预测与时间序列（PPT讲稿）Non-Seasonal Box-Jenkins Models

Non-Seasonal box-Jenkins models

1 Non-Seasonal Box-Jenkins Models

Box-Jenkins(ARIMa) Models The Box-Jenkins methodology refers to a set of procedures for identifying and estimating time series models within the class of autoregressive integrated moving average(ARIMA)models ARIMA models are regression models that use lagged values of the dependent variable and/or random disturbance term as explanatory variables ARIMa models rely heavily on the autocorrelation pattern in the data This method applies to both non-seasonal and seasonal data. In this course we will only deal with non-seasonal data

2 Box-Jenkins (ARIMA) Models  The Box-Jenkins methodology refers to a set of procedures for identifying and estimating time series models within the class of autoregressive integrated moving average (ARIMA) models.  ARIMA models are regression models that use lagged values of the dependent variable and/or random disturbance term as explanatory variables.  ARIMA models rely heavily on the autocorrelation pattern in the data  This method applies to both non-seasonal and seasonal data. In this course, we will only deal with non-seasonal data

Box-Jenkins(ARIMa) Models Three basic ARIMA models for a stationary time series yt (1)Autoregressive model of order p(ar(p) y1=0+y1+如2y12+…+yn+E1 1. e,, y, depends on its p previous values (2) Moving Average model of order g malg) y 6+8,-61E,1-6,E t-2 t-9 1. e, y, depends on g previous random error terms

3 Box-Jenkins (ARIMA) Models  Three basic ARIMA 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 + − −  − , t t 1 t 1 2 t 2 q t q y = + − − − − − − −         

Box-Jenkins(ARIMa) Models (3)Autoregressive-moving average model of order p and g (arma(p, q) y=6+y=1+n2y2+…+yp +E.-6 t-1 g t-g . e, y, depends on its p previous values and q previous random error terms

4 Box-Jenkins (ARIMA) Models (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 − − − − − − + − − − − = + + + +             

Box-Jenkins(ARIMa) Models In an arima model. the random disturbance term E, is typically assumed to be"white noise;.e, it is identically and independently distributed with a mean of o and a common variance o across all observations e write d(0,a)

5 Box-Jenkins (ARIMA) Models  In an ARIMA model, the random disturbance term is typically assumed to be “white noise”; i.e., it is identically and independently distributed with a mean of 0 and a common variance across all observations.  We write ~ i.i.d.(0, ) t  2  t  2 

a five-step iterative procedure Stationarity Checking and Differencing 2)Model Identification 3)Parameter Estimation 4)Diagnostic Checking 5) Forecasting

6 A five-step iterative procedure 1) Stationarity Checking and Differencing 2) Model Identification 3) Parameter Estimation 4) Diagnostic Checking 5) Forecasting

Step One: Stationarity checking

7 Step One: Stationarity checking

Stationarity Stationarity is a fundamental property underlying almost all time series statistical models a time series yt is said to be stationary if it satisfies the following conditions ( 1)E(=u, for all t (2)Var(y)=elo-u=o+ for all t ()Cov(y,, yi-k)=rk for all t

8 Stationarity  “Stationarity” is a fundamental property underlying almost all time series statistical models.  A time series yt is said to be stationary if it satisfies the following conditions: 2 2 (1) ( ) . (2) ( ) [( ) ] . (3) ( , ) . t y t t y y t t k k E y u for all t Var y E y u for all t Cov y y for all t   − = = − = =

Stationarity The white noise series &, satisfies the stationarity condition because (1)E(6)=0 (2)Var()=02 (3)Cov(1s)= for all s≠0

9 Stationarity  The white noise series satisfies the stationarity condition because (1) E( ) = 0 (2) Var( ) = (3) Cov( ) = for all s 0 t  2   t  t  t  t s  −

Example of a white noise series Time Series plot 100 20 Time 10

10 Example of a white noise series 4 8 12 16 20 24 28 32 36 100 80 60 40 20 0 Time Time Series Plot