^{1}

^{2}

^{1}

^{2}

We consider two well-known facts in econometrics: (i) the failure of the orthogonality assumption (i.e., no independence between the regressors and the error term), which implies biased and inconsistent Least Squares (LS) estimates and (ii) the consequences of using nonstationary variables, acknowledged since the seventies; LS might yield spurious estimates when the variables do have a trend component, whether stochastic or deterministic. In this work, an optimistic corollary is provided: it is proven that the LS regression, employed in nonstationary and cointegrated variables where the orthogonality assumption is not satisfied, provides estimates that converge to their true values. Monte Carlo evidence suggests that this property is maintained in samples of a practical size.

Two well-known facts lie behind this work: (i) the behavior of LS estimates whenever variables are nonstationary and (ii) the failure of the orthogonality assumption between independent variables and the error term, also in an LS regression.

The reappraisal of the impact of unit roots in time-series observations, initiated in the late seventies, had profound consequences for modern econometrics. It became clear that (i) insufficient attention was being paid to trending mechanisms and (ii) most macroeconomic variables are probably nonstationary; such an appraisal gave rise to an extraordinary development that substantially modified the way empirical studies in time-series econometrics are carried out. Research into nonstationarity has advanced significantly since it was reassessed in several important papers, such as those of [

The orthogonality problem constitutes another significant research program in econometrics; its formal seed can be traced back to [

This paper aims to study the consequences of using nonstationary variables in an LS regression when the regressor is related to the error term; this is done in a simple regression framework. The specification under particular scrutiny is

This work aims to study the asymptotic properties of LS estimates when neither the orthogonality nor the stationarity assumptions are satisfied. Our approach is twofold: we assume (i) the variable

Bookcase no. 1: DGP of

Bookcase no. 2: DGP of

Bookcase no. 3: DGP of

Nonstationarity and non-orthogonality case no. 1: DGP of

Nonstationarity and non-orthogonality case no. 2: DGP of

The common belief as regards the last two cases is that the failure of the orthogonality assumption induces LS to generate inconsistent estimates, even in a cointegrated relationship. In fact, when the variables are generated as in (

Let

Let

Let

See Appendix

These asymptotic results show that a relationship between the innovations of

In order to emphasize the relevance of this result, we modified the DGPs of the variables in an effort to strengthen the link between the DGPs and the literature on simultaneous equations. The modifications are twofold and appear in the following propositions. As in Theorem

Let

See Appendix

Let

See Appendix

The two systems, represented in (

Asymptotic properties of LS estimators clearly provide an encouraging perspective in time-series econometrics. Notwithstanding, we should bear in mind that asymptotic properties may be a poor finite-sample approximation. In order to observe the behavior of LS estimates in finite samples, we present two Monte Carlo experiments. Firstly, we represent graphically the convergence process of

A brief glance at Figure

Finite-sample behavior of

The second Monte Carlo is built upon the same basis. In Table

Finite-sample behavior of

DGP parameters | Sample size | ||||||

Statistic | 50 | 100 | 200 | 500 | 700 | ||

−0.9 | −1.5 | Mean | |||||

Stand.dev. | |||||||

−0.75 | Mean | ||||||

Stand.dev. | |||||||

0.75 | Mean | ||||||

Stand.dev. | |||||||

1.5 | Mean | ||||||

Stand.dev. | |||||||

0.9 | −1.5 | Mean | |||||

Stand.dev. | |||||||

−0.75 | Mean | ||||||

Stand.dev. | |||||||

0.75 | Mean | ||||||

Stand.dev. | |||||||

1.5 | Mean | ||||||

Stand.dev. |

Table ^{−3}–10^{−4}. These differences tend to diminish further as the sample size grows. In fact, when there are 700 observations, the order of magnitude of such differences oscillates between 10^{−5}–10^{−8}. We performed the same experiment with autocorrelated disturbances

Using cointegrated variables in an LS regression where the regressor is not independent of the error term does not preclude the method from yielding consistent estimates. In other words, it is proven that, under these circumstances, the regressor remains weakly exogenous for the estimation of

Notwithstanding, one should note the striking resemblance between the properties of the DGPs used in the propositions and those of variables belonging to a classical simultaneous-equation model. It may be possible that the estimation of such models, even if the macroeconomic variables they are nourished with are not stationary, would yield correct estimates. Of course, such a possibility rules out the existence of structural shifts, parameter instability, omission of a relevant variable, or any other major assumption failure.

The estimated specification in Theorem

where

To obtain the asymptotics of

The expressions needed to compute the asymptotic values of

Glossary of the Mathematica code.

Term | Represents | Term | Represents | Term | Represents | Term | Represents |
---|---|---|---|---|---|---|---|

These expressions were written as

The expressions needed to compute the asymptotic values of

First note that DGP (

As in the previous appendix, first rewrite DGP (

The previous three appendices provide the Mathematica code of

The authors would like to thank an anonymous referee for insightful comments. The opinions in this paper correspond to the authors and do not necessarily reflect the point of view of Banco de México.