1

Chap 9 Box-Jenkins Models

• Box-Jenkins 模式用於描述 stationary 序列• Stationary series (平穩序列 )

定義： The statistical properties of the time series are constant

through times.

E(Yt) =μ ， var(Yt) =σ2 ， cor(Yt ,Yt+k) =ρk for all t

• 如果手中的時序資料不是 stationary, 必須將它轉為stationary

• 如何轉換？

2

Stationary series

Nonstationary series

3

Exp 9.1 The company would like to develop a prediction model that can be used to give prediction interval forecasts of weekly sales of Asorbent Paper Towels. For the past 120 weeks the company has recorded weekly sales of Absorbent Paper Towels.

t y 1stDiff

1 15

214.406

4

-0.593

6

314.938

30.531

9

416.037

41.099

1

5 15.632

-0.405

4

614.397

5

-1.234

5

713.895

9

-0.501

6

814.076

50.180

6

9 16.3752.298

5

1016.534

20.159

2

First Differences Zt = Yt – Yt-1

The original series is not a stationary series

Y(t)

-5

0

5

10

15

20

0 20 40 60 80 100 120

4

Zt = Y(t) -Y(t-1)

-4

-3

-2

-1

0

1

2

3

4

0 120

First Differences series becomes a stationary series

2ndDiff

-4

-3

-2

-1

0

1

2

3

4

5

0 120

Second Differences series is still a stationary series

5

圖形觀察：原資料圖、差方資料圖檢定法：

如何檢測 stationarity?( 平穩性 )

Dickey-Fuller test

Phillips-Perron test

Random-walk with drift test

6

1. Backward 運算： B(Yt) = Yt-1, B2(Yt) = Yt-1

2. First difference 一階差分 :

3. Second differences 二階差分 :

1 ttt YYY

差分運算

2112 2)( tttttt YYYYYY

ttt YBBYBY )21()1( 222

tttt YBYYY )1( .4 1

5. Difference with lag k ： tk

ktt YBYY )1(

7

差分功能

一階差分消去直線 trend

二階差方消去二次 trend

4 ttt YYY

12 ttt YYY

消除季節因素

四季節差分

月季節差分

8

Fig 9.1 nonstationary series

First difference

Second difference

9

9.2 The autocorrelation and partial autocorrelation function

• autocorrelation at lag k : cor(Yt ,Yt+k) =ρk • Sample autocorrelation at lag k, rk

n

t

kn

tktt

k

YY

YYYYr

1

2

1

)(

))((

• ACF : autocorrelation function, 由 rk , k= 0,1,2,….. 組成的函數

• Standard error of rk :

2,3,....k if )(

21

1k if

2/1

2/1

1

2

)(1

2/1

n

rsk

jj

n

rk

10

Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error

0 19.162294 1.00000 |                    |********************| 0

1 18.445606 0.96260 |                .   |******************* | 0.091287

2 17.388503 0.90743 |              .     |******************  | 0.154197

3 16.349929 0.85323 |            .       |*****************   | 0.193651

4 15.343692 0.80072 |           .        |****************    | 0.222787

5 14.232902 0.74276 |          .         |***************     | 0.245601

6 13.116331 0.68449 |         .          |**************      | 0.263656

7 12.028851 0.62774 |         .          |*************       | 0.278071

8 11.088860 0.57868 |        .           |************        | 0.289639

9 10.185709 0.53155 |        .           |***********.        | 0.299119

10 9.493686 0.49544 |        .           |********** .        | 0.306890

11 8.977998 0.46852 |       .            |*********   .       | 0.313484

12 8.517382 0.44449 |       .            |*********   .       | 0.319266

13 7.970955 0.41597 |       .            |********    .       | 0.324382

14 7.347767 0.38345 |       .            |********    .       | 0.328797

15 6.760440 0.35280 |       .            |*******     .       | 0.332503

16 6.188561 0.32296 |       .            |******      .       | 0.335608

17 5.566404 0.29049 |      .             |******       .      | 0.338187

18 4.803283 0.25066 |      .             |*****        .      | 0.340260

19 3.882712 0.20262 |      .             |****         .      | 0.341796

20 2.961125 0.15453 |      .             |***          .      | 0.342795

21 2.144619 0.11192 |      .             |**           .      | 0.343375

22 1.389010 0.07249 |      .             |*            .      | 0.343679

"." marks two standard errors

ACF for Exp9.1

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Autocorrelation Check for White Noise

