Nonlinearity Test on Time Series Data

Transcription

1 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 016 Nonlineariy Tes on Time Series Daa Case Sudy: The Number of Foreign Touriss Rahma Dwi Khoirunnisa 1, Wahyu Wibowo, Agus Suharsono 3 1 Deparmen of Saisics, Insiu Teknologi Sepuluh Nopember, ITS Deparmen of Saisics, Insiu Teknologi Sepuluh Nopember, ITS 3 Deparmen of Saisics, Insiu Teknologi Sepuluh Nopember, ITS r4hm4dwi5@gmail.com M 15 Absrac Time series daa analysis is a mehod for modeling a daa paern. Forecasing is one of he main poins in a ime series analysis. difficul o choose he mehod of parameric models ha are no linear. Before forecasing ime series daa nonlineariy esing should be done in order o explain he nonlinear relaionships in he variable and esing procedures o deec he presence of nonlinear relaionships. Some alernaive mehods ha can be used o es he nonlineariy is Ramsey's RESET es, Whie es and Terasvira es. Ramsey's RESET es is a es used o deec nonlineariy using general ess for specificaion error (Gujarai, 1996). Whie Tes is a es developed o deec nonlineariy of neural nework models were invened by Whie (1989). Terasvira es is a es used o deec nonlineariy were also developed from neural nework models and are included in he es group developed ype of Lagrange Muliplier wih Taylor expansion (Terasvira, 1993). The purpose of his sudy was o demonsrae ha he daa on he number of foreign ouriss is he daa nonlinear wih nonlineariy is esed using hree mehods: Ramsey's RESET es, Whie es and Terasvira es. In his sudy will use daa on he number of foreign ouriss a Juanda airpor in 000 unil 015. Keywords: nonlineariy es, ime series daa, number of foreign ouriss I. INTRODUCTION Time series daa is a series of observaions on a value aken a differen imes. Time series daa is daa in chronological order. Time Series is a series of variables ha form he observaion values observed from ime o ime and recorded in accordance wih he sequence of evens and he daa is assumed o be inerdependen wih one anoher (dependen). Such daa can be colleced periodically a cerain ime inervals, such as daily, weekly, monhly, or yearly. Auoregressive which is one of he Time Series models, firs inroduced by [8] and laer developed by [5]. Auoregressive models of order p or AR (p) saes ha he value of observaion all depend on he values of p observaions hroughou he previous period. Bu in some cases, he relaionship beween he daa have shaped nonlinear endencies. Based on hese cases, necessary o es o show ime series daa used in he model included linear or nonlinear models. The es can be used o indicae ha daa be linear or nonlinear. There are some es ha can be used o show he nonlineariy. Non lineariy es used in his sudy were Whie es, Ramsey s RESET es and Terasvira es. Ramsey's RESET es is a es used o deec nonlineariy using general ess for specificaion error [1]. Whie Tes is a es developed o deec nonlineariy of neural nework models were invened by [7]. Terasvira es is a es used o deec nonlineariy were also developed from neural nework models and are included in he es group developed ype of Lagrange Muliplier wih Taylor expansion [3]. several sudies using non-linear models such, esing for nonlineariy in ime series: he mehod of surrogae daa [4] indicaing ha correcly idenifies nonlineariy in several well-known examples of lowdimensional chaoic ime series, even when conaminaed wih dynamical and observaional noise. M - 93

2 ISBN lineariy es daa ime series wih rese es [6] showed he resuls of he daa generaed from nonlinear models produce nonlinearias significan a 5%. based on previous sudies, he es nonlinearias acually quie imporan in idenifying he ime series daa ha is used for helping o sor he daa in he model is a linear or nonlinear models. if included in he linear model can be regressed using a parameric regression oherwise if included in he nonlinear model can be regressed using nonparameric regression or semiparameric regression. Therefore in his sudy, he es will be conduced non-lineariy in he ime series daa. ime series daa used is daa on he number of foreign ouriss a he airpor juanda in 000 o 015. II. AUTOREGRESSIVE MODEL Auoregressive model is a model ha describes he dependen variable influenced by he dependen variable iself in periods previously, or auocorrelaion can be inerpreed also as a linear correlaion sequence periodically wih ime series iself wih a ime difference (lag) 0, 1, or more periods. The general form auoregressive model wih order p or wrien wih AR (p) has he following equaion: Y = he value of a variable a i = auocorrelaion parameer i-h wih i= 1,,...,p e = error value in Y Y Y Y e 1 1 p p A. Auoregressive 1 AR order are ofen used in ime series analysis is p = 1 is a model AR (1). AR (1) saed ha he observed values o depends on he values of he observaions hroughou he previous period. The general form auoregressive model wih order 1 or wrien wih AR (1) has he following equaion: Y = he value of a variable a 1 = auocorrelaion parameer e = error value in III. Y Y e 1 1 NONLINEARITY TEST According o [] some ess o deec non-linear relaionships beween variables in ime series analysis. in his secion he discussion focused on he deecion nonlinearias on a ime series model, paricularly Ramsey's RESET es, es and es Terasvira Whie. The following is an explanaion for each of he nonlinearias es. A. Ramsey s RESET es Ramsey has proposed a general es of specificaion error called RESET (regression specificaion error es). The general shape models describing he relaionship among he independen variables (predicors) and he dependen variable (response) can be wrien: Y f X (3) Hypohesis esing used in he es are he nonlinearias: H f X is a linear funcion of he X or he linear model 0 : H f X is a nonlinear funcion of he X or he nonlinear model 1 : H 0 is rejeced, which means non-linear model is appropriae, if he value of he F es mees namely he rejecion region F F ; df, df aau p - value (4) 1 (1) () M - 94

