## 1. Introduction

As one of the fundamental industry inputs, the electricity market is unique in its instantaneous settlement process and extra difficulty encountered during the storage process. To satisfy the fast-changing market demand and supply, as well as to fully utilize the generator’s power, the electricity market is increasingly deregulated to promote its efficiency and response time to market demand. The electricity market generally has a higher level of deregulation compared to the other commodity markets. Its price movement demonstrates volatile behaviors and peculiar patterns [

1]. The market is perceived to have a high level of exposure to external shocks, and contains significant market risk level [

1,

2]. Therefore, accurately measuring the downside market risk exposure represents a pivotally important and difficult practical and research problem for investors and researchers in the electricity market [

1].

So far, risk measurement research is rather limited in the electricity field [

3]. For example, on the design and use of financial derivatives in the electricity markets, Shenoy and Gorinevsky [

4] proposed a new data-driven stochastic model to price the forward contract in the Pennsylvania-New Jersey-Maryland (PJM) electricity market. References [

5,

6] analyzed the effectiveness of forward and futures contracts to manage market risk in electricity markets [

5,

6]. On the construction and use of Value at Risk (VaR) as the important risk assessment technique in the electricity markets, Dahlgren et al. [

7] conducted a critical literature review on the use of VaR as an important risk assessment technique and demonstrated its effectiveness for energy trading risk assessment in the electric power markets [

7]. Both [

8,

9] used the Extreme Value Theory (EVT) to estimate VaR in electricity markets, and found improved estimation accuracy [

8,

9].

However, prevalent methodologies in risk measurement are constructed based on homogeneous market assumptions and the Efficient Market Hypothesis (EMH). It views the market investors as consistent, rational, and homogeneous players in the fast-changing and volatile market environment per se. These assumptions provide an insufficient level of approximations when describing the complex electricity market environment. They need to be relaxed in order to account for the heterogeneous nonlinear market dynamics, where the price movements demonstrate fractal and multiscale behaviors in empirical studies [

10,

11]. To model these data characteristics, multiscale models (such as the popular wavelet analysis, etc.) have recently attracted significant research attention in the risk measurement literature. For example, [

12] combined the wavelet analysis and regime switching model to estimate electricity VaR. However, the performance of the wavelet-based approach is constrained by the limited amount of wavelet basis available in the literature. Thus, the Empirical Mode Decomposition (EMD) model was developed as a new data-driven empirical approach to model the multiscale data features. The basis is not pre-defined in the EMD model, but rather is defined adaptively during the model fitting process [

13,

14,

15]. In recent years, the EMD model was introduced from the engineering field into the economic and finance field, and we have witnessed wider applications. For example, Premanode et al. [

16] proposed the average intrinsic noise function to obtain more smoothed exchange rate data, which were modeled and forecasted using the multi-class support vector regression. Premanode and Toumazou [

15] proposed the differential EMD model to improve the exchange rate forecasting accuracy of the support vector regression model. Wu [

17] used the EMD model to explore the phase correlation of foreign exchange rates [

17]. Zhang et al. [

18] used Ensemble EMD (EEMD) to analyze crude oil price [

18]. EMD model has also been combined with different neural network models to improve its forecasting accuracy effectively. An et al. [

19] showed that the EMD model improves the forecasting accuracy of the Feed-Forward Neural Network model [

19]. Dong et al. [

20] showed that the EMD model effectively separated volatility and daily seasonality in electricity prices, leading to improved forecasting accuracy [

20]. However, as the performance of neural networks is sensitive to the parameters chosen and the initial parameter values, and it is difficult to appropriately assign the performance improvement contributions to either the neural network model or the EMD model. The contribution of the combined EMD model to performance improvement is not conclusive in the literature.

