Don’t Try to Forecast Everything: Predictability of Time Series
Most people start directly modeling and forecasting their time series.
No one asks whether that series is predictable or not.
We’ll look at a few handy tools that give more information about our time series.
Lyapunov Exponent
Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. There is a spectrum of Lyapunov exponents. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE) because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic¹.
Hurst Exponent
The Hurst exponent is referred to as the “index of dependence” or “index of long-range dependence”. It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction:
- Trending (Persistent) series: If 0.5< H ≤1, then the series has long-term positive autocorrelation, so a high value in the series will probably be followed by another high value and the future will also tend to be high;
- Random walk series: if H=0.5, then the series is a completely uncorrelated series, so it can go either way (up or down);
- Mean-reverting (Anti-persistent) series: if 0≤H<0.5, then the series has mean-reversion, so a high value in the series will probably be followed by a low value and vice versa².
Detrended Fluctuation Analysis
DFA is a method for determining the statistical self-affinity of a signal. It is the generalization of Hurst exponent, it means²:
- for 0<a<0.5, then the series is anti-correlated;
- for a=0.5, then the series is uncorrelated and corresponds to white noise;
- for 0.5<a<1, then the series is correlated;
- for a≈1, then the series corresponds to pink noise;
- for a>1, then the series is nonstationary and unbounded;
- for a≈1.5, then the series corresponds to Brownian noise.
Variance Ratio Test
This test is often used to test the hypothesis that a given time series is a collection of i.i.d. observations or that it follows a martingale difference sequence.
We will use Chow and Denning’s multiple variance ratio test. There are two tests:
- CD1 — Test for i.i.d. series,
- CD2 — Test for uncorrelated series with possible heteroskedasticity.
If test statistics are bigger than critical values, the null hypothesis is rejected which means the series is not a random walk.
References
¹ Contributors to Wikimedia projects. “Lyapunov exponent — Wikipedia.” 7 July 2022, https://en.wikipedia.org/w/index.php?title=Lyapunov_exponent&oldid=1096875011.
² Contributors to Wikimedia projects. “Hurst exponent — Wikipedia.” 12 June 2022, https://en.wikipedia.org/w/index.php?title=Hurst_exponent&oldid=1092814465.
This article was first published at https://muratkoptur.com/MyDsProjects/TimeSeriesPredictability/Analysis.html
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