3 PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY 016 The following seps in he RESET es by [1]: 1. Regression of Y on 1, x1, x,..., x p and calculae he esimaed values of he response variable Y ˆ, so: Y x x ˆ p p Calculae he coefficien of deerminaion of he regression, he R and furher denoe he. Regression of Y on 1, x1, x,..., x p and addiional predicors ha esimaed values of he response variable Y ˆ, so: Y x x ˆ p p Yˆ and ˆ 3 Calculae he coefficien of deerminaion of he regression, he R and furher denoe he 3. Calculae F es score m: addiional predicors p: early predicors n: number of daa in used F Rnew Rold m 1 Rnew n p 1 m R old. Y calculae he R old. 4. Based on he hypohesis of lineariy, shows F es values approaching F disribuion wih degrees of freedom of m and (n-p-1-m). Conclusion Ho is rejeced if F > F (α, m, n-p-1-m) or p-value < α (ypically use he alpha value of 0.05). B. Whie es Whie es is non lineariy deecion es developed from neural models nework raised by [7]. This es is included in he es group of ype Lagrange Muliplier (LM). Hypohesis esing used in he es are he nonlinearias: H f X is a linear funcion of he X or he linear model 0 : H f X is a nonlinear funcion of he X or he nonlinear model 1 : H 0 is rejeced, which means non-linear model is appropriae, if he value of he F es mees namely he rejecion region F F ; df, df aau p - value 1 The following seps in he Whie es by [3]: 1. Regression of Y on 1, x1, x,..., x p, calculae he residual value uˆ and calculae residual sum of squares: SSR0 uˆ (8). Regression of Y on 1, x1, x,..., x p, m addiional predicors so calculae residual v ˆ and calculae residual sum of squares: 3. Calculae F es score SSR 1 vˆ SSR SSR m F SSR n p m (5) (6) (7) (9) (10) M - 95

4 ISBN m: addiional predicors p: early predicors n: number of daa in used 4. Based on he hypohesis of lineariy, shows F es values approaching F disribuion wih degrees of freedom of m and (n-p-1-m). Conclusion Ho is rejeced if F > F (α, m, n-p-1-m) or p-value < α (ypically use he alpha value of 0.05). C. Terasvira es Terasvira es included in he group Lagrange Muliplier (LM) es wih a Taylor expansion approach ha uses a es saisic wih degrees. Terasvira es procedure is described as follows [3]: 1. Regression of Y on 1, x1, x,..., x p, and calculae he residual value u ˆ.. Regression of Y on 1, x1, x,..., x p, and m addiional predicors which is he resul of Taylor expansion approach. 3. Calculae he coefficien of deerminaion ( ) and regression in he previous sep. 4. Calculae saisics es nr wih n is number of daa. Hypohesis esing used in he es are he nonlinearias: 0 : H f X is a linear funcion of he X or he linear model H f X is a nonlinear funcion of he X or he nonlinear model 1 : 5. Based on he hypohesis of lineariy, shows Conclusion Ho is rejeced if p-value from 0.05). IV. es values approaching V disribuion. es values < α (ypically use he alpha value of METHODOLOGY Daa used in his sudy are secondary daa he number of foreign ouriss a he airpor Juanda obained from BPS. Daa used in his sudy is he monhly daa, he period o be examined is uary 000 o December 015. This sudy begins wih a descripion of he daa ha will be used o deermine he amoun of daa o be used as well as oher descripions of he daa. Then he daa will be ploed using a ime series plo o show daa on he number of foreign ouriss paern a Juanda airpor and final esing will be performed on he daa using nonlinear hree nonlinearias es is Ramsey s RESET es, Whie es adn Terasvira es. This research using miniab program o descripive daa and ime series plo, whereas for he non lineariy es using he assisance program R. V. RESULTS AND DISCUSSION The firs sep in his research was o deermine he amoun of daa on he number of foreign ouriss a he airpor Juanda sared uary 000 o December 015. The following descripion is shown in he able daa o be used: TABEL 1. DESCRIPTIVE STATISTICS Variable Toal coun Mean SE Mean S Dev Juanda Based on Table 1, indicaed ha he daa used in his sudy as many as 19 daa wih mean 1193 and sardard deviaion Since deermining much of he daa used, he nex sep is o see paerns in he daa on he number of foreign ouriss. hen be shown a paern of daa using ime series plo in he figure below: M - 96

