# Study based on real data from the French electricity market

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## Efficient estimation of # when d = 2

We look for the best attainable variance among rate-optimal estimators of # that are asymptotically normal. However, we do not have a statistical model in the classic sense, in which the parameter would simply be (#; ; ), because of the nuisance parameters and which are random processes themselves! In order to bypass this difficulty, we first restrict our attention to the case where and are deterministic functions, which enables us to identify our data within a semiparametric regular statistical model. Thanks to classic bounds on semiparametric estimation, we can explicitly compute the optimal (best achievable) variance V opt # (; ). In a second step, allowing and to be random again, we build a one-step correction of our preliminary estimator ^#2;n which has the property of being asymptotically mixed normal, with (conditional) variance equal to V opt # (; ), i.e. thus achieving the optimal variance along deterministic paths.

### Context of electricity forward contracts

The prices of existing forward contracts in the electricity markets are characterized by three time components: the quotation date t and the dates Ts and Te of respectively starting and ending power delivery. Therefore, a forward contract F(t; Ts; Te) will deliver to the holder 1 MWh of electricity, in a continuous way between dates Ts and Te. Such a contract may be bought during a quotation period [t0; T] with T < Ts and it is no more available once t > T. The classic observed contracts are of various delivery periods: one week, one month, one quarter (three months), one season (6 months) or one year. Table 2.1 shows an example of available forward contracts in the French Market on May 23rd, 2015. For example, the contract called “June 2015” will deliver to the holder 1 MWh of electricity for all the hours between June 1st (this is Ts) and June 30th of 2015 (Te). This table also introduces the contracts of relative maturity (denoted by the “ahead” formulation). A “ahead” contract is a contract with constant delivery period but with changing delivery dates. For example, the 2-month-ahead contract is the forward contract “July 2015” when it is quoted on May 31th of 2015 (2 months ahead from the quotation date), and becomes the forward contract “August 2015” on June 1st, 2015 (a jump of contract to stay 2 months ahead from the quotation date). In this study we will only consider the 6 observable monthly contracts (i.e. Te􀀀Ts = 1 month) to estimate # and the volatility processes and . Also, for simplicity, we will drop Te from the notation. In the context of simulated data, we will simulate prices of F(t; Ts) = F(t; Ts; Te), the forward delivering 1 MWh during the period [Ts; Te]. In the context of real data, the price F(t; Ts) is observable.

#### Study based on real data from the French electricity market

The data used for estimation are the 6 available month-ahead forward contracts on the French market (www.eex.com) from December 6th, 2001 to December 31st, 2013. On this history, we get 145 periods of 1 month (n ≃ 20) where 6 processes (the 6 month-ahead contracts) are jointly observed, whereas we get 141 periods of 5 months (n ≃ 100) where 2 processes (the 1 month-ahead and the 2 month-ahead contracts) are jointly observed. These numbers of periods are given in Table 2.6 for all the configurations described in Section 2.3.2. In the same column, Table 2.6 also precises the number of periods on which the estimator converges, convergence meaning that the value of ^#d;n or ~#2;n is not zero (see the definitions of the estimators). And the same table gives the estimation results of # for all the possible configurations, with the average value and the standard deviation of the estimators.