In this paper, we propose an EMD-Exponential Weighted Moving Average (EWMA) VaR estimation model to measure the market risk level. The introduced EMD algorithm is used to analyze the risk evolution in the electricity market, and has been combined with the traditional risk measurement methodologies in order to analyze the heterogeneous market structures and improve the risk measurement accuracy. Empirical studies are conducted using the Australian electricity market data. Performance evaluations against the traditional benchmark models show the superior performance of the EMD-EWMA model in dealing with heterogeneous unstationary electricity market risk data.

The contributions of the work in this paper are twofold. Firstly, we introduced the EMD model to characterize the multiscale data feature with the projection of the original risk measures into different risk factors in the multiscale domain. The distinct data patterns of different underlying data components across different scales are analyzed and modeled. Secondly, in the newly-proposed EMD-EWMA-based VaR estimation model, the heterogeneous multiscale data feature is recognized and modeled with the introduced EMD model. The time varying mixture of different Data Generating Processes (DGPs) is modeled with the EWMA model in the projected EMD domain. The heterogeneity of the investment strategies among different investors is taken into account. Since we employ the simple and robust EWMA model to construct the EMD-EWMA model, the improved risk estimate accuracy can be attributed to the introduced EMD model.

The organization of the rest of the paper is as follows.

Section 2 reviews the VaR theory and provides a detailed account of the EMD-based VaR estimation model. The performance of the proposed model has been evaluated using the extensive Australian electricity market data sets. Results have been reported and analyzed in

Section 3.

Section 4 provides some summarizing remarks.

## 3. Empirical Studies

In this paper, we use the extensive empirical data in the Australian electricity markets to conduct the experiments to evaluate the performance of the proposed model. The Australian electricity markets are chosen since it is one of the most deregulated markets in the world, representing some geographically diverse regions. The data sets are constructed using the daily observations from five sub-regions, including New South Wales (NSW), Queensland (QLD), South Australia (SA), Victoria (VIC), and Tasmania (TAS). The data is obtained from the website of the Australian National Energy Market (NEM). Except for NSW, negative and empty value are spotted in the other four markets due to potential recording errors. They are replaced with the smoothed values calculated using the interpolation method. The time period for the data set is from 1 January 2004 to 6 November 2014, except for the TAS market, which starts from 16 May 2005. The total number of observations is 3963, except for the TAS market, where the number of observations is 3462. The data set is divided based on a 70% ratio to be used for different purposes during the experiment. The daily price data have been transformed to scale free return data as in ${y}_{t}=ln\frac{{p}_{t}}{{p}_{t-1}}$. In this paper, VaR is estimated at daily frequency, and a one day holding period is assumed during VaR calculation.

To obtain the descriptive idea about the statistical characteristics of the data, we calculate the statistical moments and conduct the statistical tests. Four statistical moments include the mean, standard deviation, skewness, and kurtosis. The statistical tests include the Jarque-Bera test for normality and the Brock-Dechert-Scheinkman (BDS) test of independence [

28,

29,

30].

Table 1 lists the descriptive statistics of returns for the five electricity markets.

From the results in

Table 1, the electricity market returns deviate from the standard normal distribution, and there is significant risk exposure for investors in the market. There is significant standard deviation. The market returns lean towards loss on average, indicated by the negative skewness value. More importantly, we observe significantly large kurtosis value, deviating from the normal level, and indicating that the return changes significantly, partly because of an abnormal event. Besides, since both JB and BDS tests reject the null hypothesis, we can conclude that the market return distribution does not conform to a normal distribution and deviates from the independence.

In the meantime, different markets also exhibit their own unique characteristics. This is due to the limited physical transfer capability among different regions, as well as different degrees of market development from more deregulated NSWand QLD to the less deregulated SA market. Most notable is the positive skewness value for the SA market and the negative skewness value for other markets. The SA market had the highest level of standard deviation. This implies that among the five markets, the SA market is the most volatile and profitable, on average. This stylized fact is consistent with the unique characteristics of SA markets. The SA region is known for its very hot summer, with the peak electricity demand. The size of the market is relatively small, with limited coal and natural gas supply [

31]. Thus, it behaves significantly differently from the other four markets. Meanwhile, the NSW market has the highest level of kurtosis. This implies that the degrees of extreme event influences vary among the five markets, where the impact of the extreme events is the most significant in NSW markets. It is the most deregulated and developed market, subject to external shocks and extreme events.

