Stochastic cointegration: estimation and inference
Introduction
In this paper we study the properties of regression-based estimators in a new environment for analysing relationships between nonstationary time series. We consider a class of nonlinear, nonstationary time series that we term stochastically integrated processes which encompasses heteroscedastic and conventional I(1) integration. We define the concept of stochastic cointegration, as a means of capturing interrelationships between stochastically integrated processes. Associated with stochastic cointegration is the stochastically cointegrating vector and procedures for successfully estimating this vector are the main concern of this paper. Our framework encompasses the heteroscedastic cointegration model proposed by Hansen (1992), and the conventional model of Engle and Granger (1987) (EG). As in Hansen's model, our concept of cointegration is nonlinear and also much weaker than that of EG cointegration as we require only that I(1) behaviour is absent, rather than requiring the presence of I(0) stationarity.
In regression model format, our framework is readily contrasted with that of Hansen (1992). The fundamental difference is that his model allowed only the regressand to be a heteroscedastically integrated process, with all the regressor variables required to be I(1). This distinction imposes an asymmetry upon the model which may be impossible to justify on a priori grounds in a given empirical modelling situation. Our approach, on the other hand, imposes no asymmetry conditions. However, this weakening of Hansen's assumptions has far from trivial consequences as regards the estimation of, and subsequent inference on, the stochastically cointegrating vector. Hansen demonstrated, within the context of his model, that ordinary least-squares (OLS) estimation would yield consistent estimates of the stochastically cointegrating vector. We show that his asymmetry assumption is crucial in this regard and that unless this is imposed, OLS estimation will, in general, be inconsistent.
Because of the inconsistency of OLS outside an asymmetric model structure, we propose a new type of estimator. It is based on a set of instrumental variables (IVs) whose composition is defined and whose validity is required to hold only in an asymptotic sense. This new estimator, which we term the AIV estimator, is designed to remove endogeneity bias suffered by OLS estimators. The AIV procedure is shown to yield consistent estimates of the stochastically cointegrating vector under very general conditions that do not require the asymmetry assumption; thus the possibility that both the regressand and some (or all) of the regressors are heteroscedastically integrated processes is catered for. Hence, the practical problems of prespecifying the behaviour of each variable in the model is no longer an issue. Furthermore, it is extremely straightforward to construct, being computationally little more burdensome than the corresponding (and inconsistent) OLS estimator.
The asymptotic distribution of the AIV estimator is also considered. In general, it contains nuisance parameters and it is not asymptotically mixed normal (MN). However, if we are prepared to make certain, reasonably plausible, exogeneity assumptions within our model, then we can establish the MN property. In fact, Hansen demonstrated a similar MN result for the OLS estimator in his asymmetric model, again under exogeneity assumptions. Of course, the difference here is that we can establish the MN property for the AIV estimator in situations where the OLS estimator is not even consistent. We also show, via Monte Carlo simulation, that even if the exogeneity assumptions do not hold, the asymptotic distribution of the AIV estimator, when appropriately standardized, can still be reasonably well approximated by a standard normal distribution.
The plan of the paper is as follows. Section 2 sets out the model and defines the concepts of stochastic integration and cointegration, which nests a regression model framework that allows us to identify the stochastically cointegrating vector. In Section 3, we show that OLS estimates of the stochastically cointegrating vector are inconsistent, unless the model is specialized to that considered by Hansen. We subsequently demonstrate consistency of the new AIV estimator in the general case. Also in this section, we derive the asymptotic distribution of the AIV estimator, establishing the MN property under exogeneity. Using Monte Carlo simulation, the final section of the paper examines the finite sample size and power properties of a t-statistic based on the AIV estimator. Despite the new estimator's simplicity, it is shown to perform well, even in quite complex modelling situations.
Section snippets
The model
We assume that the (observable) vector time series zt satisfiesfor t=1,…,T. Here zt,wt,εt,ηt and μ are m×1 vectors while Πt, Π and Vt are m×m matrices. The disturbances and Vt are mean zero stationary processes, which may be correlated, wt is a vector integrated process with w0=η0 and μ is a vector of constants. For this model we define what is meant by a stochastically integrated vector and define stochastic cointegration for a linear combination of such a
Statistical inference
It is clear from Eq. (1) that zt is a nonlinear (correlated) unobserved components model. In many circumstances, therefore, there will be insufficient information in the problem at hand to specify distributions for the T×(m2+2m) disturbances that could be justified with any degree of confidence. In addition, there is no likelihood theory currently available for stochastically integrated vectors. In view of these considerations we seek a semi-parametric method of estimation that relies less
Monte Carlo results
In this section we examine, via Monte Carlo simulation, the finite sample size and power of the t-statistic given in (10). We consider model (3), specialized to the case where m=2. Specifically, our DGP isNote that only the first element of the stochastic trend vector , actually appears in the generating model. This model implies the heteroscedastically cointegrating relationshipwith β=πy/πx, where yt is I(1) and xt is
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