We have the following equation: \begin{align} ln\left( \frac{\pi_{t+1}}{\pi_t} \right) = \alpha_{t+1} - \gamma R_{t+1} - \gamma_j y_{t+1} \end{align} In this equation, $\alpha_{t+1}$ is a normalizing constant chosen to ensure that: \begin{align} E_t \left[ \left( \frac{\pi_{t+1}}{\pi_t} \right) \right] = exp(r). \end{align}
I got this out of an article in finance and I cannot figure out how to prove that the value of $\alpha_{t+1}$ must be obey \begin{align} exp(\alpha_{t+1}) = E_t \left[ exp( r + \gamma R_{t+1} + \gamma_j y_{+1} )\right]. \end{align}
It seems to work, but it's sort of weird how it worked out, as if the inverse of the mean was the mean of the inverse. Anyone sees what is going on? The article is Christoffersen, Jacobs and Ornthanalai (2012), but the context isn't very important here.
EDIT \begin{equation} E_t \left( x_{t+1} \right) = E \left( x_{t+1} | F_t \right) \end{equation} where $F_t$ is a filtration. You have to define it so it contains all relevant information up to the time $t$.