EKONOMETRIA ECONOMETRICS 1(39) 2013

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1 EKONOMETRIA ECONOMETRICS 1(39) 2013 Publishing House of Wrocław Universiy of Economics Wrocław 2013

2 Copy-ediing: Agnieszka Flasińska, Elżbiea Macauley, Tim Macauley Layou: Barbara Łopusiewicz Proof-reading: Aleksandra Śliwka Typeseing: Beaa Mazur Cover design: Beaa Dębska This publicaion is available a The Cenral European Journal of Social Sciences and Humaniies hp://cejsh.icm.edu.pl and in The Cenral and Easern European Online Library as well as in he annoaed bibliography of economic issues of BazEkon hp://kangur.uek.krakow.pl/bazy_ae/bazekon/nowy/index.php Informaion on submiing and reviewing papers is available on he Publishing House s websie All righs reserved. No par of his book may be reproduced in any form or in any means wihou he prior wrien permission of he Publisher Copyrigh by Wrocław Universiy of Economics Wrocław 2013 ISSN The original version: prined Prining: Prining House TOTEM Prin run: 200 copies

3 Conens Preface/Wsęp... 9 Maria Balcerowicz-Szkunik: Space-ime analysis of he phenomenon of unemploymen in he group of new EU member saes Jerzy W. Wiśniewski: Forecasing saffing decisions Jerzy Zemke: Forecasing risk of decision making processes Jan Acedański: Forecasing indusrial producion in Poland a comparison of differen mehods Paweł Dimann: Demand forecasing in a business based on expers opinions an applicaion of Weibull disribuions Paweł Dimann, Adam Michał Sobolewski: Demand forecasing in an enerprise he forecased variable selecion problem Barbara Namysłowska-Wilczyńska, Arur Wilczyński: Srucural analysis of variaion of elecriciy ransmission marginal coss Barbara Namysłowska-Wilczyńska, Arur Wilczyński: Geosaisical model (2D) of he surface disribuion of elecriciy ransmission marginal coss Pior Korzeniowski, Ireneusz Kuropka: Forecasing he criical poins of sock markes indices using Log-Periodic Power Law Alicja Ganczarek-Gamro: Forecas of prices and volailiy on he Day Ahead Marke Iwona Dimann: Forecas accuracy and similariies in he developmen of mean ransacion prices on Polish residenial markes Krzyszof Basiaga, Tomasz Szkunik: The applicaion of generalized Pareo disribuion and copula funcions in he issue of operaional risk Alicja Wolny-Dominiak: Zero-inflaed claim coun modeling and esing a case sudy Joanna Perzyńska: Comparaive analysis of accuracy of seleced mehods of building of combined forecass and mea-forecas Helmu Maier: On measuring he real value of producion. Reflecions on he economic order of he real world on occasion of he financial crisis Anna Nicińska, Małgorzaa Kalbarczyk-Sęclik: End of life in Europe: An empirical analysis Pior Tarka: Geomerical perspecive on roaion and daa srucure diagnosis in facor analysis Mara Targaszewska: An aemp a idenificaion sources of variaion in monhly ne incomes among persons wih eriary educaion

4 8 Spis reści Sreszczenia Maria Balcerowicz-Szkunik: Analiza przesrzenno-czasowa zjawiska bezrobocia w grupie pańsw nowych członków UE Jerzy W. Wiśniewski: Prognozowanie w decyzjach kadrowych Jerzy Zemke: Prognozowanie sanów ryzyka procesów decyzyjnych Jan Acedański: Prognozowanie produkcji przemysłowej w Polsce porównanie alernaywnych meod Paweł Dimann: Prognozowanie popyu w przedsiębiorswie na podsawie opinii eksperów zasosowanie rozkładu Weibulla Paweł Dimann, Adam Michał Sobolewski: Prognozowanie popyu w przedsiębiorswie problem wyboru zmiennej prognozowanej Barbara Namysłowska-Wilczyńska, Arur Wilczyński: Analiza srukuralna zmienności koszów marginalnych przesyłu energii elekrycznej Barbara Namysłowska-Wilczyńska, Arur Wilczyński: Model geosaysyczny (2D) powierzchniowego rozkładu koszów marginalnych przesyłu energii elekrycznej Pior Korzeniowski, Ireneusz Kuropka: Prognozowanie punków zwronych indeksów giełdowych przy użyciu funkcji log-periodycznej Alicja Ganczarek-Gamro: Prognozowanie cen i zmienności na Rynku Dnia Nasępnego Iwona Dimann: Podobieńswa kszałowania się średnich cen ransakcyjnych na rynkach mieszkaniowych w Polsce a rafność konsruowanych prognoz Krzyszof Basiaga, Tomasz Szkunik: Zasosowanie uogólnionego rozkładu Parea i funkcji łączących w zagadnieniu ryzyka operacyjnego Alicja Wolny-Dominiak: Modelowanie liczby szkód z uwzględnieniem efeku nadmiernej liczby zer oraz nadmiernej dyspersji sudium przypadku. 151 Joanna Perzyńska: Analiza porównawcza dokładności wybranych meod budowy prognoz kombinowanych i meaprognoz Helmu Maier: O pomiarze rzeczywisej warości produkcji. Refleksje o porządku ekonomicznym świaa realnego w dobie kryzysu finansowego Anna Nicińska, Małgorzaa Kalbarczyk-Sęclik: Koniec życia w Europie: analiza empiryczna Pior Tarka: Ujęcie geomeryczne w analizie czynnikowej meody roacji i diagnoza srukuralna danych Mara Targaszewska: Próba idenyfikacji źródeł zróżnicowania miesięcznego dochodu neo wśród osób z wykszałceniem wyższym

