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Identification of persistent cycles in non-Gaussian long-memory time series. (English) Zbl 1198.62089

The author deals with the autoregressive model φ(B)y t =ε t , where y t is the t-th observation on the dependent variable, y t =0 for t<0, φ(B)=1-φ 1 B--φ p B p is the characteristic polynomial, B is the backward shift operator, i.e., By t =y t-1 , and the disturbance process ε t is given by ε t = jt b t-j u j , where {u j } is a sequence of i.i.d. (0,1) random variables, {b j } is a sequence which decays hyperbolically, i.e., b j =j H 3 /2 L 1 (j), 0<H<1, j=0 b j 2 <, and L 1 (j) is a slowly varying function, bounded on every finite interval. The unknown parameter Φ=(φ 1 ,,φ p ) is estimated by the least squares estimate (LSE):

Φ ^= k=1 n 𝐲 k-1 𝐲 k-1 ' -1 k=1 n 𝐲 k-1 y k ,

where 𝐲 k1 =(y k ,,y k-p+1 ) ' . The least squares error satisfies

Φ ^-Φ= k=1 n 𝐲 k-1 𝐲 k-1 ' -1 k=1 n 𝐲 k-1 ε k ·

The asymptotic distribution is derived for the least squares estimates (LSE) only in the case when the characteristic polynomial φ(z) is unstable with complex-conjugate unit roots, i.e., an appropriate non-stationary model to identify persistent cycles in non-Gaussian long-memory time series. The behaviour of the LSE when φ(z) has stable roots, roots equal to -1 and 1, and explosive roots remains an open problem. In order to describe the limiting distribution of the LSE, two limit theorems involving long-memory processes are proved. The first theorem gives the limiting distribution of the weighted sum, k=1 n c n,k ε k , where ε k = jk b k-j u j is a non-Gaussian long-memory moving-average process and {c n,k ,1kn} is a given sequence of weights. The second theorem is a functional central limit theorem for the sine and cosine Fourier transforms k=1 [nt] sin(kθ)ε k , k=1 [nt] cos(kθ)ε k , where θ(0,π) and t[0,1].

MSC:
62M10Time series, auto-correlation, regression, etc. (statistics)
62E20Asymptotic distribution theory in statistics
60F17Functional limit theorems; invariance principles