Do jumps and cojumps matter for electricity price forecasting? Evidence from the German-Austrian day-ahead market

Electric Power Systems Research 212 (2022) 108144 Available online 22 June 2022 0378-7796/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Do jumps and cojumps matter for electricity price forecasting? Evidence from the German-Austrian day-ahead market Aitor Ciarreta a,∗, Peru Muniain b, Ainhoa Zarraga c aDepartment of Economic Analysis, University of the Basque Country, UPV/EHU. Avda. Lehendakari Aguirre, 83. 48015 Bilbao, Spain bDepartment of Applied Mathematics, University of the Basque Country, UPV/EHU. Torres Quevedo Ingeniaria Plaza, 1. 48013 Bilbao, Spain cDepartment of Quantitative Methods, University of the Basque Country, UPV/EHU. Avda. Lehendakari Aguirre, 83. 48015 Bilbao, Spain ARTICLE INFO Keywords: Price forecasting Jumps Cojumps Elastic net Variance stabilizing transformations ABSTRACT This paper analyzes the potential for including jumps and cojumps in electricity price forecasting models. The study is carried out on the German–Austrian day-ahead electricity market with a multivariate framework in which each hour of the day is treated as an individual time series. Three models are specified: The ARX model, the ARX-J model (which includes jumps), and the ARX-J-CJ model (which also includes cojumps). Prices are transformed using several variance stabilizing transformations. The forecasting performance of the three models with original and transformed prices is compared using several forecast horizons running from one day-ahead to one week-ahead. Results show that the forecast horizon is crucial in determining whether jumps and cojumps should be included in electricity price forecasting. Jumps and cojumps add important information to forecast prices for horizons longer than 4 days, but there is no gain in forecast accuracy for shorter horizons. The results are of interest to market participants for taking optimal decisions and pricing base week futures contracts. 1. Introduction With the liberalization of electricity markets, a need to understand electricity price formation has arisen. Once prices are modeled appropriately, they are predicted. Market participants need different price forecast horizons depending on the economic decisions that they have to make at any given time. A better understanding of electricity price dynamics is therefore crucial. However, electricity prices have unique characteristics not found in other commodities that make forecasting them difficult. Electricity cannot be stored, at least on a large scale, and it must be available and managed upon demand. This makes electricity prices very volatile and leads to frequent spikes. Recent large scale deployment of smart grid communication networks enables suppliers and customers to manage their power based on metered data, thus reducing volatility and shaving spikes [2]. Taking into account these features, many research papers have tackled the modeling and forecasting of electricity prices in different markets. Modeling approaches can be divided into four categories: Fundamental models, artificial intelligence-based models, hybrid models, and statistical models [3]. Given the nature of electricity price formation, incorporating jumps into these categories is a leap forward towards improving the forecasting ability of the models. [4] analyze ∗Corresponding author. E-mail addresses: [email protected] (A. Ciarreta), [email protected] (P. Muniain), [email protected] (A. Zarraga). 1Properties of different jump tests are analyzed in [1]. the role of jumps in electricity price modeling. Since their paper, jump tests have been widely used to detect spikes in electricity price series data.1Fundamental models require an extensive representation of the electricity system to simulate the market clearing mechanism before estimation and forecasting take place [5]. Artificial intelligence models include machine-learning models and deep-learning models (examples are [6], and [7]). Hybrid models combine different models with the goal of achieving higher accuracy in forecasting (examples are [8,9], and [10]). Finally, statistical models use historical price data and some external price-related information (mainly weather and load forecasts) to predict prices for different time frames. Examples include [3]. Our work belongs to this last category. Statistical models have the advantage of enabling an interpretation of their components to be obtained. They therefore help market participants to understand the relationship between electricity prices and their determinants, and to take their decisions accordingly. In particular, we estimate autoregressive models with exogenous variables (ARX), where prices depend on their past and other exogenous factors. The role of cojumps, defined as jumps occurring at the same time in different time series, has also been studied in the literature, especially in financial markets. For instance, [11] find evidence of cojumps in https://doi.org/10.1016/j.epsr.2022.108144 Received 12 January 2022; Received in revised form 4 May 2022; Accepted 26 May 2022 Electric Power Systems Research 212 (2022) 108144 2 A. Ciarreta et al. stock prices using different approaches, and [12] show that the forecast accuracy of asset returns variance improves when jumps and cojumps are considered. However, to the best of our knowledge, the relevance of cojumps in electricity markets has not been analyzed. This paper analyzes whether jumps and cojumps add relevant information to electricity price forecasting. To that end, univariate and multivariate frameworks can be adopted. The former considers one time series for all prices, while the latter divides the whole time series into several series, one for each load period. As an example, [13] focus on ARX models to compare univariate and multivariate frameworks and conclude that there is a slight gain in forecast accuracy using the latter. We use German–Austrian electricity day-ahead auction prices in a multivariate framework, so 24 price series are constructed, one for each hour. Following [14], we transform prices applying different variance stabilizing transformations, which are intended to smooth series and improve forecasts. We specify three ARX models: First, the ARX model (which includes no jumps or