Microsoft Word - 00_tresc.docx DYNAMIC ECONOMETRIC MODELS Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 Aneta Włodarczyk, Marcin Zawada Politechnika Częstochowska The Use of Weather Variables in the Modeling of Demand for Electricity in One of the Regions in the Southern Poland A b s t r a c t. The main objective of the paper is the verification of usefulness of the ARFIMA- FIGARCH class models in the description of tendencies in the energy consumption in a selected region of the southern Poland taking into consideration weather variables. K e y w o r d s: weather variables, the ARFIMA-FIGARCH class model, weather risks. 1. Introduction The companies specializing in the production or distribution of power are particularly exposed to the weather risk, understood as the possibility of change in the financial result of a company caused by the variability of daily weather conditions: air temperature, rainfall and snowfall, sun light exposure, wind speed and humidity. Furthermore, the inability to store the power leads to the necessity of a precise measurement of the future demand for electricity by the companies specialising in its sale. Therefore the search for statistical and eco- nometrical tools enabling the modeling and forecasting of the demand for power in varying weather conditions has become such an important research problem. 2. Review of Research in the Scope of the Impact of the Climatic Factors on the Electrical Energy Consumption Identification and measurement of the weather risk are connected with the necessity to isolate from the observable electrical energy consumption a part which is sensitive to the effects of climatic factors. While analyzing historical time series relating to the demand for electrical energy, containing daily, weekly or monthly data from a dozen years, one may notice a strong long-term tenden- cy, whose occurrence has been affected by social, demographic and economic Aneta Włodarczyk, Marcin Zawada 100 factors. In order to isolate the demand for electrical energy which is sensitive to weather factors, various ways of data filtration can be used. In empirical re- search on modeling the above relation the following methods are used: 1. the method of the decomposition of time series into the trend component, the calendar component, the periodic component and the irregular compo- nent (Moral-Carcedo, Vicéns – Otero, 2005; Bessec, Fouquau, 2008): , 1 ,0 ∑ = ++++= m j tttaug j jt FEWDItE κδαα (1) or ,33 2 210 ttt FEYtttE +++++= δαααα (2) where Et is the demand for electricity, Iaug,t is a dummy variable taking the value 1 if the observation of the demand corresponds to the month of Au- gust, WDt is the variable describing working day effect, Yt is the seasonal unadjusted production in total manufacturing at time t, FEt is the electricity demand with the deterministic component filtered out. 2. the index-related equalization of the long term tendencies which do not re- sult from weather conditions in terms of the demand for electrical energy (Sailor, Muñoz, 1997; Valor, Meneu, Casellles, 2001): , j ij ij E E MSVI = (3) , jk ijk ijk E E DSVI = (4) where MSVIij is the index value for month i in year j, Eij is the monthly electricity consumption for month i in year j, jE is the monthly average electricity load for year j, DSVIijk is the index value for day i of week j of year k, Eijk is the electricity consumption for this same day, jkE is the daily average electricity load for week j in year k. After the estimation of the demand for electrical energy which is sensitive to climatic factors, the strength and nature of the relations between the weather variables and the electrical energy consumption should be assessed. Different types of models were used in the previous research: 1. Pardo, Meneu, Valor (2002) estimated the following model: , )()( 11 1 6 1 10 tt k ktk i itittt HM DCDDLHDDLtLE εϖϕ δγβαα +⋅++ ++++= ∑ ∑ = = (5) The Use of Weather Variables in the Modeling of Demand for Electricity … 101 ,)1( 33 9 9 2 21 ttLLL ξεφφφ =−−−− K (6) where Dit is dummy variable for daily data (D1t = 1 for Monday, D1t = 0 for other days of the week), Mit is dummy variable for monthly data (M1t = 1 for January, M1t = 0 for other months of the year), Ht is dummy variable for holidays (Ht = 1 for holidays, Ht = 0 for other days of the year). 