Holistic Localized Autocovariance

The concept of localized autocovariance, c(t, k), as a function that measures the covariance between X_t and X_t+k, has been around for a long time See, for example, the excellent book by Priestley (1981), but the idea goes way back. There are several ways it can be estimated.

One method is to obtain a time-varying local frequency spectrum and applying a windowed Fourier transform of the spectrum. Nason, von Sachs and Kroisandt (2000) introduced another method based on using wavelet processes (wavelet processes are models for time series that use small oscillations, wavelets, as building blocks). The idea is similar to the Fourier method above. One can form a time-varying wavelet spectrum which provides information on how the time series' power is distributed over time and scale. The wavelet spectrum can be converted into the localized autocovariance using a type of wavelet, not Fourier, transformation.

The idea of estimating localized autocovariance using wavelet methods has recently been developed further by Nason (2013). The new work presents a new estimator as well as mathematically rigorous confidence intervals permitting, for the first time, confidence intervals for the autocovariances. The confidence intervals permit the user to assess whether a given autocovariance at a given time point is statistically significantly different from zero or not. The localized autocovariance figures below are produced using this new work in conjunction with the associated R software package locits.

Some examples:

Simulated Autoregressive Example>

Real FTSE log returns Example>