cojumps); second, the ARX-J model (which includes jumps); and third, the ARX-J-CJ model (which includes jumps and cojumps). Jumps and cojumps are detected in the residuals of the ARX model because they are seasonally adjusted. This prevents spikes that are purely seasonal effects from being flagged as jumps or cojumps. As far as we know, there are no other analyses that use this approach to detect jumps and cojumps. Among the several widely-used jump tests applied in the literature, we use the one proposed by [15], hereinafter LM, which is applicable for daily data. Cojumps are constructed following [11]. We assess the role of jumps and cojumps in electricity price forecasting. To that end, jumps and cojumps are embedded in ARX statistical models. To the best of our knowledge, in electricity markets jumps have mostly been used to forecast electricity price volatility and there is little research into their use for predicting electricity prices directly. Furthermore, cojumps have not been analyzed in electricity markets to date. ARX models are usually over-parameterized, which makes them hard to estimate using OLS. Estimation methods with a shrinkage property have therefore been applied in the literature. [16] proposes the lasso estimation method, which allows variable selection, and [17] propose the elastic net estimation method, which imposes the so-called lasso and ridge penalties on the OLS estimation to reduce the number of variables. [18] apply different estimation methods with the shrinkage property in electricity price forecasting, and conclude that the elastic net is the best-performing method. Therefore, we estimate the three ARX models using the elastic net for the original and transformed price series using a rolling window. Forecasting is then carried out with the following seven days of each window being forecast for all 24 h of each day. Finally, the mean absolute error (MAE) and the root mean squared error (RMSE) out-of-sample criteria are used to assess the forecasting performance. The difference in forecasting performance between models in pairs is compared using a multivariate approach via the [19] test. Interesting results are obtained regarding the role of jumps and cojumps and price transformations in electricity price forecasting depending on the forecast horizon. The day-ahead market is the one with the highest liquidity in the German–Austrian zone of the European Power Exchange (EPEX). The participating agents need signposts to decide their bidding strategies optimally. Those signposts are the forecasts made for the following days’ prices. This highlights the importance of forecasting as accurately as possible. Moreover, in the European Energy Exchange (EEX) there is trading of base, peak and off-peak products for electricity with cash settlement in the German–Austrian zone known as Phelix (Physical Electricity Index).2The underlying prices of these future products are based on the German–Austrian EPEX day-ahead hourly prices.3The 2See https://www.eex.com/en/markets/trading-ressources/indices. 3In EPEX, hourly, half-hourly and quarter-hourly prices are set in the intraday continuous market. Phelix base price product is calculated as the mean of all hourly prices in the delivery period. For weekly products it is calculated as the mean of the 168 hourly prices. Therefore, we reasonably concentrate on forecasting the corresponding underlying prices for different time horizons. Note that to forecast weekly products it is necessary to forecast prices for intermediate horizons. It is in this context that we explore the role of jumps and cojumps in price forecasting. In summary, our contribution to the literature is the following: •We build an ARX model that embeds jumps or/and cojumps. •We detect jumps and cojumps in the seasonally adjusted residuals of the ARX model. •We measure the accuracy of the models in forecasting prices from one day to one week ahead. •We analyze the forecast for Phelix base week futures contracts including jumps and cojumps in different forecasting horizons. The rest of the paper is organized as follows. Section 2explains the methodology used. Section 3describes the data used in the analysis. Section 4shows the estimation and forecasting results for all models and transformations. Section 5summarizes and concludes. 2. Methodology Price forecasting follows several steps. In the first, the price series is divided into 24 series, one for each hour, which are transformed using several variance stabilizing transformations (VST). These transformations make the time series smoother, thus improving forecast performance. In the second step, an ARX model is specified and estimated using the elastic net method. Next, jumps are detected in the residuals of the estimated model by applying the LM test. Once jumps are detected in each of the 24 time series of the original and transformed prices, cojumps are detected as per [11]. The ARX model is then expanded including only jumps and both jumps and cojumps, resulting in the ARX-J and the ARC-J-CJ models, respectively, also estimated via the elastic net. Finally, price forecasting accuracy is assessed in each model and for original and transformed prices using RMSE and MAE criteria. The forecasting performance of the different pairs of models is compared using the multivariate approach in the [19] test, hereinafter DM. The subsections below provide a detailed explanation of each step. 2.1. Variance stabilizing transformations Based on [14], different VSTs are applied. All the transformations used are applicable with negative prices. The objective of these transformations is to obtain transformed price series which are easier to forecast and then to apply the inverse of the transformation to recover the forecast prices. In total 6 different transformations are used: 3𝜎, logistic, area hyperbolic sine, mirror-logarithmic, and probability integral transformation using both normal and Student-t cumulative distributions. In the first four transformations standardized prices are obtained before the transformation is applied. In addition to the common standardization that uses the standard deviation of prices, a second standardization that uses the median absolute deviation of prices and is more robust to outliers, is also applied.4Once prices are forecast, the standardization process is undone. 