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?

  • $\begingroup$ What can be gleaned from transforming your equation into ECM form? i.e rearranging after replacing yt and xt with ∆yt + y(t-1) and ∆xt + x(t-1) respectively. $\endgroup$
    – OldSchool
    Commented Jun 17, 2020 at 12:34

2 Answers 2


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.

  • $\begingroup$ Thanks for this Alecos. I've run a simulation where I assume an error correction model generates $\Delta y_t$ with one cointegrating variable, and one 'extra' stationary driver variable. I then fit the suspect model shown above and find that the estimated coefficient on the stationary variable is unstable for realistic sample sizes (changes sign). Even when I generate 5000 observations, the parameter still 'meanders' when I run parameter stability regressions, adding observations up to 5k.But I did assume an error correction model to start, so I don't know how meaningful my result is. $\endgroup$
    – JPErwin
    Commented Nov 9, 2014 at 17:11

+1 for the question! It can be shown that if $y_{t}$ and $x_{1t}$ are cointegrated then the OLS estimator of that equation will be super consistent with a rate of convergance of $T^{-2} $ compared to the stationary $T^{-1} $ case. Notice further, that even if you have misspecified stationary terms terms in your regression or haven't captured all the dynamics of the true DGP the estimator will still be consistent since the stochastic trends will dominate asymptotically for $T\rightarrow\infty $ so any misspecification of stationary terms will not affect the estimator. The problem with running that regression is that the estimator can be severely biased in small samples if the stationary terms are misspecified and you cannot test hypothesis on the parameters since the estimator is not normal and the distribution depends on unknown parameters. In short: you will get super consistent estimates but you cannot test parameter significance nor say anything else about them.

An easy way to test if the static relationship, (1) below, is stationary or not would be to test your estimated residuals for a unit root using an ADF-test. Remember to not include a constant since we assume a mean zero process and remember that the distribution is not the regular Dickey-Fuller distribtuion but depends on the no. of parameters in the static regression, (1) below.

I guess you have heard of the Engle-Granger 2-step approach and your problem seems very similar to that approach.

1) Estimate the static regression: $y_{t}=\beta_{0}+\beta_{1}x_{1t}+\beta_{2}x_{2t}+\varepsilon_{t} $ (1), where $y_{t}\sim I\left(1\right) $, $x_{1t}\sim I\left(1\right) $ and $x_{2t}\sim I\left(0\right) $ and test the estimated residuals for a unit root. If the variables cointegrate then the residuals should be stationary. Note that the null hypothesis of a unit root corresponds to the null hypothesis of no-cointegration.

2) Run the dynamic regression: ${\Delta y_{t}=\beta_{0}+\beta_{1}\Delta x_{1t}+\beta_{2}\Delta x_{1t-1}+\beta_{3}\Delta x_{2t1}+\beta_{4}\Delta x_{2t-1}+\beta_{5}\Delta x_{3t1}+\beta_{6}\Delta x_{3t-1}+ecm_{t-1}}+u_{t} $ (2), where $ecm_{t-1}=\hat{\varepsilon}_{t-1} $, i.e. the estimated residuals from the static regression. Note that I included another stationary variable $x_{3t} $ to show that in this second step you can include whatever stationary variables you will. $ecm_{t-1} $ will show if and by how much $y_{t} $ error corrects to $x_{1t} $. Notice further that the t-values in (2) follow a normal distribution and regular inference applies.

Some notes to think about: A) If $y_{t} $ and $x_{1t} $ cointegrate and you want to test that there is no need to include $x_{2t} $ in the static regression, (1), since you can include it in, (2), instead. B) Notice that the biased parameters which arise in small samples in the static regression, (1) are not a problem for an ECM, which is a rewritten ADL model, since it should be dynamically complete (this can be shown by MC simulations). C) Why would you want to estimate the static regression (1) on its own? You do not know if the variables cointegrate (if they do not cointegrate you'll get a spurious regression) and inference is invalid, hence we cannot test any hypothesis on our estimated parameters.

In short: Estimate your ECM or use the Engle-Grange approach unless you want to expand your analysis to CVAR's.

  • $\begingroup$ Thank you for the thorough response Dan. I agree completely that it's weird to want to estimate (1). Why not just go for (2) and stick with the well-understood framework from the literature? However, my task is to explain why (1) shouldn't be fit. I should clarify that, for the sake of argument, I am assuming that the cointegrating relationship is already established in an ADF test. The bigger concern is on $\beta_2$, with $y_t ~ I(1)$ and $x_2 ~ I(0)$ can we recover $\beta_2$? If I follow, the conclusion is yes, but not with realistic (applied macro/stress testing) sample sizes. $\endgroup$
    – JPErwin
    Commented Nov 12, 2014 at 20:10
  • $\begingroup$ I should perhaps clarify my answer a bit. Regression 1) and 2) are step 1) and 2) in the Engle-Granger 2-step approach to cointegration. Let me make it short here: You assume cointegration between $y_{t}$ and $x_{1t}$. By estimating equation 1) we have: A) Super consistent estimates of $\beta_{1}$ and $\beta_{2}$ although these may be severely biased in small samples. B) Inference is invalid so you cannot say anything about the statistical significance of the estimated parameters. $\endgroup$
    – Plissken
    Commented Nov 12, 2014 at 21:15

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