(Structural Time Series Models for Describing Trend in All India Sunflower Yield Using SAS

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1 (Srucural Time Series Models for Describing Trend in All India Sunflower Yield Using SAS Himadri Ghosh, Prajneshu and Savia Wadhwa I.A.S.R.I., Library Avenue, New Delhi Sunflower occupies fourh posiion among oilseed crops nex o groundnu, rapeseed-musard and soybean boh in erms of area and producion in he counry. This crop has conribued subsanially o he oal oilseed kiy due o developmen of high yielding hybrids and improved crop managemen pracices. For modeling ime series daa colleced over ime, ARIMA mehodology is generally employed. One disadvanage of his mehodology is ha he ime series under consideraion should be saionary or should be capable of becoming so by means of differencing or derending. Anoher alernaive approach ha does no make use of saionariy assumpion is he srucural ime series modeling. In his approach various componens of ime series, viz. rend, cyclical flucuaions, and seasonal variaions are included in he model depending upon heir presence in he daa under consideraion. Thus he idenified model is likely o provide more sensible predicions. Here esimaion of parameers is done by firs puing he model in sae space form and hen applying Kalman filer. Srucural ime series models can be uilized o model daa from various fields of Indian agriculure. Our aim is o highligh he imporance of his echnique for modeling all India sunflower yield daa using Proc ucm (Procedure Unobserved Componen Models) available in SAS. Model Idenificaion A Srucural ime series model is se up in erms of is various componens, like rend, cyclical flucuaions, and seasonal variaions, i.e. Y = T +C + S +, (1) where Y is he observed ime series a ime, T, C, S, are he rend, cyclical, seasonal and irregular componens. As menioned above a Srucural μime series model is ψobained by using various ime series componens, like rend (), cyclical flucuaions, seasonal νε)variaion () and irregular erm(), i.e. Y=μ+ψ+ν,T, () +ε=all four componens are sochasic and disurbances driving hem are muually uncorrelaed. Since sunflower yield daa are published on an annual basis, i is no possible o include seasonaliy componen in he model. Furher, a graphical represenaion of he daa does no indicae presence of any cyclical paern. Thus, we shall confine our aenion o ha class of Srucural ime series models in which only rend and irregular flucuaions are presen. (i) Local level model (LLM) In he absence of seasonal and cyclical componens, Eq. (1) reduces o Y, ~ N0,, 1,,..., T, (3) when he rend componen does no show a seady upward or downward movemen, i becomes a permanen componen called level. Someimes, i is assumed o vary according o a random walk, i.e., ~ N0,. (4) 1 1,,.,

2 N,.,. Eqs. (3) and (4) ogeher from he LLM. I may be noed ha hese equaions are already in sae space form. However, level is no direcly observable bu can be esimaed. Furher, forecas values are weighed average of he daa poins. When 0, forecas is jus he las observaion and when 0, level is consan and he bes forecas is sample mean. Level of ime series vary over ime depending on signal o noise raio q /. Esimaion of, condiional on and, is done recursively using Kalman filer and smooher (Harvey, 1996). The parameer and, are unknown and are reaed as hyperparameers. Likelihood funcion can be evaluaed by Kalman filering via predicion error decomposiion (Shumway and Soffer, 000). Once and, are known, one-sepahead predicion of level, i.e. esimaor of 1 given Y = {Y 1, Y,, Y }, viz. a +1 = E( +1 Y ), (5) is evaluaed recursively by Kalman filer. Predicion error variance P +1 = Var ( +1 Y ), = Var (a +1 ) (6) is also obained recursively. Reduced form of LLM is ARIMA (0,1,1) model. (ii) Local linear rend model () As described by Harvey (1996), is given by Eq. (3) along wih he following wo equaions: μ=μ+β11β=β+ξ,=1,1(8) η Nσση,ξεwhere ηξ. I may be menioned ha and are σ=σ=0independen of one anoher. If ηξequaions (7) and (8) collapse o μ=μ+β,=1,t1(9) which can equivalenly μbe =wrien α+β,as,t(10) showing ha he deerminisic linear rend () is a limiing case. α=μ,β is in sae space form wih sae vecor. Updaing and predicion are +η,(7) --.,-,0,1-0,ancarried ou using Kalman filer by assuming ha dξ0,,.,-=1,, and are known. Oherwise hese can be esimaed using maximum likelihood mehod for sae space models (de Jong, 1988; Koopman and Shephard, 199). Reduced form of a is ARIMA (0,,) model. Forecas funcion of perform beer han ha of corresponding ARIMA model. (iii) Local linear rend model wih inervenion effec () Inervenion analysis is concerned wih making inference abou effecs of known evens. These effecs are measured by including inervenion, or dummy variables in a dynamic model (Harvey and Durbin, Y=μ1986). +λw +εis described by he following equaions:, (11) μ=μ+β+λw+η-1-1, (1)

