Dasyam et al.

ofȱwheatȱduringȱ2012.ȱTheȱmajorȱwheatȱproducingȱ replacedȱ byȱ theȱ medianȱ ofȱ respectiveȱ seriesȱ (Sahuȱ

statesȱinȱIndiaȱareȱplacedȱinȱtheȱnorthernȱpartȱofȱtheȱ 2010).ȱTheȱstatisticȱZȱisȱcalculatedȱasȱabsoluteȱvalueȱ

countryȱ withȱ Uttarȱ Pradesh,ȱ Punjabȱ andȱ Haryanaȱ ofȱ differenceȱ betweenȱ theȱ observationȱ andȱ meanȱ

contributingȱ toȱ nearlyȱ 80%ȱ ofȱ theȱ totalȱ wheatȱ dividedȱbyȱtheȱStandardȱDeviationȱ(SD)ȱasȱshownȱinȱ

production.ȱWheatȱprovidesȱmoreȱnourishmentȱforȱ Eq.ȱ1:

humansȱthanȱanyȱotherȱfoodȱsource.ȱGlobally,ȱwheatȱ

isȱtheȱleadingȱsourceȱofȱvegetableȱproteinȱinȱhumanȱ

X − X i

foodȱ havingȱ higherȱ proteinȱ contentȱ thanȱ maizeȱ orȱ

Z =

SD X

ȱ

(1)

rice,ȱ theȱ otherȱ majorȱ cereals.ȱ So,ȱ properȱ forecastȱ isȱ

veryȱimportantȱinȱanȱeconomicȱsystemȱforȱsuchȱanȱ HereȱSD x ȱisȱtheȱStandardȱdeviationȱofȱvariableȱXȱandȱ

essentialȱcropȱasȱitȱwouldȱbeȱeasierȱtoȱformulateȱandȱ iȱindicatesȱtheȱnumberȱofȱobservation.ȱIfȱZȱisȱgreaterȱ

initiateȱ appropriateȱ policyȱ measuresȱ ifȱ dataȱ withȱ thanȱ1.96,ȱthenȱitȱisȱimpliedȱthatȱoutlierȱexistsȱinȱtheȱ

regardȱtoȱtheȱtrendȱofȱwheatȱproductionȱisȱobtainedȱ presentȱsample.

andȱanalyzedȱinȱadvance.ȱTimeȱseriesȱforecastingȱisȱ

anȱimportantȱstatisticalȱtechniqueȱusedȱasȱaȱbasisȱforȱ

ARIMA Model

manualȱandȱautomaticȱplanningȱinȱmanyȱapplicationȱ AccordingȱtoȱBoxȱandȱJenkinsȱ(1976),ȱaȱnonȱseasonalȱ

domainsȱ(GooijerȱandȱHyndmanȱ2006;ȱSonawaneȱ et

ARIMAȱmodelȱisȱdenotedȱbyȱARIMAȱ(p,d,q)ȱwhichȱ

al. 2013).ȱInȱthisȱpresentȱstudy,ȱtimeȱseriesȱmodelingȱ isȱ aȱ combinationȱ ofȱ Autoȱ Regressiveȱ (AR)ȱ andȱ

wasȱdoneȱforȱproductionȱofȱwheatȱinȱIndiaȱbyȱusingȱ MovingȱAverageȱ(MA)ȱwithȱanȱorderȱofȱintegrationȱ

parametricȱ regression,ȱ exponentialȱ smoothingȱ andȱ orȱdifferencingȱ(d),ȱwhereȱpȱandȱqȱareȱtheȱorderȱofȱ

ARIMAȱmodelsȱandȱoutȱofȱsampleȱaccuracyȱforȱeachȱ autocorrelationȱ andȱ movingȱ averageȱ respectivelyȱ

ofȱ theȱ modelsȱ hasȱ beenȱ computed.ȱAsȱ theȱARIMAȱ (Gujarati et al. 2012).

modelȱoutperformsȱotherȱmethodsȱforȱthisȱparticularȱ

TheȱAuto-regressiveȱ modelȱ ofȱ orderȱ pȱ denotedȱ byȱ

dataset,ȱtheȱfinalȱforecastingȱofȱwheatȱproductionȱforȱ

AR(p)ȱisȱasȱfollows:

theȱyearȱ2014-15ȱtillȱ2017-18ȱhasȱbeenȱestimatedȱbyȱ

usingȱthisȱapproach.

Z _tȱ =ȱcȱ+ȱØ ₁ ȱZ t-1 ȱ+ȱØ 2 ȱZ t-2 ȱ+ȱ…ȱ+ȱØ Z +ȱe tȱ

pȱ t-pȱ

(2)

Materials and Methods

whereȱcȱisȱconstantȱterm,ȱ Ø _p ȱisȱtheȱp-thȱautoregressiveȱ

parameterȱandȱe _t ȱisȱtheȱerrorȱtermȱatȱtimeȱt.

DataȱwithȱrespectȱtoȱwheatȱproductionȱofȱIndiaȱforȱ

periodȱofȱ1961-2013ȱwasȱcollectedȱfromȱDirectorateȱofȱ

TheȱgeneralȱMovingȱAverageȱ(MA)ȱmodelȱofȱorderȱqȱ

EconomicsȱandȱStatistics,ȱDepartmentȱofȱAgricultureȱ

orȱMA(q)ȱcanȱbeȱwrittenȱas:

andȱCooperation,ȱGovernmentȱofȱIndia.ȱInȱthis,ȱlastȱ Z _tȱ =ȱcȱ+ȱe _tȱ –ȱθ ȱe ȱ–ȱθ e ȱ–ȱ……–ȱθ e

1 t-1

2ȱ t-2

qȱ t-qȱ

(3)

threeȱyearsȱdataȱwereȱusedȱforȱmodelȱvalidationȱandȱ

whereȱ cȱ isȱ constantȱ term,ȱ θ ȱ isȱ theȱ q-thȱ movingȱ

q

remainingȱforȱmodelȱbuilding.ȱBeforeȱanalysis,ȱasȱtheȱ

averageȱparameterȱandȱe ȱisȱtheȱerrorȱtermȱatȱtimeȱ

t-k

studyȱ isȱ dealingȱ withȱ timeȱ series,ȱ presentȱ dataȱ setȱ

t-k.

haveȱbeenȱverifiedȱinitiallyȱforȱexistenceȱofȱoutlier.

ARIMAȱinȱgeneralȱformȱisȱasȱfollows:

Test for Outlier

Δ d Z _t = c + ( Ø ₁ Δ d Z t-1 + … + Ø p Δ d Z t-p ) – ( θ 1 e t-1 + … + θ q

Forȱdetectingȱtheȱoutlierȱinȱtheȱtimeȱseries,ȱGrubbsȱ

e ) + e

t-q

t

(4)

testȱ wasȱ usedȱ inȱ theȱ currentȱ scenarioȱ asȱ theȱ testȱ isȱ

particularlyȱusefulȱinȱcaseȱofȱlargeȱsampleȱandȱeasyȱ

where Δ denotes difference operator like

toȱfollow.ȱGraphȱpadȱsoftwareȱwhichȱisȱwidelyȱused,ȱ

hasȱbeenȱemployedȱinȱtheȱpresentȱstudyȱtoȱidentifyȱ Δ Z = Z - Z

t

t

t-1

(5)

theȱ existenceȱ ofȱ outliersȱ andȱ ifȱ found,ȱ haveȱ beenȱ

Δ 2 Z _t-1 = ΔZ _t - ΔZ t-1

(6)

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