5 EKONOMETRIA ECONOMETRICS 1(39) ISSN Alicja Ganczarek-Gamro Universiy of Economics in Kaowice, Poland FORECAST OF PRICES AND VOLATILITY ON THE DAY AHEAD MARKET Absrac: The subjec of his paper is he forecas of prices and volailiy on he Day Ahead Marke (DAM). The analysis was made for wo porfolios of four conracs from o for wo fixings on DAM. Four ou of 24 conracs noed on DAM were chosen by PCA. Prices were forecas by he SARIMA models incorporaing auocorrelaion and seasonaliy. Value-a-Risk calculaed hrough he DCC model was used o forecas volailiy. These models describe well he prices and volailiy on he DAM and may be used for forecasing purposes. Prices on fixing 2 are characerized by higher volailiy han prices on fixing 1. Keywords: principal componen analysis (PCA), SARIMA model, DCC model, Value-a- -Risk, porfolio. 1. Inroducion Invesors on he Polish Power Exchange may paricipae in he Day Ahead Marke (DAM, spo marke), Commodiy Derivaives Marke (CDM, fuure marke), Elecriciy Aucions, Propery Righ Marke, Emission Allowances Marke (CO 2 spo) and Inraday Marke. All hese markes differ wih respec o he invesmen horizon s lengh and he raded commodiy. The mos popular marke is DAM where he horizon of he invesmen is one day. Conracs for elecric energy on DAM are characerized by hree ypes of prices: fixing prices, aucion prices and inraday prices. Every conrac on DAM is a conrac on physical delivery of elecric energy he nex day. DAM offers: 24 conracs for every hour of he day, four block conracs (BASE, PEAK, OFPEAK, MOR) and four indexes (IRDN, IRDN24, SIRDN, SIRDN24), which represen average prices of elecric energy on DAM during a day. The fixing price of elecric energy on DAM is esablished hree imes a day (a 8:00 fixing 1, a 10:30 fixing 2 and from o fixing 3 (since )). Several papers [Ganczarek 2008; Ganczarek-Gamro 2009; 2010] show ha prices on DAM are characerized by seasonaliy, auocorrelaion in mean and variance as well as long memory. In his paper, prices are forecased and risk of price change on

6 112 Alicja Ganczarek-Gamro DAM is esimaed. The resul of his research was used o build he composie porfolio of conracs on elecric energy. All 24 ime series of daily linear raes of reurn of elecric energy fixing prices from o were considered. 2. Mulivariae linear and nonlinear models Mulivariae linear model of expeced value raes of reurn is as follows: r = µ + ε, (1) where: r vecor of raes of reurn N 1, µ vecor of condiional expeced values N 1, ε vecor of residuals N 1 (whie noise), N number of ime series. The vecor of condiional expeced value of rae of reurn of elecric energy prices µ is described by Seasonal Auo-Regressive Inegraed Moving Average SARIMA (Eng.) (p, d, q) (P, D, Q) according o Brockwell and Davis [1996]: where: s d s pbp ( ) s( B) s r = qbq ( ) s( B) ε, (2) p P i i s si i= 1 i= 1 q Q i i s si i= 1 i= 1 i p( B) = 1 - pb, P( B) = 1- P B, i q( B) = 1 - qb, Q( B) = 1- Q B, s seasonal lag, d inegraed rank, s B shif operaor Br= r - s, s s differencing operaor r = r - r = (1 - B ) r, -s This model describes seasonal rend, auoregression and moving average so i is appropriae o analyze expeced value of prices and raes of reurn of prices from he elecric energy marke. In empirical research, vecor ε very ofen does no have he whie noise propery. Firs of all he vecor of residuals has varying variance. These properies may be described by he nonlinear auoregression GARCH model: ε = H 0,5 u, (3) where: H condiional covariance marix N N: r ~D(µ, H ), ε ~D(0, u vecor N 1 wih zero mean and 2 D (u ) = 1. H ), For he esimaion of he marix H Engle (2002) and Tse and Tsui (2002) proposed independenly Dynamic Condiional Correlaion (DCC). Model DCC proposed by Engle is as follows:

