STAD57 Time Series Analysis. Lecture 17

Transcription

1 STAD57 Time Series Analysis Lecure 17 1

2 Exponenially Weighed Moving Average Model Consider ARIMA(0,1,1), or IMA(1,1), model 1 s order differences follow MA(1) X X W W Y X X W W Very common model for economic / financial daa I is also known as Exponenially Weighed Moving Average (EWMA) model EWMA has simple forecass as weighed averages of series using exponenial weighs 2

3 Exponenially Weighed Moving Average Model EWMA model can be re-wrien as: (1 ) j1 j1 j X X W Proof: 3

4 Exponenially Weighed Moving Average Model Using runcaed predicion, he 1-sep-ahead forecass from he EWMA become: X (1 ) X X, n 1, wih X X n n1 0 n 1 n n 1 1 Proof: 4

5 Example Fi EWMA in R using HolWiners() funcion: To check parameer: daa fixed opions fi.ewma = HolWiners( x, bea=false, gamma=false ) fi.ewma$alpha (his is equal o 1 λ) Using IMA(1,1) sarima(x,0,1,1) Coefficiens: ma1 consan s.e

6 Example (con d) Series 1-sep-ahead EWMA forecass

7 Uni Roo Tesing I is ofen difficul o disinguish Random Walk from AR(1) wih φ 1 (especially for small n) Using confidence inerval for φ from AR(1) model X X W 1 has poor performance Uni roo ess are designed o es H 0 : φ=1 vs H 0 : φ <1, for X X W model 1 Mos common es is he augmened Dickey-Fuller (ADF) es 7

8 Example ACF plo (n=200) for Random Walk AR(1) X.9X W

9 Example ADF es in R using ADF.es() funcion load series package daa library(series); adf.es( x ) Resuls: for Random Walk (n=200) Augmened Dickey-Fuller Tes Dickey-Fuller = , Lag order = 5, p-value = alernaive hypohesis: saionary Augmened Dickey-Fuller Tes Dickey-Fuller = , Lag order = 5, p-value = alernaive hypohesis: saionary for AR(1) w/ φ=.9 (n=200) 9

10 Seasonaliy Seasonaliy is periodic paern in TS E.g. Increase in summer sales of sunan loions Period of seasonal paern denoed by s E.g. monhly daa w/ annual paern s=12, daily daa w/ weekly paern s=7, ec 2 ways o model seasonaliy, depending on is ype: Deerminisic Seasonaliy Sochasic Seasonaliy 10

11 Deerminisic Seasonaliy Consider series Z is saionary series, where: S() is deerminisic seasonal componen S( ) S( s), i.e. periodic funcion w/ period s: To esimae deerminisic S(), fi #s separae means (one for each ime wihin he period) E.g. X S() Z ˆ 1, for 1,1 s,1 2 s, ˆ 2, for 2,2 s,2 2 s, S ˆ( ) ˆ s, for s,2 s,3 s, Can use ANOVA for his 11

12 Example Consider series wih s=7 Series ACF For seasonal series, ACF will ypically exhibi paern wih same period (s) 12

13 Example Series wih esimaed ˆ( ) : S ˆ 1 ˆ 2 ˆ 3 ˆ S ( ) : ˆ 4 ˆ 5 ˆ 6 ˆ Furher analysis (e.g. fi ARMA model) uses de-seasonalized series: Zˆ X Sˆ ()

14 Sochasic Seasonaliy More common and flexible siuaion is when seasonaliy is sochasic: X S Z, where: S is RV, ypically dependen on pas of X E.g. monhly sales w/ annual paern, where his year s Jan sales depend on las year s Jan sales Simple way o model his behavior is o look a auo-regression a muliple lags of period s E.g. For monhly daa w/ annual paern (s=12), can use: X X W 12 14

15 Pure Seasonal ARMA Model More generally, a pure seasonal ARMA model of order (P,Q) and period s, denoed by SARMA(P,Q) s, is given by: X X X X 1 s 2 2s P Ps W W W W 1 s 2 2s Q Qs s s ( B ) X ( B ) W, where: seasonal seasonal AR( P) polynomial: ( B ) 1 B B B s s 2s Ps 1 2 P MA( Q) polynomial: ( B ) 1 B B B s s 2s Qs 1 2 Q 15

16 Pure Seasonal ARMA Model The pure seasonal SARMA(P,Q) s model is causal / inverible if and only if he roos of Φ(z s ) / Θ(z s ) lie ouside he uni circle: ( s s z ) 0 / ( z ) 0, z 1 ACF / PACF of pure seasonal SARMA(P,Q) s behaves similarly o ACF / PACF of usual ARMA(p,q) a muliple lags of s, and is 0 elsewhere 16

17 Example Find ACF of X X W (SAR(1) s ) s 17

18 Pure Seasonal ARMA Model Similarly, for SMA(1) s he ACF is: 1, h 0 2 ( h) / (1 ), h s 0, oherwise Following able describes behavior of pure SARMA(P,Q) s models ACF PACF SAR(P) s SMA(Q) s SARMA(P,Q) s Tails off a lags k s, k 1 Cus off a lag P s Cus off a lag Q s Tails off a lags k s, k 1 Tails off a lags k s, k 1 Tails off a lags k s, k 1 18

19 Example SAR(2) 4 model: ACF PACF

20 Example SMA(2) 4 model: ACF PACF

21 Example SARMA(2,2) 4 model: ACF PACF

22 Muliplicaive Seasonal ARMA Model Can also combine seasonal SARMA(P,Q) s wih simple ARMA(p,q) model, by muliplying heir corresponding polynomials: s s ( B ) ( B) X ( B ) ( ) W Called muliplicaive SARMA(p,q) (P,Q) s model E.g. monhly sales depend on las year s sales and las monh s sales use muliplicaive SARMA(1,0) (1,0) 12 model: ( B ) ( B) X W (1 B )(1 B) X W X X X X W

23 Example Wrie general form of SARMA(1,1) (1,0) s model 23

24 Example Consider SARMA(1,1) (1,1) 4 model: 4 4 (1.6 B )(1.3 B) X (1.6 B )(1.6 B) W ACF PACF Even for muliplicaive SARMA models, ACF will ypically have paern wih period (s) 24