A Smooh Transiion Auoregressive Model for Elecriciy Prices of Sweden Apply for Saisic Maser Degree Auhor:Xingwu Zhou Supervisor:Changli He June, 28 Deparmen of Economics and Social Science, Dalarna Universiy
Absrac Absrac In his paper, nonlinear feaures of he daily elecriciy prices of Sweden are sudied. Following from Chow Tes, an evidence of a daa break poin is found beween he warm season and he cold season during he period from May s, 27 o April 3 h, 28. A es of he null hypohesis of lineariy agains nonlineariy for he firs difference of daily prices of Sweden is carried ou by using a Muliplier (LM)-ype es given by Teräsvira (994). Modeling for his elecriciy prices is considered and a logisic smooh ransiion auoregressive (LSTAR) model allowing o have a regime swiching can characerize nonlinear feaures of elecriciy prices of Sweden. Keywords: Nonlineariy, Smooh Transiion, LM-es, Chow es I
Conen CONTENT INTRODUCTION... 2 DATA... 2 3 MODELS... 6 3. General... 6 3.2 Tes daa breaks... 6 3.3 Tes lineariy again nonlineariy... 7 3.3. LSTAR model... 7 3.3.2 Arificial example for LM-ype es... 9 3.3.3 The resuls for he really daa... 9 3.4 Esimaion... 3.5 Evaluaions... 3 4 CONCLUSIONS... 4 REFERENCES... 6 APPENDIX A: THE CHARACTER OF ST FUNCTION... 7 APPENDIX B: ESTIMATION OF THE FIRST DIFFERENCE OF THE LOG PRICE... 7 II
A STAR model for elecriciy prices of Sweden A STAR model for elecriciy prices of Sweden Inroducion As he world s firs inernaional elecriciy power exchange marke, Nord Pool began operaing officially in 993 in Olso, Norway. Sweden joined in January 996; Finland was fully inegraed in March 999. Wes-Denmark became a separae price area in July 999 (Hjalmarsson, 2). Nord pool is a free marke, basically, i organizes wo markes, a physical marke (Elspo) and a financial marke (Elermin and Elopion). In he Elspo marke, hourly power conracs are raded daily for physical delivery in he nex day's 24-hour period. The price calculaion is based on he balance beween bids and offers from all marke paricipans finding he inersecion poin beween he marke s supply curve and demand curve. Every conrac in Elspo refers o a load, in megawa-hours (MWh, MWh equals,kwh), during a given hour, and a price per MWh. Since elecriciy is flow commodiy wih limied sorabiliy and ransporabiliy, he demand will change in differen seasons. The price is srongly dependen on he elecriciy demand. For example, in winer, he demand is larger and he price is higher. In Sweden, approximaely half of he oal insalled capaciy is hydropower. Nuclear power has he nex highes capaciy share of approximaely 3 percen. Excep for a small amoun of renewable generaion capaciy, oil and gas comprise he ohers pars
A STAR model for elecriciy prices of Sweden Haldrip (26) and Lucia (22) have ouched he issue abou Nordic Pool spo price sysem, bu boh of hem used linear model o analyze he characers of he price sysem. Since here are significanly difference beween cold season and warm season, nonlineariy model maybe is a beer choice. Despie ha lineariy is now a sandard procedure in he characerizaion of he ime series properies of any process, nonlineariy has been applied in some areas like unemploymen flucuaions (Skalin and Terävira, 22), inflaion (Arango and Gonzalez, 2) and sock marke reurns (Bredin, 28). The aim of he paper is o sudy he feaures of Sweden elecriciy prices, o es wheher here have daa break poin beween he cold season and he warm season and hen o use ime series smooh ransiion auoregressive (STAR) models o specify he changes of he daily elecriciy prices. The reminder of his paper is se ou as follows. Secion 2 presens he daa, secion 3 shows he models and secion 4 presens he conclusions and he final remarks. 