7 Forecas of prices and volailiy on he Day Ahead Marke 113 Γ = diag( q ;...; q ) Q diag( q ;...; q ), 0,5 0,5 0,5 0,5 11, NN, 11, NN, H = DΓD, (4) 0,5 0,5 0,5 where: D = diag( h1, h2,..., hn ) diagonal marix N N, every elemen of his marix is univariae GARCH model, Γ condiional correlaion marix, Q = ( qij ) symmeric, posiive marix N N: - ' Q = (1 -α - β) Q+ αu- 1u-1 + βq -1, εi ui = 0,5 h - i Q uncondiional variance marix of u αβ, posiive parameers, α + β < 1. More informaion abou univariae and mulivariae GARCH models is given by: Osiewalski, Pipień [2002], Zivo, Wang [2006], Fiszeder [2009], Ganczarek [2008], Trzpio [2010] and Ganczarek-Gamro [2010]. 3. Risk esimaion On he elecric energy marke, where sudden and dramaic changes of prices are very frequen, one of he appropriae risk measures is Value-a-Risk (VaR). VaR is given by he formula [Pionek 2001; Weron, Weron 2000]: VaR = + α (R μ )P (5) 1 R = F ( α) H. (6) α where: R α vecor of quaniles of order α for porfolio raes of reurn, P 0 vecor of prices, µ porfolio expeced value, H condiional covariance marix, - F 1 ( α) quanile of order α for sandardized disribuion. The Kupiec [1995] es is used o esimae he effeciveness of VaR. The esing hypoheses are as follows: H0 : ω = α, H : ω α 1. α 0 where ω is a proporion of he number of resuls exceeding VaRα o he number of all resuls. Assuming ha he null hypohesis is rue, he saisic: T-K K T-K K K K LRuc =-2ln[(1 - α) α ] + 2ln 1 -, (7) T T where: K a number of excesses, T a lengh of ime series, α he given probabiliy of he loss of value no exceeding VaR, has an asympoic χ 2 -disribuion wih 1 degree of freedom.

8 114 Alicja Ganczarek-Gamro 4. Empirical analysis In Figure 1, he ime series of Index Day Ahead Marke (IRDN PLN/MWh) was presened. Afer 2008, prices on DAM are characerized by a posiive rend, and a clearly lower volailiy han prices in So in he analysis he ime series from o were used. Figure 1. IRDN (PLN/MWh) noed on DAM from o To forecas prices and volailiy, daily raes of reurn of 24 fixing prices from fixing 1 and fixing 2 are used. Prices of elecric energy during a day are characerized by srong dependence. Based on he resul of Principal Componen Analysis (PCA) o forecas conracs of elecric energy in hour: 2, 6, 10 and 22 (Figure 2) were used. So on fixing 1 conracs: K1.2, K1.6, K1.10, K1.22 were analyzed and on fixing 2 conracs: K2.2, K2.6, K2.10, K2.22 were analyzed. For each of eigh conracs he SARIMA(1,0,1)(1,1,1) 7 model was used o describe he mean of ime series for linear raes of reurn. For every conrac he parameers of SARIMA models are significan (on significance level 0.05). In Figure 3 ACF and PACF funcions of residuals SARIMA(1,0,1)(1,1,1) 7 model for conrac K1.2 were presened. In Tables 1 and2 real fixing prices and forecas fixing prices wih Relaive Roo Mean Square Error (RRMSE) for fixing 1 and 2 during one week are presened.

