Inference of Seasonal Long‐memory Time Series with Measurement Error

Abstract

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.

Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

“…When we have identified the location parameters with cycles of integer numbers, we can estimate the GG model for ∆y t with the Whittle likelihood (WL) estimator. Given ω l , it is straightforward to show asymptotic normality of the WL estimator based on the results of Chan and Tsai (2012) and Tsai, Rachinger, and Lin (2015).…”

“…As Ramachandran and Beaumont (2001) and Smallwood and Norrbin (2008) estimated one factor GG process, without checking the sample spectral densities of interest rates, so there are opportunities to improve upon these results. We can extend their work by estimating the GG process, using the techniques developed in Chan and Tsai (2012), Hidalgo and Soulier (2004), Holan (2012, 2016), and Tsai, Rachinger, and Lin (2015). In addition, the previous literature considers tests of unit roots against long memory processes, but we may also consider that the differenced series may have long memory properties such as a stationary multifactor GG 1 process.…”

Realized stochastic volatility models with generalized Gegenbauer long memory

“…As discussed in Asai, Chang, and McAleer (2017), we can obtain the asymptotic results of the WL estimator by checking the conditions of Hosoya (1997). If the vector of frequency parameters, ω, is known, we can apply the approach which was used to prove Theorem 2 in Chan and Tsai (2008) and Theorem 1 in Tsai, Rachinger, and Lin (2015), in order to verify Assumptions A, C, and D of Hosoya (1997) to show the consistency and asymptotic normality of the WL estimator.…”

Estimation of cyclic long‐memory parameters

“…Several parametric and semiparametric methods were proposed for the case when poles of spectral densities are unknown; see Arteche and Robinson (2000), Giraitis et al (2001), Hidalgo (1996), Whitcher (2004), and the references therein. Various problems in statistical inference of random processes and fields characterized by certain singular properties of their spectral densities were investigated in Leonenko (1999); Tsai, Rachinger, and Lin (2015); and Dehling, Rooch, and Taqqu (2013). Some methods for estimating a singularity location were suggested by Arteche and Robinson (2000), Giraitis et al (2001), and Ferrara and Guígan (2001).…”