9 box jenkins

Time series analysisBox-Jenkins method

Chanon Chingchayanurak

Faculty of Business Administration

C.M.U.

Objective

Choose the appropiate model Analyze by using SPSS Diagnosis the models Use model for forecasting

Overview of Box-Jenkins method

Complicate forecasting method, but high accuracy.

Developed by two statisticians, George E.P. Box and Gwilym M. Jenkins.

Published in Time Series Analysis: Forecasting and Control

(1970) The model is called ARMA (p,q) or ARIMA (p,d,q)


The Price of Stock A

Date Price (Yt)1 42.35212 43.12563 41.87164 39.41255 39.4589… …

Time series data analysis need >30 samples


ARMA (p,q) model

Stationary time series

ARIMA (p,d,q) model

Non-stationary time series

In this session we will focus on ARMA (p,q)


Identify Model to Be Tentatively Entertained

Estimate Parameters In Tentatively Entertained

Diagnostic Checking

Use Model for Forecasting


Stationary Time Series E(Yt), V(Yt) and Probability distribution

function of Yt is constant at different time period.

Consider

Yt, Yt-1, Yt-2, …, Yt+T-1

and Yt+j, Yt+j-1, Yt+j-2, …, Yt+j+T-1


Stationary Time Series Plot the time series data

if its movement has trend Non-stationary

Consider the correlogram of rk

(coefficient autocorrelation at lag K) and K

if rk die down when K is increasing Stationary

ARMA (p,q) Models

p is an order of Autoregressive Models (AR)

q is an order of Moving Average Models (MA)

Parameter in the model equal to p+q+1

(not more than 3)

AR(p) Models

Autoregressive model of order p

where

Yt = dependent variable at time t

= constant

= coefficient (autoregressive parameter)

= error term

tptp2t21t1t Y...YYY

AR(p) Models

AR(1)

or

and

AR(2)

or

and

t1t1t YY tt1 Y)B1( 11

t2t21t1t YYY

t2

21 )BB1( 1,1,1 21221

MA(q) Models

Moving average model of order q

where

Yt = dependent variable at time t

= constant

= coefficient (parameter of moving average)

= error term

qtq2t21t1tt ...Y

MA(q) Models

MA(1)

and

MA(2)

and

1t1ttY 11

1,1,1 21221

2t21t1ttY

ARMA(p,q) Models

Autoregressive and moving average model of

order p and q

ARMA (1,1)

and ,

tptp2t21t1t Y...YYY

qtq2t21t1t ...

11t1t1t YY

11 11

rk and rkk

Consider rk( Coefficient auto correlation) and rkk( Partial coefficient auto correlation)

to identify the appropiate ARMA (p,q) model AR(1) AR(2) MA(1) MA(2) ARMA(1,1)

Coefficient auto correlation: rk

The correlation of the same time series data but one with lagging K period

The variance of rk

n

1t

2t

Kn

1tKtt

k

)YY(

)YY)(YY(r

1k, n

1)r(V k ,...4,3,2k, )r2(1

n

1)r(V

1-k

1j

2

jk


Considert 1 2 3 4 5 6 7 8 9 10

Yt 8 5 10 7 12 14 9 15 16 4

Calculate r1,r2,r3

V(r1),V(r2),V(r3)


Hypothesis Testing

Test Statistics

Zcalc=

Accept H0 when

0:H

0:H

k1

k0

krn

n

1.96.1r

n

1.96.1 K

Partial coefficient auto correlation: rkk

The correlation of the same time series data but different period (e.g. Lag K) when assume the other different periods remain constant.

,...4,3,2K,rr1

rrrr 1K

1jjj,1K

1K

1jjKj,1Kk

kk

1K, ,rr 1kk

jK,1KKKj,1Kj,K r.rrr n

1)r(V KK and

Partial Coefficient auto correlation: rKK

Hypothesis Testing

Test Statistics

Zcalc=

Accept H0 when

0:H

0:H

KK1

KK0

krn

n

1.96.1r

n

1.96.1 KK

Identify Model with SPSS

Use the SPSS Graphs Time Series Autocorrelation Select the appropiate model from

correlogram.




Diagnostic Checking


Estimate Parameter with SPSS

Use the SPSS Analyze Time Series Arima

Or Calculate from the table




Diagnostic Checking


Diagnostic Testing

Hypothesis Testing

Test Statistics

Accept H0 when

0:H

0:H

1

0

1n,2

1n,2

ttt

ˆS

ˆt




Diagnostic Checking


9 box jenkins

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