2. Moral-Carcedo, Vicéns – Otero (2005) have constructed the following models in order to describe the non-linear relation between the energy con- sumption and air temperature: a) switch regression model ,tSttStt TMPFE εβμ ++= (7) b) threshold regression model [ ] [ ] ( ) , )Pr( Pr )Pr( 2 1 ∑ = == == St ttt ttt tt SSDFf iSiSDFf iS ψ (8) ⎪ ⎩ ⎪ ⎨ ⎧ >++ >=>++ <++ = .2 ,12 ,1 3 2 1 ThTMPTMP ThTMPThTMP ThTMPTMP DF ttt ttt ttt t εημ εγμ εβμ (9) In a warmer climate the relation between air temperature and energy con- sumption has a non-linear character with the form resembling letter U (Valor, Meneu, Casellles, 2001; Sailor and Muñoz, 1997); i.e. the maximum demand for energy is observed at the low and high temperatures. The introduction of the HDD and CDD indices, which separate the winter and summer seasons, enables better quantification of the analysed relation. Furthermore, the research by other authors (Bessec and Fouquau, 2008; among others) proves that in the climate zone, which includes Poland, the effect of a bigger demand for energy in the summer season connected with the use of air conditioning equipment is not significant. 3. Statistical Analysis of Characteristics of Analysed Time Series For the purposes of this paper the authors used information concerning: power consumption (in kWh), air temperature (in oC) and wind speed (in m/s) in one of the regions in the southern Poland in the period from September 1, 2005 to June 30, 2008. In the analysis and further calculations daily data was used in the following way: − the HDD index (heating degree days) was calculated on the basis of the re- lation: HDD = max (0,180C – Ti),where Ti – average daily air temperature on day number i; 102 − the C lation The f weather v cycles of grouping Figure 1. C C a In ord consumpt were dete One o (with var power. M 2009). 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D index have nce of wind f ng the dispers air temperatur ndex (lower pa cs of time se c modelling, s brings effec onal mean a other words ng of Demand f es, which is iables ture W 9 3.4 2 1.5 80 0.4 0 11.9 1 0.9 2 1.3 00** 539.6 00** 498.9 ** 0.87 01 level. Calcula e the biggest force and of sion of points e (upper pane anel) eries made i as the identi cts in the for nd the condi , it enables t for Electricity … characterise ind 495 1 5052 583 9170 3 086 241 6100** 191 9290** 121 796** 1 ations made in G@ t impact on the CDD ind for the energy el), energy co n this part c fication of re rm of the rele itional varian the construct … 103 ed by the HDD 10.6700 8.0799 0.0000 38.2080 0.5307 -0.3965 103.6000** 120.5000** 1.0585** @RCHTM. the power dex is sig- y consump- nsumption, constitutes egularities evant spe- nce of the tion of the Aneta Włodarczyk, Marcin Zawada 104 congruent econometric model according to the concept of Z. Zieliński (Zieliński, 1984). 