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 autoregressive-moving average model of order (p,q). <var>T<SUB>BP</SUB></var> is called the classical portmanteau test. Under the null hypothesis that the autoregressive-moving average model of order (p,q) is adequate, they suggested that the distribution of <var>T<SUB>BP</SUB></var> is approximated by a chi-square distribution with (m-p-q) 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 autoregressive-moving 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 chi-square distribution with (m-p-q) degrees of freedom in distribution, and that if we assume Bloomfield's exponential spectral density, <var>T<SUB>PW</SUB></var> is asymptotically chi-square 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 chi-square 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 chi-square 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), 177-192, 2009-12-01

    THE JAPAN STATISTICAL SOCIETY

References:  18

Codes

  • NII Article ID (NAID)
    10025992037
  • NII NACSIS-CAT ID (NCID)
    AA1105098X
  • Text Lang
    ENG
  • Article Type
    ART
  • ISSN
    18822754
  • NDL Article ID
    10579185
  • NDL Source Classification
    ZD43(経済--統計)
  • NDL Call No.
    Z76-A259
  • Data Source
    CJP  NDL  J-STAGE 
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