9 Forecas of prices and volailiy on he Day Ahead Marke 115 1,0 0,8 Fixing ,6 6 Facor 2 0,4 0, ,0-0,2 0,0 0,2 0,4 0,6 0,8 1,0 Facor 1 Figure 2. Resul of PCA for conracs on fixing 1 ACF K1.2 : ARIMA (1,0,1)(1,1,1) residuals Opóźn Kor. S.E 1 +,007, ,011, ,073, ,039, ,081, ,109, ,002, ,030, ,094, ,054, ,043, ,013, ,043, ,005, ,016, ,060, ,023, ,010, ,041, ,118, ,033, ,042, ,052, ,098, ,007, ,015, ,051, ,019, ,0-0,5 0,0 0,5 1,0 Q p,05,8225,16,9240 5,16,1606 6,55, ,68, ,96, ,96, ,80, ,13, ,91, ,65, ,81, ,55, ,58, ,81, ,25, ,74, ,83, ,44, ,79, ,81, ,52, ,08, ,28, ,34, ,56, ,11, ,46, P. ufności Opóźn Kor. S.E PACF K1.2 : ARIMA (1,0,1)(1,1,1) residuals 1 +,007, ,011, ,073, ,038, ,083, ,105, ,011, ,024, ,074, ,054, ,063, ,009, ,039, ,016, ,039, ,038, ,031, ,003, ,071, ,105, ,029, ,046, ,067, ,087, ,000, ,008, ,040, ,006, ,0-0,5 0,0 0,5 1,0 Figure 3. ACF and PACF for SARIMA residual of conrac K1.2 RRMSEs on fixing 1 are lower han on fixing 2. Firsly, his is he resul of beer fiing SARIMA model for fixing 1 han for fixing 2, and secondly he greaer volailiy a fixing 2 (Figure 3) [Ganczarek-Gamro 2009]. The lowes errors are obained for conracs in hour 2 (prices of elecric energy by nigh are low, and have low volailiy), he highes errors are obained for conracs in hours 6 and 10 (prices of elecric energy during he day are high and are characerized by very high volailiy). The conracs K1.22 and K2.22 represen he evening peak, and were prices which exhibi somehow lower errors hen conracs for an hour of a day peak.

10 116 Alicja Ganczarek-Gamro Table 1. Real and forecas prices on fixing 1 from o Dae Real prices Forecas prices (RRMSE) K1.2 K1.6 K1.10 K1.22 K1.2 K1.6 K1.10 K (0.0453) (0.0464) (0.0469) (0.0471) (0.0472) (0.0472) (0.0472) (0.0662) (0.0686) (0.0688) (0.0688) (0.0688) (0.0688) (0.0688) (0.0646) (0.0682) (0.0692) (0.0694) (0.0694) (0.0694) (0.0694) (0.0486) (0.0515) (0.0523) (0.0525) (0.0525) (0.0525) (0.0525) Table 2. Real and forecas prices on fixing 2 from o Dae Real prices Forecas prices (RRMSE) K2.2 K2.6 K2.10 K2.22 K2.2 K2.6 K2.10 K (0.0581) (0.0618) (0.0623) (0.0624) (0.0624) (0.0624) (0.0624) (0.1038) (0.1097) (0.1103) (0.1103) (0.1103) (0.1103) (0.1103) (0.0778) (0.0839) (0.0848) (0.0849) (0.0849) (0.0849) (0.0849) (0.0664) (0.0712) (0.0718) (0.0719) (0.0719) (0.0719) (0.0719)

11 Forecas of prices and volailiy on he Day Ahead Marke 117 Small errors sugges he good fi of he SARIMA model o he empirical ime series, bu residuals of hese models indicae he presence of volailiy clusering effec and fa ail effec. This means ha RRMSEs changes in ime. So GARCH models should be used o analyze risk and o forecas volailiy. The risk analysis and volailiy forecas were made for wo porfolios. Prices and raes of reurn from fixing 1 and 2 are srongly correlaed, so porfolios consising of conracs from boh fixings were omied in he analysis. Based on PCA, wo porfolios were proposed. The firs one for fixing 1 and he second for fixing 2. The share of conracs in porfolios was calculaed based on relaive profi measure. For every conrac and for porfolio Value-a-Risk was esimaed using he Dynamic Auocorrelaion Model (DCC). This model is esimaed in wo seps. In he firs sep univariae residual variances for he SARIMA model were esimaed. These variances were modeled by IGARCH(1,1) models wih -Suden disribuion. In Figure 3 univariae variances of K1.2 and K2.2 were presened. They were modeled by IGARCH(1,1) models wih -Suden disribuion. Figure 4. Condiional univariae variances for conracs K1.2 and K2.2 In he second sep of DCC model esimaion, condiional correlaion marix Γ was calculaed. In Figure 4 elemens of marix Γ for conracs on fixing 1 were presened. Nex, VaR for every conrac and for he porfolios were esimaed using equaion (4). In Table 3, he resuls of VaR esimaion and resuls of he Kupiec es are presened. Based on he Kupiec es, he number of excesses of VaR is no significan (on significance level 0.01) for conracs K1.2, K1.22, K2.22 and for he porfolio of conracs on fixing 1. The high values of VaR on fixing 2 are he consequence of higher volailiy on fixing 2 han on fixing 1.

