Instrumental variables estimation of stationary and non‐stationary cointegrating regressions

Abstract

Summary Instrumental variables estimation is classically employed to avoid simultaneous equations bias in a stable environment. Here we use it to improve upon ordinary least-squares estimation of cointegrating regressions between non-stationary and/or long memory stationary variables where the integration orders of regressor and disturbance sum to less than 1, as happens always for stationary regressors, and sometimes for mean-reverting non-stationary ones. Unlike in the classical situation, instruments can be correlated with disturbances and/or uncorrelated with regressors. The approach can also be used in traditional non-fractional cointegrating relations. Various choices of instrument are proposed. Finite sample performance is examined.

First Stage Estimation of Fractional Cointegration

“…As an alternative, Robinson and Gerolimetto's (2006) proposed an IV estimator. Unlike NBLS or DT, this is expressed in the time domain, which might be viewed as an advantage for some practitioners.…”

“…Among the former estimators, we nd the ordinary least squares (OLS), narrow band least squares (NBLS) (see, e.g., Robinson, 1994, the difference tapered NBLS (DT) (Chen and Hurvich, 2003a), and the instrumental variables (IV) (Robinson and Gerolimetto, 2006). Among the latter, we nd the full system parametric estimation (Jeganathan, 1999, Robinson and Hualde, 2003, Hualde and Robinson, 2007, fully modi ed OLS or NBLS (Dolado and Marmol, 1996, Kim and Phillips, 2000, Nielsen and Frederiksen, 2011, and the full system nonparametric frequency domain approach (Hualde and Robinson, 2010).…”

“…Denoting by it the ith element of a vector t , we say, as in Robinson and Gerolimetto (2006), that a scalar process t is integrated of order d (I .d/) if for any l 1 zero mean covariance stationary process t with spectral density nite and nonsingular at all frequencies,…”

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In a fractionally cointegrated model, we analyze, both theoretically and by means of a Monte Carlo experiment, the performance of the most popular first stage estimation methods, including ordinary and narrow band least squares (Robinson, 1994), difference taper narrow band least squares (Chen and Hurvich, 2003a), instrumental variables (Robinson and Gerolimetto, 2006), and compare it with the behavior of a new proposal, the integrated ordinary least squares. An appropriate version of this latter estimator (and also of the instrumental variables one) achieves in all circumstances the fastest convergence rate (among other first stage methods) and performs well in finite samples.

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Semiparametric inference in multivariate fractionally cointegrated systems

“…Robinson and Gerolimetto (2006) refer to each summand of (1) as ''basic fractional'', so our I(d) process, like theirs, is a linear combination of ''basic fractional'' processes, with maximal order d. We say that a vector process is I(d) if at least one of its components is I(d), the rest having integration orders no greater than d. This definition is identical to that of Hualde (2008) and resembles that of Johansen (2008), which allows I(d) vectors to have individual components of smaller order than d.…”