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7/24/2019 ARIMA Model
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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
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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)
: / /
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S U D D H I P A R I T A D 103
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
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S U D D H I P A R I T A D104
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)
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S U D D H I P A R I T A D 105
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
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S U D D H I P A R I T A D106
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
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S U D D H I P A R I T A D 107
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
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S U D D H I P A R I T A D108
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
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S U D D H I P A R I T A D 109
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)
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S U D D H I P A R I T A D110
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
| 1 | < 1 Invertible or Stationary
. 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
| 1 | < 1 Invertible or Stationary
. 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)
| 1 | < 1 Invertible or Stationary
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
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S U D D H I P A R I T A D 111
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
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S U D D H I P A R I T A D112
(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)
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S U D D H I P A R I T A D 113
=
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
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S U D D H I P A R I T A D114
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
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S U D D H I P A R I T A D 115
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
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S U D D H I P A R I T A D116
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
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S U D D H I P A R I T A D 117
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
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S U D D H I P A R I T A D118
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
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S U D D H I P A R I T A D 119
. . , 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.
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S U D D H I P A R I T A D120
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