2 Daa The daa employed in his sudy is daily observaions of Sweden elecriciy prices from May s 27 o April 3 h 28 (366 observaions). The daa was obained from he Nordic Pool s FTP sever files. The sample period includes wo differen pars: warm season and cold season. Warm season is from s May o 3 h Sepember of he same year, and cold season is from s Ocober o 3 h April of he nex year. Table presens summary saisics for he four differen series P, log (P ), P -P -, log (P)-log (P - ). From his able, we can see he significan difference exiss beween he cold season and he warm season. Noing ha warm season had a daily mean abou 4% lower han ha of cold season. However, he sandard deviaion of log-price changes show ha warm season is abou.6 imes as volaile as cold sea- 2
A STAR model for elecriciy prices of Sweden son (.99 or 89% annualized and.566 or 298% annualized). The raio of sandard deviaion over mean of he daily price is.24 for warm season and.7 for cold season, indicaing a significanly higher sabiliy of he mean price for cold seasons as compared o warm seasons. Changes of prices have very similar sandard deviaions for boh warm and cold seasons. The reason why he price in winer is higher and more sabile han in summer is ha he demand in winer is higher and abou 5% of insalled capaciy of his counry is hydropower, however, in winer he lakes and rivers are freezing. 3
A STAR model for elecriciy prices of Sweden Table : he descripive saisics for he daily sysem price and oher relaed ime series(275-2843) series number of Mean Median Minimum Maximum Sand Deviaion Skewness Kurosis observaion Panel A all seasons ( 275-2843) P 366 326.8 325.5 42.4 55.6 94.82758.364,.85969 P -P - 365.4398-2.39-7.3 42.6 37.9562.6772 5.23862 Log(P ) 366 5.744 5.785 4.958 6.3.382654 -.3569 2.6582 Log(P )-log(p - ) 365.839 -.6296 -.3553.6229.25832.734 5.6387 Panel B Warm season ( 275-2793) P 53 24.2 23.9 42.4 42. 53.697.7768, 3.3963 P -P - 52.33 -.98-85. 42.6 38.343.879, 4.82876 Log(P ) 53 5.462 5.442 4.958 6.2.246573.2844 2.562724 Log(P )-log(p - ) 52.4337 -.4862 -.3286.6229.55787.764 4.4229 Panel C cold season (27-2843) P 23 388.3 39. 234.2 55.6 65.83965 -.2935, 2.48666 P -P - 22.7-2.65-7.3 4.7 37.864.535, 5.59445 Log(P ) 23 5.946 5.967 5.456 6.3.79288 -.6552, 2.7642 Log(P )-log(p - ) 22 3.576e-5-6.443e-3-3.553e- 3.823e-.99934.3779 5.62696 Noe: This able displays descripive saisics for he daily spo sysem price (denoed P ) and oher relaed series as indicaed in he firs column. The sample period is from s May, 27 o 3 h April, 28. Panel A displays he resuls for he whole se of daa, panel B for warm season from s May. o 3 h Sep of he same year and panel C for he cold season from s Oc. o 3 h Apr. of he nex year. Using he model y = ( c + βy + βy 2 + + β py p) + ( c2 + β2δ y + β2δ y 2 + + β2 pδ y p) ( ) + u, where Δy ( y c) e γ Δ sands for P P - and log + (P )-log(p - ). 4
A STAR model for elecriciy prices of Sweden Figure. he Daily Sysem price Time Series (27.5.-28.4.3) 5
A STAR model for elecriciy prices of Sweden 3 Models 3. General As discussed in secion and secion 2, since he series exhibi non-linear feaures such as regime swiching, smooh raher han abrup changes are expeced. A STAR model allows ha he changes of elecriciy price alernaives smoohly beween wo regimes. The firs hing we should do is o es wheher here have daa break poins during he process. Then we will es lineariy agains he alernaive ha he process is nonlineariy for he changes of he daily prices series. If lineariy is rejeced agains such an alernaive, we specify, esimae, and evaluae smooh ransiion auoregressive model for he elecriciy and discuss he implicaions of he esimaed model. 3.2 Tes daa breaks Le a model for a ime series process {y }( =,2,, T) a a poin (T ) and before i ( =,2,,T ) be a ph order AR(p) model as follow. y = β + β y + β y + + β y + u ( T ) () 2 2 p p Similarly, le a model afer a poin (T ) of a srucural change (=T B+,T B+2, T) y = β + β y + β y + + β y + u ( T T) (2) 2 2 22 2 2 p p For he whole period, suppose anoher AR (p) model y = β + βy + β2y 2 + + βpy p + u ( T) (3) In hese models, he null hypohesis ha a srucural change does no exis and he alernaive hypohesis ha i does are shown as follows. H : β = β2, β = β2, β2 = β22,, β p = β2 p 6
