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Tests for the presence of trends in linear processes

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Zaremba S. K.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction.............................................................................................................................................5 2. Assumptions and notations................................................................................................................6 3. An important asymptotic distribution..................................................................................................8 4. The asymptotic distribution of $K*_{μ, ν}$ under the null hypothesis.........................................14 5. The denominators of the test functions under the alternative hypothesis..................................18 6. The behaviour of $K*_{μ, ν}$ under the alternative hypothesis....................................................22 7. A class of alternative hypotheses for which the $K*_{μ, ν}$ test is consistent..........................23 8. The asymptotic distribution of under the null hypothesis..............................................................26 9. The behaviour of $J*_{μ, ν}$ under alternative hypotheses..........................................................28 10. Orthogonal polynomial.......................................................................................................................33 11. Polynomial trends...............................................................................................................................35 12. Comparisons between the powers of the three tests..................................................................38 13. The asymptotic independence of the three test functions under the null hypothesis............46 14. Numerical illustrations.......................................................................................................................53 References..................................................................................................................................................58

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1 Zaremba S. K.

1 1972

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Tests for the presence of trends in linear processes