Forecasting locally stationary time series Rebecca Killick [email protected]Joint work with Idris Eckley (Lancaster), Marina Knight (York) & Guy Nason (Bristol) June 30, 2014 Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 1 / 20
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Rebecca Killick [email protected] Joint work with · This leads to a localised measure of autocovariance c(t;˝). Rebecca Killick (Lancaster University) Forecasting locally stationary
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Joint work withIdris Eckley (Lancaster), Marina Knight (York) & Guy Nason (Bristol)
June 30, 2014
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 1 / 20
What do I mean by Nonstationary Time Series?
I mean NOT second-order stationary.
So, unconditional variance changes with time.
Autocovariance, spectrum, etc. change with time.
Typically assume EXt = 0 (assume mean removed).
More interested in Variability and CI
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 2 / 20
What do I mean by Nonstationary Time Series?
I mean NOT second-order stationary.
So, unconditional variance changes with time.
Autocovariance, spectrum, etc. change with time.
Typically assume EXt = 0 (assume mean removed).
More interested in Variability and CI
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 2 / 20
Structure of Presentation
Motivation
Nonstationary forecasting
The local partial autocorrelation function
Forecasting using the lpacf
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 3 / 20
Motivation
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 4 / 20
Motivation - ABML
ABML consists of gross value added amountsComponent in the estimate of GDP223 observations from Q1 1955 to Q3 2010We use second differences to remove trendTests of stationarity reject H0.
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1947 1954 1961 1968 1975 1982 1989 1996 2003 2010
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 5 / 20
Motivation - ABML
ONS currently use ARIMA models to forecast this data
What is the danger in doing this?
Red - Full series forecast
Blue - Last 30 obs forecast
Full fits ARMA(1,1) non-zeromean
Last 30 obs fits AR(2)
Overconfident in forecast?
2004 2006 2008 2010
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Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 6 / 20
Motivation - ABML
ONS currently use ARIMA models to forecast this data
What is the danger in doing this?
Red - Full series forecast
Blue - Last 30 obs forecast
Full fits ARMA(1,1) non-zeromean
Last 30 obs fits AR(2)
Overconfident in forecast?2004 2006 2008 2010
−60
00−
4000
−20
000
2000
4000
6000
Year
AB
ML
Sec
ond
Diff
eren
ces
(£m
)
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 6 / 20
Recall: Forecasting Stationary TS— Notation
Suppose have data x1, . . . , xT from stationary series.
Want to make forecast x̂T (h) made at time T for horizon h.
Want forecast using linear combination of past:
x̂T (1) =T−1∑i=0
ϕixT−i
= ϕ0xT + ϕ1xT−1 + ϕ2xT−2 + · · ·
Note: ϕ sequence DOES NOT depend on time (stationary xt)
Theory can tell us optimal least-squares forecast (Box-Jenkins).
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 7 / 20
Extension to Nonstationary Time Series
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 8 / 20
Modelling nonstationary time series
Modelling in the face of non-stationarity is no easy task!
Various approaches have been explored, built on models that fitparticular types of non-stationarity:
assume piecewise stationarity;
use parametric models with time-changing coefficients,e.g. Time Varying AR (tvAR).
Processes with a slowly time-varying second order structure areknown as locally stationary (LS).
Advanced LS models (ARCH) (Dahlhaus and Subba Rao, 2006).
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 19 / 20
Summary
Motivated why forecasting nonstationary time series is important.
Proposed a new measure – the local partial autocorrelation function –and associated theoretical justification.
Used the lpacf to choose p for the localised Yule-Walker equations.
Showed increased forecasting performance when using the lpacf.
We have used the lpacf as a tool for forecasting but it can be used ina variety of settings.
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 20 / 20
References I
R. Dahlhaus and S. Subba Rao.Statistical inference for time-varying ARCH processes.Annals of Statistics, 34(3):1075–1114, 2006.
R. Dahlhaus.Fitting Time Series Models to Nonstationary Processes.Annals of Statistics, 25(1):1–37, 1997.
G.P. Nason, R. von Sachs and G. Kroisandt.Wavelet Processes and Adaptive Estimation of the EvolutionaryWavelet Spectrum.JRSSB, 62(2):271–292, 2000.
P. Fryzlewicz, S. Van Bellegem and R. von Sachs.Forecasting non-stationary time series by wavelet process modelling.Ann. Inst. Statist. Math., 55(4):737–764, 2003.
Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, 2014 21 / 20