Autocorrelation Estimates of Locally Stationary Time Series Srshti - - PowerPoint PPT Presentation

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study

Autocorrelation Estimates of Locally Stationary Time Series

Srshti Putcha Supervisor: Jamie-Leigh Chapman 4 September 2015

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study

Contents

1

Problem Motivation Problem Motivation

2

Autocovariance Estimation Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows

3

Comparison of Methods Simulated Example ONS Economic Data

4

Simulation Study Simulation Study Conclusions Further Work

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Problem Motivation

Problem Motivation

Quarterly Savings Data − Households & NPISH saving ratio

Time % 1970 1980 1990 2000 2010 4 6 8 10 12 14 16

Figure: Quarterly Savings Data UK (ONS).

We often assume that time series are second-order stationary, with their statistical properties remaining the same over time. In reality, many time series are not stationary and we shouldn’t try to apply traditional methods. This aim of this project was to explore and compare alternative methods of estimating time-varying autocovariance.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows

Rolling Windows

For a window length w, a time series can be segmented into T rolling windows. Then, the sample autocovariance can be computed by:

Figure: Rolling windows for a time series of length T.

Windowed Autocovariance At lag k and a window length w, rk = 1 w

w−k

• j=1

(xj − ¯ x)(xj+k − ¯ x).

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows

Exponentially Weighted Windows

Using the same rolling window structure, we can instead weight the contributions of each observation within a window. Exponentially Weighted Autocovariance A value of β is selected such that 0 < β < 1. Then, qk =

w−k

• j=1

βj−1 Ck (xj − ¯ x)(xj+k − ¯ x) where Ck = w−k−1

i=0

βi.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows

Kernel Weighted Windows

For each individual window of the time series and lag k, the values (xj − ¯ x)(xj+k − ¯ x) from j = 1, 2, ..., (w − k) are weighted using a Gaussian Kernel. Gaussian Kernel The Kernel function is given by K(z) = 1 √ 2π exp(−z2 2 ). K(z) is nonnegative and integrates to 1. A weighted average is created and this is a localised estimate of the autocovariance of the time series at lag k.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulated Example ONS Economic Data

Change in Variance

This plot compares the various lag zero autocovariance estimates

• f a time series that exhibits a change in variance.

200 400 600 800 1000 5 10 15

Change in Variance (lag zero autocovariance)

Windows Autocovariance Rolling Window Exponential Kernel

Figure: Change in Variance Comparison (lag zero).

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulated Example ONS Economic Data

Quarterly Savings Data

Differenced Quarterly Savings Data

Time Differenced Percentage Data 50 100 150 200 −4 −2 2 4 50 100 150 200 5 10 15

Comparison of Local Autocovariance Estimates (WL = 5)

Window Estimated Autocovariance Kernel Exponential (0.9) Rolling Window

Figure: Left: Differenced data with changepoints, Right: Comparison of autocovariance methods.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulated Example ONS Economic Data

Quarterly Savings Data

5 10 15 20 −0.5 0.0 0.5 1.0

Autocorrelation Estimates of Quarterly Savings Data (diff.)

Lag Aucorrelation

Figure: Different autocorrelation estimates (t=55).

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Simulations

The purpose of this simulation study was to compare the performance of the three methods. The study was conducted across seven different examples of simulated time series data, where changes were made to the second-order structure. Two important criteria were: Accuracy of the method (MSE). How robust the method was to window length.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Simulation Example 1

A step change in variance was created halfway through the time

• series. The lag zero autocovariance estimates were compared to

the simulated change in variance.

Change in Variance − Simulation Average

Time Data 200 400 600 800 1000 −0.2 −0.1 0.0 0.1 0.2 20 40 60 80 0.005 0.010 0.015 0.020

Change in Variance Comparison (Lag 0)

Window Length / Binwidth Average MSE Rolling Window Exponential (Beta=0.9) Kernel

Figure: Left: Change in variance, Right: Comparison of methods (lag zero autocovariance).

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Simulation Example 1

20 40 60 80 0.005 0.010 0.015 0.020 0.025

Change in Variance − Exponential (Lag 0)

Window Length Average MSE Beta = 0.8 Beta = 0.85 Beta = 0.9 Beta = 0.95

Figure: Comparison across different beta values.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Simulation Example 2

The simulated TVAR(1) model can be written as Xt = αtXt−1 + Zt, where Zt ∼ N(0, 1) and αt evolves linearly over time (between −0.5 and 0.5). The estimated lag one autocovariance was compared to a time-dependent theoretical equivalent γt, γt = αt 1 − α2

t

.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Simulation Example 2

TVAR(1) − Simulation Average

Time Data 200 400 600 800 1000 −0.3 −0.1 0.0 0.1 0.2 0.3 20 40 60 80 0.00 0.05 0.10 0.15 0.20 0.25

TVAR(1) − Comparison (Lag 1)

Window Length Average MSE Rolling Window Exponential (Beta=0.9) Kernel

Figure: Left: TVAR(1) simulation, Right: Comparison of methods (lag

• ne autocovariance).

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Conclusions

When statistical properties are constant for large sections of the time series, all methods perform in a similar manner.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Conclusions

When statistical properties are constant for large sections of the time series, all methods perform in a similar manner. The kernel weighted windows produce the most accurate estimates of nonstationary processes.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Conclusions

When statistical properties are constant for large sections of the time series, all methods perform in a similar manner. The kernel weighted windows produce the most accurate estimates of nonstationary processes. While dependent on β, the exponentially weighted windows were more robust to changes in window length, with flatter error curves.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Conclusions

When statistical properties are constant for large sections of the time series, all methods perform in a similar manner. The kernel weighted windows produce the most accurate estimates of nonstationary processes. While dependent on β, the exponentially weighted windows were more robust to changes in window length, with flatter error curves. Longer window lengths provide no significant reduction in error and produce less localised estimates.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Further Work

Changes for future work, include:

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Further Work

Changes for future work, include: Establishing a metric to determine what window length to take.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Further Work

Changes for future work, include: Establishing a metric to determine what window length to take. Forming confidence intervals for the estimates using varying window lengths as a parameter.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

Further Work

Changes for future work, include: Establishing a metric to determine what window length to take. Forming confidence intervals for the estimates using varying window lengths as a parameter. Comparisons to current wavelet methods for local autocovariance estimation.

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series

Problem Motivation Autocovariance Estimation Comparison of Methods Simulation Study Simulation Study Conclusions Further Work

References

Chatfield, C. (2003) The Analysis of Time Series: An Introduction, Chapman and Hall/CRC Press, Sixth edition Shumway, R.H., & Stoffer, D.S. (2010) Time series analysis and its applications: with R examples, Springer Science & Business Media

Srshti Putcha Autocorrelation Estimates of Locally Stationary Time Series