การประยุกต์ใช้ arima model เพื่อการวิจัย

7/24/2019 ARIMA Model

1/20

S U D D H I P A R I T A D 101

ARIMA Model

Application of ARIMA Model for Research

*Jindamas Sutthichaimethee

**Lecturer Plan and Policy Analyst, Ministry of Science and Technology


2/20

S U D D H I P A R I T A D102

The Best Model ARIMAModel (Structure Variable) ARIMAX Model Statistics Model

Non Stationary integration Error Correction Mechanism (ECM)

(The Best Model)

: / /


3/20


Abstract

This article is intended to create The Best Model ARIMAapplies to the variable structure called ARIMAX Model. Stasteps and can take the model used for forecasting the maximum

For information on the system economy, most will lo Stationary, so researchers need to be updated to look as Statiothe data is Co-integration parties is essential to introduce the EMechanism assembly in that model and The Best Model to creestimating the correct and appropriate for that type of infor

result in the forecast errors are low and can be used to accuratKeyword : Structure Variable / Time Series Data / The Best


4/20


1. (Model)

Box-Jenkins

(Model) BoxJenkins George E.P. Box Gwilym M.Jenkins . . 1970

ARIMA Model . . 1994

BoxJenkins (Time Series Data)

Stochastic Process Stationary Time Series

Nonstationary Time Series

(Stationary)

Stationary

(Actual Value)

2. ARIMAModel

BoxJenkins

ARIMA Model

(1) Stationary (2) Cointegration Error Correction Mechanism (ECM)

1.Stationary Y tStochastic Variable Time Series

Stationary 3

Mean : E(Y t ) = E(Y t-k ) =

Variance :Var (Y

t ) = E(Y

t - ) 2 = E(Y

t -k - ) 2= 2

Covariance :E [ (Y t - )

2 (Y t -k - )2] = k

Covariance Y t (Time)

3 StationaryStationary Stochastic Process

Stationary (Mean or

Value) (Va (Covariance)

(Constant Over Time)

(Distance or Lag)


5/20


Nonstationary Nonstationary

Mean : E(Y t ) = E(Y t-k ) = t

Variance :Var (Y t ) = E(Y t - ) 2 = E(Y t -k - ) 2= t 2

Covariance : E [ (Y t - )

2 (Y t -k - )2] = t k

Nonstationary Stochastic Process

1 Stationary

Nonstationary

Random Walk

2 Nonstationary


6/20


Stationary

Dickey Fuller (DF), Augmented Dickeyand Fuller (ADF) Nonstationary

Unit Root

Nonstationary Unit Root RegressionModel Ordinary Least Square (OLS)

(Signi cance) Spurious

Regression ( , 2549) Non

stationary Stationary WeakStationary First Moment Second Moment Strictly Stationary

Moment Moment Moment

2 Stationary

Nonstationary

Observation (Shock)

Stationary

Nonstationary

Model

Nonstationary (Long Run Mean Level)

, 2544) Stationar

Dickey Fuller (DF) AugmentedDickey Fuller (ADF)

Dickey Fuller TestAugmented Dickey Fuller Test Unit

Root Test DickeyFuller Nonstationary

(Difference Regression) First Order Autoregressive

Process 3

. Y t = Y t-1 + t (Random WalkProcess Pure Random Walk)

. Y t = 1+Y t-1 + t (Random Walkwith Drift Intercept)

. Y t = 1+ 2T + Y t-1 + t (RandomWalk with Drift Linear Time Trend Drift Term T T )

Y t = =

(Coef cient of Lagged) t = Error Term t , Mean = 0,

Variance = 2 (Hypothesis) Unit R

Test

H 0 : = 0, Nonstationary H : < 1, Stationary


7/20


Y t Nonstationary Accept H 0 = 0

Exponential Explosive

. Y t = Y t-1 + t (1) (1)

(Mean)

(Drift Term) . Y t = 1+Y t-1 + t (2)

1 = (Drift Term) (2) Unit Root Test

Trend Stationary (TS) Difference Stationary (DS)

. Y t = 1+ 2T + Y t-1 + t

T = (Time Trend)

2 = t Stationary

0 2 t ~ IID,(0, 2)

(Time Series) First Difference

Stationary

Difference Stationary

Y t =Y t-1

Y t = 1+ 2T + Y t - 1 + t (4) (4)

Level H Accept H 0 Nonsta-tionary = 0

Tau Absolut DF Critical Absolute Term

t White Noise Autocorrelation

Augmented Dickey Fuller (AD Goodness of Fit

Dickey (DF)

