Weak convergence of sample covariance matrices and testing for seasonal unit roots
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Date
2020
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Abstract
The paper has two main contributions. First, weak convergence results are derived from
sampling moments of processes that contains a unit root at an arbitrary frequency, where,
in contrast to the previous literature, the proofs are mainly based on algebraic manipulations
and well known weak convergence results for martingale difference sequences. These
convergence results are used to derive the limiting distribution of the ordinary least squares
estimator for unit root autoregressions. As as second contribution, a Phillips-Perron type
test for a unit root at an arbitrary frequency is introduced and its limiting distributions are
derived. This test is further extended to a joint test for multiple unit roots and seasonal
integration. The limiting distributions of these test statistics are asymptotically equivalent
to various statistics presented earlier in the seasonal unit root literature.
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Keywords
invariance principle, unit root test, seasonal unit root, weak convergence