688 Panel Methods to Test for Unit Roots and Cointegration
- and would reach very similar characterizations of the limiting densities (with
different correction terms in each case).
Second, the asymptotic moments of the form 2 and! 2 can be approximated
by Monte Carlo simulation for all the different models (for all the different tests.)
Banerjee and Carrion-i-Silvestre compute these moments for up to seven stochastic
regressors andT=1, 000.
Third, since the large sample correction terms (or moments) may perform poorly
in finite samples, the moments of the test statistics for different values ofT, specifi-
callyT={30, 40, 50, 60, 70, 80, 90, 100, 150, 200, 250, 300, 400, 500}are calculated.
To generalize the applicability of these techniques yet further, some response sur-
faces are also computed to approximate the critical values for different values
ofT.
13.3.1.4 Allowing for cross-sectional dependence
In some sense it is worth thinking of (13.18)–(13.22) above as the most general
formulation (within the framework adopted by this chapter) of the elements nec-
essary for testing unit roots and cointegration in a panel, and we propose this
structure as one that encompasses almost all of the issues involved. There are diffi-
culties which are not addressed in sufficient detail, for example to do with multiple
cointegrating vectors, but in terms of the three key elements of (i) testing for unit
roots or cointegration, (ii) cross-sectional dependence, and (iii) instability in the
deterministic or stochastic processes, (13.18)–(13.22) offers all the generality that
is needed.
One can regard the problem as a series of switches which, when on, introduce
the technology relevant to the investigator. For example, the switch from a co-
integration to a unit root framework is more or less immediate by abstracting from
thexi,tvector, formally setting its dimension equal to zero. The switch for structural
breaks in the process is given by the vectors (or scalars in the case of a single
break of parameters)(θi′,γi′,b′i)′, which if set to zero (together or in part) removes
structural instability from the relevant parts of the process.^26 Finally, theπivectors
multiplying the common factorsFtintroduce cross-sectional dependence. Cross-
sectional dependence can also be introduced via correlation in the idiosyncratic
components.
This sub-section deals with testing for cointegration when all the remaining
switches (governing dependence and breaks) are, in principle, also on. Thus, in
addition to structural breaks, we reintroduce cross-sectional dependence through
the factors. We still work within the frameworks of Models 1–6 above, but with the
additional element of cross-sectional dependence.
The assumptions governing theui,tprocesses in (13.18′)and (13.22′)are as given
in section 13.2.2.2 (as they apply to (13.5)–(13.7)). The factor structure (that is,
equation (13.19)) is now switched on. An important restriction (applying also
to the simpler Pedroni tests described in section 13.3.1.2) should be considered
explicitly concerning the relationship between theei,tprocess in (13.19) andνi,t
in (13.22′′).^27 IfE(ei,t|νi,t)=0, the regressors are said to be (strictly) exogenous and