\section{Results}

Using the models introduced, the goal is to analyze the ten years of daily spot price data\index{spot price data} from the Nord Pool\index{Nord Pool}. As an introduction, let us start off with an unconditional tail fitting\index{tail fitting!unconditional}, that is, apply the POT and ACER methods directly to the return series. Then, the conditional approach\index{tail fitting!conditional} will be introduced, as detailed in \cite{McNeil2000271}, where both the ACER and POT methods will be used to fit the residuals produced by the AR-GARCH filtering\index{AR-GARCH filtering}. 


\subsection{Unconditional approach}

As mentioned, the unconditional approach will be to apply the POT and ACER method directly to the return series. To compare the performance of these two methods, both in sample and out of sample fit will be considered. For the in sample fit, the methods are simply used to estimate the tail quantiles of the return distribution, and these quantiles will be compared to what is actually observed. For the out of sample test, the first five years of the data will be used to estimate the model and then predict the tail quantiles for the following day. The rolling window is then moved to the next day and the model is re-estimated and the quantiles for the next day is calculated from the estimated parameters. This is repeated for the last five years of the data set.

For the POT method, it is necessary for the data used to be iid, which from the Ljung--Box test results is clearly not the case. There is both significant serial correlation and volatility clustering. To deal with this problem, the data will be declustered by extracting peaks with enough lags in between that it is reasonable to assume independence between the observations. For the ACER method, there is no need to decluster the data as the correlation between lags is accounted for in the choice of ACER function.

In Table~\ref{tab:insampuncon}, the number of empirical exceedances over the estimated in sample quantiles is presented for both methods, along with the number of expected exceedances over these quantiles. It is observed from this table that there is not a great difference between the number of exceedance over the estimated quantiles for the two methods. For the out of sample performance the number of exceedances over the estimated quantiles can be observed in Figure~\ref{tab:outsampuncon}. Again, there is no great difference between the results of the two methods, but the ACER method is again slightly more accurate than the POT method. A problem with doing such out of sample prediction, is that the last five years of the data are used with no more emphasis on what happened yesterday than five years ago. This means that the estimated quantiles, and thus risk measures, will need a long time to be able to incorporate a rise or fall in volatility. This again leads to the effect that most of the exceedances over the predicted quantiles will be observations from the periods with high volatility, while one would ideally like the exceedances to be uniformly distributed over the period in question.

\begin{table}
	\centering
	\caption{In sample performance of the unconditional methods.}
	\begin{tabular}{c c c c}
	\hline
	Probability & Expected & POT & ACER \\
	\hline
	$.95$ & $183$ & $175$ & $179$ \\
	$.99$ & $37$ & $38$ & $38$ \\
	$.995$ & $18$ & $19$ & $19$ \\
	$.999$ & $4$ & $3$ & $4$ \\
	$.9995$ & $2$ & $3$ & $2$ \\
	$.9999$ & $0$ & $1$ & $0$ \\
	\hline
	\end{tabular}
\label{tab:insampuncon}
\end{table}

\begin{table}
	\centering
	\caption{Out of sample performance of the unconditional methods.}
	\begin{tabular}{c c c c}
	\hline
	Probability & Expected & POT & ACER \\
	\hline
	$.95$ & $91$ & $99$ & $99$ \\
	$.99$ & $18$ & $29$ & $24$ \\
	$.995$ & $9$ & $17$ & $11$ \\
	$.999$ & $2$ & $2$ & $2$ \\
	$.9995$ & $1$ & $1$ & $1$ \\
	$.9999$ & $0$ & $0$ & $0$ \\
	\hline
	\end{tabular}
\label{tab:outsampuncon}
\end{table}



\subsection{Conditional approach}

To be able to accommodate for sudden changes in volatility, at least to some extent, the conditional approach is used. The first step is to filter the data with an AR-GARCH process. For this, a GARCH(1,1) process with AR parameters for the 1st and 7th lag is applied before using the POT and ACER methods for fitting to the standardized residuals. The residuals are standardized with the current volatility. The parameters for the AR-GARCH model with both normal and $t$ distributed errors are presented in Table~\ref{tab:garchparam}, with the standard errors in brackets. For the model with normally distributed errors, all parameters are significant at a $0.01$ significance level, except $\alpha_0$ which is significant at a $0.1$ significance level and $\mu$ which is non-significant. For the model with $t$ distributed errors, all parameters are significant at $0.01$ significance level, except $\mu$ which is non-significant.

