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In this question, I'm continuing to explore the tools used/presented in Lars Hansen's Econometrica paper "Dynamic Valuation Decomposition within Stochastic Economies" (2012).

I'm trying to compute an example of the change of measure associated with multiplicative decomposition proposed in Hansen (2012). This question continues from the same example discussed in this previous question.

$\newcommand{\E}{\mathbb E}$ Suppose that $$ X_{t+1} = A X_t + B W_{t+1} $$ where A has stable eigenvalues and $\{W_{t+1} : t = 0,1,... \}$ is an iid sequence of multivariate standard normally distributed random vectors. Suppose that $$ \log M_{t+1} - \log M_t = D \cdot X_t + F \cdot W_{t+1}. $$ I want to show that there exists a solution to the equation $$ \E \left [ \frac{M_{t+1}}{M_t} e(X_{t+1}) \mid X_t = x \right ] = \exp(\eta) e(x) \tag{1} $$ for $\log e(x) = H \cdot x.$

From the previous question we see that $H' = D'(A - I)^{-1}$ and $\eta = (F + H'B)'(F + H'B)$. How would I compute the change of probability measure discussed in section 6.3 of Hansen's paper? Under this change of measure, what is the conditional distribution of $X_{t+1}$ conditioned on $X_t = x$?

Progress: In order to compute the proposed change of measure, I first need to complete the factorization described in equation (20) of the paper. Then I need to identify the factor $\tilde M_t$. In order to do this, I need to clarify how equation (1) here fits into the framework outlined in the paper.

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