Experiment on model errors using simulated data

In this part, inspired from the study on real data, we evaluate the impact of error terms on the estimation of # thanks to simulation. More precisely, we consider an additive Gaussian noise ϵ with mean 0 and variance 1. Because the configurations differ in the number of observations, we propose to add a noise of which standard deviation is proportional to some power of n. It will be a function of parameters (; ) : for all j = 1; : : : ; d and i = 1; : : : ; n, the standard deviation of nj ϵj i is equal to nj = 10􀀀 n . To simulate data, we set = 2:1865 y􀀀1/2 and = 0:23550 y􀀀1/2, which are the averages of the values that we estimated on the 30 last days in the experiment on real data of Section 2.3.3. Moreover, we used two different values for #, 2.0936 y􀀀1 and 26.065 y􀀀1 (coming from estimation on real observations, see Table 3.1) and 2 values of the exponent (1 and 2).
Figures 3.1, 3.2, 3.3 and 3.4 show the estimators of # in the four cases that are defined by the two values of # and the two values of . In each of those four cases, we have performed 100,000 simulations for various values of . Each graph represents the estimation results (average and empirical confidence interval) of the instances of ^#2;n, ^#3;n and ^#6;n that converged, together with the ones of #3;n, with respect to different values of . In a given configuration, it may happen that a great percentage of the estimators do not converge (as they all rely on inverting some function, which is not always possible). Most of the time, such a situation is met for = 1 and for low values of . We do not plot the points corresponding to cases in which less than 15% of the instances of the considered estimator converged. When the noise variance is small ( = 2) and for the smallest of the two values of #, we see in Figure 3.1 that the averages of the four estimators are close to the true value, plotted in solid black line, when the parameter is high. As it decreases, we see that all the estimators( ^#2;n (plotted in green), #3;n (in thin black), ^#3;n (red) and ^#6;n (blue)) take a wider range of values, which is, yet, still centered at the true value.
For the highest value of #, we see in Figure 3.2 that while the values of ^#2;n and #3;n are not very sensitive to the increasing of error, the ones of ^#3;n and ^#6;n decrease: it appears that these two estimators tend to be biased when is low.

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Résumé
Abstract
1 Introduction
1.1 Introduction
1.2 Première partie : estimation efficace dans un modèle à deux facteurs
1.3 Deuxième partie : estimation en présence d’erreurs de modèle
1.4 Troisième partie : exécution optimale sur le marché de l’électricité
1.5 Description des marchés de gros de l’électricité
1.6 Perspectives
2 Efficient estimation in a two-factor model from historical data
2.1 Introduction
2.1.1 Motivation
2.1.2 Setting
2.1.3 Main results and organization of the chapter
2.2 Construction of the estimators and convergence results
2.2.1 Rate-optimal estimation of #
2.2.2 Rate-optimal estimation of the volatility processes
2.2.3 Efficient estimation of # when d = 2
2.2.4 Discussion on the case d 3
2.3 Numerical implementation
2.3.1 Context of electricity forward contracts
2.3.2 Results on simulated data
2.3.3 Study based on real data from the French electricity market
2.4 Proofs
2.4.1 Preliminaries : localization
2.4.2 Proof of Theorem 2.1
2.4.3 Proof of Theorem 2.2
2.4.4 Proof of Theorem 2.3
2.4.5 Proof of Theorem 2.4
2.5 Appendices
2.5.1 Proof of Lemma 2.4.1
2.5.2 Technical lemmas
2.5.3 Some histograms from the numerical experiments
3 Estimation in a two-factor model incorporating model errors
3.1 Introduction
3.1.1 Motivation
3.1.2 Setting
3.1.3 Main results
3.2 Construction of the estimators and convergence results
3.2.1 Estimators of #
3.2.2 Nonparametric estimation of the volatility processes
3.2.3 Efficient estimation of # in presence of model errors, when d = 2
3.3 Numerical implementation
3.3.1 Results on real data
3.3.2 Experiment on model errors using simulated data
3.3.3 Impact of model errors on nonparametric estimation
3.4 Proofs
3.4.1 Preliminaries : localization
3.4.2 Proof of Theorem 3.1
3.4.3 Proof of Theorem 3.2
3.4.4 Proof of Theorem 3.3
3.4.5 Proof of Theorem 3.4
3.4.6 Proof of Theorem 3.5
3.4.7 Proof of Theorem 3.6
3.5 Appendices
3.5.1 Technical lemmas
3.5.2 Plots of nonparametric estimators with model errors
4.1 Introduction
4.2 Problem formulation
4.3 Optimal execution without delay in production
4.3.1 Auxiliary optimal execution problem
4.3.2 Approximate solution
4.3.3 Numerical results
4.4 Jumps in the residual demand forecast
4.4.1 Auxiliary optimal execution problem
4.4.2 Approximate solution
4.4.3 Numerical results
4.5 Delay in production
4.5.1 Explicit solution with delay
4.5.2 Numerical results
4.6 Appendices
4.6.1 Proof of Theorem 4.1
4.6.2 Proof of Theorem 4.2

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