4For the rest of the paper we use subscripts 1 and 2 after the name of the VST to indicate that prices have been standardized using the standard deviation and the median absolute deviation, respectively. For the sake of simplifying the notation, we denote as 𝑝both the original and standardized prices. Electric Power Systems Research 212 (2022) 108144 3 A. Ciarreta et al. The 3𝜎transformation smooths the series, thus decreasing the effect of outliers in price forecasting. Following [14], the transformation is made as follows: 𝑦𝑑,ℎ =⎧ ⎪ ⎨ ⎪ ⎩ 3sign(𝑝𝑑,ℎ)if |𝑝𝑑,ℎ|>3 𝑝𝑑,ℎ if |𝑝𝑑,ℎ|≤3 where 𝑝𝑑,ℎ denotes the price at day 𝑑of the time series corresponding to hour ℎ. By construction, the 3𝜎transformation does not have an inverse. The logistic transformation has often been applied in data analytics, but as far as we know, it has only been applied as a VST in electricity price forecasting by [14]. The transformation is: 𝑦𝑑,ℎ =(1 + 𝑒−𝑝𝑑,ℎ )−1 After forecasting, the inverse transformation is used to recover the forecast of the original price as: 𝑝𝑑,ℎ = log (𝑦𝑑,ℎ 1 − 𝑦𝑑,ℎ ) The area hyperbolic sine (asinh) has been used as a VST in electricity data when modeling negative prices (see [13,20], and [14]). It preserves the behavior of the logarithmic transformation for positive prices but is also defined for negative prices. The transformed prices are calculated as: 𝑦𝑑,ℎ =asinh (𝑝𝑑,ℎ)=𝑙𝑜𝑔 (𝑝𝑑,ℎ +√𝑝2 𝑑,ℎ + 1) with the corresponding inverse transformation: 𝑝𝑑,ℎ =sinh (𝑦𝑑,ℎ) The mirror-logarithmic (mlog) transformation is a generalization of the logarithmic transformation to make it applicable for negative prices (see [14]). The transformation is constructed as: 𝑦𝑑,ℎ =sign (𝑝𝑑,ℎ)[log (|𝑝𝑑,ℎ|+1 𝑐)+ log (𝑐)] The mlog transformation depends on the constant 𝑐, which is set to 𝑐=1 3following [14]. Consequently, the inverse transformation is: 𝑝𝑑,ℎ =sign (𝑦𝑑,ℎ)[𝑒|𝑦𝑑,ℎ|−log (𝑐)−1 𝑐] The last transformation considered is based on the so-called probability integral transformation (PIT), constructed using the empirical cumulative distribution as an approximation of the unknown true distribution of the time series (see [14]): 𝑦𝑑,ℎ =𝛷−1( 𝐹𝑝(𝑝𝑑,ℎ)) where 𝛷−1 is the inverse cumulative distribution and  𝐹𝑝is the empirical cumulative distribution of the price series 𝑝. Both the normal (N-PIT) and the Student-t with eight degrees of freedom (T-PIT) cumulative distributions are considered. The inverse of the transformation is: 𝑝𝑑,ℎ = 𝐹𝑝 −1(𝛷(𝑦𝑑,ℎ)) 2.2. Models Three different ARX-type models are estimated. The first is the ARX model, based on the fARX model proposed by [13]: 𝑝𝑑,ℎ =𝛽 ⏟⏟⏟ Constant + 24 ∑ ℎ=1 7 ∑ 𝑖=1 𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ Autoregressive effects + 6 ∑ 𝑗=1 𝛾0,𝑗W𝑗 𝑑 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ Day-of-the-week effects +𝜖𝑑,ℎ (1) where 𝑝refers to the original price or the transformed price (𝑦), W𝑗 𝑑is a dummy variable for day 𝑗of the week, and 𝜖𝑑,ℎ is the error term with mean 0 by construction. The second term accounts for up to seventh order autoregressive and cross-period effects (effects of each hour from up to 7 days ago). The third term accounts for seasonality. Jumps are included in model (1), resulting in the ARX-J model. The sign of the jumps might differ depending on the hour of the day, so this model considers both positive and negative jumps: 𝑝𝑑,ℎ =𝛽 ⏟⏟⏟ Constant + 24 ∑ ℎ=1 7 ∑ 𝑖=1 𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ Autoregressive effects + 6 ∑ 𝑗=1 𝛾0,𝑗W𝑗 𝑑 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ Day-of-the-week effects + 7 ∑ 𝑖=1 𝜃𝑝 𝑖𝑃𝐽𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ Positive jumps + 7 ∑ 𝑖=1 𝜃𝑛 𝑖𝑁𝐽𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ Negative jumps +𝜖𝑑,ℎ (2) where 𝑃𝐽𝑑,ℎ is a dummy variable that takes a value of one if there is a positive jump on day 𝑑in time series ℎ, and zero otherwise. Analogously, 𝑁𝐽𝑑,ℎ is a dummy variable that takes a value of one if there is a negative jump on day 𝑑at hour ℎ, and zero otherwise. The ARX-J model is expected to capture the behavior of prices more accurately at the tails of the distribution. The third model proposed, ARX-J-CJ, includes jumps and cojumps: 𝑝𝑑,ℎ =𝛽 ⏟⏟⏟ Constant + 24 ∑ ℎ=1 7 ∑ 𝑖=1 𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ Autoregressive effects + 6 ∑ 𝑗=1 𝛾0,𝑗W𝑗 𝑑 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ Day-of-the-week effects + 7 ∑ 𝑖=1 𝜃𝑝 𝑖𝑃𝐽𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ Positive jumps + 7 ∑ 𝑖=1 𝜃𝑛 𝑖𝑁𝐽𝑑−𝑖,ℎ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ Negative jumps +𝜃𝑐𝐶𝐽𝑑−1 ⏟⏞⏟⏞⏟ Cojumps +𝜖𝑑,ℎ (3) where 𝐶𝐽𝑑is a dummy variable that takes a value of one if a cojump is detected on day 𝑑and 0 otherwise. Note that it is equal for all time series. The ARX-J-CJ model accounts for correlation between jumps by considering cojumps, which are jumps that occur on the same day across different hours. The ARX-J-CJ model not only accounts for correlation by cross-period effects; it also takes into account correlation in the tails, through the cojump variable. Finally, we also consider the following naive model as a benchmark: 𝑝𝑑,ℎ =𝑝𝑑−1,ℎ +𝜖𝑑,ℎ, where price at a given hour is determined by the price at the same hour of the previous day. It should be noted that other factors such as load and weather forecasts might also affect prices. However, these factors are not included in the models because the focus of the paper is to forecast prices up to seven days ahead and data are available day-ahead, so they cannot be used to predict prices beyond horizon one. 2.3. Jump and cojump detection The LM jump test is applied to the residuals of the ARX model estimated (Eq. (1)), thus avoiding spikes that could be explained by seasonal effects. The test compares the size of a standardized observation to a threshold so that it can be assessed whether a significant jump has occurred or not. First, a window size must be selected. According to LM, the optimal choice of the window size is 𝐾= 20.5Thus, the local variation at day 𝑑and for hour ℎis estimated as: 𝜎𝑑,ℎ2=1 𝐾− 2 𝑑−1 ∑ 𝑗=𝑑−𝐾+2 |𝜖𝑗,ℎ||𝜖𝑗−1,ℎ|, where 𝜖𝑗,ℎ is the residual from the estimated ARX model for day 𝑗and hour ℎ. 5Each hour of the day is analyzed separately and the data frequency is daily, so 𝐾=⌈√365⌉. Electric Power Systems Research 212 (2022) 108144 4 A. Ciarreta et al. The standardized residual is 𝑧𝑑,ℎ =𝜖𝑑,ℎ 𝜎𝑑,ℎ . The asymptotic distribution of the maximums of the test statistic in the absence of jumps converges to a Gumbel variable.6The LM test identifies significant jumps but does not indicate their sign. Hence, the sign of the corresponding price is checked to determine the jump sign. The jump detection procedure is usually applied in a single iteration. However in this paper an iterative jump detection procedure is followed, because if jumps are close together the detection of the second jump may be affected. Jumps detected in the iteration are therefore set to the mean of the previous 𝐾observations and the jump test is rerun until no more jumps are detected or a maximum of five iterations is reached. Cojumps over different hours of the same day are detected following the approach proposed by [11].7Specifically, day 𝑑is classified as a cojump day, i.e. 