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From Tsay's Analysis of Financial Time Series,

For a univariate weakly stationary time series $r_t$, its sample autocorrelation function $\hat{\rho}_l$ is defined as:

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and the Ljung-Box test is

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For a weakly stationary multivariate time series $r_t$, the sample lag-$l$ cross-covariance matrix $\hat{\Gamma}_l$ and sample cross-correlation matrix $\hat{\rho}_l$ of $r_t$ is defined as

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The multivariate Ljung-Box test is:

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I wonder how to see the test statistic for a multivariate time series is a generalization of the one for a univariate time series.

  • Why do the coefficients $T(T+2)$ become $T^2$?
  • Why does $\hat{\rho}_l^2$ become the trace of something?
  • How is the representation in Kronecker product equivalent to (8.7)?
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