Modelling a nonstationary variable with stationary and nonstatianary variables

I am very confused about time series analysis. Let $y$ is the dependent variable, which has an increasing trend. Let $x1$ is a price index for a group of goods. I know that $x1$ creates the general trend of $y$. Both $y$ and $x1$ are $I(1)$. So, I can regress $y$ on $x1$ (cointegration).

However, I also know that there are some other variables that affect $y$. For example, variable $x2$, affects $y$ in short term - it creates some seasonal effects around the trend. And I have also other such variables, which explain the variability of $y$ around the trend.

I am confused about this situation. I can regress $y$ on $x1$, because they are both $I(1)$. However, $x2$ is $I(0)$. So, I can't regress $x2$ on $y$.

How can I model $y$ using $x1$ and $x2$? I will appreciate help or advice on articles on this subject.

• You could check back later if my answer gets any comments or votes -- I am not 100% confident in the answer myself, although I think I am right. In your case I would use a VECM instead because I feel more comfortable with it. – Richard Hardy Apr 21 '15 at 19:35

I believe it is OK to have a regression of the form

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\varepsilon$$

where $y$ and $x_1$ are cointegrated and $x_2$ is stationary. The OLS coefficient estimates will be consistent; in fact, $\hat \beta_1^{OLS}$ will be superconsistent -- that's even better.

But that will only be useful if you need just the point estimates of the $\beta$'s. The OLS standard error for $\beta_1$ will not give you the right measure of uncertainty because of the superconsistency (not sure about $\beta_0$ and $\beta_2$). Hence, you will not be able to test the statistical significance of $\beta_1$, at least in the standard way. To be able to do that, you should go for a vector error correction model (VECM) instead. In general, I find VECM to be a "cleaner" modelling approach for a case like yours.

A usual definition of co-integration (there are alternatives) is that, given a random vector, say

$$\mathbf z = (y,x_1,x_2)'$$ then there exists a non-zero coefficient vector of fixed numbers $$\mathbf \alpha = (a_1,a_2,a_3)'$$ such that

$$u= \mathbf \alpha'\mathbf z \;\; \text{is covariance stationary}$$

In your case, let's say that indeed $y$ and $x_1$ are indeed co-integrated (the fact that they are both $I(1)$ does not make them automatically co-integrated, co-integration implies a relationship between the two). Setting $\tilde \alpha = (a_1,a_2)$ we have that

$$\tilde u = \tilde \alpha' \left [\begin{matrix} y \\ x_1\end{matrix} \right]$$ is covariance stationary. Now since $x_2$ is covariance stationary, then

$$u = \tilde u + a_3x_2$$ will also be covariance-stationary. But

$$\tilde u + a_3x_2 = \alpha' \left [\begin{matrix} y \\ x_1 \\ x_2 \end{matrix} \right]$$

which is your full specification. In short, if two $I(1)$ variable are co-integrated, then adding any number of $I(0)$ variables to the specification does not create any methodological issues.