Abstract
A new comprehensive approach to nonlinear time series analysis and modeling is developed in the present paper. We introduce novel data-specific mid-distribution-based Legendre Polynomial (LP)-like nonlinear transformations of the original time series {Y(t)} that enable us to adapt all the existing stationary linear Gaussian time series modeling strategies and make them applicable to non-Gaussian and nonlinear processes in a robust fashion. The emphasis of the present paper is on empirical time series modeling via the algorithm LPTime. We demonstrate the effectiveness of our theoretical framework using daily S&P 500 return data between 2 January 1963 and 31 December 2009. Our proposed LPTime algorithm systematically discovers all the ‘stylized facts’ of the financial time series automatically, all at once, which were previously noted by many researchers one at a time.
Highlights
When one observes a sample Y(t), t = 1, . . . , T, of a time series Y(t), one seeks to nonparametrically learn from the data a stochastic model with two purposes: (a1) scientific understanding; (a2) forecasting
This article provides a pragmatic and comprehensive framework for nonlinear time series modeling that is easier to use, more versatile and has a strong theoretical foundation based on the recently-developed theory of unified algorithms of data science via Legendre Polynomial (LP) modeling (Mukhopadhyay 2016, 2017; Mukhopadhyay and Fletcher 2018; Mukhopadhyay and Parzen 2014; Parzen and Mukhopadhyay 2012, 2013a, 2013b)
From the theoretical standpoint, the unique aspect of our proposal lies in its ability to simultaneously embrace and employ the spectral domain, time domain, quantile domain and information domain analyses for enhanced insights, which to the best of our knowledge has not appeared in the nonlinear time series literature before
Summary
(1) Marginal modeling: The identification of the marginal probability law (in particular, the heavy-tailed marginal densities) of a time series plays a vital role in financial econometrics. (2) Correlation modeling: Covariance function (defined for positive and negative lag h) R(h; Y) = Cov[Y(t), Y(t + h)]. (3) Frequency-domain modeling: When covariance is absolutely summable, define spectral density function f (ω; Y) = ∑ R(h; Y) e−2πiωh, −1/2 < ω < 1/2. Autoregressive scheme of order m, a predominant linear time series technique for modeling conditional mean, is defined as (assuming E[Y(t)] = 0): Y(t) − a(1; m)Y(t − 1) − . With the spectral density function given by:. We aim to develop a parallel modeling framework for nonlinear time series
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