4. Estimation and Verification of Models of the Demand for Electricity At the first stage of the research the authors identified a deterministic trend connected with the impact of demographic, economic and social factors on the demand for power in a region in the southern Poland. From the estimated vari- ous models of trend for the daily power consumption the author selected a third degree polynomial trend, taking into account the value of determination coeffi- cient and the significance of the estimates of structural parameters of the mod- els. Due to the object of the research, which was the description of the relation between the impact of weather factors on the energy consumption, in the equa- tion of demand for power the author included also the analysed weather va- riables- giving them a dynamic structure. Additionally the equation includes also dummies, whose task is to describe a weekly periodocity, annual seasonali- ty and holiday effect in the shaping of demand for energy. Finally the authors proposed the following specification of the model of energy consumption, ex- pressed in logarithms: , ln 6 1 11 1 13121 5 0 5 0 11 3 3 2 210 ∑ ∑ ∑ ∑ = = +− = = −+−+ ++++++ ++++++= i j ttttjtjiti k k ktkktkt uSSSMD windtemptttE κκκϕδ βλαααα (10) where: Dit – dummy variable equals one for the day i, and zero otherwise, Mjt – dummy variable equals one for the month j , and zero otherwise, St – dummy variable is equal to one for the holiday, and zero otherwise, St-1 – dummy variable equals one for the day preceding the holiday, and zero otherwise, St+1 – dummy variable is equal to one for the day following the holiday, and zero otherwise. The models of power consumption without weather variables, including the impact of air temperature, wind, as well as the HDD index were estimated with the OLS method. On the basis of information criteria, tests for model residuals and the parameter significance test the authors selected the following models with the weather variables (Table 2). The results of parameter estimation of the model (10) indicate that the cur- rent and one period lagged air temperature as well as one day lagged wind force have the significant impact on the power consumption in a given day. Moreo- The Use of Weather Variables in the Modeling of Demand for Electricity … 105 ver, estimates of parameters which stand by dummy variables and model peri- odicity in the weekly cycle on demand on energy indicate that on Mondays, Saturdays and Sundays energy consumption is lower than the average level and higher in the other days of the week. In the case of dummy variables associated with monthly seasonal effects all estimates of parameters are significant and negative for summer months (May, June, July, August, and September). It is connected with the impact of seasonal factors, such as, air temperature, length of the day, level of sun light exposure on the demand for energy. All parameters standing by dummy variables associated with holidays and neighbouring days are statistically significant and negative which indicates that the energy con- sumption on holidays and neighbouring days is significantly lower in compari- son with regular working days (as indicated by results of the Wald test for equality of parameters). Table 2. Estimates of the parameters of the model (10) Parameter Coefficient p-value Parameter Coefficient p-value α0 15.3739 0.0000*** φ1 0.0579 0.0000*** α1 -0.0002 0.0000*** φ2 0.0261 0.0000*** α2 4.336e-07 0.0000*** φ3 0.0262 0.0000*** α3 -1.729e-010 0.0089*** φ4 0.0073 0.0667** λ1 -0.0009 0.0358** φ5 -0.0568 0.0000*** λ2 -0.0036 0.0000*** φ6 -0.0830 0.0000*** β2 0.0024 0.0040*** φ7 -0.0491 0.0000*** δ1 -0.0067 0.0235** φ8 -0.0478 0.0000*** δ2 0.0293 0.0000*** φ9 -0.0185 0.0000*** δ3 0.0392 0.0000*** φ10 0.0199 0.0000*** δ4 0.0408 0.0000*** φ11 0.0500 0.0000*** δ5 0.0404 0.0000*** κ1 -0.1845 0.0000*** δ6 -0.0059 0.0442* κ2 -0.0369 0.0000*** Adjusted R2 0.88326 κ3 -0.0859 0.0000*** AIC -3785.4470 - - - BIC -3652.0610 - - - Hannan-Quinn -3734.8310 - - - Note: The symbol *** indicates the significance of the result at the 0.001 level. Calculations made in Gretl. In order to identify the autocorrelation effect, Box-Pierce test (the lag level: 5, 10, 20, 50) has been used for residuals of model (10) – all