Palgrave Handbook of Econometrics: Applied Econometrics

718 Panel Methods to Test for Unit Roots and Cointegration

process with varianceσi^2. The OLS estimates and correspondingt-statistics ofρiare
given by:

ρˆi=

∑

yit− 1 yit

∑

y^2 it− 1

(C.1)

tρi=

∑

yit− 1 yit

σˆi

(∑

y^2 it− 1

) 1 / 2 , (C.2)

withσˆi^2 =T^1

∑(

yit−ˆρiyit− 1

) 2

. Under the null hypothesis thatρi=0, it holds
under rather general assumptions onεitforT→∞that:

Tρˆi→

∫

W(r)dW(r)

∫

W^2 (r)dr

(C.3)

tρi→

∫

W(r)dW(r)

(∫

W^2 (r)dr

) 1 / 2. (C.4)

For example, Nabeya (1999) shows the existence of moments up to order six for the
above two limiting quantities (C.3) and (C.4) under the assumption that the inno-
vationεitis an i.i.d. sequence. Denote the expected value, respectively the variance,
of the limit in (C.4) byμtρandσt^2 ρ. Now, if we assume that the individual series are

cross-sectionally independent and that it holds for all cross-section members that
ρi=0, then it immediately follows forN→∞afterT→∞that:

̄tρ=^1

N

∑N

i= 1

tρi−μtρ

σtρ

⇒N(0, 1).

This result also holds true in the sequential limit if one allows for heterogeneity in
the individual series, for example, by allowing for different autoregressive orders
and performing corresponding ADF regressions since, as is well known, the same
time series limit (C.4) prevails also in this case.
If one wants to consider a joint limit whereNandTtend to infinity together,
matters become more complicated even if we stick to the assumption of cross-
sectional independence. Two properties need to be established to allow for the
applicability of joint central limit theorems, which we illustrate again for thet-
test. First, the existence of the necessary moments of the finiteTquantities (C.2)
has to be established. Such results have been derived only under relatively strong
assumptions, mainly relying upon normality of the innovationsεitand often only
for simple DGPs (see, for example, Evans and Savin, 1981, or Larsson 1997). In this
respect it is also important to note that the finite sample distributions of, for exam-
ple, thet-values depend, in the case of higher-order autoregressive processes, upon
all autoregressive coefficients, since the dependence upon these nuisance parame-
ters vanishes only in the limit forT→∞. This dependence upon characteristics
of the DGPs of the cross-section members is often the reason for joint limits being