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\noindent Statistica Sinica {\bf 13}(2003), 827-852

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\centerline{\large\bf ROBUST TESTS FOR INDEPENDENCE OF}
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\centerline{\large\bf TWO TIME SERIES}
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\author{Pierre Duchesne and Roch Roy}
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\centerline{\it %HEC-Montr\'eal and
Universit\'e de Montr\'eal}
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\begin{quotation}
\noindent {\it Abstract:}
This paper aims at developing a robust and omnibus procedure
for checking the independence of two time series.
Li and Hui (1994) proposed a robustified version of Haugh's (1976)
classic portmanteau statistic which is based on a fixed number of lagged
residual cross-correlations.
Hong (1996a) introduced a class of consistent test statistics
which are weighted sums of residual cross-correlations.
These tests provide a generalized Haugh's test statistic.
Hong's tests are sensitive to outliers, since they are based
on the usual cross-correlation function and least-squares estimators.
Here, we introduce a robustified version of Hong's test statistics.
We suppose that for each series,
the true ARMA model is estimated by a $n^{1/2}$-consistent method.
If outliers are suspected, robust estimators of the parameters are obtained
and the new test statistics rely on the sample robust
cross-correlation function
introduced in Li and Hui (1994).
Under the null hypothesis of independence, the new tests
asymptotically follow a $N(0,1)$ distribution.
Using the truncated uniform kernel,
our test provides a generalized version of the robust test statistic
of Li and Hui (1994).
We also propose a robust procedure for checking independence
at individual lags and a descriptive causality in mean analysis in
the Granger sense
is discussed.
From simulation results, we find that Hong's and Haugh's tests
can be severely affected by additive outliers in the time series.
The new robust statistics and the test of Li and Hui have reasonable
levels when outliers are present.
However, using a kernel different from the truncated uniform kernel,
our test statistics may be substantially more powerful than the test
of Li and Hui.
Finally, the proposed robust procedures are applied to a set of
financial data. \par
\vspace{9pt}
\noindent {\it Key words and phrases:}
ARMA model, causality in mean, coherency, independence,
 robust estimation, robust serial correlation.\par
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