(Lagged) (Dependent Variable) Autocorrelation

(Hypothesis) Unit Test

H 0 : = 0, Nonstationary H : < 1, Stationary

Level Reject H 0 Accept H Stationary 0

Tau Statistics Absolute Term

ADF Critical Absolute Term t White Noise

Stationary Y tIntegrated d


8/20


Y t ~ I (d)

Y t = (5) Y t = (6)

Y t = (7)

p =

(Lagged Values of First Difference of theVariable) (5), (6)

(7) Aug-

mented Dickey Fuller (6)

Y t =

DF ADF

ADF Error Term White Noise Error Term Mean

2. Cointegration Eagle and Granger Cointegration (Time Series) 2

(Steady State)

Stationary

Engle Granger Cointegration

(Error) Cointegra

Regression (Hypothesis) H 0

Stationary (Linear Combination)

Cointegration DickeyFuller (DF)

Augmented DickeyFuller (ADF) Cointegration 1

Integrated (Dependent Variable :Y t ) (Independent Variable : X t)

Unit Root Test Integrated

Cointegratio Integrated

2 CointegratingParameter (Error Term) OrdinaryLeast Squares (OLS)

u t =Y t - - X t (8) 3 u t

Station-ary u t

(Line-

tion) White Noise A

/=pi2


9/20


Fuller (ADF) Autocorrelation

Reject H 0 Accept H Tau Test (Absolute)

Tau Critical MacKinnon u t

Stationary Unit Root Y t X t

(Cointegration) Reject H Accept H 0 u t Nonstationary Unit Root

Y t X t (NonCointegration)

3. Error Correction Mechanisms (ECM)

Cointegration

(Short RunDynamic Adjustment)

(Model) (Macro Model)

ECM ECM

ECM Model Co integration

Stationary Cointegration ECM

Y t = (9)

(9) (ECM Model)

(Error Team :u t - i )

Model

Y t X t ECM Model () Y t

() Y t

3. ARIMA Model AR BoxJenkins

4 (1) (Iden-ti cation) (2) (Pa-rameter Estimator) (3) (Diagnostic Checking) (4) (Forecast)

1. (Identi cation)

Box Jenkins Stationary invertible

. Autoregressive ModelOrder p AR(p) Y t = + 1Y t - 1 + 1Y t - 2 + ... + pY t - p + t (10)

(10)


10/20


l AR (1) Y t = + 1Y t - 1 + t (11)

| 1 | < 1

Stationaryl AR (2) Y t = + 1Y t - 1 + 2Y t - 2 + t (12)

1 2 2 - 1 < 1 Stationary

q(Moving Average Model of Order q) MA(q) Y t = + t - 1 t - 1- 2 t - 2- ... - 2 t - 1

(13)

(13) l MA(1)

Y t = + t - 1 t - 1 (14)

| 1 | < 1 Invertible or Stationary

l MA(2) Y t = + t - 1 t - 1 - 2 t - 2 (15)

1 + 2 < 1, 2 + 1< 1


. Autoregressive p q (Mixed

Autoregressive and Moving AveraModel of Order p and q) ARMA (p, q) Y t = + 1Y t - 1 + 2Y t - 2 + ... + pY t - p + t

- 1 t - 1 - 2 t - 2 - ... - q t - ql ARMA(1, 1)

Y t = + 1Y t - 1 + t (16)

| 1 | < 1


. Integrated Autoregressive (Autoregressive Integ

Moving Average) ARIMA(p, d (Different Term)

l ARIMA(0,1,1) IMA(1,

Y t - Y t - 1 = + t - 1 t - 1 (17)


l ARIMA(1,1,0) ARI(1,1

Y t - Y t - 1 - 1 (Y t - 1 + Y t - 1 ) = + t (18)

| 1 | < 1 Stationary


11/20


l ARIMA(1,1,1) Y t - Y t - 1 - 1 (Y t - 1 + Y t - 1 ) = + t - 1 t - 1

(19)

| 1 | < 1,| 1 | < 1 Invertible or

Stationaryl ARIMA(0,1,0) Y t - Y t - 1 = t (20)

. Integrated Autoregressive (SeasonalAutoregressive Integrated Moving Average)

SARIMA(p, d, q)L d L

Y t - Y t - 12 = t - * t - 12

| 1 | < 1Y t - Y t - 12 = 12

* = (Parameter) (Seasonal Mov-

ing Average Model) 2. (Param-

eter Estimation) (Parameter Estimation) 1

(Or-dinary Least Square : OLS)

3. (DiChecking)

2

.