Then, we conducted empirical studies using the Australian electricity data set to evaluate the performance of the proposed model. To evaluate the model’s generalizability, we limit the models tested to EWMA model with elliptical distributions, including the normal and Student’s

t distributions. Although different distributions exist in the literature, normal and Student’s

t distributions are the most commonly used normal and non-normal distributions in the literature [

32,

33]. There have been research results reported on the use of non-normal distributions, such as skewed normal, skewed Student’s

t distributions, etc. [

34], but no consensus exists for their robustness and accuracy in capturing the empirical data distributions. In the meantime, the EWMA model is the most robust model, whose parameters optimization is less sensitive to the data set. Thus, the performance improvement with the proposed model can be attributed to the EMD model employed. The results and findings obtained can generalize to more complex risk measurement models such as the GARCH model with different underlying distributions. The performance improvements may vary, but are expected to be significant in different market circumstances.

In this paper, we use different kinds of forecasting measures to evaluate the model performance. These include the number of VaR exceedances, the

p value for the Kupiec backtesting procedure, and the Mean Squared Error (MSE).The performance of different models under standard normal distribution are listed in

Table 2.

From results in

Table 2, it can be seen that the exceedances of the proposed EMD-EWMA model under all the confidence levels are higher than that of the EWMA model. As far as

p value is concerned, our proposed model achieved mixed performance under different confidence levels against the benchmark EWMA. It performs better at the

$95\%$ confidence level, but it performs worse at the

$99\%$ confidence level. Overall, the EMD-EWMA model does not show significant improvement in risk coverage. Results in

Table 2 show that MSEs of the proposed EMD-EWMA model are lower than that of the EWMA model under all three confidence levels(

$95\%$,

$97.5\%$,

$99\%$). The predictive accuracy of the proposed EMD-EWMA model largely improves upon the traditional EWMA model.

To further improve the model performance, we noted that electricity market return distribution may not satisfy the normal distribution during the VaR estimation. Since the electricity market returns do not conform to standard normal distribution, we use the Student’s

t distribution to estimate VaR and conduct the empirical studies, and we use

${T}_{\alpha}$ representing Student’s

t distribution instead of

${Z}_{\alpha}$. The corresponding exceedances,

p values, and MSEs are listed in

Table 3.

Results in

Table 3 show that the performance of the proposed EMD-EWMA model using the Student’s

t distribution improve significantly upon the benchmark models, in terms of both risk coverage and predictive accuracy.

Firstly, the exceedances of EMD-EWMA model using Student’s t distribution are higher than that of EWMA using Student’s t distribution. Secondly, traditional EWMA model tends to overestimate VaR, shown by the relatively lower Kupiec p values. On the contrary, most p values of the EMD-EWMA model are over 0.05, which shows great risk coverage. Moreover, except in TAS and NSW under the $99\%$ confidence level, all the other p values of the EMD-EWMA model are higher than that of EWMA. Thirdly, as for MSE, the EMD-EWMA model demonstrates the improved predictive accuracy by a large margin compared to the EWMA model. For SA, TAS, VIC, and QLD markets, the MSEs of the EMD-EWMA model are lower than that of the EWMA model. For the NSW market, the MSE is only slightly higher than that of the EWMA model. The proposed model performs competently in terms of predictive accuracy in the NSW market.

More importantly, the utilized EWMA model in the proposed EMD-EWMA model is widely recognized as the most robust and stable model, due to its simple form. It is nested within the proposed EMD-EWMA model. Thus, the performance improvement of the proposed model can be attributed to the introduction of the EMD model to analyze and model the additional data features refreshed in the multiscale data domain.