A STAR model for elecriciy prices of Sweden H : a leas one equaion above does no hold. The classical es for his kind of srucural change is ypically aribued o Chow (96). This es was popular for many years and was exended o cover mos economeric models for ineres. For a recen reamen, see Andrews and Fair (988). KF = K( SSRR SSR SSR2)/ K ( SSR + SSR )/( T 2 K) 2 2 ~ ( K) χ (4) Where K is he number of parameers; SSR R is he sum of he squared residuals of (3); SSR and SSR 2 are he sum of he squared residuals of () and (2). Chow KF saisics follows Chi-square disribuion wih he degree of freedom K. In his sudy, le he T = 53, which poin is he boundary of he warm season and he cold season. Afer esimaion of he AR (p) model (here choose p = 7), he main es resuls a he 5% level are summarized in Table 2. Null hypohesis H is rejeced for series y and log( y ) series. Table 2 : he resuls of Chow es abou daa breaks series Number of observaion (T) K KF-value Rejec or Accep H P 365 8 24.28 R Log(P ) 365 8 26.5 R χ 2 (8) = 6.9.95 3.3 Tes lineariy again nonlineariy 3.3. LSTAR model The LSTAR model of order p [STAR (p)] is as below 7
A STAR model for elecriciy prices of Sweden y = ( c + β x ) + ( c + β x ) G( s, γ, c ) + ε (5) 2 2 ε 2 ~ iid(, σ ) (6) x = ( y,..., y ), β = ( β β ), j =, 2 (7) p j j,..., jp Gs ( ; γ, c) ( exp{ ( s c)}) = + γ (8) The main propery of his model is he Smooh ransiion beween regimes insead of a sudden jump from one regime o he oher. G is a ransiion funcion bounded by zero and uniy. s is he ransiion variables and c is he hreshold value. γ represens he speed of he ransiion process. When γ is large, he shape of he ransiion funcion G(.) is very seep in he neighborhood of he hreshold value c (see appendix A). As γ, and s > c hen G=; when s c, G=; so equaion (5) become a TAR (p) model. Whenγ, equaion (5) becomes an AR (p) model. The Lagrange Muliplier (LM)-ype es lineariy agains nonlineariy were described by Granger and Teräsvira (993, Ch.6), Luukkonen (988) and Teräsvira (994) is used here. Afer expanding he ransiion funcion (8) ino a hird order Taylor series around γ =, merging erms and reparameerizing. 3 3 y = ( c + β x) + ( c2 + β 2x)[ r( s c ) r ( s c ) ] + η (9) 4 48 = c + β x + β xs + β xs + β xs + η () 2 3 2 3 4 Tes lineariy agains non-lineariy Hypohesis H : β = β = β = γ = () 2 2 3 4 H : β, β, β no be zero a he same ime 2 2 3 4 8
A STAR model for elecriciy prices of Sweden As Teräsvira (994) and van Dijk and Teräsvira (22) shown ha LM = T SSR SSR SSR p (2) 2 3 ( )/ ~ χ (3 ) SSR is he sum of he residual squares of he esimaed model under he null hypohesis of lineariy of y on x SSR T 2 = ε =. SSR is he sum of he squared residual squares of he esimaed model under he auxiliary regression of y on x i and xs, i=, 2,3. Compue he squared residuals e and he sum of he squared residuals freedom. SSR T 2 e = =.The ess saisic is disribued as 2 χ wih 3p degree of 3.3.2 Arificial example for LM-ype es Le c, c2, c equal o zero, p= and s =, changing γ, use (5) o generae series, and hen use (2) o es ( replicaions). The resul is in Table 3 is he successful frequency. Table 3: LM-es on daa generaed wih nonlinear model (5) y ( replicaions ) γ =.5 γ = γ =2 γ =5 γ = γ = T=5 2 7 9 2 5 T= 7 24 29 32 35 38 T=2 47 5 65 75 78 86 T=5 97 99 T= As shown above, when γ and he number of observaions are more large, i is more easy o es he nonlineariy successfully. 3.3.3 The resuls for he really daa Since he mehod above requires saionary variables, we es for uni roos on he four series (P, log (P ), P -P -, log (P )-log (P - )) boh under ADF and KPSS proce- 9
