SYSTEMATIC APPROACH FOR PORTMANTEAU TESTS IN VIEW OF THE WHITTLE LIKELIHOOD RATIO
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Abstract
Box and Pierce (1970) proposed a test statistic <var>T<SUB>BP</SUB></var> which is the squared sum of <var>m</var> sample autocorrelations of the estimated residual process of an autoregressivemoving average model of order (p,q). <var>T<SUB>BP</SUB></var> is called the classical portmanteau test. Under the null hypothesis that the autoregressivemoving average model of order (p,q) is adequate, they suggested that the distribution of <var>T<SUB>BP</SUB></var> is approximated by a chisquare distribution with (mpq) degrees of freedom, ``if <var>m</var> is moderately large". This paper shows that <var>T<SUB>BP</SUB></var> is understood to be a special form of the Whittle likelihood ratio test <var>T<SUB>PW</SUB></var> for autoregressivemoving average spectral density with <var>m</var>dependent residual processes. Then, it is shown that, for any finite <var>m</var>, <var>T<SUB>PW</SUB></var> does not converge to a chisquare distribution with (mpq) degrees of freedom in distribution, and that if we assume Bloomfield's exponential spectral density, <var>T<SUB>PW</SUB></var> is asymptotically chisquare distributed for any finite <var>m</var>. From this observation we propose a modified <var>T<SUP>†</SUP><SUB>PW</SUB></var> which is asymptotically chisquare distributed. In view of the likelihood ratio, we also mention the asymptotics of a natural Whittle likelihood ratio test <var>T<SUB>WLR</SUB></var> which is always asymptotically chisquare distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of <var>T<SUB>PW</SUB></var>, <var>T<SUP>†</SUP><SUB>PW</SUB></var>, and <var>T<SUB>WLR</SUB></var>. Because many versions of the portmanteau test have been proposed and been used in a variety of fields, our systematic approach for portmanteau tests and proposal of tests will give another view and useful applications.
Journal

 JOURNAL OF THE JAPAN STATISTICAL SOCIETY

JOURNAL OF THE JAPAN STATISTICAL SOCIETY 39(2), 177192, 20091201
THE JAPAN STATISTICAL SOCIETY