## Semiparametric inference in seasonal and cyclical long memory processes.(English)Zbl 0974.62079

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: $f(\omega +\lambda)\sim C_1\lambda^{-2d_1} \text{as} \lambda \to 0^+,\quad f(\omega -\lambda)\sim C_2\lambda^{-2d_2} \text{as} \lambda \to 0^+ ,$ 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 $f(\lambda) = C |1 - \cos(s\lambda)|\sum_{n=-\infty}^\infty\biggl|n+ s\lambda/2\pi\biggr|^{-2(1+d)},$ 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: $\hat d_i =2^{-1}\sum_{j=1}^m\nu_j \log I(\omega +(-1)^{i+1} \lambda_j)(\sum_{j=1}^m\nu_j^2)^{-1},\quad i=1,2,$ 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.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and financial mathematics
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