Autocorrelation Estimates of Locally Stationary Time Series

Transcription

1 Autocorrelation Estimates of Locally Stationary Time Series Supervisor: Jamie-Leigh Chapman 4 September 2015

2 Contents 1 2 Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows 3 Simulated Example ONS Economic Data 4

3 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. % Quarterly Savings Data Households & NPISH saving ratio Time Figure: Quarterly Savings Data UK (ONS).

4 Rolling Windows Rolling Windows Exponentially Weighted Windows Kernel Weighted 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, r k = 1 w w k (x j x)(x j+k x). j=1

5 Exponentially Weighted Windows Rolling Windows Exponentially Weighted Windows Kernel 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, q k = w k j=1 β j 1 (x j x)(x j+k x) C k where C k = w k 1 i=0 β i.

6 Kernel Weighted Windows Rolling Windows Exponentially Weighted Windows Kernel Weighted Windows For each individual window of the time series and lag k, the values (x j x)(x j+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.

7 Change in Variance Simulated Example ONS Economic Data This plot compares the various lag zero autocovariance estimates of a time series that exhibits a change in variance. Change in Variance (lag zero autocovariance) Autocovariance Rolling Window Exponential Kernel Windows Figure: Change in Variance Comparison (lag zero).

8 Quarterly Savings Data Simulated Example ONS Economic Data Differenced Quarterly Savings Data Comparison of Local Autocovariance Estimates (WL = 5) Differenced Percentage Data Estimated Autocovariance Kernel Exponential (0.9) Rolling Window Time Window Figure: Left: Differenced data with changepoints, Right: Comparison of autocovariance methods.

9 Quarterly Savings Data Simulated Example ONS Economic Data Autocorrelation Estimates of Quarterly Savings Data (diff.) Aucorrelation Lag Figure: Different autocorrelation estimates (t=55).

10 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.

11 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 Change in Variance Comparison (Lag 0) Data Average MSE Rolling Window Exponential (Beta=0.9) Kernel Time Window Length / Binwidth Figure: Left: Change in variance, Right: Comparison of methods (lag zero autocovariance).

12 Simulation Example 1 Change in Variance Exponential (Lag 0) Average MSE Beta = 0.8 Beta = 0.85 Beta = 0.9 Beta = Window Length Figure: Comparison across different beta values.

13 Simulation Example 2 The simulated TVAR(1) model can be written as X t = α t X t 1 + Z t, where Z t N(0, 1) and α t evolves linearly over time (between 0.5 and 0.5). The estimated lag one autocovariance was compared to a time-dependent theoretical equivalent γ t, γ t = α t 1 αt 2.

14 Simulation Example 2 TVAR(1) Simulation Average TVAR(1) Comparison (Lag 1) Data Average MSE Rolling Window Exponential (Beta=0.9) Kernel Time Window Length Figure: Left: TVAR(1) simulation, Right: Comparison of methods (lag one autocovariance).

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

16 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.

17 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.

18 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.

19 Changes for future work, include:

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

21 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.

22 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.

23 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