Long range dependence modelling in Julia

This talk presents a package to analyse long-range dependence (LRD) in time series data. LRD is shown by the fact that the effects from previous disturbances take longer to dissipate than what standard models can capture. Failing to account for LRD dynamics can perversely affect forecasting performance: a model that does not account for LRD misrepresents the true prediction confidence intervals. LRD has been found in climate, political affiliation and finance data, to name a few examples.

Long-range dependence has been a topic of interest in time series analysis since Granger's study on the shape of the spectrum of economic variables. The author found that long-term fluctuations in data, if decomposed into frequency components, are such that the amplitudes of the components decrease smoothly with decreasing periods. This type of dynamics implies long-lasting autocorrelations; that is, they exhibit long-range dependence. Long-range dependence has been estimated in temperature data, political affiliation data, financial volatility measures, inflation, and energy prices, to name a few. Moreover, it has been shown that the presence of long-range dependence on data can have perverse effects on statistical methods if not included in the modelling scheme.

This talk presents a package for modelling long-range dependence in the data. We develop methods to model long-range dependence by the commonly used fractional difference operator and the theoretically based cross-sectional aggregation scheme. The fast Fourier transform and recursive implementations of the algorithms are used to speed up computations. The proposed algorithms are exact in the sense that no approximation of the number of aggregating units is needed. We show that the algorithms can be used to reduce computational times for all sample sizes.

Moreover, estimators in the frequency domain are developed to test for long-range dependence in the data. A broad range of estimators are considered: the original Geweke and Porter-Hudak (GPH) estimator, local Whittle (LW) variants that allow for non-de-meaned data, bias-reduced versions of both GPH and LW methods, and Maximum Likelihood Estimators (MLE) in the frequency domain for the fractional differenced and cross-sectional aggregated data. For the latter, the profile likelihood is obtained for efficiency.

The proposed package is simple to implement in real applications. In particular, we present an exercise using temperature data modelled using standard and long-range dependence models. The experiment shows that the standard model misrepresents the prediction confidence intervals of future global temperatures. The misrepresentation can potentially explain some of the previous underestimations of temperature increases in the last decades.