Time Series Modeling for Trend Analysis and Forecasting Wheat Production of India

Table 4. Result of ADF test

Critical values at

Test

ADF statistic

Prob.

Decision

1%

5%

10%

ADF at level

-2.978

-4.161

-3.506

-3.183

0.148

Data Non-Stationary

ADF at first difference

-11.016

-4.161

-3.506

-3.183

0.0001

Data Stationary

0.4477ȱZ _t-1ȱ +ȱe _t

FromȱtheȱresidualȱACFȱandȱPACFȱplotsȱofȱARIMAȱ

(1,1,0),ȱ itȱ wasȱ clearȱ thatȱ allȱ autocorrelationsȱ andȱ

partialȱ autocorrelationsȱ lieȱ betweenȱ 95%ȱ controlȱ

limitsȱasȱshownȱinȱFigureȱ4.ȱThisȱalsoȱconfirmedȱtheȱ

‘goodȱfit’ȱofȱthisȱselectedȱmodel.

Figure 3 Correlogram of ACF and PACF for first differenced

wheat production

Inȱthisȱpresentȱwork,ȱpossibleȱARIMAȱ(p,d,q)ȱmodelsȱ

suchȱasȱ(1,1,1),ȱ(0,1,1)ȱandȱ(1,1,0)ȱwereȱcomparedȱtoȱ

eachȱ other.ȱ Amongȱ allȱ possibleȱ models,ȱ ARIMAȱ

(1,1,0)ȱwasȱselectedȱasȱoptimalȱandȱmostȱappropriateȱ

modelȱ dueȱ toȱ modelȱ selectionȱ criteriaȱ suchȱ asȱ

minimumȱvaluesȱofȱRMSE,ȱMAPE,ȱMAE,ȱMSE,ȱAIC,ȱ

Figure 4 Residual ACF and PACF of ARIMA (1,1,0)

SBCȱandȱhighȱR-squaredȱvalueȱ(Tableȱ5).

Forȱ checkingȱ normalityȱ andȱ randomness,ȱ Shapiro-

Table 5. ARIMA Model Fit statistics

Wilkȱ andȱ Runȱ testsȱ wereȱ appliedȱ respectivelyȱ toȱ

residualsȱofȱARIMA(1,1,0)ȱandȱresultsȱwereȱpresentedȱ

Model

R-squared

RMSE

MAPE

MAE

MSE

AIC

SBC

inȱ Tableȱ 7.ȱ Theȱ probabilityȱ valuesȱ forȱ theȱ bothȱ theȱ

(1,1,1)

0.980

3.141

7.866

2.585

9.861

118.027 123.702

testsȱwereȱgreaterȱthanȱ0.05ȱindicatingȱresidualsȱwereȱ

(0,1,1)

0.978

3.172

8.002

2.631 10.049 117.069 120.852

distributedȱnormallyȱandȱindependently.ȱHistogramȱ

(1,1,0)

0.981

3.136

7.791

2.576

9.838

116.145

119.928

ofȱ residualsȱ isȱ depictedȱ inȱ Figureȱ 5ȱ whichȱ furtherȱ

confirmedȱtheȱnormalityȱforȱtheȱresiduals.

Itȱ wasȱ foundȱ thatȱARIMAȱ modelȱ performedȱ betterȱ

thanȱtheȱearlierȱselectedȱmodelsȱviz.ȱQuadraticȱandȱ

Table 7. Tests of Normality and Randomness of residuals

Holt.ȱ Theȱ parametersȱ wereȱ estimatedȱ forȱ theȱ bestȱ

Shapiro-Wilk

Run test

selectedȱmodelȱi.e.,ȱARIMAȱ(1,1,0)ȱasȱmentionedȱinȱ

Statistic df

Sig. Z-value No of Runs Sig.

Tableȱ6.

Residuals 0.984

49 0.741 -0.801

22

0.423

Table 6. ARIMA (1,1,0) Model Parameters estimation

Finally,ȱforecastingȱwasȱdoneȱforȱwheatȱproductionȱ

Model Parameter Estimate Std. Error

t

Sig.

ofȱIndiaȱfromȱ2011-12ȱtillȱ2017-18ȱbyȱusingȱARIMAȱ

Intercept

1.46373

0.3177

4.6074

0.0001

(1,1,0)ȱ withȱ keepingȱ firstȱ threeȱ yearsȱ dataȱ forȱ

Autoregressive,

-0.44770

0.1295

-3.4562 0.0012

Lag 1

validation.ȱPredictedȱvaluesȱwithȱ95%ȱUpperȱcontrolȱ

limitsȱ (UCL)ȱ andȱ Lowerȱ controlȱ limitsȱ (LCL)ȱ wereȱ

Fromȱ Tableȱ 6,ȱ equationȱ ofȱ theȱ ARIMAȱ modelȱ wasȱ presentedȱinȱTableȱ8.ȱ

formulatedȱ as:ȱ Wheatȱ production _t ȱ (Z _t ) =ȱ 1.4637ȱ –ȱ

307