0

tstatistic H 0 : = 0 H : 0

t = / S (21)

= S =

. Box PierceSquare Test (Q ) Box Pierce

H 0 : 1 (e t ) = ... = k (e t ) = 0 Box Pierce Chi Square

(Q) t e t , t = 1, 2,, n

e t

Q =( n - d ) r j 2 (e t ) n = d =

Stationary r j

2 (e t ) = j


12/20


(22) Q ChiSquare

(Degree ofFreedom) k - n p Q

Q

4. (Forecast)

(Point Forecast) (Interval Forecast)

4. ARIMA Model ARIMA Model Statistics

Model

Structure Variables ARIMAX Model

ARIMAX Model

ARIMA Model

ARIMAX Model 1

1

2538-2547 ARIMA

Model 1-4 2548

- Autoregressive Integrated M

Average X (ARIMAX) ARIMA

ARIMA

3 Stationary

(Determine Order of Integration

Cointegra-tion

ARIMAX

( )

2 / 2,(k - np)


13/20


=

t = t - i

= t - i

= t - i

= t - i

ECM = Error Correction MechanismMA(i) = Moving Average

i = (GDP)

t - it = t

= (First Difference)

=

( )

=

t

= t - i= t - i =

t - i=

t ECM = Error Correction MechanismMA(i) = Moving Average i

= (GDP) t - i

t = t = (First Difference)

=

( )

= t

= t - i

= t - i =

t - iECM = Error Correction MechaMA(i) = Moving Average

i = (GDP) t - i

t = t = (First Difference)

3

Stationary


14/20


9 ( ),

( ), ( ), ( ),

( ), ( ), ( ), ( E t) (GDP)

( I t ) ARIMAX Sta-

Lag ADF TestMacKinnon Critical Value

Status1% 5% 10%

1 -3.2138 -4.2191 -3.5331 -3.1983 I(0

1 -2.4634 -4.2191 -3.5331 -3.1983 I(0

1 -1.3101 -4.2191 -3.5331 -3.1983 I(0

1 -3.2676 -4.2191 -3.5331 -3.1983 I(0

1 -2.6385 -4.2191 -3.5331 -3.1983 I(0

1 -1.4694 -4.2191 -3.5331 -3.1983 I(0

1 -1.7578 -4.2191 -3.5331 -3.1983 I(0

1 -1.9339 -4.2191 -3.5331 -3.1983 I(0

1 -8.9689 4.2191 -3.5331 -3.1983 I(0

tionary Unit Root Test Augmented Dickey Fuller Test (ADF)

Non stationary Unit Root Difference Stationary

1 URoot (At Level)

: Logarithm


15/20


I t Trend Stationary( I t )

1 ADF TestStatistic (Level) Nonstationary

ADF (Critical) 1%

5% Box Jenkins

Nonstationary Stationary (Differencing)

FirsDifferencing Two

Unit Root Unit Root First Differenci

Lag ADF TestMacKinnon Critical Value

Status1% 5% 10%

1 - 6.2169 - 4.2268 - 3.5366 - 3.2003 I(

1 - 6.0058 - 4.2268 - 3.5366 - 3.2003 I(

1 - 4.3705 - 4.2268 - 3.5366 - 3.2003 I(1 - 5.2999 - 4.2268 - 3.5366 - 3.2003 I(

1 - 6.6846 - 4.2268 - 3.5366 - 3.2003 I(

1 - 4.8247 - 4.2268 - 3.5366 - 3.2003 I(

1 - 4.6358 - 4.2268 - 3.5366 - 3.2003 I(

1 - 3.4325 - 4.2268 - 3.5366 - 3.2003 I(

2 Unit Root (At First Difference)

: Logarithm


16/20


2 Stationary (Unit Root Test)

(At First Difference) ADF TStatistic

(MacKinnon Critical Value) Stationary

1%, 5% 10% Differencing

ARIMAX

Model

(Cointegration Test) Stationary

Cointegration

integration

Cointegration

Stationary Integrated (I(d))

Cointegration AD Statistic) (MacKinnon

Critical Value) 3

1%, 5% 10%

Residual Stationary

-

Error Correction Mechanism

integration 3 Cointegration Engle Granger

ADF Test StatisticMacKinnon Critical Value

Status1% 5% 10%

Residual x -3.3441 - 2.6272 -1.9499 -1.6115 I(

Residual y -3.5094 - 2.6272 -1.9499 -1.6115 I(0

Residual z -8.2431 - 2.6272 -1.9499 -1.6115 I(0 : Residual x = Residual

Residual y = Residual Residual z = Residual


17/20


3. ARIMAX ( ), ( )

( ) = 0.86181 + 0.32296

0.95722 +(23.12164)***(5.08672) ***(42.50558) ***1.52574 +0.60955 0.12175+ (23.76910)***(8.60323) ***