test statistics indi- cate for the significant autocorrelation in residuals. To verified the ARCH ef- fect, two different test have been used: Engle test (for 1, 2, 5, 10, and 20 lags) for residuals and Box-Pierce test for squared residuals (level of lag: 5, 10, 20, 50). Similarly, in this case all test statistics indicate for the significant autocor- relation in squared residuals. Using the Geweke-Porter-Hudak test, the long memory effect in residuals and squared residuals of the electricity demand Aneta Włodarczyk, Marcin Zawada 106 model has been captured2. With regard to the verification of residuals proper- ties, the model of ARFIMA (P, D, Q)-FIGARCH (p, d, q) class can be used for description of correlation between weather variables and energy consumption3: ,)()()( ttt D BuB εθμφ =−Δ (11) ),1,0(~z , t IIDhz ttt ⋅=ε (12) ),)]((1[)( 2 1 , 2 tt r k tkkt d hBxB −−++=Δ ∑ = εβωωεϕ (13) where: jj j DD B j D B )1()1( 0 −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =−=Δ ∑ ∞ = - filter difference of order D, ss s dd B s d B )1()1( 0 −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =−=Δ ∑ ∞ = - filter difference of order d, -1< D < 0,5, 0 < d < 1, 0 1 , >+∑ = r k tkk xωω , P P BBB φφφ −−−= ...1)( 1 , Q Q BBB θθθ +++= ...1)( 1 , q q BBB ϕϕϕ −−−= ...1)( 1 , p p BBB βββ ++= ...)( 1 . Introduction to the equation of conditional variance of regressor, which is a variability of weather factors or dummy variables which model periodicity of variance enables to connect dynamics of variability of energy consumption with variability of weather conditions of different structure of energy consumers in working days and holidays. In the current framework the following descriptive variables have been introduced to the equation of conditional variance of the process:4 dummy variables which model the effect of week day, dummy va- riables which model the month effect in the year, dummy variables which mod- el holidays, square of increment of daily average temperature in subsequent days, square of increment of wind power in subsequent days. Orders of models ARFIMA(P,D,Q)-FIGARCH(p,d,q) were chosen on the basis of information criteria and significance of the model parameters. The best models in this class are presented in Table 3. 2 Because of limited size of this framework, results of conducted tests for model residuals have not been presented. 3 In order to guarantee stationarity of analysed models of time series, it is assumed that roots of polynomial 0(B) ,0)( == ϕφ B lie outside the unit circle (Preś, 2007, p. 206; Laurent, 2007, p. 55–74). 4 Because of large number of model parameters and problems associated with estimation, proposed variables were separately attached to the equation of conditional variance. Table 3. Parameter estimates of ARFIMA(1,1)-GARCH(1,1) models Parameter ARMA(1,1)-GARCH(1,1)+R ARFIMA(1,1)-GARCH(1,1)+R ARFIMA(1,1)-GARCH(1,1) Cst(M) 0.001950 [0.4575] 0.0020 [0.5154] 0.0023 [0.4644] D-ARFIMA - 0.0426 [0.6560] 0.04460 [0.6341] AR(1) 0.7198 [0.0000] 0.6904 [0.0000] 0.6774 [0.0000] MA(1) -0.0929 [0.0925] -0.1104 [0.0694] -0.0983 [0.1168] Cst(V) 0.0004 [0.0000] 0.0003 [0.0000] 0.0003 [0.0032] Dif(temp) 0.87e-5 [0.0000] 0.55e-5 [0.0000] - ARCH1 0.2611 [0.0000] 0.2091 [0.0000] 0.2314 [0.0049] GARCH1 0.2709 [0.0032] 0.4439 [0.0000] 0.3931 [0.0299] Skewness -0.1157 [0.0173] -0.1217 [0.0130] -0.1202 [0.0165] Df-Student 5.2049 [0.0000] 5.0735 [0.0000] 5.1046 [0.0000] AIC -4.3980 -4.3939 -4.3926 SC -4.3549 -4.3460 -4.3494 H-Q -4.3816 -4.3757 -4.3762 Shibata -4.3981 -4.3941 -4.3927 Note: p-values have been presented in the brackets. Calculations made in G@RCHTM. Table 4. Summary statistics for model residuals of models ARFIMA-GARCH Statistic ARMA(1,1)- GARCH(1,1)+R ARFIMA(1,1)- GARCH(1,1)+R ARFIMA(1,1)- GARCH(1,1) Q (Box-Pierce) Statistics on Standardized