We consider a common-components model for multivariate fractional
cointegration, in which the $s\geq1$ components have different memory
parameters. The cointegrating rank may exceed 1. We decompose the true
cointegrating vectors into orthogonal fractional cointegrating subspaces such
that vectors from distinct subspaces yield cointegrating errors with distinct
memory parameters. We estimate each cointegrating subspace separately, using
appropriate sets of eigenvectors of an averaged periodogram matrix of tapered,
differenced observations, based on the first $m$ Fourier frequencies, with $m$
fixed. The angle between the true and estimated cointegrating subspaces is
$o_p(1)$. We use the cointegrating residuals corresponding to an estimated
cointegrating vector to obtain a consistent and asymptotically normal estimate
of the memory parameter for the given cointegrating subspace, using a
univariate Gaussian semiparametric estimator with a bandwidth that tends to
$\infty$ more slowly than $n$. We use these estimates to test for fractional
cointegration and to consistently identify the cointegrating subspaces.Comment: Published at http://dx.doi.org/10.1214/009053606000000894 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
We present a goodness-of-fit test for time series models based on the discrete spectral average estimator+ Unlike current tests of goodness of fit, the asymptotic distribution of our test statistic allows the null hypothesis to be either a short-or long-range dependence model+ Our test is in the frequency domain, is easy to compute, and does not require the calculation of residuals from the fitted model+ This is especially advantageous when the fitted model is not a finite-order autoregressive model+ The test statistic is a frequency domain analogue of the test by Hong~1996, Econometrica 64, 837-864!, which is a generalization of the Box and Pierce~1970, Journal of the American Statistical Association 65, 1509-1526! test statistic+ A simulation study shows that our test has power comparable to that of Hong's test and superior to that of another frequency domain test by Milhoj~1981, Biometrika 68, 177-187!
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