Applying Time Series Analysis Builds Stock Price Forecast Model

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1 Vol. 3, No. 5 Modern Applied Science 15 Applying Time Series Analysis Builds Sock Price Forecas Model Jun Zhang (Corresponding auhor) Deparmen of Science, Yanshan Universiy, Hebei , China Tel: zhangjunmah@163.com ui Shan &Wenfang Su Deparmen of Science, Yanshan Universiy, Hebei , China Absrac Time series analysis is a heory ha used random process and mahemaical saisics heory o analyze ime.i is apply comprehensive o naional economy macroeconomic adjusmen and conrol, area complex developmen plan, enerprise operaing managemen, marke poenial forecasing, weaher hydrology predicion. I is an imporan means for esimaion and forecas. The sock price has very deep effec o he economic benefis of he naion and he macro-economy policy. So people pay close aenion o i. In his aricle, SSE composie index of one year is fied wo kinds of ime series models, hen forecas in shor-ime. Comparing he esimaed valve wih he rue valve, he resul is he relaive error is small. So I hink he model is suied o he daa. A las, compare he wo models. Keywords: SSE composie index, Time series analysis, AIMA model, ACH model In he producion and scienific sudy, hrough carrying on observaion and measuremen o some group of or a variable, obained a series of discree digi in a series of imes, he series is called ime series. Pu he series of discree digials form a series se. The series se is called ime series. The ime series analysis is a mahemaical echniue ha hrough curve fiing and parameer esimaion esablish mahemaical model wih he sysem observaion daa and forecas he fuure rend. Sock price flucuaions has refleced he sae economy change o a cerain degree and affeced he naional macroeconomic policy. Therefore hrough economic indicaor's change forecas sock price rend is he opic which hroughou he people explore. Taken of ime series analysis mehod no only is he cos low bu also accuracy. Bu he sock price flucuaions is influenced by many kinds of he economic agen and he non-economic agen facors, for example he economic agen of marke rae, fiscal levy, price level, domesic inernaional poliical siuaion, war disaser, he non-economic agen facor of he invesors mood and he ransacion limis sipulaion. This paper uses he ime series analysis wo kind of model fiing sock price rend. 1. AIMA model AIMA (p, d, ) he model has he following srucure: d ϕ( B) x = θ( B) ε E( ε) = 0 var( ε) = σε E( εεs) = 0 s E( xsε) = 0 s > The abbreviaed formula is d θ ( B) x = ε, { ε } ϕ ( B) ( ) 1 ϕ B = ϕ B ϕ B L ϕ B 1 ( ) 1 1 p p zero average value whie noise series. is he auoregressive coefficien polynomial. θ B = θ B θ B L θ B is he moving average coefficien polynomial. ( 1 B) d d =, is he difference operaor, d is he difference order, B is backward shif operaor Bx = x. 1. AIMA model modeling sep.1 Daa processing Firs judge he series wheher o be seady or no by observing he auocorrelaion coefficien figure and he parial auocorrelaion coefficien figure. Second if he series is no seady, carries on he series difference or he season

2 Modern Applied Science May, 009 difference for eliminaing he endency flucuaion and he season flucuaion. If he difference can no make he series o be seady, aking he logarihm o he series o eliminaes he differen variance, hen make he series difference. Third he seady ime series is carried on he whie noise check o judge wheher i o be relaed. Fix he sep of he model Firs observe he auocorrelaion coefficien figure and he parial auocorrelaion coefficien figure of seady non-whie noise series o fix model auoregressive sep p and moving average sep. Bu he model is no necessarily only. Second selecs he suiable esimaion mehod o esimae he unknown parameer he value..3 Opimize model Carries on he residuals whie noise check and he parameer significance check o he fiing model, selecs superior model in he hrough he check models..4 Model predicion Using fiing model forecas he series shor-erm rend. 3. ACH model The ACH model full ile is auo regressive condiional differen variance model. Is complee srucure is x = f (, x 1, x L) + ε ε = he h = α0 + αε i j j = 1 f (, x 1, x L ) + ε is he { } residuals series and have differen variance ( ) series: ( ) h iid.. x Auo-egressive model; e ~ N ( 0,1).Series{ ε } is zero average value random var ε = h,differen variance eual o average value of residual suare E ε =.Under normal disribuion supposiion ~ N ( 0,1) ε h,wih residual suare series auocorrelaion coefficien inspecs he differen variance funcion he auocorrelaion. If here is some auocorrelaion coefficien is no zero, he residual suare series auocorrelaion coefficien is impermanen zero and differen variance funcion exis auocorrelaion. Then i is possible ha hrough consrucing residual suare series auoregressive model o fi he differen variance funcion h = α αε. + 0 i j j = 1 4. Example analyses The figure 1 shows he Shanghai Composie Index closing price daa from January 18h, 007 o January 13, Esablish he AIMA model The figure demonsraes ha he series is no seady. The series has a cerain rend, bu does no have he season effec. Because of ha reason carries he firs order difference on his seuence. The following is auocorrelaion and he parial auocorrelaion coefficien char and ADF Uni oo ess diagram. Because figure and figure 3 shows auocorrelaions and he parial auocorrelaion coefficien of he series afer firs order difference are firs order runcaion, he firs order difference seuence is seady. Figure 4 show all of he P value is smaller han I is also explained ha he seuence is seady. Boh he wo checks have confirmed he firs order difference series is seady. Figure 5 is he whie noise check figure. I shows ha under he level of =0.05, P value is all smaller han α.the resul indicaed ha firs order difference series is non-whie noise seuence. Because he firs order difference seuence is seady non-whie noise seuence, i is can be used o esablished he AIMA model. Because he auocorrelaion and he parial auocorrelaion coefficien of he difference series is firs order runcaion ry o esablish AIMA(1,1,0) model and AIMA(0,1,1) model. Make comparison he wo models Table 1 shows he AIC value and he SBC value and erroneous esimaed value of he A (1) model are smaller han hose of MA (1) model. I can be deermined ha AIMA (1,1,0) model is beer han AIMA (0,1,1) model. Therefore designae AIMA (1,1,0) model o fi he seuence. The char 6 is he model parameer esimaed value and he significance check and he residual whie noise check resul. Because he P value is bigger han 0.05 in he residual 153

