey-Fuller regression. As an illustration, we examine whether the Law of One Price holds in the European car market since the start of stage three of the EMU in 1999. We find strong evidence of price convergence in the EMU countries. Journal The Econometrics we find K 11 = tr(C L MLC C L MLC) = tr(D L MLD D L MLD) + tr(rb L MLbr rb L Mbr ) = tr(D L MLD D L MLD) + r (4) * (b L MLb) 2 ,where r (4) * ≡ lim T 0 →−∞ (r r) (r r) = (1 − ρ 4 ) −1 . Similar calculations givetr (C L MS C L MS) = tr (D L MF D L MF ), K 55 = tr (D MF D MF ), K 14 = tr (D L MLD D L MF ), K 15 = tr (D L MLD D MF ), K 24 = tr (D MLD D L MF ), K 25 = tr (D MLD D MF ), K 34 = tr (D MD D L MF ), K 35 = tr (D MD D MF ), K 45 = tr (D L MF D MF ) and J 11 = 2tr (L MLA) 2 , J 22 = tr MLA(ML + L M)A, J 33 = 2tr (MA) 2 , J 44 = tr (L MF D ) 2 + tr (L MLA), J 55 = tr (MF D ) 2 + tr (MA), J 12 = tr L MLA(ML + L M)A, J 13 = 2tr L MLAMA, J 14 = 2tr L MLAL MF D , J 15 = 2tr L MLAMF D , J 23 = 2tr MLAMA, J 24 = tr (ML + L M)AL MF D , J 25 = tr (ML + L M)AMF D , J 34 = 2tr MAL MF D , J 35 = 2tr MAMF D , J 45 = tr L (MF D ) 2 + tr L MA.

Does the law of one price hold for hedonic prices?

“…In the second stage, I adopt a standard test in the LOP literature which is to examine the price differences for stationarity using a panel unit root test. In particular I employ a test which exhibits good properties for short panels (Blander & Dhaene, 2012). Across multiple specifications I consistently fail to reject the null hypothesis of no unit root and accept the alternative hypothesis that LOP holds.…”

“…Blander & Dhaene, 2012;Funke & Koske, 2008;Goldberg & Verboven, 2004Parsley & Wei, 1996). 5 The test provided by Blander and Dhaene (2012) is of particular relevance to this paper, since it is suitable for short panels. This is the test I will use in the empirical section.…”

“…To do this I test the matrix of estimated price differences ‫ݍ‬ ௧ stationarity using the unit root test described by Blander and Dhaene (2012):…”

Unit Root Tests for Dependent Micropanels

“…There are several panel unit root tests designed for short T and large N . Breitung and Meyer (), De Blander and Dhaene (), De Wachter et al . (), Harris and Tzavalis (), Karavias and Tzavalis (2014a,b) and Kruiniger () belong to this category.…”

The Price Convergence of the Czech Republic and Euro Zone Countries

“…In this case we can speak about a non-stationary process with a unit root. The overview of unit root tests verifying the existence of unit root can be found in Green (2008), Levin et al (2002), De Blander et al (2007), Harris et al (1999). The previous relation (2) can be transformed to the equation with no incremental form:…”