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I'm having trouble validating if following procedure to test for predeterminedness is plausible. Given the linear model:

$y_t=\beta_1+\beta_2x_{1t}+\beta_3x_{2t}+\epsilon_t$

Having $x_{2t} = y_{t+1|t}+v_t$ and $\epsilon_t$ uncorrelated with $x_{1t}, x_{2t}$. And $\epsilon_t$ i.i.d. with $E(\epsilon_t)=0, E(\epsilon_t^2)=0$. Given $v_t$ i.i.d. and also uncorrelated with $\epsilon_t$.

\begin{equation} \label{eq1} \begin{split} E[x_{1t}\epsilon_t] & = E[E[x_{1t}\epsilon_t|x_{1t}]] \\ & = E[x_{1t}\underbrace{E[\epsilon_t|x_{1t}]]}_{=0} = 0 \\ E[x_{2t}\epsilon_t] & = E[E[x_{2t}\epsilon_t|x_{2t}]] \\ & = E[x_{1t}\underbrace{E[\epsilon_t|x_{2t}]]}_{=0} = 0 \\ \end{split} \end{equation}

Can anyone confirm this?

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