# Stationary and non-stationary variables in time series - how to difference?

I want to predict a multivariate daily time series, the target output is the volume of packages that is send and the covariates are day specific information as weather, the distance to holidays but as well lagged values of the target variable. The target output time series is not stationary, when I difference it, it is. So my intention was to just difference every variable. However, some of my covariates are already stationary, so differencing makes them non-stationary.

I am not really sure if I should difference everything or nothing or just some variables, where the latter sounds not really reasonable to me. Could you please help?

• $\Delta y_{t} = \beta_{0} + \beta_{\text{c}}\left[y_{t-1}-\left(w_{t-1}\right)\right] + \beta_{\Delta w}\Delta w_{t} + \beta_{w}w_{t-1} + \beta_{\Delta h}\Delta h_{t} + \beta_{h}h_{t-1} + \varepsilon_{i}$. Instantaneous short run effect of change in, say, $w$ on $y: \beta_{\Delta w}$. Lagged short run effect of previous level of, say, $w$ on $y: \beta_{w}-\beta_{\Delta w} - \beta_{\text{c}}$. Long run equilibrium effect of previous level of, say, $w$ on trend in $y: \left(\beta_{\text{c}} - \beta_{w}\right)/\beta_{\text{c}}$. – Alexis Mar 19 '19 at 15:29
• Boef, S. D., & Keele, L. (2008). Taking Time Seriously. American Journal of Political Science, 52(1), 184–200. $$\phantom{0}$$Banerjee, A., Dolado, J. J., Galbraith, J. W., & Hendry, D. F. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press, USA. – Alexis Mar 19 '19 at 15:31
• The de Boef and Keele article give a nice bestiary of how many different time series models are the model above, but with specific constraints imposed on them a prior. – Alexis Mar 19 '19 at 15:33
• I have spent a lot of time reading on this topic and I have found various authors disagree. I think the most common view is that you make all the variables stationary unless you have cointegration. – user54285 Mar 19 '19 at 22:11