# Ex post predictability of "mps" in Bauer, Swanson (2022)

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?