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I am reading the paper "A Reassessment of Monetary Policy Surprises and High-Frequency Identification" by Bauer and Swanson (2022). In Section 2, there is a Bayesian updating model that goes as follows. Suppose $x_t$ is a random variable observed by an agent at time $t$ (an output gap). Suppose the interest rate $i_t$ is determined as $$i_t = \alpha_t x_t + \epsilon_t,$$ where $\epsilon_t$ follows an iid Gaussian process, and the coefficient $\alpha_t$ follows a random walk.

The problem of the agent is to form beliefs on $\alpha_{t+1}$ based on the history $$\mathcal H_t = \left(x_t, i_t, x_{t-1}, i_{t-1}, \cdots\right)$$. Updating is done through a Kalman filter. Let $a_{t+1}\equiv\mathbb E[\alpha_{t+1}\mid\mathcal H_t]$. Define now $$mps_t = \left(\alpha_t - a_t \right)x_t + \epsilon_t,$$ equation (4) on page 9.

On page 9, last paragraph, the authors say that "$mps_t$ can be correlated with $x_t$ ex post if $\alpha_t > a_t$ for several periods in a row. From equation (4), $Cov(mps_t, x_t) = (\alpha_t - a_t)Var(x_t)$ (...)".

I do not understand what type of covariance this might be. The notation seems to indicate that it is the unconditional covariance, but $\alpha_t$ and $a_t$ seem to be treated as constants. What type of covariance is this and why does it take that form?

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