Residuals Q(5) 4.76412 [0.1899] 4.9544 [0.1752] 4.7439 [0.1915] Q(10) 15.8704 [0.0443] 15.9747 [0.0427] 15.5973 [0.0485] Q(20) 29.4930 [0.0427] 30.0496 [0.0370] 30.0668 [0.0368] Q(50) 57.5181 [0.1633] 57.7321 [0.1586] 58.5882 [0.1407] Q (Box-Pierce) Statistics on Squared Standardized Residuals Q(5) 0.9382 [0.8162] 1.4152 [0.7020] 1.4080 [0.7037] Q(10) 3.9715 [0.8597] 4.8275 [0.7758] 5.0532 [0.7519] Q(20) 6.1725 [0.9954] 7.2110 [0.9882] 7.2413 [0.9879] Q(50) 27.6562 [0.9919] 27.9235 [0.9909] 27.3902 [0.9927] Engle’s LM ARCH Test ARCH(1-2) 0.2695 [0.7638] 0.4347 [0.6476] 0.4418 [0.6430] ARCH(1-5) 0.1866 [0.9677] 0.2834 [0.9224] 0.2819 [0.9231] ARCH(1-10) 0.3822 [0.9547] 0.4635 [0.9137] 0.4853 [0.9003] Nyblom Stability Test Nyblom Statistic for parameter vector 1.4860 stability 1.6558 stability 1.2591 stability Nonstability parame- ter by Nyblom test Nonstability parameter MA(1) Nonstability parameter : D-ARFIMA, AR(1), MA(1) Nonstability parame- ters : D-ARFIMA, MA(1) Sign Bias Test SB 1.6800 [0.0930] 1.7303 [0.0836] 1.7104 [0.0872] NSB 1.2895 [0.1972] 1.0867 [0.2772] 1.1378 [0.2552] The Joint Test 4.3732 [0.2239] 5.3313 [0.1491] 5.1391 [0.1619] Adjusted Pearson Goodness-of-fit Test Empirical distribution is congruent with theoreti- cal distribution Empirical distribution is congruent with theoreti- cal distribution Empirical distribution is congruent with theoret- ical distribution Note: p-values have been presented in the brackets. Calculations made in G@RCHTM. Aneta Włodarczyk, Marcin Zawada 108 When model estimates are assessed with regard to its quality the following results of tests conducted on its standardized residuals should be analysed: veri- fication of uncorrelated standardized residuals (Box-Pierce test), lack of ARCH effect (Box-Pierce test for squared residuals and Engle’s test), testing parame- ters stability in the model (Nyblom test), lack of diversity of influence made by negative and positive innovations on the level of variability (SB test), lack of diversity of influence made by large and small negative (positive) innovations on the variability (NSB test), fit of a distribution of empirical standardized resi- duals with assumed distribution (Pearson’s chi-square goodness-of-fit test).5 Each time, the introduction of GARCH structure with conditional skewed distribution of t-Student has been made, the result was that the effect of group- ing variances, which was present in residuals of ARFIMA model has been elim- inated. In the case of different estimated models of ARFIMA-FIGARCH class the estimate of fractional integration parameter d in conditional variance equa- tion was statistically insignificant. Even when dummy variables which model the effect of week day, month, and holidays in the equation of conditional va- riance of the process were considered, the characteristics of the model were not improved significantly. Next, the authors introduce the variability of the weath- er factors as the regressor to the conditional variance equation of the electricity demand. The result is that the autocorelation effect, which is found in standar- dized residuals of ARFIMA-GARCH model, has been decreased or eliminated. 5. Summary Demonopolization in energy industry in Poland has forced companies from energy industry to work out and implement internal procedures of risk man- agement, because the risk is present in energy trade. Companies from this in- dustry more and more often use weather derivatives to hedge against effects of weather risk, because this activity allows to make financial results independent of changing weather conditions. Analysis of influence of particular weather factors on energy consumption conducted by the Authors concerned only a particular region of southern Pol- and. Unfortunately, Polish conditions does not allow straight-forward