Forecasting locally stationary time series
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1 Forecasting locally stationary time series Rebecca Killick 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, / 20
2 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 EX t = 0 (assume mean removed). Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
3 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 EX t = 0 (assume mean removed). More interested in Variability and CI Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
4 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, / 20
5 Motivation Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
6 Motivation - ABML ABML consists of gross value added amounts Component in the estimate of GDP 223 observations from Q to Q We use second differences to remove trend Tests of stationarity reject H0. ABML ( million) ABML Second Differences ( m) Year Year Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
7 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-zero mean Last 30 obs fits AR(2) ABML Second Differences ( m) Year Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
8 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-zero mean Last 30 obs fits AR(2) Overconfident in forecast? ABML Second Differences ( m) Year Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
9 Recall: Forecasting Stationary TS Notation Suppose have data x 1,..., x T from stationary series. Want to make forecast ˆx T (h) made at time T for horizon h. Want forecast using linear combination of past: ˆx T (1) = T 1 i=0 ϕ i x T i = ϕ 0 x T + ϕ 1 x T 1 + ϕ 2 x T 2 + Note: ϕ sequence DOES NOT depend on time (stationary x t ) Theory can tell us optimal least-squares forecast (Box-Jenkins). Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
10 Extension to Nonstationary Time Series Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
11 Modelling nonstationary time series Modelling in the face of non-stationarity is no easy task! Various approaches have been explored, built on models that fit particular 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 are known as locally stationary (LS). Advanced LS models (ARCH) (Dahlhaus and Subba Rao, 2006). Locally stationary fourier processes (Dahlhaus, 1997). Locally stationary wavelet processes (Nason et al., 2000). Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
12 Locally stationary models If you take a small enough region, it will appear stationary as the structure varies slowly over time Application areas include; medicine, finance environmental processes, e.g. wind speeds. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
13 Locally stationary wavelet (LSW) processes LSW processes (Nason et al., 2000): X t,t = 1 j= J(T ) k Z w j,k;t ψ j,k (t)ξ j,k, t = 1,..., T. {ψ j,k } is a collection of discrete non-decimated wavelets. {ξ j,k } j,k is a sequence of zero-mean, orthonormal random variables. Smoothness of wavelet amplitudes w j,k;t as a function of k controls the degree of non-stationarity. LSW processes encapsulate other models and represent processes whose variance and autocorrelation function vary over time. This leads to a localised measure of autocovariance c(t, τ). Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
14 Forecasting in LSW framework Given observations x 0,..., x t 1, we: Predict ˆx t = t 1 s=t p b t 1 i,t X s, where p is the number of latest observations used for prediction and b is the solution to localized Yule-Walker equations. c(t, 1) c(t 1, 0) c(t 2, 1) c(t, 2). = c(t 1, 1) c(t 2, 0)..... c(t, p) c(t 1, p) c(t 2, p 1) b t 1 b t 2. b t 1 p Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
15 Forecasting in LSW framework Given observations x 0,..., x t 1, we: Predict ˆx t = t 1 s=t p b t 1 i,t X s, where p is the number of latest observations used for prediction and b is the solution to localized Yule-Walker equations. c(t, 1) c(t 1, 0) c(t 2, 1) c(t, 2). = c(t 1, 1) c(t 2, 0)..... c(t, p) c(t 1, p) c(t 2, p 1) b t 1 b t 2. b t 1 p These require knowledge of the covariance structure at time t which is the point we are trying to predict. The covariance at time t is similar to that at time t 1. We extrapolate at the smoothing step. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
16 Previous nonstationary forecasting work Fryzlewicz et al. (2003) propose a method for LSW forecasting: smooth and extrapolate the covariances directly by kernel smoothing; choose p (and bandwidth) by in sample optimization. For a practitioner this leaves many questions. How much data do I train on? Do I update p and bandwidth simultaneously or another method? When updating, how many alternative options do I consider? What kernel smoother should I use? Ultimately this p is hard to choose. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
17 Our approach Our work: introduces localised partial acf shows that local partial ACF is an interesting tool in its own right gives encouraging forecasting results Thus we, mirror the stationary process and produce a data driven approach for practitioners. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
18 The local partial autocorrelation function (lpacf) We define the lpacf at time t and lag τ as: q(t, τ) = Corr(X t, X t τ {X t 1..., X t τ+1 }) q T (t, τ) = c( t 1,1) T c( t t 1,0)c( T T,0), for τ = 1 ϕ MSPE( ˆX t,x t X t 1...,X t τ+1 ) t,τ,τ MSPE( ˆX, for τ 2. t τ,x t τ X t 1...,X t τ+1 ) For stationary models the square root equals 1 and the ϕ t,τ,τ is the usual estimate of the pacf. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
19 ABML lpacf LPACF of ABML second differences Year Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
20 Forecasting Simulations Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
21 Comparisons with ARMA Range of models considered: TVAR(1) TVAR(2) TVAR(12) TVMA(1) TVMA(2) Uniformly modulated white noise LSW process Summary lpacf greatly improves on ARMA in 2/3 of cases comparable results (ratio ±0.05) in 1/3. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
22 AMBML - lpacf forecasting RMSE: lpacf=3430m, B-J=4290m, Fryzlewicz=8830m, 11/15, 8/15, 6/ Year.Quarter (most recent first) Forecast and Prediction Intervals 10.Q3 10.Q1 09.Q3 09.Q1 08.Q3 08.Q1 07.Q3 07.Q1 Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
23 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 in a variety of settings. Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
24 References I R. Dahlhaus and S. Subba Rao. Statistical inference for time-varying ARCH processes. Annals of Statistics, 34(3): , R. Dahlhaus. Fitting Time Series Models to Nonstationary Processes. Annals of Statistics, 25(1):1 37, G.P. Nason, R. von Sachs and G. Kroisandt. Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum. JRSSB, 62(2): , P. Fryzlewicz, S. Van Bellegem and R. von Sachs. Forecasting non-stationary time series by wavelet process modelling. Ann. Inst. Statist. Math., 55(4): , Rebecca Killick (Lancaster University) Forecasting locally stationary time series June 30, / 20
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