Hi, everybody;
LW and GPH estimators are consistent for d=<1 and asymptotically normal for d<0.75 (see Velasco, 1999a, 1999b, Phillips and Shimotsu, 2004 and Phillips, 2007), but LW is more efficient. Besides, the asymptotic variance depends on the bandwidth (and, normally, the bigger the sample size the bigger the bandwidth). If you want to test the hypothesis of unit root you should apply the test on the first differences of the series and test if the memory parameter is equal zero. If you don't reject that hypothesis, the series seems to have a unit root. To avoid differenciation, there are other posibilities: tapering (Velasco, 1999a, 1999b), ELW of Shimotsu and Phillips (2005), non-local estimators (Abadir et Al., 2007), etc.
Cheers
Javi
References:
Velasco, C., 1999a. Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20, 87-127.
Velasco, C., 1999b. ‘Non-Stationary Log-Periodogram Regression’, Journal of Econometrics, 91, 325-371
Phillips, P.C.B., Shimotsu, K. (2004). Local Whittle estimation in nonstationary and unit root cases. Ann. Stat. 32, 656-692. PHILLIPS, P.C.B. (2007): Unit root log periodogram regression. Journal of Econometrics 138(1), 104-124. ABADIR, K.M., DISTASO, W. and GIRAITIS, L. (2007): Nonstationary-extended local Whittle estimation. Journal of Econometrics 141, 1353-1384. SHIMOTSU, K. and PHILLIPS, P.C.B (2005): Exact local Whittle estimation of fractional integration. The Annals of Statistics 33(4), 1890-1933.
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