15.4 Semiparametric Estimation of Difference Parameter $ d$

These estimators are semiparametric estimators. The estimators involve the unknown parameter of interest d, in the parametric relation

$\displaystyle {(1 - L)}^d y_t =x_t \qquad t = 1, 2, \ldots , $

where L is the lag operator. The spectra density $ f_y(\lambda$) is estimated nonparametrically imposing the condition that $ 0 < fy(0) < \infty$, with mild regularity assumptions in a neighbourhood of zero frequency, and its behaviour away from zero is unrestricted. The three estimators of interest are:
(i)
GPH (Geweke, and Porter-Hudak; 1983)
(ii)
Average Periodogram (Robinson; 1994)
(iii)
Semiparametric Gaussian (Robinson; 1995)
Further discussions can be found in Chapter 14. We shall use the default bandwidth for estimation given by the quantlets. Further discussions regarding bandwidth selection can be found in Delgado and Robinson (1996).