3 β=β+λw+-13 ξ, (13) where w is inervenion variable and is is coefficien. The quaniy w depends on he form which inervenion is assumed o ake. As for, esimaion of sae vecor, for is similarly carried ou by puing he model in sae space form and applying Kalman filer recursively by reaing w as an explanaory variable. Goodness of fi of he above models is assessed based on various informaion crieria such as esimaes of error variances, Akaike informaion crierion (AIC), Schwarz-Bayes informaion crierion (SBC), and Sandard error (S.E.). I may be reminded ha he lower he values of hese saisics, beer is he fied model. Fiing of Srucural ime series models o all India sunflower yield Daa on all-india sunflower yield (Kg/Hec) during o is obained from agricoop.nic.in and uilized for he sudy. I may be recalled ha Technology Mission on Oilseeds (TMO) was esablished in 1986 wih a view o inegraing all he faces and secors of oilseeds under a single umbrella for breaking sagnaion in oilseeds producion. As a firs sep model is fied o he daa. Then using he informaion of addiive ouliers in he year model is fied wih inervenion a year Forecased he yield for nex eigh years ( o ) and SAS codes are given below. SAS Codes: Tile 'Sunflower Yield (Kg/Hec) o '; Daa sunflower; inpu number yield; year = innx( 'year', '1jan1985'd, _n_-1 ); forma year year4.; cards; ; /* Prin daa*/ proc prin daa=sunflower; Tile 'All India Sunflower Yield (Kg/Hec) o '; proc sgplo daa=sunflower;

4 4 scaer x=year y=yield; qui; /* Save resul file*/ ods rf file='resulsunflower'; /* Unobserved Componen Models (ucm) for */ ods graphics on; proc ucm; id year inerval=year; model yield; /*yield is response variable*/ irregular ; /*Error in he model*/ level ; /*Trend in he model*/ slope; esimae plo=(panel residual); forecas plo=forecass lead=8 ; /*plo acual prediced(i.e o ) if lead=0 */ ods graphics off; /* Include Years o */ daa nsunflower; do number= 14 o 1; yield =.; oupu; end; proc prin daa= nsunflower; daa nsunflower; merge sunflower nsunflower; by number; year = innx( 'year', '1jan1985'd, _n_-1 ); forma year year4.; proc prin daa=nsunflower; /*Include Time-Inervenion */ daa nsunflower; se nsunflower; ls1989=(year>='1jan1989'd); /*Level Shif*/ proc prin daa=nsunflower; /* */ ods graphics on; proc ucm daa=nsunflower; id year inerval=year;

5 5 model yield=ls1989; irregular; level; slope; esimae plo=(panel residual); forecas plo=forecass ; ods graphics off; ods rf close; Parial Oupu is given below: Obs year Break Type Esimae Oulier Summary Sandard Error Chi-Square DF Pr > ChiSq Addiive Oulier Using he above informaion of addiive oulier we fied model and resuls of boh models ( and ) are given below. Likelihood Based Fi Saisics Value Saisic AIC (smaller is beer) BIC (smaller is beer) AICC (smaller is beer) HQIC (smaller is beer) CAIC (smaller is beer)

6 6 Final Esimaes of he Free Parameers Componen Parameer Esimae Approx Sd Error Esimae Approx Sd Error Irregular Error Variance Level Error Variance Slope Error Variance ls1989 Coefficien Significance Analysis of Componens (Based on he Final Sae) Componen DF Chi-Square Pr > ChiSq Chi-Square Pr > ChiSq Irregular Level < <.0001 Slope Trend Informaion (Based on he Final Sae) Sandard Sandard Name Esimae Error Esimae Error Level Slope Forecass of All India Sunflower Yield (Kg/Hec.) for o MODEL YEARS Forecased Yield Sandard Error Forecased Yield Sandard Error Acual Yield

7 7 Local linear rend model () Local linear rend model wih inervenion effec () Conclusion: model may be used o describe rends in yield of oher Oilseed crops.