To Lag

Chi-Square

DF Pr > ChiSq

Autocorrelations

6 518.57 6 <.0001 0.963 0.907 0.853 0.801 0.743 0.684

12 739.59 12 <.0001 0.628 0.579 0.532 0.495 0.469 0.444

18 836.62 18 <.0001 0.416 0.383 0.353 0.323 0.290 0.251

24 848.87 24 <.0001 0.203 0.155 0.112 0.072 0.033 0.002

Test H0 : ρj = 0, j=1,2, … k

註： White noise (純雜訊 ) 是一獨立常態分佈的序列 εt ~ NID(0, σ2) , then εt is a white noise

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Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error

0 1.208715 1.00000 |                    |********************| 0

1 0.370658 0.30665 |                .   |******              | 0.091670

2 -0.078249 -.06474 |                .  *|   .                | 0.099919

3 -0.086619 -.07166 |                .  *|   .                | 0.100271

4 0.126391 0.10457 |                .   |** .                | 0.100700

5 0.101691 0.08413 |                .   |** .                | 0.101609

6 0.027608 0.02284 |                .   |   .                | 0.102192

7 -0.160292 -.13261 |                .***|   .                | 0.102235

8 -0.143891 -.11904 |                . **|   .                | 0.103671

9 -0.210121 -.17384 |                .***|   .                | 0.104813

10 -0.142910 -.11823 |                . **|   .                | 0.107209

11 -0.062396 -.05162 |                .  *|   .                | 0.108299

12 0.025252 0.02089 |                .   |   .                | 0.108505

13 0.049984 0.04135 |                .   |*  .                | 0.108539

14 0.023417 0.01937 |                .   |   .                | 0.108672

15 -0.073248 -.06060 |                .  *|   .                | 0.108701

16 -0.0029263 -.00242 |                .   |   .                | 0.108984

17 0.154399 0.12774 |                .   |***.                | 0.108985

18 0.259741 0.21489 |                .   |****                | 0.110236

19 0.067449 0.05580 |               .    |*   .               | 0.113701

20 -0.054839 -.04537 |               .   *|    .               | 0.113931

21 -0.084327 -.06977 |               .   *|    .               | 0.114083

ACF for Exp9.1with 一次差分

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Autocorrelation Check for White Noise

To Lag Chi-Square DF Pr > ChiSq Autocorrelations

6 14.96 6 0.0206 0.307 -0.065 -0.072 0.105 0.084 0.023

12 25.27 12 0.0136 -0.133 -0.119 -0.174 -0.118 -0.052 0.021

18 34.95 18 0.0096 0.041 0.019 -0.061 -0.002 0.128 0.215

24 37.22 24 0.0416 0.056 -0.045 -0.070 -0.035 -0.052 -0.038

Test H0 : ρj = 0, j=1,2, … k

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In general, for nonseasonal data

1. If the ACF either cuts off fairly quickly or dies

down fairly quickly, then the time series

shoud be considered stationary.

2. If the ACF dies down extremely slowly, then

the time series should be considered

nonstationary.

以 ACF 判斷平穩性

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• Sample partial autocorrelation at lag k is

2,3,...k if 1

1k if

1

1,1

1

1,1

1

k

jjk

k

jjkjkk

kk

rr

rrr

r

r

• PACF : partial autocorrelation function, 由 rkk , k= 0,1,2,….. 組成的函數

• Standard error of rkk : 2/1)(1nrkks

•ACF 及 PACF 是辨識 Box-Jenkins 模式的重要工具