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Consider the random variables $Y,X$. Can we always write $$ (1) \quad Y=E(Y|X)+\epsilon $$$$ Y=E(Y|X)+\epsilon\tag 1 $$ with $\epsilon$ independent of $X$?

Note: from this answer here, we know that we can always write (1) with $E(\epsilon|X)=0$. However, here I am asking if we strengthen the relationship between $\epsilon $ and $X$ to stochasticstochastic independence.

Consider the random variables $Y,X$. Can we always write $$ (1) \quad Y=E(Y|X)+\epsilon $$ with $\epsilon$ independent of $X$?

Note: from this answer here, we know that we can always write (1) with $E(\epsilon|X)=0$. However, here I am asking if we strengthen the relationship between $\epsilon $ and $X$ to stochastic independence.

Consider the random variables $Y,X$. Can we always write $$ Y=E(Y|X)+\epsilon\tag 1 $$ with $\epsilon$ independent of $X$?

Note: from this answer here, we know that we can always write (1) with $E(\epsilon|X)=0$. However, here I am asking if we strengthen the relationship between $\epsilon $ and $X$ to stochastic independence.

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Can we always write a random variable as conditional expectation plus independent error?

Consider the random variables $Y,X$. Can we always write $$ (1) \quad Y=E(Y|X)+\epsilon $$ with $\epsilon$ independent of $X$?

Note: from this answer here, we know that we can always write (1) with $E(\epsilon|X)=0$. However, here I am asking if we strengthen the relationship between $\epsilon $ and $X$ to stochastic independence.