Box-Jenkins Methodology

BJ models use only current and past values of the time series to produce forecasts (no other independent variables)

Steps in Box-Jenkins Modeling

Prepare Raw Data

Identify Model

Estimate Parameters

Model Good ?

Forecast

Revise the Model

No

Data PreparationData has to be transformed to stationarity before applying BJ technique. Stationarity consists of three parts.

Stationary in Mean Fluctuates about a fixed level. Detectable by scatter plot and ACF. Usually enforced by differencing a suitable

number of times d. Mathematically, E(Yt) = .

Data Preparation

Stationary in Variance Fluctuation constant over time. Detectable by scatter plot. Usually enforced by taking loge or square root. Mathematically, Var(Yt) = 2.

Covariance Stationary Not detectable by scatter plot. Mathematically, for any k 0, Cov(Yt ,Yt-k) depends on k only.

Variance StabilizationYt = tt where t ~ iid N(5,1)

Before Taking Log

0

100

200

300

400

500

600

700

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

t

Y

(

t

)

After Taking Log

0

1

2

3

4

5

6

7

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

t

L

o

g

(

Y

(

t

)

)

Backshift Operator : B

kt t kB y = y

1 21 1t tB y = B B ... y( ) ( )

11 t t tB y = y y( ) 2

1 21 2t t t tB y = y y y( )

Autoregressive (AR) Models

Typical model :

General AR(p) model : (stationarity assumed)

where does not contain

the factor .

1 26 1 2 0 8t t ttY . Y . Y

0 1 1t t p t p tY Y ... Y 1 01

pp t t B ... B Y ( )

11p

p B ... B 1 B

Moving Average (MA) Models

Typical model :

General MA(q) model : (stationarity assumed)

1 20 8t t t tY .

0 1 1t t t q t qY ... 0 11

qt q tY B ... B ( )

Typical model :

General ARMA(p,q) model : (stationarity assumed)

where does not contain the factor

Autoregressive Moving Average (ARMA) Models

1 120 2 0 8 0 1t t t ttY Y Y . . .

1 1 0 1 1t t p t p t t q t qY Y ... Y ...

11p

p B ... B 1 0 11 1

p qp t q tB ... B Y B ... B ( ) ( )

1 B.

Autoregressive Integrated Moving Average (ARIMA) Models

1

0 1

1 11

p dp t

qq t

B ... B B Y B ... B ( ) ( )

( )

11

ppB ... B

1 .B

These are ARMA models fitted to data that need to be differenced to ensure stationarity in mean.

General ARIMA(p,d,q) model :

where does not contain the factor

ARIMA(p,d,q) Models

ARIMA(2,1,1) = ARMA(2,1) fitted to data differenced once

ARIMA(0,2,1) = MA(1) fitted to data differenced twice

ARIMA(1,0,1) = ARMA(1,1)

Model Identification

First transform data to stationarity by differencing suitable number of times, taking logs, etc

Choose those models (there may be more than one) with (1) the theoretical ACF most closely matches the sample ACF and (2) the theoretical PACF most closely matches the sample PACF

What is PACF ?

For given k, regress Yt against Yt-1,,Yt-k :

The lag-k partial autocorrelation coefficient (PAC) is the coefficient bk of Yt-k

It measures the strength of correlation between Yt-k and Yt when the effects of other time lags : 1, 2, ,(k-1) are removed

The collection of bk (k1) constitutes the PACF

0 1 1 1 1t k t k tk ktY a +a Y +...+ ba Y + Y

Plots for Yt = 0.7Yt-1+ t ; t ~ iid N(0,1)

-5.000

-4.000

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

5.000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Y(t)

-.6000

-.4000

-.2000

.0000

.2000

.4000

.6000

.8000

1.0000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ACF

Upper Limit

Low er Limit

-.4000

-.2000

.0000

.2000

.4000

.6000

.8000

1.0000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

PACF

Upper Limit

Low er Limit

Typical PACFs for AR Models

Y(t) = -0.7Y(t-1) + e(t)

-1-0.8-0.6-0.4-0.2

00.20.4

1 2 3 4 5 6 7 8 9 10 11 12

Y(t) = 0.5*Y(t-1) - 0.4*Y(t-2) + e(t)

-.4000

-.2000

.0000

.2000

.4000

.6000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Y(t) = -0.5Y(t-1) + 0.4Y(t-2) + 0.3Y(t-3) + e(t)

-.6000

-.4000

-.2000

.0000

.2000

.4000

.6000

1 2 3 4 5 6 7 8 9 10 11 12

Typical PACFs for MA Models

Y(t) = -0.7e(t-1) + e(t)

-.5000-.4000-.3000-.2000-.1000.0000.1000.2000

1 2 3 4 5 6 7 8 9 10 11 12

Y(t) = -0.9e(t-1) + 0.8e(t-2) + e(t)

-.8000

-.6000

-.4000

-.2000

.0000

.2000

.4000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Y(t) = -0.4e(t-1) + 0.5e(t-2) + 0.6e(t-3) + e(t)

-.6000

-.4000

-.2000

.0000

.2000

.4000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

AR(1) Model : Examples AR(2) Model : Examples

MA(1) Model : Examples MA(2) Model : Examples

ARMA(1,1) Model : Examples ARMA(1,1) Model : Examples

Guidelines for Model Identification

MODEL ACF PACF

AR(p) Decays rapidly Truncates after lag p

MA(q) Truncates after lag q Decays rapidly

ARMA(p,q) Decays rapidly Decays rapidly

In most cases, 0 p,d,q 2 and 0 p+q 2

Case : S&P Monthly Closing

S&P Monthly Closing : Differenced Once S&P Closing : One Step Ahead Forecast

1 t tB Y ( )1t ttY Y

The ARIMA(0,1,0) model is :

Forecast for t = 234 :

= 1482.37234

1t ttY Y

Case : Transportation Daily Closing Index Closing : Identify Model

Closing : ACF of Differenced Data Closing : PACF of Differenced Data

Closing : SPSS Closing : Choosing (p, d, q)

Closing : Error Measures Closing : Residual ACF

Closing : Saving Residuals Closing : Residuals Saved

Closing : Error Measures Closing : Parameter Estimates

Closing : ACF of Residuals Closing : Normality of Residuals

Closing : One Step Ahead Forecast

1 10 438 t tB B Y ( )( ).21 1 438 0 438 t tB B Y ( . . )

The ARIMA(1,1,0) model is :

Forecast for t = 66 :

= 1.438(288.57) 0.438(286.33) = 289.55 66

1 21 438 0 438t t- t- tY Y Y . .

Case : Paper Towel Weekly Sales

Towel : Identify Model Towel : Identify Model

Towel : Parameter Estimates (1) Towel : Residual ACF (1)

Towel : Residual Saved (1) Towel : Q-Q Plot (1)

Towel : Parameter Estimates (2) Towel : Residual ACF (2)

Towel : Q-Q Plot (2) Towel : Parameter Estimates (3)

Towel : One-Step Ahead Forecast

1 10 351t t- t- tY Y .

The ARIMA(0,1,1) model is :

Forecast for t = 121 := 15.65 + 0.351(0.69) = 15.89121

31 0 11 5t tB Y B .( ) ( )

Steps in Model Building Transform data to stationarity

Based on the ACF & PACF, determine the values of pand q

From the computer printout, determine whether ALLfitted parameter values are significant; if not re-fit using other values of p and/or q

Check whether the residuals appear random

If there are more than one tentative model, choose the best one by considering their error measures