A stationary coupled multivariate time series is defined by x t = λ y t − 1 + σ z t y t = λ x t − 1 + σ w t where  λ  and  σ  are constants, and  z t  and  w t  are independent, uncorrelated standard Gaussian random variables. That is, w t , z t ∼ N ( 0 , 1 ) E [ z t w s ] = 0 ∀   t , s E [ z t z s ] = E [ w t w s ] = δ t s now What are the lag- l  cross-covariances  Cov ( x t , y t − l ) ? Express your answer in terms of  λ , σ , l .  Your answer will depend on the parity of  l . Fill in the answer accordingly below:  For even  l , Cov ( x t , y t − l ) =? give me an answer no steps

For even \( l \),

\[ \text{Cov}(x_t, y_{t-l}) = \frac{\lambda^l}{1 - \lambda^2} \]

Do you want details or have any questions? Here are some questions you might ask:

1. How do we find the covariance for odd \( l \)?
2. Can you explain the derivation of this covariance result?
3. What is the significance of the parity of \( l \) in this context?
4. How do \(\lambda\) and \(\sigma\) affect the stability of the time series?
5. How does the Gaussian nature of \( z_t \) and \( w_t \) influence the covariance?

Tip: In time series analysis, understanding the properties of the covariance function can help in identifying the underlying processes and their interactions over time.

Explore how to calculate cross-covariances for a stationary coupled multivariate time series defined by given equations. Understand the impact of parameters λ and σ on the covariance structure. Suitable for advanced students and researchers in time series analysis.

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