A STAR model for elecriciy prices of Sweden dure. Accordingly, since saionary is needed, we use he firs difference of he daily prices or he firs difference of he log prices. According he values of AIC and ACF, we choose he lap p =7, he lineariy and nonlineariy es using (2) is shown in Table 4, under he 5% level. According he resul of Table 4, he null hypohesis of H 2 is rejeced for boh he series Δ P = P P and Δ log( P) = log( P) log( P ) Table 4 : he resuls of LM-ype es linear agains nonlinear series Number of observaion P LM 3 -value Rejec or Accep H 2 P P - 365 7 89.6 R Log(P )- Log(P - ) 365 7 96. R χ 2 (3 7) = 32.7.95 3.4 Esimaion The final model is esimaed by non-linear leas squares (NLS) including esimaion of he parameers γ and c. Esimaion of he slope parameerγ is no easy. Teräsvira (994) and ohers sugges ha he ransiion funcion Gs (, γ, c) should be sandardized o make γ scale-free, which implies dividing he exponen in Gs (, γ, c) by he sandard deviaion of s. Grid search mehod is used here and he final γ and c and oher parameers are chosen according he AIC. Gs (,, c) ( exp{ γ γ = + ( s c)}) (3) σ s As he number of observaions exends o infinie, he parameers exend o he
A STAR model for elecriciy prices of Sweden real value, which shows his mehod is quie good. Here we only give he resuls of he firs difference of he daily price sysem, p= 7 and s =Δ. The firs difference of he naural log price sysem is shown in he appendix B. y Δ y = (.95Δy.893Δy 2 3 (.) (.2).4Δy.638Δ y +.844 Δy ) 5 6 7 (.) (.3) (.8) + (.94Δy.896 Δy ) 4 (4) (.5) (.6) [ + exp{ 22 / σ ( Δ y ) ( Δy 5.675)]}] + u ( <. ) σ Δy = 38.47 S = 4.892 SK = -.622 EK = -.842 JB = 2.4(.2) AIC =27.59 S/S AR =.46 Where figures in parenheses below he parameer esimaes denoe he p-values of he ess. S is he esimaed sandard deviaion of he residuals, SK is skewness, and EK is excess kurosis. Jarque-Bera (JB) es rejecs normaliy. S/S AR is he raio beween he residuals sandard deviaion of he LSTAR model and he AR model. Here he raio is less han uniy (=.46), which means ha he former ouperforms he laer. The esimaed value for he hreshold value c, c =5.675, shows he inermediae poin beween increasing or decreasing daily price. c is wihin he range of { Δ y }, which is a good sympom of his model. The esimaion of γ, γ = (22 / σ Δy ), sugges a quick ransiion from one regime o he oher o he ex-
A STAR model for elecriciy prices of Sweden en ha i could mimic a TAR model. The oher esimaes can be inerpreed by analyzing he limi values ha describe he local dynamics of he high (G = ) and low (G=) elecriciy prices. For doing so, we use he roos of he LSTAR model which can be obained as usual from: p p p j ( j + 2 j G) j= λ β β λ (5) Where G=, (see Table 5). In he lower regime i can be seen ha he process is convergen wih modulus.77 and he upper regime here is also a convergen wih modulus.7568. The dynamic properies depiced by he LSTAR (7) model means ha eiher he dynamics sars close o he lower regime or he upper regime, he elecriciy price convers o a posiive one very easily. Table 5: Characerizaion of exreme regimes polynomials and dominan roo Model Regime Roo Modulus Lower(G=) -.234+.76i.77 Upper(G=).523+.557i.7568 Figure 2. Transiion funcion for he smooh ransiion auoregressive (STAR) models Lef hand panels plo he ransiion funcion G( Δy ; γ, c) agains ime. Righ hand 2 panels plo he ransiion funcion G( Δy ; γ, c) agains he ransiion variable Δ y.
