International macroe.. - Free

62 CHAPTER 2. SOME USEFUL TIME-SERIES METHODScians view as being too restrictive. 26 Levin and Lin derive the asymptoticdistribution of their test statistic by allowing both N and T simultaneouslytogotoinÞnity.A remarkable feature of the Maddala—Wu—Fisher test is that it avoids issues of sequential or joint N,T asymptotics.(2.86) gives the exact distribution of the test statistic.The IPS test is based on the sum of τ i , whereas the Maddala—Wutest is based on the sum of the log p-values of τ i . Asymptotically, thetwo tests should be equivalent, but can differ in Þnite samples. Anotheradvantage of Maddala—Wu is that the test statistic distribution does notdepend on nuisance parameters, as does IPS and LL. The disadvantageis that p-values need to be calculated numerically.Potential Pitfalls of Panel Unit-Root TestsPanel unit-root tests need to be applied with care. One potential pitfallwith panel tests is that the rejection of the null hypothesis does notmean that all series are stationary. It is possible that out of N timeseries,only 1 is stationary and (N-1) are unit root processes. This isan example of a mixed panel. Whether we want the rejection of theunit root process to be driven by a single outlier or not depends on thepurpose the researcher uses the test. 27A second potential pitfall is that cross-sectional independence isa regularity condition for these tests. Transforming the observationsby subtracting off the cross-sectional means will leave some residualdependence across individuals if common time effects are generated bya multi-factor process. This residual cross-sectional dependence canpotentially generate errors in inference.A third potential pitfall concerns potential small sample size distortionof the tests. While most of the attention has been aimed at26 That is, they deduce the limiting behavior of the test statistic Þrst by lettingT → ∞ holding N Þxed, then letting N → ∞ and invoking the central limittheorem.27 Bowman [17] shows that both the LL and IPS tests have low power againstoutlier driven alternatives. He proposes a test that has maximal power. Taylor andSarno [131] propose a test based on Johansen’s [80] maximum likelihood approachthat can test for the number of unit-root series in the panel. Computational considerations,however, generally limit the number of time-series that can be analyzedto 5 or less.