Solved – Modeling an I(1) process with a cointegrating I(1) and an I(0) variable

Question

A colleague says that estimating the following time series model is statistically sound:

$$y_t = \beta_0 + \beta_1 x_{1t} + \beta_2 x_{2t} + e_t$$

where $y_t$ is nonstationary $I(1)$, $x_{1t}$ is nonstationary but cointegrated with $y_t$, $x_{2t}$ is stationary, $e_t$ is a white noise residual and $\beta{_*}$ are parameters.

I'm not so sure. The safe approach would just be to fit an Error Correction Model to these variables, but in this case there is resistance to doing that (long story).

My intuition is that because the dependent variable is nonstationary and because $x_{2t}$ is stationary, that the covariance$(y_t,x_{2t})$ will be undefined and $\beta_2$ subject to change as the data set grows.

So an ECM is the straightforward/classic way to model these series, but is the equation above legitimate?

Answer 1

It is true that in the original papers on co-integration, all variables involved were assumed to be individually $I(1)$, and this is usually the case presented and used. But this is not restrictive. For example, Lutkepohl (1993), defines co-integration as follows:

A K-dimensional process $\mathbf z_t$ is integrated of order $d$ if $\Delta^d \mathbf z_t$ is stable and $\Delta^{d-1} \mathbf z_t$ is not.

("stable" as is defined in the context of time-series analysis). In p. 351-354 , he presents co-integration for systems containing variables of different order of integration, but does not pursue the issue in depth.

Hayashi (2000) ch. 10 develops the case more fully.

There are at least two theoretical variants here, the one called "polynomial co-integration", the other "mutli-cointegration". I don't feel proficient enough on the issue to respond to the specific example you give in your question, but I hope these will be useful leads for you to search the literature.