# Don’t Try to Forecast Everything: Predictability of Time Series

Author

Murat Koptur

Published

September 1, 2022

# Introduction

Most of time series analyses start with investigating series, autocorrelation and partial autocorrelation plots. Then one estimates different time series models (like ARIMA, GARCH, State-space models) and performs model checks.

But no one asks whether that series is predictable or not.

# Data

We’ll use some example time series:

• Monthly Airline Passenger Numbers 1949-1960 (AirPassengers) $$^{6}$$

• Level of Lake Huron 1875-1972 (LakeHuron) $$^{6}$$

• Simulated time-series data from the Logistic map with chaos $$^{1}$$

# Tools

Let’s look the tools.

## Lyapunov Exponent

Lyapunov exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector $$\delta Z_0$$ diverge at a rate given by

$|\delta Z(t)|\approx e^{\lambda t}|\delta Z_0|$

where $$\lambda$$ is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, 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 $$^{7}$$.

I’ll not go into detail on how to calculate the maximal Lyapunov exponent, we’ll look at practical implications.

A positive MLE is usually taken as an indication that the system is chaotic $$^{7}$$.

## 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 $$0.5 < H \leq 1$$ , then 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 series is a completely uncorrelated series, so it can go either way (up or down);

• Mean-reverting (Anti-persistent) series: if $$0 \leq H < 0.5$$, then series has mean-reversion, so a high value in the series will probably be followed by a low value and vice versa $$^{8}$$.

## Detrended Fluctuation Analysis

DFA is a method for determining the statistical self-affinity of a signal. It is the generalization of Hurst exponent, it means $$^{8}$$:

• for $$0<\alpha<0.5$$, then the series is anti-correlated;

• for $$\alpha=0.5$$, then the series is uncorrelated and corresponds to white noise;

• for $$0.5<\alpha<1$$, then the series is correlated;

• for $$\alpha\approx1$$, then the series corresponds to pink noise;

• for $$\alpha>1$$, then the series is nonstationary and unbounded;

• for $$\alpha\approx1.5$$, 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.

# Statistics of the series

## AirPassengers data

Results:

• Lyapunov exponent spectrum:

Call:
Lyapunov exponent spectrum

Coefficients:
Estimate Std. Error   z value      Pr(>|z|)
Exponent 1 -0.8398548  0.2333552 -28.33887 5.739062e-177
Exponent 2 -1.5136329  0.1937088 -61.52719  0.000000e+00
---
Procedure: QR decomposition by bootstrap blocking method
Embedding dimension: 2, Time-delay: 1, No. hidden units: 10
Sample size: 129, Block length: 62, No. blocks: 1000

There are two statistically significant exponent estimates. The largest one is -0.84 which is negative, which means the series is not chaotic.

• Hurst exponent is 0.8206234; it is bigger than 0.5, so series is trending.

• DFA is estimated as 1.2988566; it is nonstationary and unbounded.

• Variance ratio test:

$Holding.Periods [1] 2 4 5 8 10 27$CD1
[1] 24.48521

$CD2 [1] 21.22941$Critical.Values_10_5_1_percent
[1] 2.378000 2.631038 3.142756

Both of test statistics are bigger than critical values, so the series is not a random walk.

## LakeHuron data

Results:

• Lyapunov exponent spectrum:

Call:
Lyapunov exponent spectrum

Coefficients:
Estimate Std. Error    z value Pr(>|z|)
Exponent 1 -0.2245224 0.03079226  -56.00722        0
Exponent 2 -0.6465142 0.01144893 -433.74968        0
Exponent 3 -0.6696687 0.01006248 -511.18811        0
Exponent 4 -1.6931702 0.02747627 -473.33519        0
---
Procedure: QR decomposition by bootstrap blocking method
Embedding dimension: 4, Time-delay: 1, No. hidden units: 2
Sample size: 94, Block length: 59, No. blocks: 1000

There are four statistically significant exponent estimates. The largest one is -0.22 which is negative, which means the series is not chaotic.

• Hurst exponent is 0.7364948; it is bigger than 0.5, so series is trending.

• DFA is estimated as 1.1128455; it is nonstationary and unbounded.

• Variance ratio test:

$Holding.Periods [1] 2 4 5 8 10 3$CD1
[1] 11.45734

$CD2 [1] 9.407748$Critical.Values_10_5_1_percent
[1] 2.378000 2.631038 3.142756

Both of test statistics are bigger than critical values, so the series is not a random walk.

## Simulated time-series data from the Logistic map with chaos

Results:

• Lyapunov exponent spectrum:

Call:
Lyapunov exponent spectrum

Coefficients:
Estimate Std. Error   z value Pr(>|z|)
Exponent 1 -1.291195  0.1580609 -63.27662        0
---
Procedure: QR decomposition by bootstrap blocking method
Embedding dimension: 1, Time-delay: 1, No. hidden units: 2
Sample size: 99, Block length: 60, No. blocks: 1000

There is one statistically significant exponent estimate, -1.29 which is negative, which means the series is not chaotic which is a questionable result.

• Hurst exponent is 0.6255664; it is bigger than 0.5, so series is trending.

• DFA is estimated as 0.758476; it is correlated.

• Variance ratio test:

$Holding.Periods [1] 2 4 5 8 10 10$CD1
[1] 1.193817

$CD2 [1] 1.295116$Critical.Values_10_5_1_percent
[1] 2.378000 2.631038 3.142756

Both of test statistics are smaller than critical values, so the series is a random walk.

# References

$$^2$$ statcomp, https://cran.r-project.org/web/packages/statcomp/index.html

$$^4$$ tseriesChaos, https://cran.r-project.org/web/packages/tseriesChaos/index.html

$$^5$$ Daniel F. McCaffrey , Stephen Ellner , A. Ronald Gallant & Douglas W. Nychka (1992) Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression, Journal of the American Statistical Association, 87:419, 682-695

$$^7$$ Contributors to Wikimedia projects. “Lyapunov exponent - Wikipedia.” 7 July 2022, https://en.wikipedia.org/w/index.php?title=Lyapunov_exponent&oldid=1096875011.

$$^8$$ Contributors to Wikimedia projects. “Hurst exponent - Wikipedia.” 12 June 2022, https://en.wikipedia.org/w/index.php?title=Hurst_exponent&oldid=1092814465.

$$^9$$ DFA, https://cran.r-project.org/package=DFA

$$^{10}$$ Contributors to Wikimedia projects. “Detrended fluctuation analysis - Wikipedia.” 19 June 2022, https://en.wikipedia.org/w/index.php?title=Detrended_fluctuation_analysis&oldid=1093832537.

$$^{11}$$ nonlinearTseries, https://cran.r-project.org/web/packages/nonlinearTseries/index.html.

## Citation

BibTeX citation:
@online{koptur2022,
author = {Murat Koptur},
title = {Don’t {Try} to {Forecast} {Everything:} {Predictability} of
{Time} {Series}},
date = {2022-09-01},
url = {https://www.muratkoptur.com/MyDsProjects/Analysis.html},
langid = {en}
}