𝐶𝐽𝑑= 1, if a jump is detected in at least two hours of day 𝑑, i.e. if 24 ∑ ℎ=1 𝐽𝑑,ℎ ≥2, where 𝐽𝑑,ℎ = 1 if a jump is detected on day 𝑑and at hour ℎ, and 0 otherwise. 2.4. Estimation Models (1),(2) and (3) are estimated using the elastic net method introduced by [17], thus solving the poor estimation of OLS when the number of parameters to be estimated is large. Estimation is carried out using a rolling window of size 𝐷. The size of the window has to be large enough to properly estimate the model but not too large, as the effect of the variables might change over time. The window is then moved one day forward and the estimation procedure is repeated. In total, there are 𝑁different windows of size 𝐷. The elastic net estimator is obtained by solving the following optimization problem (see [17]):   𝜷ℎ=argmin 𝜷∈R𝐿 ⎡⎢⎢⎢⎢⎢⎢⎣ 𝐷 ∑ 𝑑=1 (𝑝𝑑,ℎ − 𝑿𝑑,ℎ𝜷)2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ OLS estimation term +𝜆(1 − 𝛼 2 𝐿 ∑ 𝑖=1 𝛽2 𝑖,ℎ +𝛼 𝐿 ∑ 𝑖=1 |𝛽𝑖,ℎ|) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Penalty term ⎤⎥⎥⎥⎥⎥⎥⎦ , where 𝑝𝑑,ℎ and  𝑿𝑑,ℎ are the scaled price and the scaled regression matrix on day 𝑑and at hour ℎ, respectively, so that 𝑝𝑑,ℎ and each column of  𝑿𝑑,ℎ have zero mean and standard deviation one. 𝐿is the number of parameters to be estimated and 𝜆and 𝛼are the tuning parameters, which take values between zero and one, and characterize the penalty term for including variables. When 𝛼= 1 the elastic net estimation method is identical to the lasso penalty proposed by [16], while if 𝛼= 0 the elastic net results in the ridge penalty first introduced by [21]. Following [18], 𝛼is set at 0.5.8 The optimum value of 𝜆is selected by 10-fold block cross-validation (see [22]). [23] show that the number of observations, the number of parameters, the variance and the correlation are taken into consideration when selecting the tuning parameter. 6Using a 10% significance level, the threshold for the test statistic is − log(− log(0.9)) = 2.25. See [15] for more detail. 7The authors propose two different approaches to detect cojumps. However, one of them considers intraday prices and cannot therefore be applied to daily observations. Thus, only one method is included in this paper. 8We also use lasso, 𝛼= 1, but the forecast results do not change significantly. Results are available upon request. Once the parameters are estimated by solving the optimization problem, the unscaled elastic net estimations  𝜷ℎare obtained by rescaling   𝜷ℎ. Finally, we compare the goodness of fit of the models using the adjusted R-squared for each rolling window in the estimation of the three models (using original and transformed prices). 2.5. Forecast Once models (1),(2) and (3) are estimated, prices for each hour are predicted for the following 7 days. The MAE and RMSE out-of-sample criteria are used to assess forecasting performance over the 𝑁rolling windows and the 7 horizons. Both criteria are widely used in the literature on forecasting in electricity markets, for instance in [13,14,18,24], and [25]. By construction, the MAE criterion is optimal for median forecasts while the RMSE is optimal for mean forecasts. The MAE criterion for horizon 𝑘and hour ℎis calculated as follows9: 𝑀𝐴𝐸ℎ,𝑘 =1 𝑁 𝑁 ∑ 𝑑=1 |𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘|, where 𝑝𝑑,ℎ,𝑘 and 𝑝𝑑,ℎ,𝑘 are the observed and predicted price on day 𝑑, at hour ℎand horizon 𝑘, respectively. The mean error across all the hours of the day is calculated as in [14]: 𝑀𝐴𝐸𝑘=1 24𝑁 24 ∑ ℎ=1 𝑁 ∑ 𝑑=1 |𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘|(4) Analogously, the RMSE measure for the horizon 𝑘and hour ℎis calculated using the square error instead of the absolute error as10: 𝑅𝑀𝑆𝐸ℎ,𝑘 =√ √ √ √1 𝑁 𝑁 ∑ 𝑑=1 (𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘)2, and the corresponding error across all the hours of the day (see [14]) is: 𝑅𝑀𝑆𝐸𝑘=√ √ √ √1 24𝑁 24 ∑ ℎ=1 𝑁 ∑ 𝑑=1 (𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘)2(5) To determine whether the differences in forecasting performance of the models is significant the multivariate version of the DM test is applied, using the absolute errors for the MAE criterion and the squared errors for the RMSE criterion as loss functions. 3. Data description The data used in this paper are day-ahead prices from the former German–Austrian electricity market.11 This is a fully integrated market which sets a single price for both countries. On day 𝑑−1 the market sets prices for the 24 h of day 𝑑according to the following mechanism: First, market agents submit electricity sale and purchase bids up to 12 pm on day 𝑑− 1. Then the system aggregates the bids to demand and supply functions, and finally the intersection between the supply and demand curves determines the quantity traded and the market price for each hour of day 𝑑. This was the market with the highest level of liquidity in the EPEX power exchange market. The data run from 1st January 2014 to 30th September 2018, which was the last day on which the German– Austrian day-ahead market operated.12 In total, there are 41,616 hourly prices and 1734 days in the sample period. 9See for example [26]. 10 See for example [26]. 11 Data available on the ENTSOE transparency platform. 