Descriptive statistics and Ljung--Box test results for the residual series, that is, the residual series after pre-filtering with the normal AR-GARCH model, can be found in Table~\ref{tab:desstatres}. For the residual series there is still positive skewness and high excessive kurtosis. It is observed that while there is still significant serial correlation, it has been greatly reduced, and there are no significant GARCH effects in the residual series. It was observed that it is possible to remove slightly more of the serial correlation by including more AR-terms, but the difference is minimal so the model with less parameters is preferred. 
\begin{table}
	\caption{Estimated AR-GARCH parameter values.}
	\label{tab:garchparam}
	\centering
	\begin{tabular}{c | l l}
		\hline
		Parameter & \: Value-N & \: Value-$t$\\
		\hline
 		$\mu$ & $2.997\cdot10^{-3}(1.69\cdot10^{-3})$ & $-1.628\cdot10^{-3}(1.14\cdot10^{-3})$ \\
 		$a_1$ & $-0.338(2.95\cdot10^{-2})$ & $-0.283(1.57\cdot10^{-2})$ \\
 		$a_7$ &   $0.388(2.33\cdot10^{-2})$ & $0.425(1.45\cdot10^{-2})$ \\
 		$\alpha_0$ &  $2.985\cdot10^{-4}(1.62\cdot10^{-5})$ & $8.518\cdot10^{-4}(2.16\cdot10^{-4})$ \\
 		$\alpha_1$ & $6.362\cdot10^{-2}(1.43\cdot10^{-2})$ & $0.455(8.93\cdot10^{-2})$\\
 		$\beta_1$  & $0.931(1.70\cdot10^{-2})$ & $0.740(2.58\cdot10^{-2})$ \\
 		$\nu$ & - & $2.582(0.14)$ \\
 		\hline
	
	\end{tabular}
\end{table}



\begin{table}
	\centering
	\caption{Descriptive statistics for residual series.}
	\label{tab:desstatres}
	\begin{tabular}{c c c c c c c c c}
	Mean & Skew. & Ex.Kurt. & $Q(1)$ & $Q(2)$ & $Q(7)$ & $Q^2(1)$ & $Q^2(2)$ & $Q^2(7)$ \\
	\hline
	$2.90\cdot10^{-3}$ & $4.44$ & $97.72$ & $6.70$ & $102.91$ & $141.82$ & $7.20\cdot10^{-3}$ & $8.60\cdot10^{-3}$ & $3.67\cdot10^{-1}$ 
	\end{tabular}
\end{table}


After filtering the return series, the POT and ACER methods are applied to the series of standardized residuals. As this series is much closer to iid than the return series, and observations over the chosen threshold seem to be independent of each other, there is no need to decluster the data in the same way that was done with the unconditional method. Nevertheless, it should be noted that only the observations over the chosen threshold are used, which in this case will be less than $10\%$ of the data. Using the POT method to fit a GP distribution  to the data, with the threshold $u$ selected from inspection \citep{cole01}, the parameters presented in Table~\ref{tab:peaksparam} are obtained. Here $\lambda$ is the empirical estimate of $\mathrm{P}(X>u)$. Inverting Eq.~\eqref{eq:pot.4} for the desired probabilities, gives us the POT estimated quantiles, which in turn is inserted into Eq.~\eqref{eq:condquan} to get the conditional quantiles. For the ACER method, the same procedure is used, and the parameters for the extrapolated ACER function can be found in Table~\ref{tab:acerparam}.


