Non-Seasonal box-Jenkins models
1 Non-Seasonal Box-Jenkins Models
Four-step iterative procedures 1)Model Identification Parameter estimation 234 Diagnostic Checking Forecasting
2 Four-step iterative procedures 1) Model Identification 2) Parameter Estimation 3) Diagnostic Checking 4) Forecasting
Step One: Model Identification
3 Step One: Model Identification
Model identification I. Stationarit I. Theoretical autocorrelation function (TAC II. Theoretical Partial autocorrelation Function(TPAc IV. Sample partial Autocorrelation Function (SPAC V. Sample autocorrelation Function (SAC
4 Model Identification I. Stationarity II. Theoretical Autocorrelation Function (TAC) III. Theoretical Partial Autocorrelation Function (TPAC) IV. Sample Partial Autocorrelation Function (SPAC) V. Sample Autocorrelation Function (SAC)
Stationarity (D A sequence of jointly dependent random variables -oo<t<o is called a time series
5 Stationarity (I) A sequence of jointly dependent random variables is called a time series {y : − t } t
Stationarity(l) ◆ Stationary process Properties (1)E(=u, for all t (2)ar(y)=E(y,-1)]=a2ral (3)Cov(,, yk=r for all t
6 Stationarity (II) Stationary process Properties : (3) ( , ) . (2) ( ) [( ) ] . (1) ( ) . 2 2 Cov y y for all t Var y E y u for all t E y u for all t t t k k t t y y t y = = − = = +
Stationarity (ir) Example: The white noise series &G/ .Es are id as n(o, o ). Note that ()E(e=0 for all t (2)Var(&)=E()=0f for all t (3)CoV(e, Es=0 for all t
7 Stationarity (III) Example: The white noise series {et } ◼ e’s are iid as N(0,e 2 ). Note that (3) Cov( , ) 0 for all t. (2) Var( ) E( ) for all t. (1) E( ) 0 for all t. t t s 2 2 t t t = = = = + e e e e e e
Stationarity (Iv) Three basic box-Jenkins models for a stationary time series t,/ (1)Autoregressive model of order p(ar(p) y1=O+y1+n2V12+…+ p.t-p 1. e,, y, depends on its p previous values (2) Moving Average model of order g malg) y 6+E.-6 t-2 t-9 1. e, y, depends on g previous random error terms
8 Stationarity (IV) Three basic Box-Jenkins models for a stationary time series {yt } : (1) Autoregressive model of order p (AR(p)) i.e., yt depends on its p previous values (2) Moving Average model of order q (MA(q)) i.e., yt depends on q previous random error terms, t 1 t 1 2 t 2 p t p t y = + y + y + + y +e − − − , t t 1 t 1 2 t 2 q t q y = + − − − − − − − e e e e
Stationarity(v) Three basic box-Jenkins models for a stationary time series ty (3) Autoregressive-moving average model of order p and g (arma(p, q) y=+y1+n2y2+…+yp +6-661-62E12-…-6 g t-g i.e., y, depends on its p previous values and g previous random error terms
9 Stationarity (V) Three basic Box-Jenkins models for a stationary time series {yt } : (3) Autoregressive-moving average model of order p and q (ARMA(p,q)) i.e., yt depends on its p previous values and q previous random error terms , 1 1 2 2 1 1 2 2 t t t q t q t t t p t p y y y y − − − − − − + − − − − = + + + + e e e e
AR(1)(I) Simple ar(i) process without drift Py+E(where at is white noise y, =O,Ly, +E(where L is the back-shift operator) 0(L)=(1-Ly or y 1-如L E1(1+L+:L2+…) =E+1+2+ 10
10 AR(1) (I) Simple AR(1) process without drift . (1 L L ) 1 L y or (L) (1 L)y y Ly (where L is the back shift operator) y y (where is white noise ) t 2 2 t 1 t 1 1 2 2 t 1 1 1 t t 1 t t t 1 t t t 1 t 1 t t = + + + = + + + − = = − = = + − = + − − − e e e e e e e e
