Semiparametric inference in seasonal and cyclical long memory processes (Q2742769)

The authors consider the time series \(x_t\), \(t=0, \pm 1, \pm 2, \ldots\), which has an absolutely continuous spectral distribution function. Let \(f(\lambda)\) be the spectral density of \(x_t.\) It is supposed that for \(\omega \in (0, \pi)\) the following relations hold true: NEWLINE\[NEWLINEf(\omega +\lambda)\sim C_1\lambda^{-2d_1} \text{as} \lambda \to 0^+,\quad f(\omega -\lambda)\sim C_2\lambda^{-2d_2} \text{as} \lambda \to 0^+ ,NEWLINE\]NEWLINE where \(0< C_i < \infty\), \(|\lambda_i|<2{-1}\), \(i=1, 2.\) Examples of time series for which the relations hold true are presented. E.g., the seasonal fractional noise is characterized by a spectral density NEWLINE\[NEWLINEf(\lambda) = C |1 - \cos(s\lambda)|\sum_{n=-\infty}^\infty\biggl|n+ s\lambda/2\pi\biggr|^{-2(1+d)},NEWLINE\]NEWLINE where \(s\) is the number of observations per year. For this time series \(\omega = \omega_J = 2\pi j/s\), \(j=1, \ldots,[(s-1)/2].\) The authors propose two semiparametric methods for estimation of \(d_1\) and \(d_2\), namely the log-periodogram and the Gaussian or Whittle methods. For example, the following estimates are proposed: NEWLINE\[NEWLINE\hat d_i =2^{-1}\sum_{j=1}^m\nu_j \log I(\omega +(-1)^{i+1} \lambda_j)(\sum_{j=1}^m\nu_j^2)^{-1},\quad i=1,2,NEWLINE\]NEWLINE where \(\nu_j = \log|j|-m^{-1}\sum\limits_{j=1}^m \log l\), \(T(\lambda) =|w(\lambda)|^2\), \(w(\lambda) = (2\pi n)^{-1/2} \sum\limits_{t=1}^n x_t e^{it\lambda}.\) The properties of these estimates are presented. Three tests of symmetry are also given. The behaviour of these tests in finite samples is analyzed. An empirical application to a monthly UK inflation series is considered.