# Modelling in the presence of multiple cointegrating relationships

I am looking for some clarification regarding multivariable cointegration and what steps I should take to avoid spurious regressions.

I am analysing a time series $y$ as a function of independent variables $\{a,b,c,d,e,f\}$ etc.

Let's say, for example, that I test all the series for stationarity. Importantly,

• $y$ and $d$, $e$, $f$ are I(1) (nonstationary),
• $a$, $b$, $c$ are I(0) (stationary).

I know that I should therefore avoid spurious regressions by testing for cointegration between the variables. Given this set up, my questions are:

1. If I find that $d$, $e$ and $f$ are cointegrated with each other by a linear combination, should I combine them into a new stationary variable? Can I use this variable in my regression analysis? Would I lose any long term information doing this? What is the best way to deal with this scenario?

2. What if $y$ (the dependent outcome variable) is cointegrated with any of $d$, $e$, $f$? What does this mean and how should I proceed with the regression analysis?

3. What if there are multiple cointegrated relationships between multiple variables? $(d,f)$ is cointegrated, and $(e,f)$? What is the best way to deal with this?

Basically, I want to make sure that my correlations between $y$ and $\{a,b,c,d,e,f\}$ are not spurious. I am a bit confused about the implications of cointegration when there are multiple variables.

1. If $(d,e,f)$ are cointegrated but $(y,d,e,f)$ are not cointegrated, then using a stationary combination of $(d,e,f)$ (or two stationary combinations, depending on the cointegration rank of $(d,e,f)$) in the equation for $\Delta y$ seems fine to me.
2. If $(y,d,e,f)$ are cointegrated, you would build a vector error correction model (VEC model, or VECM) for them, including $(a,b,c)$ as exogenous regressors. Note that if $(d,e,f)$ are indeed independent variables, as you state in the beginning of your post, they should not adjust towards $y$ due to the error correction mechanism. Hence, the corresponding error correction terms would not be included in the equations for $(\Delta d,\Delta e,\Delta f)$.