On the Distribution of Residual Autocorrelations in Box–Jenkins Models

McLeod, A. Ian

doi:10.1111/j.2517-6161.1978.tb01042.x

Cited by 121 publications

(98 citation statements)

References 13 publications

(11 reference statements)

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“…Consequently, a second objective is to derive the asymptotic distributions of the residual autocovariance matrices in the context of SVARMA models. These results generalize a theorem of McLeod (1978) which is valid for univariate seasonal models. As applications of our results, Portmanteau test statistics are considered and we study their asymptotic distributions, which are approximately chi-square.…”

Section: Introductionsupporting

confidence: 51%

“…Theorem 2 generalizes for vector time series a result of McLeod (1978) for SARMA models. See also Li (2004, p. 13).…”

Section: Asymptotic Distributions Of the Residual Autocovariance In Tmentioning

confidence: 89%

“…(11) and (12) are approximately chi-square χ 2 d 2 M−K . These test statistics represent natural multivariate versions of test statistics originally proposed by McLeod (1978) in univariate SARMA models. See also Li (2004, p. 13).…”

Section: Applications: Portmanteau Test Statisticsmentioning

confidence: 99%

See 2 more Smart Citations

On multiplicative seasonal modelling for vector time series

Ursu

Duchesne

2009

Statistics & Probability Letters

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Section: Introductionsupporting

confidence: 51%

“…Theorem 2 generalizes for vector time series a result of McLeod (1978) for SARMA models. See also Li (2004, p. 13).…”

Section: Asymptotic Distributions Of the Residual Autocovariance In Tmentioning

confidence: 89%

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On multiplicative seasonal modelling for vector time series

Ursu

Duchesne

2009

Statistics & Probability Letters

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“…which is the result obtained by McLeod (1978). When m is large, ρ m I m − m { m m } −1 m is close to a projection matrix with m − ( p + q) eigenvalues equal to 1, and ( p + q) eigenvalues equal to 0, and we retrieve the result given by Box and Pierce (1970).…”

Section: Limiting Distribution Of the Portmanteau Statisticsmentioning

confidence: 79%

Diagnostic Checking in ARMA Models With Uncorrelated Errors

Francq

Roy

Zakoïan

2005

Journal of the American Statistical Association

138

140

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We consider tests for lack of fit in ARMA models with nonindependent innovations. In this framework, the standard Box-Pierce and Ljung-Box portmanteau tests can perform poorly. Specifically, the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration. In this article we derive the asymptotic covariance matrix ρ m of a vector of autocorrelations for residuals of ARMA models under weak assumptions on the noise. The asymptotic distribution of the portmanteau statistics follows. A consistent estimator of ρ m , and a modification of the portmanteau tests are proposed. This allows us to construct valid asymptotic significance limits for the residual autocorrelations, and (asymptotically) valid goodness-of-fit tests, when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference. A set of Monte Carlo experiments, and an application to the Standard & Poor 500 returns, illustrate the practical relevance of our theoretical results.

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“…is an adequately chosen integer and Tr C denotes the sum of the diagonal elements of a square matrix C. Following the lines ofMcLeod (1978), the asymptotic properties of this statistic are derived in the next section under the assumption that the pure feedback model equations (2.8) and (2.9) have stationary ARMA forms.…”

mentioning

confidence: 99%