The estimation of parameters in the frequency spectrum of a seasonally persistent stationary stochastic process is addressed. For seasonal persistence associated with a pole in the spectrum located away from frequency zero, a new Whittle-type likelihood is developed that explicitly acknowledges the location of the pole. This Whittle likelihood is a large sample approximation to the distribution of the periodogram over a chosen grid of frequencies, and constitutes an approximation to the time-domain likelihood of the data, via the linear transformation of an inverse discrete Fourier transform combined with a demodulation. The new likelihood is straightforward to compute, and as will be demonstrated has good, yet non-standard, properties. The asymptotic behaviour of the proposed likelihood estimators is studied; in particular, $N$-consistency of the estimator of the spectral pole location is established. Large finite sample and asymptotic distributions of the score and observed Fisher information are given, and the corresponding distributions of the maximum likelihood estimators are deduced. A study of the small sample properties of the likelihood approximation is provided, and its superior performance to previously suggested methods is shown, as well as agreement with the developed distributional approximations.
In this paper, we develop likelihood estimation of the parameters of a stationary stochastic process that exhibits seasonal persistence, that is, long memory behaviour associated with a stationary, quasi-seasonal dependence structure. We introduce a new frequency-domain likelihood approximation which is computed using demodulation and which, for the first time, facilitates maximum likelihood estimation. We consider joint estimation of the seasonality and persistence parameters, and establish the asymptotic and large sample properties of the likelihood and its associated maximum likelihood estimators. This is in direct contrast with previously suggested procedures, where the distribution of the estimator of the seasonality parameter could not be established (Giraitis et al., 2001). The estimators are demonstrated to have good small sample properties compared with estimators based on the classic Whittle likelihood, and other non-likelihood derived estimators. Our non-standard asymptotic results rely on the appropriate renormalization of the score and Fisher information, and utilize a parameter-dependent linear transformation of the data. This transformation enables an efficient approximation to the likelihood. The transformation also introduces a number of interesting and non-regular features into the likelihood surface: jumps, local oscillations, and non-regular large sample theory. Despite these issues the large sample theory can be determined, and appropriate finite large sample approximations provided, as will be demonstrated. It transpires that the small sample properties of the estimators are competitive with existing methods, as well be discussed in later sections.
The contributions of this paper thus include new theory for non-regular maximum likelihood problems. In similarly motivated work, Cheng and Taylor (1995) discussed problems associated with maximum likelihood estimation for unbounded likelihoods: in contrast we discuss problems associated with distributions of non-identically distributed, weakly dependent variables with highly compressed and for increasing sample sizes unbounded variances. Given the importance of compressed linear decompositions in modern statistical theory, our work has implications for the distribution of sparseness-inducing transformations much beyond the analysis of seasonal processes and Fourier theory, and forms a contribution to developing methodology for inference of stochastically compressible processes.
One of the concrete and substantive conclusions of our new estimation procedures is illustrated in Figure 1; this figure illustrates that whereas a standard estimation procedure, based on the Whittle likelihood (see Section 1.3), produces estimates that are, on average, biased even in large samples, our new procedure, based on a carefully constructed likelihood (see Sections 2.2 and 3), produces estimators that exhibit no such bias. Full details of this Figure are given in Section 4.1.
Stationary time-series models with long range dependence describe a wide range of physical phenomena; see for general discussion Andel (1986) and Gray et al. (1989), and also applications in econometrics (Porter-Hudak, 1990;Gil-Alana, 2002), biology (Beran, 1994) and hydrology (Ooms, 2001). Dependence in a stationary time series is parameterized via the autocovariance sequence, {γ τ }. We are concerned with the estimation of parameters that specify γ τ under an assumption of seasonal persistence. Specifically, of particular importance is the seasonality of the data characterized by a frequency, ξ, termed the pole, and an associated 16) whilst the discrete Whittle likelihood is noted in equation ( 8). On average, the demodulated likelihood has its mode at the true values, whereas the Whittle likelihood does not. See Section 4.1 for full details. degree of dependence, characterized by a persistence (or long memory) parameter δ. Whereas inference for the persistence parameter in the context of poles at frequency zero has been much studied (Beran, 1994), the theoretical behaviour of estimators of the persistence parameter remains largely uninvestigated when the underlying seasonality of the process is unknown.
Let {X t } be a zero-mean, second-order stationary time series with autocovariance (acv) sequence γ τ = cov {X t , X t+τ } = E {X t X t+τ }, and spectral density function (sdf), f (•),
(1)
The process {X t } exhibits seasonal or periodic persistence if there exist real numbers H ∈ (1/2, 1) and ξ ∈ (0, 1/2), and a bounded function c(γ) such that lim
or equivalently if there exist β ∈ (0, 1) and ξ ∈ 0, 1 2 and a bounded function c(λ) such that
Following convention, we parameterize the persistence parameter via δ = β/2. In line with this definition, a process is considered to be a seasonally persistent process
where f † (λ) ≡ c(λ) > 0, 0 < λ < 1 2 is bounded above.
Parameters (ξ, δ) determine the dominant long term behaviour of the process; typically, ξ corresponds to the loc
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