Anindya Banerjee and Martin Wagner 655πˆ =y ̃′fˆ/(T− 1 )under the normalizationfˆ′fˆ/(T− 1 )=Ir, whereIris ther×r
identity matrix.
The estimated factors can now be recovered by summation:
Fˆt=∑ts= 2fˆs,t=2,...,T.Ifr= 1, this single factor can be tested for a unit root using an augmented Dickey–
Fuller test with an intercept.
If (13.5) contains a linear trend in addition to an intercept, the method of prin-
cipal components is applied to thedemeanedanddifferenceddata and the ADF test
for the factor contains an intercept and a trend. The critical values for both these
sets of tests are as provided by Dickey and Fuller (1979).
When the number of common factorsris greater than 1, Bai and Ng (2004)
also develop tests for the number of linearly independentI(1) common trends
(equivalently the cointegrating rank) contained in the common factors. After an
appropriate basis change the number of stationary factors is given byr 0 , and there
arer 1 linearly independent integrated factors or common trends, such that, as
always in theI(1) framework,r 0 +r 1 =r.
The two test statistics (MQc(m)andMQf(m))follow, up to a transformation to
ensure real valued test statistics, Stock and Watson (1988). The test statistics for test-
ing the null hypothesis ofmcommon trends in the common factors are computed
recursively with the first test statistic based onr=mcommon factors. The statistic
MQc(m)is based on estimating a VAR(1) process forYˆt, whereYˆt=βˆORTHF ̃t, and
where, in the first step,F ̃tis them-dimensional vector of factors computed from
the demeaned, respectively demeaned and detrended observations, whileβˆORTHis
the matrix ofmeigenvectors associated with themeigenvalues of the matrix given
byT^12
∑T
t= 2 F ̃tF ̃
′
t. The statisticMQf(m)is constructed similarly, except that apth
order VAR is fitted first to getyˆt=(ˆ L)Yˆtand the test is based on the filteredyˆt
series.
The testing procedure consists of constructing a sequence of theseMQstatis-
tics, starting with testing the null hypothesism=r(that is,rstochastic trends)
against the alternative hypothesism=r−1, and testing down until the first non-
rejection of the null hypothesis occurs; for example, in the second step, the test
statistic is based on onlyr−1 eigenvectorsβˆORTHcorresponding to ther−1 largest
eigenvalues (for a detailed description see Bai and Ng, 2004, pp. 1133–4).
Two versions of these statistics are considered by the authors to allow for demean-
ing and/or detrending of the observations, depending upon the model considered.
Critical values are provided by them for up to six stochastic trends.
Similarly, the estimated idiosyncratic components can also be tested for unit
roots. These are obtained fromeˆi,t=
∑t
s= 2 zˆi,s,i=1, 2,...,N;t=2,...,T, with
zˆi,s=yˆi,s−ˆπi′fˆs. The additional complication here is that the estimates that
are available are not, in general, cross-sectionally independent in finite samples
due to their dependence upon the estimated factors and loadings. We believe that