12 After this date, the German and Austrian energy regulators agreed to split their combined day-ahead market zone. This came as a result of frequent transmission congestion between the two grids and the resulting costly redispatching to deal with it. The Luxembourg electricity market subsequently joined the German market to form a single zone. Electric Power Systems Research 212 (2022) 108144 5 A. Ciarreta et al. Table 1 Descriptive statistics for original and transformed prices. Transformation Mean Median Minimum Maximum Std. Dev. Skewness Ex. Kurtosis Original 33.50 32.34 −130.09 163.52 15.04 −0.12 6.42 3𝜎10.10 0.00 −3.00 3.00 1.17 0.15 0.30 Logistic10.52 0.50 0.00 1.00 0.23 0.05 −0.70 Asinh10.07 −0.00 −3.35 3.13 0.89 0.05 −0.37 Mlog10.02 0.00 −1.75 1.57 0.32 0.06 0.74 N-PIT 0.00 0.00 −4.06 4.06 1.00 0.00 −0.00 T-PIT 0.00 0.00 −7.89 7.89 1.15 0.01 1.39 Descriptive statistics for original and transformed price series using the standard deviation for standardization. Std. Dev. and Ex. Kurtosis stand for Standard Deviation and Excess Kurtosis, respectively. Fig. 1. Number of cojumps detected in each rolling window. Table 1 reports the main descriptive statistics for the original and transformed price series.13 As expected, the variability of prices decreases significantly when transformed data are used. Specifically, original prices range from e−130.09 to e163.52/MWh with a standard deviation of e15.04/MWh. By contrast, the largest spread in transformed prices is 15.78 and occurs for the T-PIT transformation. The original prices show negative skewness and excess kurtosis. The heavy tails of the distribution might indicate the presence of jumps. The distribution of the transformed prices is close-to-normal except for the N-PIT, whose distribution is normal. The original and transformed price series are divided into 24 time series, one for each hour of the day. All the series are stationary according to the ADF unit root test. To estimate models (1),(2), and (3), a rolling window of size 𝐷= 730 (two years) is used. The initial rolling window starts on 1st January 2014 and ends on 31st December 2015. 7 horizons are forecast in each window, and 𝑁= 998 different rolling windows are considered. Note that the first 730 observations of the sample are used to forecast the first price. According to the LM test, there are significant jumps in both original and transformed prices in most of the 24 h and each rolling window. Fig. 1 shows the number of cojumps detected in the residuals of the ARX model in each rolling window for the original and transformed prices.14 As expected, the largest number of cojumps is detected in 13 For 3𝜎, logistic, asinh, and mlog transformations the common standardization has been applied to prices. Statistics for standardized prices with the median absolute deviation do not change significantly and are not shown. They are available upon request. 14 These results are for standardized prices using the standard deviation. Results for standardized prices using the median absolute deviation are available upon request. the original price series. Observe that the number of cojumps detected rises to almost 70 at the end of the period in which the market was in place. During periods of highly variable renewable generation, negative electricity price spikes occur because prices drop sharply to feed that renewable generation into the grid. In periods of low renewable generation when thermal units fill the demand gap, prices may spike as a response to unexpected fossil fuel fluctuations, as happened in 2018. The spikes observed may happen in adjacent hours, so cojumps are also identified. This seems to be particularly so in early 2017 and late 2018. For the transformed series cojumps are detected in each rolling window, although they are fewer in number. In general, the T-PIT is the transformed price series with the largest number of cojumps, especially during the first half of the sample. This result is expected, as the probability distribution of T-PIT transformed prices is heavytailed. By contrast, the number of cojumps detected is lower in the N-PIT transformed price series. 4. Estimation and forecast results 4.1. Estimation results Models (1),(2), and (3) are estimated and the corresponding adjusted R-squared is calculated for each rolling window. Table 2 reports the mean values of the adjusted R-squared for all hours and each model, and for original and transformed prices. The comparison within each transformation shows that the best model in terms of goodness of fit is the ARX. 4.2. Forecast results Models (1),(2), and (3) are used to forecast prices for 1 to 7 days ahead for each rolling window. Electric Power Systems Research 212 (2022) 108144 6 A. Ciarreta et al. Table 2 Mean adjusted-R2for all hours. Original 3𝜎13𝜎2Logistic1Logistic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT ARX 0.655 0.683 0.675 0.686 0.688 0.688 0.688 0.683 0.679 0.686 0.680 ARX-J 0.653 0.681 0.674 0.684 0.687 0.687 0.687 0.682 0.678 0.685 0.679 ARX-J-CJ 0.653 0.681 0.674 0.684 0.687 0.687 0.687 0.682 0.678 0.685 0.679 Mean adjusted R-squared criterion for all hours, models, and price transformations. Subscripts 1 and 2 indicate that prices are standardized using the standard deviation and the median absolute deviation, respectively. A heat map is used to indicate higher (green) and lower (red) values within each transformation. Table 3 Mean MAE for all hours. Model Transf. H1 H2 H3 H4 H5 H6 H7 Naive 8.396 10.754 11.297 11.530 11.603 10.706 9.389 ARX Original 5.598 7.327 7.845 8.116 8.311 8.475 8.563 3𝜎15.584 7.153 7.639 7.908 8.119 8.290 8.382 3𝜎25.530 7.167 7.675 7.951 8.156 8.315 8.401 Logistic15.722 7.270 7.734 7.993 8.180 8.343 8.444 Logistic25.544 7.178 7.672 7.941 8.144 8.310 8.405 Asinh15.558 7.229 7.718 7.989 8.187 8.351 8.445 Asinh26.254 7.310 7.734 7.991 8.229 8.392 8.483 Mlog15.500 7.190 7.703 7.976 8.181 8.346 8.444 Mlog25.507 7.198 7.705 7.982 8.188 8.351 8.447 N-PIT 5.529 7.134 7.638 7.919 8.128 8.308 8.402 T-PIT 5.523 7.125 7.627 7.910 8.129 8.304 8.404 ARX-J Original 5.623 7.316 7.848 8.114 8.274 8.428 8.520 3𝜎15.615 7.174 7.658 7.925 8.123 8.281 8.378 3𝜎25.553 7.167 7.671 7.943 8.115 8.261 8.361 Logistic15.746 7.324 7.787 8.042 8.219 8.374 8.471 Logistic25.579 7.207 7.701 7.967 8.159 8.314 8.409 Asinh15.590 7.275 7.770 8.025 8.214 8.366 8.460 Asinh26.285 7.349 7.785 8.038 8.259 8.411 8.496 Mlog15.533 7.226 7.728 7.982 8.177 8.331 8.435 Mlog25.532 7.209 7.716 7.973 8.169 8.319 8.424 N-PIT 5.550 7.171 7.672 7.959 8.164 8.338 8.445 T-PIT 5.548 7.137 7.631 7.911 8.120 