(2.57628)** 0.93264 +0.78845 0.68147 + (3.88292)***(2.73838)***

(0.68148)***0.000134(9.10267)*** ( )

R2 = 0.811650Adjust R2 = 0.741019LM Statistic = 7.53944ARCH Test = 0.123382Ramsey RESET Test = 0.001763Jarque Bera = 0.341645

: tstatistic*** 1%** 5%* 10%

= 0.34076 0.99002 1.28113 +(1.47138)

(595634.7) ***(3.65073) *** 1.04543 + 1.30037

0.86857 + (3.57268)*** (1.79248)*

(3.48704) ***0.00003(2.91406)*** ( )

R2 = 0.342569Adjust R2 = 0.236531

LM Statistic = 5.346580ARCH Test = 0.551461Ramsey RESET Test = 0.444123Jarque Bera = 0.379525

= 0.78347 0.88538+1.03606 +(9.25121)*

(20.2410) ***(2.10172)** 1.21828 0.30323+ 0.00009 (2.12692)**(3.33231)*** (4.75533)**

R2 = 0.810356Adjust R2 = 0.776491LM Statistic = 2.363755ARCH Test = 0.709357Ramsey RESET Test = 0.171932Jarque Bera = 0.747478

The Best Model

1 - 4 2548

Root Mean Square Forecast Error 4


18/20


4 The Best Model

Root Mean Square Forecast Error 1

ARIMA Model Correlogram

5. The Best

Model

Box-Jinkins (A

ARIMA Model Model The BestModel

4 1 - 4 2548 The Best Model

. .

2548

1 52,011 55,413 48,040 52,000 54,955 49,0

2 58,324 53,981 39,281 59,945 54,080 40,

3 53,215 45,008 25,423 55,084 44,978 23,

4 50,453 47,121 35,441 49,897 46,015 32,4

Root Mean Square Forecast Error 0.05 0.02 0.01


19/20


. . , 2549 . . 1, : , 2553.

. . 1, : , 2553.

. . 1, : , 2553.

. (2553) . .Anderson, O.D.,Time Series Analysis and Forecasting The Box

Approach, Butterworths, London, 1975.Box, George and D. Piece. Distribution of Autocorrelations

Moving Average Time Series Models. Journal of the AStatistical Association 65 (1970), 1509-26

Dickey, D. A.,Likelihood Ratio Statistics for Autoregressive Tima Unit Root, Econometric (March 1987), 1981, 251-76

Dickey, and W.A. Fuller (1979),Distribution of the Estima Regressive Time Series with a Unit Root, journal of Am Association, 74 , pp.427-431.

Drapper, N.R, and Smith, H.,Applied Regression Analysis, 2nd Edition,John Wiley & Sons, New York,1981.

Granger, Clive and P. Newbold.Spurious Regressions in Econometric Journal of Econometrics 2 1974, 111-20.

Granger, E.S., JR. and Mckenzie ED.Forecasting Trends in Time Series

Management Science Vol 31 . 10 (October 1985) : 123Johansen, S. and K. Juselius , Maximum Likelihood Estimaon Co-integration: With Applications to the Demand foOxford Bulletin of Economics and Statistics 52 (Februa, 1990.

Kolb, R.A. and Stekler, H.O.,Are Economic Forecasts Signi cantly BThan Nave Predictions ? An Appropriate Test, International Jouof Forecasting, Vol.9, 1993, pp. 117 120.

Makridakis, S.,The accuracy of major extrapolation (time series) m

J. of Forecasting., 1: 1982, 111 153.


20/20


Montgomery, D.C., Johnson, L.A. and Gardiner, J.S.,Forecasting and TimeSeries Analysis, 2nd Edition, McGraw Hill Inc., New York, 19

Nelson, C.R.,Applied Time Series Analysis for Managerial Foreca Holden Day, San Francisco, 1973Newbold, P. and Granger, C.W.J.,Experience with Forecasting Univaria

Time Series and the Combination of Forecast, Journal of RoyalStatistical Society A,Vol,137, 1974, pp.131 146

Willeam W.S. Wei.Time Series Analysis : Univariate and MultivariatNew York USA: Addison Wesley Publishing company

การประยุกต์ใช้ arima model เพื่อการวิจัย

Documents