Fully modified narrow‐band least squares estimation of weak fractional cointegration

We consider estimation of the cointegrating relation in the weak fractional cointegration model, where the strength of the cointegrating relation (difference in memory parameters) is less than one‐half. A special case is the stationary fractional cointegration model, which has found important applications recently, especially in financial economics. Previous research on this model has considered a semi‐parametric narrow‐band least squares (NBLS) estimator in the frequency domain, but in the stationary case its asymptotic distribution has been derived only under a condition of non‐coherence between regressors and errors at the zero frequency. We show that in the absence of this condition, the NBLS estimator is asymptotically biased, and also that the bias can be consistently estimated. Consequently, we introduce a fully modified NBLS estimator which eliminates the bias, and indeed enjoys a faster rate of convergence than NBLS in general. We also show that local Whittle estimation of the integration order of the errors can be conducted consistently based on NBLS residuals, but the estimator has the same asymptotic distribution as if the errors were observed only under the condition of non‐coherence. Furthermore, compared to much previous research, the development of the asymptotic distribution theory is based on a different spectral density representation, which is relevant for multivariate fractionally integrated processes, and the use of this representation is shown to result in lower asymptotic bias and variance of the narrow‐band estimators. We present simulation evidence and a series of empirical illustrations to demonstrate the feasibility and empirical relevance of our methodology.