For the in sample performance of these two methods, the same procedure as for the unconditional approach is used. In Table~\ref{tab:insampleqright}, the number of exceedances over a given quantile for the different methods are presented. It is seen that for the standard AR-GARCH model with standard normally distributed errors, the quantiles are severely underestimated for all quantile levels, and for the same model with $t$ distributed errors the quantiles for the lower levels are severely overestimated. It is also observed that the AR-GARCH model, where the POT method has been applied to the standardized residuals, clearly is able to estimate the extreme quantiles much better than just a standard AR-GARCH model. This is the same as was found by \cite{BYSTROM}. 


\begin{table}
	\caption{Generalized Pareto distribution parameters from POT.}
	\label{tab:peaksparam}
	\centering
	\begin{tabular}{c c}
		\hline
		Parameter & Value \\
		\hline
		$\sigma$ & $0.5318$ \\
		$\xi$ 	 & $0.3118$\\
		$\lambda$& $9.32\cdot10^{-2}$ \\
		$u$ & $1$\\
		\hline
	\end{tabular}
\end{table}


\begin{table}
	\caption{Parameters estimated for the ACER method.}
	\label{tab:acerparam}
	\centering
	\begin{tabular}{c c}
		\hline
		Parameter & Value \\
		\hline
		$\tilde{a}$ & $0.254$ \\
		$b$ 	 & $0.010$ \\
		$c$ & $1.181$\\
		$q$ & $0.46$ \\
		$\xi$ & $0.334$ \\
		\hline
	\end{tabular}
\end{table}


\begin{table}
	\caption{Predicted in sample right quantiles for the different methods.}
	\label{tab:insampleqright}
	\centering
	\begin{tabular}{c | c c c c c}
		\hline
		Probability & Expected & AR-GARCH-N & AR-GARCH-$t$ & C-POT & C-ACER \\
		\hline
		0.95 & 182 & 128 & 32 & 187 & 182 \\
		0.99 & 37 & 57 & 9 & 34 & 35 \\
		0.995 & 18 & 43 & 4 & 19 & 18 \\
		0.999 & 4 & 28 & 2 & 4 & 4 \\
		0.9995 & 2 & 23 & 1 & 3 & 1 \\
		0.9999 & 0 & 17 & 0 & 1 & 0 \\
		\hline
	
	\end{tabular}
\end{table}


To compare the performance of the POT and ACER methods, it is important to assess the out of sample fit for the two methods. To do this, a model is estimated using the first five years of the data, and then the conditional quantiles for the next day are estimated. The model is then reestimated using what is now the last five years of the data, and again the next day conditional quantiles is predicted. This gives us a period of five years for the out of sample prediction. In Table~\ref{tab:outsampleqright}, the number of exceedances over the predicted out of sample quantiles is presented. It is seen from this, that the performance of the conditional POT and the conditional ACER method is quite similar, with the conditional ACER method giving a slightly better out of sample fit for the more extreme quantiles. In Table~\ref{tab:yeardist}, the counted exceedances over the predicted $95\%$ day ahead quantile is presented. It is seen that the distribution of the exceedances is fairly even over the years. The exceedances are well distributed and do not become more frequent in the high volatility periods. 

\begin{table}
	\caption{Exceedances over predicted out of sample quantiles. Right tail.}
	\label{tab:outsampleqright}
	\centering
	\begin{tabular}{c | c c c c  c}
		\hline
		Probability & Expected & AR-GARCH-N & AR-GARCH-$t$ & C-POT & C-ACER \\
		\hline
		0.95 & 91 & 117 & 32 & 94 & 92 \\
		0.99 & 18 & 42 & 9 & 17 & 17 \\
		0.995 & 9 & 27 & 4 & 11 & 10 \\
		0.999 & 2 & 12 & 2 & 4 & 2 \\
		0.9995 & 1 & 11 & 1 & 0 & 1 \\
		0.9999 & 0 & 4 & 0 & 0 & 0 \\
		\hline
		\end{tabular}
\end{table}


\begin{table}
	\caption{Yearly distribution of exceedances for the conditional ACER and POT method. Out of sample.}
	\label{tab:yeardist}
	\centering
	\begin{tabular}{c c c}
		\hline
		Year & C-POT & C-ACER  \\
		\hline
		1 & 16 & 16 \\
		2 & 17 & 17\\
		3 & 19 & 20\\
		4 & 24 & 21\\
		5 & 18 & 18\\
		\hline	
	\end{tabular}
\end{table}



