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.

  • $\begingroup$ 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. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.