A STAR model for elecriciy prices of Sweden Figure 2 shows he ransiion funcion Gs ( ; γ, c ) of he LSTAR model of Δ P series. The ransiion beween he wo regimes is sharp ( γ = 22 / σ Δy, see appendix A). 3.5 Evaluaions To es he model fi, need o es no residual auocorrelaion (van Dijk and Teräsvira, 22, Eirheim and Teräsvira, 996). u u iid 2 = ρ + ε ε ~ (, σ ) (6) H : ρ = ; 3 (7) H : ρ 3 (8) SS = n x SS ρ (9) Here SS u u n 2 x = ( ), SS ( u u ) ( u u )( u u ) 2 = ρ If accep H, shows ha here exis no residual auoregressive. ρ =.64, =.38 <.25 ( ) =.64. So here does no exis auocorrelaion. Figure 3 shows he residuals of he model (4). 3
A STAR model for elecriciy prices of Sweden Figure 3. he residuals of he model (4) 4 Conclusions In his paper, we sudy he feaures of he daily elecriciy prices of Sweden from he period of May s, 27 o Apr 3 h, 28. There are significan evidence ha here exis daa breaks beween he warm season and he cold season. The price in winer is higher bu more sable han in summer. We also find ha daily prices changes exhibis non-lineariy feaures such as regime swiching. Afer using Lagrange muliplier (LM)-ype es (Teräsvira, 994), we prove ha he Δ P and 4
A STAR model for elecriciy prices of Sweden Δ log( p ) series are nonlineariy. And hese wo series can be adequaely characerized by a logisic smooh ransiion auoregressive (LSTAR) model wih he resul beer han he linear models. 5
References REFERENCES Arango, L. and Gonzalez, A. (2) Some Evidence of Smooh Transiion nonlineariy in Colombian Inflaion. Applied Economics 33,55-62. Andrews, D.and Fair, R. (988) Inference in Nonlinear Economeric Models wih Srucural Change. Review of Economic Sudies, 65-39. Bredin, D. (28) Regime Change and he Role of Inernaional Markes on he Sock Reurns of Small Open Economies. European Financial Managemen.4, 35-346. Chow, G. (96) Tess of Equaliy beween Ses of Coefficiens in Two Linear Regressions. Economerica 28, 59-66. Eirheim,Φ. and Terävira, T. (996) Tesing he adequacy of smooh ransiion auoregressive models. Journal of Economerics 74, 59-75. Granger, C.W.J. and Terävira, T. (993) Modeling Nonlinear Relaionships. Oxford Universiy Press, New York. Haldrup, N. (26). A regime swiching long memory model for elecriciy prices. Journal of Economerics 35, 349-376. Hjalmarsson, E. (2) Nord Pool: A Power Marke wihou Marke Power Lucia, J. (22). Elecriciy Prices and Power Derivaives: Evidence from he Nordic Power Exchange. Review of Derivaives Research 5, 5-5. Luukkonen, R. (988).Tesing lineariy agains smooh ransiion auoregressive models. Biomerika 75,49-9. Skalin, J.and Terävira, T. (22). Modeling Asymmeries and moving equilibria in unemploymen raes. Macroeconomic Dynamics 6, 22-24. Terävira, T. (994) Specificaion, esimaion, and evaluaion of smooh ransiion auoregressive models. Journal of he American Saisical Associaion 89, 28-28. van Dijk, D. and Terävira, T. (22) Smooh ransiion auoregressive models a survey of recen developmens. Economeric Reviews 2(), -47. 6
Appendix Appendix A: he characer of ST funcion F( γ, x) =, γ =,2,5, * x + e γ r= r= Appendix B: esimaion of he firs difference of he log price x = log( y ) Δ x = (.2Δx.2544Δx.2 Δx ) 2 3 (.4) (4.22e 7) (.26e 5).69Δx.9Δ x +.24 Δx ) 5 6 7 (4.2e 5) (.) (5.27e 6) + (.676Δx.24Δx.587 Δx ) 2 4 7 (B.) (.8) (.4) (.43) [+ exp{ 42 / σ ( Δ x ) ( Δ x +.337)]}] + u σ Δx ( < 2e 6) =.2634 S =.26967 SK = -.55 EK = -.843 JB =.69(.4) AIC =98.648 S/S AR =.52 7