12 118 Alicja Ganczarek-Gamro Figure 5. Condiional correlaion marix Γ Table 3. Value of VaR and resul of Kupiec es Parameers/conracs K1.2 K1.6 K1.10 K1.22 Porfolio MIN [PLN/MWh] MEAN[PLN/MWh] MAX[PLN/MWh] p-value of Kupiec es Share of porfolio Parameers/conracs K2.2 K2.6 K2.10 K2.22 Porfolio MIN [PLN/MWh] MEAN[PLN/MWh] MAX[PLN/MWh] p-value of Kupiec es Share of porfolio

13 Forecas of prices and volailiy on he Day Ahead Marke Conclusion Prices on fixing 2 of DAM are characerized by higher volailiy han prices on fixing 1. Consequenly, he ransacions on fixing 2 are characerized by a greaer level of risk han ransacions on fixing 1. Hence invesors should prefer o execue ransacions on fixing 1. Lieraure Brockwell P.J., Davis R.A. (1996), Inroducion o Time Series and Forecasing, Springer-Verlag, New York. Engle R.F. (2002), Dynamic condiional correlaion a simple class of mulivariae GARCH models, Journal of Business and Economic Saisics 20: Fiszeder P. (2009), Modele klasy GARCH w empirycznych badaniach finansowych, Wydawnicwo Naukowe UMK, Toruń. Ganczarek A. (2008), Weryfikacja modeli z grupy GARCH na dobowo-godzinnych rynkach energii elekrycznej w Polsce, Rynek Kapiałowy. Skueczne Inwesowanie Sudia i Prace Wydziału Nauk Ekonomicznych i Zarządzania nr 9, Ganczarek-Gamro A. (2009), Analiza ryzyka na dobowo-godzinnych rynkach obrou energią elekryczną w Polsce, [in:] K. Jajuga, W. Ronka-Chmielowiec (eds.), Inwesycje finansowe i ubezpieczenia endencje świaowe a polski rynek, Prace Naukowe Uniwersyeu Ekonomicznego we Wrocławiu no. 60, Ganczarek-Gamro A. (2010), Pomiar ryzyka w sysemie ceny jednoliej na Towarowej Giełdzie Energii, [in:] T. Trzaskalik (ed.), Modelowanie preferencji a ryzyko, Prace Naukowe Akademii Ekonomicznej w Kaowicach, pp Kupiec P. (1995), Techniques for verifying he accuracy of risk managemen models, Journal of Derivaives 2: Osiewalski J., Pipień M. (2002), Mulivariae -GARCH Models-Bayesian Analysis for Exchange Raes, Modeling Economies in Transiion. Proceedings of he Sixh AMFET Conference, Łódź. Pionek K. (2001), Heeroskedasyczność rozkładu sóp zwrou a koncepcja pomiaru ryzyka meodą VaR, Prace Naukowe Akademii Ekonomicznej w Kaowicach, pp Trzpio G. (ed.), (2010), Wielowymiarowe meody saysyczne w analizie ryzyka inwesycyjnego, PWE, Warszawa. Tse Y.K., Tsui A.K.C. (2002), A mulivariae GARCH model wih ime-varying correlaions, Journal of Business & Economic Saisics 20: Weron A., Weron R. (2000), Giełda energii, Cenrum Informacji Rynku Energii, Wrocław. Zivo E., Wang J. (2006), Modeling Financial Time Series wih S-PLUS, Springer, New York.

14 120 Alicja Ganczarek-Gamro PROGNOZOWANIE CEN I ZMIENNOŚCI NA RYNKU DNIA NASTĘPNEGO Sreszczenie: Celem pracy jes prognozowanie cen i zmienności cen na Rynku Dnia Nasępnego. Analizę przeprowadzono na porfelach zbudowanych z czerech spośród 24 konraków noowanych od do wyłonionych za pomocą analizy głównych składowych niezależnie na dwóch aukcjach. Sopy zwrou opisano za pomocą modeli SARIMA uwzględniających auokorelację i sezonowość szeregów. Ryzyko zmiany ceny oszacowano w oparciu o warości VaR z uwzględnieniem zmiennej w czasie warunkowej korelacji modelem DCC. Podsumowując wyniki, można swierdzić, że zasosowane modele są dobrze dopasowane do szeregów z wybranego okresu badań, ponado konraky na aukcji drugiej charakeryzują się wyższym ryzykiem zmiany ceny niż konraky aukcji pierwszej. Słowa kluczowe: analiza głównych składowych (PCA), model SARIMA, model DCC, Value-a-Risk, porfel.