Instrumental Variable Interpretation of Cointegration With Inference Results for Fractional Cointegration

Abstract

In this paper we propose an alternative characterization of the central notion of cointegration, exploiting the relationship between the autocovariance and the crosscovariance functions of the series+ This characterization leads us to propose a new estimator of the cointegrating parameter based on the instrumental variables~IV! methodology+ The instrument is a delayed regressor obtained from the conditional bivariate system of nonstationary fractionally integrated processes with a weakly stationary error correction term+ We prove the consistency of this estimator and derive its limiting distribution+ We also show that, in the I~1! case, with a semiparametric correction simpler than the one required for the fully modified ordinary least squares~FM-OLS!, our fully modified instrumental variables~FM-IV! estimator is median-unbiased, a mixture of normals, and asymptotically efficient+ As a consequence, standard inference can be conducted with this new FM-IV estimator of the cointegrating parameter+ We show by the use of Monte Carlo simulations that the small sample gains with the new IV estimator over OLS are remarkable+ INTRODUCTIONThe concept of cointegration was coined by Granger~1981, 1983! and Engle and Granger~1987! when the data generating process~DGP! was generated by integrated processes+ The time series x t is integrated of order d, denoted

Semi‐parametric Estimation of Linear Cointegrating Models With Nonlinear Contemporaneous Endogeneity

“…The Monte Carlo simulation results given in Kitamura and Phillips () also show that the kernel estimates of the probability density functions of the crude IV and the FM‐IV estimators both become more dispersed than the OLS estimator when the I(1) instrument becomes less relevant. This observation provides intuitive support for the choice of lagged integrated covariates of Marmol et al () as I(1) IVs, based on which Marmol et al () proposed another FM‐IV estimator. Theoretically, Marmol et al () showed that their FM‐IV estimator is asymptotically equivalent to the FM‐OLS estimator.…”

“…This observation provides intuitive support for the choice of lagged integrated covariates of Marmol et al () as I(1) IVs, based on which Marmol et al () proposed another FM‐IV estimator. Theoretically, Marmol et al () showed that their FM‐IV estimator is asymptotically equivalent to the FM‐OLS estimator. For the rest of the article, the FM‐IV estimator is referred to as the FM‐IV estimator of Marmol et al ().…”

“…Theoretically, Marmol et al () showed that their FM‐IV estimator is asymptotically equivalent to the FM‐OLS estimator. For the rest of the article, the FM‐IV estimator is referred to as the FM‐IV estimator of Marmol et al ().…”

Nonlinear Error Correction: The Case of Money Demand in the United Kingdom (1878–2000)

“…Johansen (1992) considers the efficiency of the one-step maximum likelihood estimation procedure of linear EC models based on weakly exogenous variables. On the other hand, Phillips and Hansen (1990), Phillips (1991), Phillips and Loretan (1991), and Marmol et al (2002) study alternative efficient least-squares and instrumental variables estimators, always within a linear cointegration context.…”

Instrumental variables estimation of stationary and non‐stationary cointegrating regressions

“…IV estimation has previously been considered in a cointegration context, mostly in the traditional case of I (1) observables and I (0) cointegrating errors, but typically to provide representations of optimal estimates, that may also have an approximate maximum likelihood interpretation, say (see e.g. Phillips and Hansen 1990; Kitamura and Phillips 1995, 1997; Marmol et al 2002).…”