access to these type of data because of the high cost of data purchase, whereas in many countries, databases concerning weather variables are available for free on web pages of meteorological stations of national entities which collect this type of data. Introduction to the equation of conditional variance of the regressor, which is a variability of average daily temperature increase in the coming days (Dif(temp)) enables to connect the dynamics of volatility of energy consump- 5 All above-mentioned methods have been described in the econometric literature (Doman, Doman, 2004, p. 295–308; Laurent, 2007, p. 41–46). The Use of Weather Variables in the Modeling of Demand for Electricity … 109 tion with the volatility of weather conditions. Moreover, the assessment of the ARFIMA-GARCH models on the basis of the residuals of model (10) made it possible to assess the conditional volatility of the process of demand for electri- cal energy. With the use of conditional volatility one can measure the volatility of the demand for electrical energy, i.e. the risk related to unpredictable change in the energy consumption under the influence of e.g. changing weather condi- tions. While extending analysis of the impact of weather factors on the function- ing of the power energy industry branch company, one should apply the Value at Risk methodology to measure the weather risk. Such approach will make companies dealing with the energy production and sales aware of the potential losses they may suffer as a result of unexpected change of weather factors. References Benth, F. E., Benth, J. S. (2009), Dynamic Pricing of Wind Futures, Energy Economics, 31, 16–24. Bessec, M., Fouquau, J. (2008), The Non-linear Between Electricity Consumption and Tempera- ture in Europe: A Threshold Approach, Energy Economics, 30, 2705–2721. Doman, M., Doman, R. (2004), Ekonometryczne modelowanie dynamiki polskiego rynku finan- sowego (Econometric modeling of the dynamics of the Polish financial market), Wydaw- nictwo AE w Poznaniu, Poznań. Laurent, S. (2007), Estimating and Forecasting ARCH Models Using G@RCH™5, Timberlake Consultants Ltd, London. Moral-Carcedo, J., Vicéns – Otero, J. (2005), Modelling the Non-linear Response of Spanish Electricity Demand to Temperature Variations, Energy Economics, 27, 477–494. Pardo, A., Meneu, V., Valor, E. (2002), Temperature and Seasonality Influences on Spanish Electricity Load, Energy Economics, 24, 55–70. Preś, J. (2007), Zarządzanie ryzykiem pogodowym (Management of Weather Risk), Wydawnictwo CeDeWu, Warszawa. Sailor, D. J., Muñoz, J. R. (1997), Sensitivity of Electricity and Natural Gas Consumption to Climate in the U.S.A. – Methodology and Results for Eight States, Energy, 22, 987–998. Valor, E., Meneu, V., Casellles, V. (2001), Daily Air Temperature and Electricity Load in Spain, Journal of Applied Meteorology, 40,1413–1421. Włodarczyk, A., Zawada, M. (2006), Behavior of Prices in the Polish Power Exchange and Euro- pean Power Exchanges. Statistical – Econometric Analysis, 3rd International Conference: The European Electricity Market EEM06. Challenge of the Unification, Warsaw, 313–321. Zieliński, Z. (1984), Zmienność w czasie strukturalnych parametrów modelu ekonometrycznego (Time Variability of Structural Parameters in Econometric Model), Przegląd Statystyczny (Statistical Survey), 1/2, 135–148. Zastosowaniem zmiennych pogodowych w modelowaniu zapotrzebowa- nia na energię elektryczną w jednym z regionów Polski południowej Z a r y s t r e ś c i. Głównym celem opracowania jest zweryfikowanie przydatności modeli klasy ARFIMA-FIGARCH do opisu kształtowania się zużycia energii elektrycznej w wybranym regio- nie południowej Polski z uwzględnieniem zmiennych pogodowych. S ł o w a k l u c z o w e: zmienne pogodowe, model ARFIMA-FIGARCH, ryzyko pogodowe.