8.305 8.413 ARX-J-CJ Original 5.626 7.316 7.847 8.116 8.281 8.436 8.536 3𝜎15.614 7.178 7.666 7.931 8.129 8.288 8.385 3𝜎25.553 7.169 7.673 7.946 8.122 8.268 8.368 Logistic15.747 7.326 7.786 8.039 8.218 8.373 8.472 Logistic25.579 7.211 7.700 7.969 8.164 8.314 8.410 Asinh15.593 7.277 7.770 8.027 8.216 8.370 8.466 Asinh26.281 7.354 7.785 8.040 8.257 8.411 8.497 Mlog15.534 7.218 7.727 7.986 8.176 8.336 8.442 Mlog25.539 7.217 7.725 7.989 8.180 8.331 8.438 N-PIT 5.555 7.175 7.674 7.964 8.166 8.341 8.451 T-PIT 5.547 7.138 7.624 7.904 8.117 8.296 8.405 Mean MAE criterion (Eq. (4)) for all hours per horizon (H1 to H7), model, and price transformation. Subscripts 1 and 2 indicate that prices are standardized using the standard deviation and the median absolute deviation, respectively. A heat map is used to indicate lower (green) and higher (red) forecast errors within each horizon. Model selection involves two criteria. First, we measure the forecasting performance of the models by sorting them according to the value of the MAE and RMSE criteria and then we choose the ones with the lowest values for each criterion and forecast horizon. Second, we run the multivariate approach of the DM test to determine whether forecasts for each pair of models are significantly better in one of them. Tables 3 and 4show the forecasting errors of each model for original and transformed prices, for the naive model and for the seven horizons (H1 to H7) using the MAE and RMSE criteria across all hours of the day, i.e. Eqs. (4) and (5), respectively.15 Regardless of the forecasting horizon, results show that the naive model provides the largest errors, which means that at least terms accounting for correlation between prices at a given hour and day and their lags should be considered as explanatory variables. For the 15 The results of the DM test for the MAE criterion are reported in Appendix and those using the RMSE criterion are available from the authors upon request. rest of the models, the VST results show that none of the models are selected with the original prices, so it is important to smooth price series so as to obtain more accurate forecasts. However, not all the transformations are equally good: logistic1, logistic2, and asinh1 transformations do not provide better forecasts as they are not selected for the best models. By contrast, 3𝜎2, mlog1, mlog2, and T-PIT are, in general, the transformations with the best forecast performances. These are the transformations for which price distribution has excess kurtosis, so it is important to accurately capture the behavior at the tails because jumps are observations that fall at the tails of the distribution. Market participants take decisions that depend on the time horizon under consideration. Forecasting is relevant in day-to-day market operations of EPEX and risk management in the EEX futures markets for different delivery periods. We discuss results ranging from the closest-to-delivery one day-ahead forecast to a one week-ahead price forecast. •Horizon 1: Day-ahead forecasting is important for electricity trading and plant operation scheduling decisions. The ARX model Electric Power Systems Research 212 (2022) 108144 7 A. Ciarreta et al. Table 4 Mean RMSE for all hours. Model Transf. H1 H2 H3 H4 H5 H6 H7 Naive 13.351 16.535 17.444 17.803 17.905 16.727 15.195 ARX Original 8.959 11.235 11.859 12.163 12.397 12.558 12.636 3𝜎19.397 11.225 11.801 12.133 12.376 12.530 12.613 3𝜎29.121 11.141 11.776 12.113 12.360 12.513 12.592 Logistic19.506 11.399 11.981 12.313 12.537 12.688 12.773 Logistic29.148 11.234 11.840 12.178 12.423 12.574 12.660 Asinh19.030 11.245 11.850 12.195 12.446 12.601 12.695 Asinh29.623 11.252 11.769 12.084 12.349 12.527 12.603 Mlog18.891 11.150 11.771 12.106 12.359 12.514 12.613 Mlog28.894 11.143 11.755 12.094 12.342 12.505 12.599 N-PIT 9.081 11.211 11.830 12.185 12.435 12.598 12.688 T-PIT 8.981 11.159 11.786 12.133 12.388 12.543 12.642 ARX-J Original 9.007 11.277 11.900 12.187 12.380 12.541 12.631 3𝜎19.445 11.264 11.840 12.159 12.386 12.525 12.614 3𝜎29.161 11.166 11.785 12.114 12.324 12.461 12.557 Logistic19.550 11.490 12.057 12.373 12.580 12.719 12.806 Logistic29.205 11.289 11.886 12.216 12.442 12.583 12.673 Asinh19.083 11.318 11.922 12.245 12.481 12.621 12.715 Asinh29.676 11.316 11.824 12.131 12.379 12.544 12.612 Mlog18.941 11.212 11.813 12.123 12.355 12.506 12.617 Mlog28.941 11.191 11.796 12.101 12.338 12.486 12.594 N-PIT 9.116 11.264 11.879 12.226 12.470 12.625 12.729 T-PIT 9.032 11.195 11.801 12.133 12.383 12.540 12.659 ARX-J-CJ Original 9.022 11.273 11.904 12.196 12.390 12.558 12.651 3𝜎19.448 11.269 11.846 12.164 12.391 12.534 12.619 3𝜎29.163 11.165 11.789 12.120 12.329 12.467 12.568 Logistic19.569 11.499 12.057 12.373 12.582 12.720 12.813 Logistic29.207 11.292 11.888 12.221 12.449 12.584 12.670 Asinh19.085 11.322 11.924 12.251 12.488 12.626 12.725 Asinh29.655 11.324 11.826 12.134 12.376 12.542 12.617 Mlog18.940 11.207 11.814 12.126 12.362 12.508 12.624 Mlog28.942 11.198 11.810 12.122 12.357 12.503 12.612 N-PIT 9.126 11.273 11.886 12.237 12.477 12.632 12.740 T-PIT 9.039 11.200 11.799 12.133 12.378 12.538 12.650 Mean RMSE criterion (Eq. (5)) for all hours per horizon (H1 to H7), model, and price transformation. Subscripts 1 and 2 indicate that prices are standardized using the standard deviation and the median absolute deviation, respectively. A heat map is used to indicate lower (green) and higher (red) forecast errors within each horizon. with mlog transformed prices outperforms the rest. Furthermore, models that include jumps and/or cojumps are not selected. Thus, the inclusion of jumps as explanatory factors does not improve the forecast. •Horizon 2: The results are not that conclusive. Under the MAE criterion, ARX, ARX-J, and ARX-J-CJ are candidate models for selection using T-PIT transformation. However, the DM test finds no significant differences between them or with respect to mlog and N-PIT transformations. For the RMSE criterion, in general the ARX model gives the lowest error and the DM test never selects models with jumps and cojumps. These results are in line with those for horizon 1, so information on jumps does not help to forecast prices two days ahead. •Horizon 3: The results differ depending on the criterion. Under MAE the best performing model is the ARX-J-CJ with T-PIT transformed