\section{Conditional Approach}


Following the conditional approach\index{tail fitting!conditional} of \cite{McNeil2000271}, and the use of a similar model for the electricity spot prices as given by \cite{BYSTROM}, the data were modelled with the use of an AR-GARCH model, and the ACER method was then applied for estimation of the residual tail quantiles. To accommodate the heavy tailed data, which are typically observed in finance, the proposed procedure of Section~\ref{sec:Q} was used.
For these returns, the goal was to fit an AR-GARCH model\index{AR-GARCH model} where the seasonality in the process was modeled by an AR process\index{AR process}. The terms included in this AR process were the AR(1) and AR(7) terms because of the clear seasonality over the day and over the week. For modeling the volatility, a GARCH(1,1) process\index{GARCH(1,1) process} \citep{Bollerslev1986307} was chosen. This provides a model that should be able to capture the serial correlation over the week and the observed heteroscedasticity\index{heteroscedacity}. The model then assumes the form,
\begin{eqnarray}
	R_t &=& a_0 +  a_{1}R_{t-1} + a_{7}R_{t-7} + \sigma_t Z_t, \\
	\sigma_t^2 &=& \alpha_0 + \alpha_1 \sigma_{t-1}^2 Z_{t-1}^2 + \beta_1 \sigma_{t-1}^2,
\end{eqnarray}
where $R_t$ denotes daily log return rates of spot prices $X_t$, that is, $R_t = \log (X_t / X_{t-1})$. The $Z_t$ are iid random variables of mean value zero and variance equal to 1.0, while $a_0, a_{1}, a_{7}, \alpha_0, \alpha_1, \beta_1$ are all non-negative constants. In this chapter, it will be assumed that $Z_t$ is Normal or Student's $t$ distributed, scaled to unit variance. The conditional quantiles, for these two models, can then be calculated as,
\begin{equation}
	q_{t,\alpha}^* = a_0 + a_{1}R_{t-1} + a_{7}R_{t-7} + \sigma_t q_{\alpha},
\end{equation}
where $q_{\alpha}$ is the standard $\alpha$-quantile of the Normal- or Student's $t$-distribution\index{student's $t$-distribution}. 

For the heavy tails observed in this type of data, this standard AR-GARCH process will not be sufficient to model the tails of the return distribution accurately. The error distribution simply cannot be assumed to be Normal or Student's $t$, as will be observed later. Introduced by \cite{McNeil2000271}, and applied to Nord Pool spot prices by \cite{BYSTROM}, the extreme value theory approach has proven itself superior to the use of a standard AR-GARCH approach. This approach was used here, but instead of using only a POT fitted GP distribution for estimation of the tail quantiles for the standardized residual distribution, the ACER method was also applied. The performance of this method was then compared to the performance of the conditional model where the POT method was used. After using the ACER and POT methods to estimate the tail quantiles of the residual distribution, the conditional quantiles for these models were calculated as,
\begin{equation}
	q_{t,\alpha}^* = a_0 + a_1 R_{t-1} + a_7 R_{t-7} + \sigma_t \hat{q}_{\alpha},
	\label{eq:condquan}
\end{equation}
where $\hat{q}_{\alpha}$ is the quantile of the residual distribution, associated with probability $\alpha$, estimated by the POT or ACER method. The standardized residuals obtained by the Normal AR-GARCH filter were still slightly serially correlated, but most of the heteroscedacity was removed. These conditional quantiles may also be regarded as the $\mathrm{VaR}(\alpha)$ estimate when considering the distribution of the lower tail. For the in sample performance test, the conditional tail quantiles were estimated for all observations, and then the number of empirical exceedances over these quantiles were compared to what was to be expected. When dealing with the out of sample performance, the initial step was estimating a model using the first five years, and then this model was used to predict the tail quantiles for the next day. The model was then re-estimated with the last five years of the data, or a rolling window of length five years, for each day, and the tail quantiles for the next day were predicted. As for the in sample performance test, the empirical exceedances over the predicted tail quantiles are compared to what is expected.

The reason for the pre-filtering of the data set, was the desire to use a GARCH process to model the volatility and to accommodate for sudden changes in the volatility. The use of an AR process for the autocorrelation is mainly because of the POT method's need of iid observations. There is no such iid requirements for the ACER method, so the use of an AR process is strictly not needed in this case.