3 Vol. 3, No. 5 Modern Applied Science whie noise check resul, he residual is he whie noise series B 1 B x = ε. The esablishmen model is( )( ) Figure 7 shows fiing figure of AIMA (1,1,0) model. The figure shows ha he fiing figure and he original figure superpose basically. So we can decide he model build successfully. The modeling goal is he forecas. Table shows he model shor-erm prediced resuls. Comparing he prediced values wih he rue values, he relaive errors are minor. elaive error's average value is Esablish he ACH model Check he residual series which have rejeced he endency iem o judge wheher i o be he auocorrelaion or he differen variance. Figure 8 shows residual series DW check amoun P value is smaller han 0.001; i explained ha he residual series is he auocorrelaion. And he PormaneaQ saisics and Lagrange number muliplicaion LM check amoun P value is smaller han 0.001; i explained ha residual series is differen variance Se up he model afer screening: x = u u = 0.857u u + ε ε = he 1 h = ε 1 Afer esing he parameers of he model are significan. The whole model's value is I is o mee he reuiremens. Figure 9 shows ACH model fiing figure. The red is he fied curve, and he black is he original curve. Table 3 is he shor-erm prediced of model resuls. Comparing he prediced values wih he rue values, he relaive errors are minor. elaive error's average value is Conclusion Comparing he wo models, judging form he prediced oucome, he ACH model relaive error is smaller, so ACH model fiing beer han he AIMA model. I is mainly because of he sock price change misalignmen behavior; AIMA model for ime series predicion only considers he characerisics of he ime series, wihou aking ino accoun he sock price iself is affeced by many unpredicable and complex facors; hese facors can only indicaed by he sochasic disurbing erm in he AIMA model ha is acually unable o display in he forecas expecaion value. In addiion, AIMA models generally assume ha he model residuals are zero average value and same variance, bu in fac sock index series of china ofen are differen variance. eferences Brockwell P, J, & Davis., A. (001).Time series: heory and mehod.springer Press. GAO, H., X. (1998).SAS sysem SAS/ETS he sofware user manual. China Saisics Press. Hen, B., L. (008).Sock price predicion model for he opimal selecion, Saisics and Decision, (6), KITAGAWA,G,A (1981).nonsaionary ime series model and is fiing by recursive echniue. J Time Ser Anal.(): Wang, Y. (005).Applicaion ime series analysis. People's Universiy of China publishing house. WONG, C, S. (000).On a mixure auoregressive model, J of he oyal Saisical Sociey (6) par1: Table 1. model comparison form AIC SBC Sd Error AIMA(1,1,0) AIMA(0,1,1)

4 Modern Applied Science May, 009 Table. AIMA (1,1,0) model prediced oucome Dae True values Prediced values elaive errors Table 3. ACH model prediced oucome Dae True values Prediced values elaive errors x Figure 1. Shanghai Composie Index series figure Auocorrelaions Lag Covariance Correlaion STD Error ******************** ****** * * * * * * ** * Figure. Auocorrelaion coefficien figure 155

5 Vol. 3, No. 5 Modern Applied Science Parial Auocorrelaions Lag Correlaion ****** * * ** * **. Figure 3. Parial auocorrelaions coefficien figure Augmened Dickey-Fuller Uni oo Tess Type Lags ho Pr < ho Tau Pr < T au F Pr > F Zero Mean < <.0001 Single Mean < < Trend < < Figure 4. ADF Uni oo Tess figure Auocorrelaion Check for Whie Noise To Chi- Pr > Lag Suare DF Chi S Auocorrelaions Figure 5. whie noise inspecion figures Uncondiional Leas Suares Esimaion Sandard Approx Parameer Esimae Error Value Pr > Lag MU A1, < Auocorrelaion Check of esiduals To Chi- Pr > Lag Suare DF Chi S Auocorrelaions Figure 6. AIMA (1,1,0) model oupu resul and residual check 156

6 Modern Applied Science May, 009 x Figure 7. AIMA (1,1,0) model fiing figure Q and LM Tess for ACH Disurbances Order Q Pr > Q LM Pr > LM < < < < < < < < < < < < < < < < < < < < < < < <.0001 Figure 8. Differen variance check figure x Figure 9. ACH model rendering 157