prices, followed by the ARX model with the same transformation. However, the DM test finds no significant differences between them. By contrast, RMSE selects the ARX model for the mlog2transformed prices. DM results show that for the ARX model the difference in error measures between the mlog2 and the asinh2transformations is not significant. These results are qualitatively similar to previous horizons, so there is no clear gain from including jumps or cojumps. •Horizon 4: Under the MAE criterion, ARX-J-CJ with the T-PIT transformation is the best model, but the error difference with respect to the ARX model for 3𝜎1and T-PIT transformations is not significant according to the DM test results. The results for the RMSE criterion differ because the model with the smallest error is the ARX with the asinh2transformed prices. However, DM results show no significant differences between the forecasting accuracy in this case and that of the three models for 3𝜎2, mlog1and mlog2transformations. Therefore, results are qualitatively similar to those for previous horizons. •Horizon 5: MAE and RMSE criteria select the ARX-J model with the 3𝜎2transformation as the best model. However, the DM test using MAE shows that it does not outperform the ARX-J-CJ model with the T-PIT transformation. Moreover, the same test using RMSE shows no clear evidence of superiority for any of the three models. Five days ahead there is some evidence that models that incorporate jumps and/or cojumps as explanatory variables provide better price forecasts. Factors that contribute to the occurrence of price shocks are expected to become more likely as the forecast horizon becomes longer. •Horizon 6: MAE and RMSE criteria select the ARX-J model with the 3𝜎2transformation as the best model. This result is confirmed by the DM test using MAE, which shows this model to be significantly superior to the ARX-J-CJ model. However, the DM test using RMSE shows that it does not outperform the ARX-J-CJ model with the same transformation. Nor is it superior to any of the models with the mlog transformation. Information on jumps and/or cojumps is therefore more significant in forecasting. These results reinforce the findings for horizon 5 and show that hedging is important in longer horizons too. •Horizon 7: MAE and RMSE criteria select the ARX-J model with the 3𝜎2transformation as the best model. This result is confirmed by DM testing using either criterion, which shows it to be significantly superior to the ARX and ARX-J-CJ models. As expected, these results are in line with those for horizons 5 and 6, and Electric Power Systems Research 212 (2022) 108144 8 A. Ciarreta et al. highlight the gain from taking into account the information on jumps and/or cojumps in managing risk. Given that the Phelix base product is calculated as the mean of all 168 hourly forecast prices, including jumps in the models improves the Phelix base weekly product. To summarize, in terms of forecast accuracy, the ARX model outperforms models that incorporate jumps and/or cojumps for the shortest horizons. Therefore, electricity trading and plant operation scheduling decisions do not benefit from information on jumps and/or cojumps. However, as the forecast horizon lengthens, incorporating jumps and/or cojumps into the estimation of the models improves forecasting accuracy. This could be because jumps are extremely rare, short-lived events, so the likelihood of their occurring increases with time. These results have implications for pricing weekly products in futures markets. For instance, the way in which the Phelix base product for one-week delivery is calculated shows the importance of models that incorporate jumps and cojumps as explanatory variables. 5. Summary and conclusions Price modeling and forecasting have become challenging since electricity markets were liberalized. This is especially relevant with the large-scale deployment of renewable energy production, integration with neighboring markets, and increased use of financial products. Electricity prices also exhibit unique characteristics that make these tasks more complex. One of those characteristics is the presence of spikes. We use day-ahead prices from the German-Austrian electricity market for the period from January 1, 2014 to September 30, 2018 to analyze the role of jumps and cojumps in price forecasting. It should be noted that this is the market with the greatest liquidity in Germany and Austria, even after market decoupling. It is therefore important to model price dynamics accurately to forecast prices several periods ahead. Price series for each hour of the day are considered, leading to a multivariate framework. We specify three models: The ARX model; the ARX-J model, which includes jumps; and the ARX-J-CJ model, which includes jumps and cojumps. Cojumps are defined as jumps that occur on the same day. We also transform the price series using several variance stabilizing transformations. Our results show that using 3𝜎2, mlog1, mlog2, and T-PIT variance stabilizing transformations provides more accurate forecasts of prices than considering original price data. Furthermore, including jumps and cojumps as covariates further improves price forecasting only for horizons beyond four days. These conclusions are also of interest to participants in the futures market. Electricity markets around the world are encouraging market agents to participate in futures markets, and the decision to do so is taken after profitability analyses. In particular, Phelix futures are traded on the EEX market. Hence, day-ahead price forecasting helps participants to optimize their bidding strategies for the following days and decide whether to participate in the futures market or not. Finally, we consider the following lines for future research. First, it should be noted that there is an intraday auction that sets prices every 15 min, and a continuous market with several products (every 15 min, every 30 min and hourly). The incorporation of large-scale intermittent renewable generation and the integration of the European market increase the importance of these markets. Our framework of analysis could also be extended to these markets. However, the different frequencies of price formation would need to be carefully considered in specifying the statistical models. Second, models