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 the Tutorial on Long-Memory. 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).



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