Deregulated Wholesale Electricity Prices in Italy: An Empirical Analysis

Abstract

In this paper we analyze a time series of daily average prices in the Italian electricity market, which started to operate as a Pool in April 2004. Our objective is to model the high degree of autocorrelation and the multiple seasonalities in electricity prices. We use periodic time series models with GARCH disturbances and leptokurtic distributions and compare their performance with more classical ARMA-GARCH processes. The within-year seasonal variation is modelled using the low-frequency components of physical quantities, which are very regular throughout the sample. Our results reveal that much of the variability in the price series is explained by the interactions between deterministic multiple seasonalities. Periodic AR-GARCH models seem to perform quite well in mimicking the features of the stochastic part of the price process.

Appendix

In order to filter the low frequencies of the daily time series of electricity demand, we designed a partially model-based low-pass filter with time varying cut-off frequency. We used the model

$$ \begin{array}{*{20}c} {y_{t} = \mu _{t} + y_{t} ^{{{\left( 1 \right)}}} + y_{t} ^{{{\left( 2 \right)}}} + y_{t} ^{{{\left( 3 \right)}}} + \varepsilon _{t} } \\ {\mu _{t} = \mu _{{t - 1}} + \beta _{{t - 1}} } \\ {\beta _{t} = \beta _{{t - 1}} + \zeta _{t} } \\ \end{array} $$

$$ {\left[ {\begin{array}{*{20}c} {{\gamma ^{{{\left( i \right)}}}_{t} }} \\ {{\widetilde{\gamma }^{{{\left( i \right)}}}_{t} }} \\ \end{array} } \right]} = {\left[ {\begin{array}{*{20}c} {{\cos \omega _{j} }} & {{\sin \omega _{j} }} \\ {{ - \sin \omega _{j} }} & {{\cos \omega _{j} }} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{\gamma ^{{{\left( j \right)}}}_{{t - 1}} }} \\ {{\widetilde{\gamma }^{{{\left( j \right)}}}_{{t - 1}} }} \\ \end{array} } \right]} + {\left[ {\begin{array}{*{20}c} {{k^{{{\left( j \right)}}}_{t} }} \\ {{\widetilde{k}^{{{\left( j \right)}}}_{t} }} \\ \end{array} } \right]} $$

with $\omega _{j} = j \times {2\pi } \mathord{\left/ {\vphantom {{2\pi } 7}} \right. \kern-\nulldelimiterspace} 7{\text{, j = 1, 2, 3, }}\varepsilon _{t} \tilde{}N{\left( {0,\sigma ^{2}_{\varepsilon } } \right)}{\text{,}}\zeta _{t} \tilde{}N{\left( {0,\sigma ^{2}_{\varepsilon } } \right)}{\text{, }}\kappa ^{{{\left( j \right)}}}_{t} {\text{, }}\widetilde{\kappa }^{{{\left( j \right)}}}_{t} \tilde{}N{\left( {0,\sigma ^{2}_{\varepsilon } } \right)}$ μ _t captures the low frequency movements in which we are interested, the γ _t ’s take care of the within-week seasonality and ɛ _t is white noise. Since the cut-off frequency of the low-pass filter for extracting μ _t is determined by the signal-to-noise ratio $\rho = {\sigma ^{2}_{\zeta } } \mathord{\left/ {\vphantom {{\sigma ^{2}_{\zeta } } {\sigma ^{2}_{\varepsilon } }}} \right. \kern-\nulldelimiterspace} {\sigma ^{2}_{\varepsilon } }$, we fixed it to 1,600 for “normal” days and to 100 for Christmas and summer vacations time (24th December–6th January and July–September). The other unknown variances have been estimated by ML. The filtered series was produced by the Kalman smoother.

The gain of the filter is given by

$$ G{\left( \lambda \right)} = \frac{{{\left[ {\rho {\left( {2 - 2\cos \lambda } \right)}^{2} } \right]}^{{ - 1}} }} {{1 + {\left[ {\rho {\left( {2 - 2\cos \lambda } \right)}^{2} } \right]}^{{ - 1}} + S{\left( \lambda \right)}}} $$

where

$$ S{\left( \lambda \right)} = {\sum\limits_{j - 1}^3 {{\left[ {r{\left( {\frac{{4{\left( {\cos \lambda - \cos \omega _{j} } \right)}^{2} }} {{1 - 2\cos \omega _{j} \cos \lambda + \cos ^{2} \omega _{j} }}} \right)}^{2} } \right]}} }^{{ - 1}} , $$

ρ is defined as above, r is the signal-to-noise ratio relative to the seasonal component (the estimated value is 48,660,207, meaning that the weekly seasonality is practically time-invariant) and $\omega _{j} = {2\pi j} \mathord{\left/ {\vphantom {{2\pi j} 7}} \right. \kern-\nulldelimiterspace} 7$.

The resulting cutoff frequency for normal times is 0.05π corresponding to a period of approximately 40 days. The cutoff frequency for vacation days is 0.10π (approximately 20 days).