Regress first-difference values of Y on lagged values of X

Question

I am struggling with an econometric theory problem. I have a time series of yield daily data on government bond:

\{ y \}_{t=0}^{T}

which has clearly a unit root (DF test confirmed). Provided that the first-difference \Delta y_t = y_t - y_{t-1} is instead stationary, I would like to run the following regression analysis:

\Delta y_t = \gamma (\Delta z_t) + \beta x_{t-1} + \epsilon_t

where x_{t-1} and \epsilon_t are both i.i.d. and z_t is itself a time series with unit root.

Can I run this regression consistently and interpret the coefficient \beta as the correlation between x_{t-1} and \Delta y_t, or is there anything I am missing here?

I have searched for references on the following textbooks, without finding anything which clarified my doubt:

• Wooldridge, Jeffrey M. (2001). Econometric Analysis of Cross Section and Panel Data
• Wooldridge, Jeffrey M. (2013). Introductory Econometrics: A Modern Approach
• Angrist, Joshua D. and Pischke, Jorn-Steffen (2008). Mostly Harmless Econometrics: An Empiricist's Companion
• Hayashi, Fumio (2000). Econometrics