662 Panel Methods to Test for Unit Roots and Cointegration
13.2.2.4 Nonlinear instrumental variables unit root test – Chang (2002)
Chang (2002) allows for cross-sectional dependence through correlation in the
noise processes. Thus, the starting point is correlation between the seriesui,tin
yi,t = ρiyi,t− 1 +ui,t, abstaining here from deterministic components for sim-
plicity. Chang assumes that allui,tseries are stationary autoregressive processes
of some orderspigenerated by some innovationsεi,t. In particular, Chang (2002,
Assumption 2.2, p. 264) assumes thatεt=
(
ε1,t,...,εN,t)′
is an i.i.d. sequencewith (non-diagonal) positive definite covariance matrix, which precludes cross-unit
cointegration. The unit root test itself is based on instrumental variable estimation
with the instrument given by integrable functions of the lagged levels ofyi,t. The
test statistic, labeled NL, for the null hypothesisH 0 :ρi=1 fori=1,...,Nagainst
the heterogeneous alternative, is given by an appropriately weighted sum of the
individualt-statistics. Chang (2002, p. 277) proposes the following instrument
generating functionF(yi,t− 1 )=yi,t− 1 e−ci|yi,t−^1 |, whereciis related to the sample
standard error ofyi,t. Im and Pesaran (2003) show that the asymptotic behavior
established in Chang (2002) holds only whenNlnT/
√
T→0, which suggests that
Nhas to be quite small compared toTin practice.
13.2.2.5 Summary of section 13.2.2
We have described above a broad class of methods developed to deal with cross-
sectional dependence. These are largely dependent on using factor models of
varying degrees of generality to model dependence across the units. It is thus of
interest to investigate the size and power of these methods within the context of
a simulation study, and to apply these methods to real-world datasets (although
a direct comparison of the results cannot be made since the tests operate under
different assumptions).
Some general principles can be identified. First, cross-sectional dependence is
mainly modeled by resorting to factor models, which allows us to model both
short- and long-run dependence, albeit with some restrictions. Second, for cer-
tain special configurations – as, for example, the one considered by O’Connell
(1998) – short-run correlation can be relatively easily accounted for in simple
testing approaches by resorting to some corrections; for example, feasible GLS.
Third, under appropriate assumptions, many of the test statistics are, under the
null hypothesis, asymptotically standard normally distributed (after appropriate
centering and re-scaling). This is most easily established for cross-sectionally inde-
pendent panels and sequential limits withT→∞followed byN→∞, but can
also be established (under appropriate assumptions) in panels with cross-sectional
dependencies.
We turn next to the most general formulation (13.5)–(13.7), which allows not
only for cross-sectional dependence but also for the presence of structural breaks in
the deterministic components of the series comprising the panel. These structural
breaks are typically assumed to take the form of changes in the intercept or linear
trend of the process (in a given unit or a set of units) at a potentially unknown
date in-sample. As in time series unit root problems, inference about unit roots in