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How to decompose the covariance with two conditioning random variables?

For example, there is a law of total variance with two conditioning variables in the Wikipedia $$ \text{Var}(Y)=\text{E}[\text{Var}(Y|X_{1},X_{2})]+\text{E}[\text{Var}(\text{E}[Y|X_{1},X_{2}]|X_{1})]+\text{Var}(\text{E}[Y|X_{1}]) $$

What is it's counter part for the covariance? Is the following expression (especially the middle RHS term) correct? $$ \text{Cov}(Y,W)=\text{E}[\text{Cov}(Y,W|X_{1},X_{2})]+\text{E}\{\text{Cov}(\text{E}[Y|X_{1},X_{2}]|X_{1},\text{E}[W|X_{1},X_{2}]|X_{1})\}+\text{Cov}(\text{E}[Y|X_{1}],\text{E}[W|X_{1}]) $$

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  • $\begingroup$ Covariance is variance. Therefore there's nothing new to be learned. $\endgroup$ – whuber Oct 8 '17 at 15:17
  • $\begingroup$ Clarified the question, Not sure about the middle RHS term $\endgroup$ – den2042 Oct 9 '17 at 13:00

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