I am trying to find information (without success) regarding serially correlated regressors in linear time series regression setting.

The topics covered are either correlation between regressors, or serial correlation of the error, but not serially correlated regressors.

To give an example, in the following model

$y_t = \beta x_{t-1} + \epsilon_t$

where $y_t$ and $x_t$ are $I(0)$ - order of integration is 0.

If I replace $x_t$ with

$z_t = (x_t + x_{t-1} + ... + x_{t-n+1})/n$, some averaging scheme

$z_t$ is still $I(0)$ but it is autocorrelated up to lag $n$. Now the model is

$y_t = \gamma z_{t-1} + \nu_t$

What are the pitfalls in this situation?

Thanks.