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Time-varying spectral analysis: theory and applications

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Abstract

Non-stationary time series are a frequently observed phenomenon in several applied fields, particularly physics, engineering and economics. The conventional way of analysing such series has been via stationarity inducing filters. This can interfere with the intrinsic features of the series and induce distortions in the spectrum. To avert this possibility, it might be a better alternative to proceed directly with the series via the so-called time-varying spectrum. This article outlines the circumstances under which such an approach is possible, drawing attention to the practical applicability of these methods. Several methods are discussed and their relative advantages and drawbacks delineated.

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Notes

  1. These and related concepts are discussed at length in Nachane (2006), chapters 3 and 4.

  2. The concept of Lebesgue measure can be found in any standard text on real analysis such as e.g. Royden and Fitzpatrick (2010). A measure υ is said to be absolutely continuous w.r.t. a measure μ if for every measurable set A, with μ(A) = 0, we have υ(A) = 0.

  3. Although the evolutionary spectrum defined by (51) is not invariant to the choice of the family ℑ, the integral \(\int_{ - \infty }^{\infty } {{\text{d}}H_{t} (\omega )\, = \,\text{var} [X(t)]}\) is independent of this choice.

  4. The problem of resolution in spectral analysis refers to the ability of estimators to distinguish fine structure in the spectrum. The resolution of an estimator can be improved by decreasing its bandwidth [see Koopmans (1995), p. 303–306].

  5. Reducing bandwidth (to improve resolution), however, increases the variance of an estimator—the so-called Grenander Uncertainty Principle (see Grenander (1958) and Priestley (1981), p. 527).

  6. The term sufficiently apart means that either (i) \(\left| {\lambda_{1} \pm \lambda_{2} } \right| > > {\text{bandwidth}}\;{\text{of}} \left| {{\Gamma}({\lambda})} \right|^{2}\) or (ii) \(\left| {t_{1} - t_{2} } \right| > > {\text{width}}\;{\text{of}} \left\{ {w_{q} } \right\}.\).

  7. Cumulants are defined and discussed in Brillinger (1975), p. 19–21.

  8. The notion of weak dependence is discussed in Billingsley (1968), p. 363–367.

  9. The notions of small o and big O are explained in Nachane (2006), p. 131.

  10. The suggested test has reasonable size properties and the power of the test increases substantially with sample size. Other properties of the test are noted in Breitung and Candelon (2006).

  11. The degrees of freedom are however adjusted to F[2, N − 3p].

  12. The term BRICS is now a common acronym for the following group of countries—Brazil, Russia, India, China and S. Africa.

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  1. Indira Gandhi Institute of Development Research, General Vaidya Marg, Goregaon (East), Mumbai, 400 065, India

    D. M. Nachane

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Nachane, D.M. Time-varying spectral analysis: theory and applications. Ind. Econ. Rev. 53, 3–27 (2018). https://doi.org/10.1007/s41775-018-0030-2

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