Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 681

13.3.1.1 Testing for the null hypothesis of no cointegration – Pedroni (1999, 2004)

Write:

ui,t=ρiui,t− 1 +υi,t, (13.23)

whereυi,tis a stationary ARMA process for all the units. As in the tests for unit
roots, two combinations of the null and alternative hypotheses may be considered

• the first applying to pooled tests, the second to group mean tests. Under the
  former, the null hypothesis is given byH 0 :ρi= 0 ∀i=1, 2,...,Nagainst the
  homogeneous alternative hypothesisHA:ρi=ρ< 0 ∀i=1, 2,...,N, where we
  again restrict attention to stationarity under the alternative. The group mean tests

are based onHA^1 :ρi<0 fori=1, 2,...,N 1 andρi=0 fori=N 1 +1,...,N, where
limN→∞NN^1 =k>0. The analogy with the unit-root testing framework is obvious
and many of the same estimation and testing principles apply. Ifui,twere known
the procedures would be exactly the same – in practice, since we must estimate
ui,t, the analogy is not exact and the tests must be based on estimating equations

of the form (13.18′)instead.
Let us denote byuˆi,tthe regression residuals from (13.18′)estimated by OLS. Two
adjustments to the OLS coefficient are needed, first to account for the endogeneity
of the regressors and second to account for the ARMA structure inυi,t.

The first adjustment requires an estimate ofωu^2 .ν,i, denotedωˆ^2 u.ν,i. This may be
done by first estimating the OLS regression ofyi,ton the deterministic compo-
nents andxi,t, extracting the residualsυˆi,tand fitting an ARMA or AR model to
this derived process (and computing its long-run variance.) Alternatively, a non-
parametric estimator of the form prescribed by Newey and West (1987) may also
be used.
The correction for serial correlation can also be dealt with either parametrically,
by means of ADF regressions on the residualsuˆi,t, or nonparametrically from the
nonaugmented regressions given by (13.23), withuˆi,treplacingui,t.
In order to compute the nonparametric correction for serial correlation, denote
the estimated variance of the residuals in (13.23) byσˆυ^2 ,i and the correspond-

ing long-run variances byωˆ^2 υ,i. The serial correlation correction factors areψi=

1

2

(

ωˆ^2 υ,i−ˆσυ^2 ,i

)

and letωˆ^2 N,T=N^1

∑N

i= 1

ωˆ^2 υ,i

ωˆu^2 .ν,i

. The following test statistics for “no

cointegration” can now be defined, laid out in four different categories. Each of the
statistics requires re-scaling and centering in order for the asymptotic distributions
to hold.

(i) Pooled tests (nonparametric corrections)

variance ratio: N^1 /^2

⎛

⎝N−^1

∑N

i= 1

ωˆu−.^2 ν,i

⎛

⎝T−^2

∑T

t= 2

uˆi^2 ,t− 1

⎞

⎠

⎞

⎠

− 1