5 he number of foreign ouriss PROCEEDING OF 3 RD INTERNATIONAL CONFERENCE ON RESEARCH, IMPLEMENTATION AND EDUCATION OF MATHEMATICS AND SCIENCE YOGYAKARTA, MAY Juanda Airpor Monh Year FIGURE 1. TIME SERIES PLOT OF JUANDA Shown in Figure 1, daa on he number of foreign ouriss have a paern ha is up and down. The nex will be esed for nonlinearias because such daa is no always a linear paern. before conducing he es, will be deermined in advance ime series model o be used in esing. he model used in his sudy is a model AR (1), his model is a ime series model wih a univariae predicor variables. The following models of he AR (1) o be used in his sudy: Y afer deermining he models o be esed, las sep he model will be esed using hree es program nonlinearias using R. afer he es is done using R obained he following resuls: Y 1 (11) FIGURE. SYNTAX NONLINEARITY TEST The synax is obained based on [], wih some simple changes. Afer he es is done using R obained he following resuls: M - 97

6 ISBN FIGURE 3. RESULT NONLINEARITY TEST Synax in R is based on he seps [1], [7] and [3]. Based on he resuls of running using he R found ha he hree es produces a value less han he p-value of 0.05.Ramsey s RESET es he values obained daa on he number of foreign ouriss a he airpor juanda of , wih a p-value of 0.05 so ha i can be seen ha he number of foreign ouriss a he airpor juanda smaller han he p-value. Whie es he values obained daa on he number of foreign ouriss a he airpor juanda of , wih a p-value of 0.05 so ha i can be seen ha he number of foreign ouriss a he airpor juanda smaller han he p- value. Terasvira es he values obained daa on he number of foreign ouriss a he airpor juanda of , wih a p-value of 0.05 so ha i can be seen ha he number of foreign ouriss a he airpor juanda smaller han he p-value. VI. CONCLUSIONS Based on he resuls and discussions can be concluded ha he ime series daa, especially daa on he number of foreign ouriss a Juanda airpor is a nonlinear model. Because i is based on hree rials showed nonlineariy p-value less han as shown in he able below: Ramsey s RESET es Whie es Terasvira es p-value Afer finding ou ha he ime series daa can be non-linear form, his daa can be used for regression semiparameric or nonparameric regression. REFERENCES [1] Gujarai, D.N., Basic Economeric 5h Ediion, New York: Mc Graw Hill Inernaional, page 51-53, [] Suharono, Saisical Daa Analysis wih R, Analisis Daa Saisik dengan R, Surabaya:ITS, 008. [3] Terasvira,T., Linc, F and Granger, C.W.J., Power of The Neural Neworks Lineariy Tes, Journal of Time Series Analysis, vol.14 pp , [4] Theiler, J., Eubank, S., Longin, A., Galdrikian, B., and Farmer, D., Tesing for Nonlineariy in Time Series: The Mehod of Surrogae Daa Norh Holland: Physica D, vol.58 pp.77-94, Sepember 199. [5] Walker, G., On Periodiciy in Series of Relaed Terms, procceding of he royal sociey of london, ser.a, vol. 131, pp , [6] Warsio and Ispriyai, Lineariy Tes Daa Time Series Wih Rese Tes, Uji Linierias Daa Time Series Dengan Rese Tes, journal of mahemaic and compuer, vol.3 no.3, pp , December 004. [7] Whie,H., An addiional hidden uni es for negleced nonlineariy in mulilayer feedforward neworks, Proceedings of The Inernaional Join Conference on Neural Neworks, Washingon, DC, pp , [8] Yule, G. Udny, Why Do We Someimes Ge Nonsense-Correlaions Beween Time Series? A Sudy In Sampling And The Naure Of Time Series, journal of he royal saisical sociey vol.89, no.1, pp.1-43, uary 196. M - 98