including other factors that may affect electricity prices, such as load, weather forecasts and reserve margin, could also be considered as more system operators begin disclosing such information. In this case, the day-ahead forecast should be considered to ensure that these factors give the information closest to the time of the forecast. Third, taking into account the integration of the European market, price forecasting could also be assessed for several European electricity markets to check for cojumps between them. CRediT authorship contribution statement Aitor Ciarreta: Conceptualization, Methodology, Formal analysis, Writing – review & editing. Peru Muniain: Conceptualization, Methodology, Formal analysis, Software, Writing – review & editing. Ainhoa Zarraga: Conceptualization, Methodology, Formal analysis, Writing – review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to thank two anonymous reviewers for valuable comments and suggestions that helped to improve the paper. Financial support from Dpto. de Educación del Gobierno Vasco, Spain under research grant IT1336-19 and from Ministerio de Ciencia e Innovación, Spain under research grant PID2019-108718GB-I00 is acknowledged. Open access funding provided by the University of the Basque Country. The authors are grateful for valuable comments from participants in the Workshop on Forecasting in Electricity Markets held in Bilbao in 2019. The authors also thank Susan Orbe, Luiggi Grossi and Rafał Weron for helpful comments. Appendix. Results of the multivariate DM test for the MAE criterion See Tables A.1–A.7. Electric Power Systems Research 212 (2022) 108144 9 A. Ciarreta et al. Table A.1 DM using MAE for H1. –ARX– ARX-J –ARX-J-CJ– Orig. 3𝜎13𝜎2Logistic1Logistic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT Orig. 3𝜎13𝜎2Logistic1Logistic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT Orig. 3𝜎13𝜎2Logistic1Logistic2Asinh1Asinh2Mlog1Mlog2N-PIT 3𝜎1−0.18 (0.43) ARX 3𝜎2−1.16 (0.122) −1.99 (0.023) Logistic11.5 (0.933) 3.88 (>0.999) 4.16 (>0.999) Logistic2−0.83 (0.205) −1.47 (0.071) 0.57 (0.715) −6.55 (<0.001) Asinh1−0.75 (0.228) −0.56 (0.289) 0.75 (0.774) −4.28 (<0.001) 0.57 (0.715) Asinh28.81 (>0.999) 6.39 (>0.999) 8.01 (>0.999) 5.13 (>0.999) 7.68 (>0.999) 8.21 (>0.999) Mlog1−2.96 (0.002) −1.51 (0.066) −0.75 (0.227) −3.92 (<0.001) −1.16 (0.123) −2.6 (0.005) −9.78 (<0.001) Mlog2−3.34 (<0.001) −1.31 (0.095) −0.55 (0.291) −3.49 (<0.001) −0.88 (0.189) −1.8 (0.036) −9.9 (<0.001) 0.95 (0.829) N-PIT −1.16 (0.123) −1.35 (0.089) −0.01 (0.496) −4.72 (<0.001) −0.53 (0.298) −1.1 (0.135) −8.1 (<0.001) 0.85 (0.803) 0.58 (0.718) T-PIT −1.51 (0.065) −1.21 (0.113) −0.19 (0.423) −3.62 (<0.001) −0.54 (0.294) −1.05 (0.147) −8.73 (<0.001) 0.73 (0.767) 0.47 (0.681) −0.35 (0.361) Orig. 2.22 (0.987) 0.5 (0.691) 1.59 (0.944) −1.2 (0.116) 1.22 (0.888) 1.22 (0.889) −8.31 (<0.001) 3.68 (>0.999) 4.17 (>0.999) 1.6 (0.945) 2.03 (0.979) 3𝜎10.21 (0.583) 3.78 (>0.999) 2.94 (0.998) −2.93 (0.002) 2.47 (0.993) 1.18 (0.882) −6.04 (<0.001) 2.01 (0.978) 1.79 (0.964) 2.05 (0.98) 1.78 (0.962) −0.11 (0.456) ARX-J 3𝜎2−0.76 (0.225) −1.16 (0.124) 2.62 (0.996) −3.68 (<0.001) 0.31 (0.622) −0.15 (0.44) −7.64 (<0.001) 1.28 (0.899) 1.06 (0.856) 0.72 (0.763) 0.82 (0.794) −1.19 (0.116) −2.29 (0.011) Logistic11.77 (0.962) 4.7 (>0.999) 4.71 (>0.999) 2.18 (0.986) 7.41 (>0.999) 4.79 (>0.999) −4.82 (<0.001) 4.29 (>0.999) 3.85 (>0.999) 5.13 (>0.999) 3.97 (>0.999) 1.48 (0.931) 3.82 (>0.999) 4.28 (>0.999) Logistic2−0.28 (0.389) −0.2 (0.421) 1.84 (0.967) −5.12 (<0.001) 3.6 (>0.999) 0.75 (0.773) −7.15 (<0.001) 1.94 (0.974) 1.6 (0.946) 1.64 (0.95) 1.35 (0.911) −0.66 (0.253) −1.38 (0.084) 1.03 (0.847) −6.59 (<0.001) Asinh1−0.15 (0.441) 0.13 (0.55) 1.58 (0.943) −3.46 (<0.001) 1.84 (0.967) 4.09 (>0.999) −7.72 (<0.001) 3.72 (>0.999) 2.77 (0.997) 2.25 (0.988) 1.93 (0.973) −0.62 (0.268) −0.53 (0.297) 0.98 (0.836) −4.15 (<0.001) 0.44 (0.669) Asinh29.25 (>0.999) 6.76 (>0.999) 8.45 (>0.999) 5.48 (>0.999) 8.11 (>0.999) 8.66 (>0.999) 2.85 (0.998) 10.25 (>0.999) 10.38 (>0.999) 8.53 (>0.999) 9.18 (>0.999) 8.79 (>0.999) 6.42 (>0.999) 8.1 (>0.999) 5.2 (>0.999) 7.62 (>0.999) 8.22 (>0.999) Mlog1−1.84 (0.033) −0.94 (0.174) 0.09 (0.536) −3.42 (<0.001) −0.31 (0.38) −1.14 (0.126) −9.22 (<0.001) 3.7 (>0.999) 2.2 (0.986) 0.11 (0.545) 0.34 (0.631) −2.63 (0.004) −1.49 (0.068) −0.5 (0.308) −3.87 (<0.001) −1.21 (0.114) −2.62 (0.004) −9.78 (<0.001) Mlog2−2.27 (0.012) −0.9 (0.185) 0.05 (0.521) −3.11 (<0.001) −0.3 (0.382) −0.92 (0.178) −9.47 (<0.001) 2.92 (0.998) 2.89 (0.998) 0.06 (0.525) 0.27 (0.605) −3.28 (<0.001) −1.41 (0.08) −0.5 (0.308) −3.51 (<0.001) −1.09 (0.139) −2.02 (0.022) −10.03 (<0.001) −0.15 (0.439) N-PIT −0.78 (0.217) −0.85 (0.198) 0.64 (0.739) −4.22 (<0.001) 0.2 (0.581) −0.3 (0.383) −7.72 (<0.001) 1.39 (0.917) 1.06 (0.856) 2.49 (0.994) 1.31 (0.905) −1.22 (0.112) −1.61 (0.053) −0.08 (0.47) −4.78 (<0.001) −1.01 (0.157) −1.48 (0.069) −8.18 (<0.001) 0.51 (0.696) 0.47 (0.682) T-PIT −0.98 (0.164) −0.73 (0.233) 0.51 (0.694) −3.19 (<0.001) 0.08 (0.532) −0.31 (0.377) −8.29 (<0.001) 1.44 (0.925) 1.14 (0.873) 0.97 (0.835) 2.68 (0.996) −1.5 (0.067) −1.33 (0.091) −0.14 (0.444) −3.6 (<0.001) −0.78 (0.219) −1.23 (0.109) −8.79 (<0.001) 0.47 (0.681) 0.46 (0.679) −0.14 (0.445) Orig. 2.5 (0.994) 0.54 (0.705) 1.65 (0.95) −1.17 (0.122) 1.27 (0.897) 1.29 (0.901) −8.29 (<0.001) 3.8 (>0.999) 4.33 (>0.999) 1.66 (0.951) 2.11 (0.982) 0.63 (0.735) 0.15 (0.558) 1.24 (0.893) −1.45 (0.073) 0.71 (0.761) 0.68 (0.751) −8.77 (<0.001) 2.73 (0.997) 3.42 (>0.999) 1.27 (0.898) 1.56 (0.941) 3𝜎10.2 (0.579) 3.73 (>0.999) 2.9 (0.998) −2.95 (0.002) 2.44 (0.993) 1.17 (0.879) −6.04 (<0.001) 2 (0.977) 1.78 (0.962) 2.03 (0.979) 1.76 (0.961) −0.12 (0.453) −0.42 (0.338) 2.25 (0.988) −3.84 (<0.001) 1.34 (0.911) 0.52 (0.697) −6.42 (<0.001) 1.47 (0.93) 1.39 (0.918) 1.59 (0.944) 1.32 (0.906) −0.16 (0.438) ARX-J-CJ 3𝜎2−0.76 (0.224) −1.15 (0.125) 2.69 (0.996) −3.66 (<0.001) 0.31 (0.621) −0.15 (0.439) −7.65 (<0.001) 1.29 (0.901) 1.07 (0.858) 0.72 (0.763) 0.82 (0.794) −1.2 (0.115) −2.27 (0.012) −0.01 (0.497) −4.26 (<0.001) −1.02 (0.155) −0.98 (0.163) −8.11 (<0.001) 0.5 (0.693) 0.5 (0.693) 0.07 (0.53) 0.14 (0.556) −1.25 (0.105) −2.23 (0.013) Logistic11.76 (0.961) 4.7 (>0.999) 4.69 (>0.999) 2.21 (0.986) 7.19 (>0.999) 4.71 (>0.999) −4.78 (<0.001) 4.24 (>0.999) 3.81 (>0.999) 5.11 (>0.999) 3.96 (>0.999) 1.47 (0.93) 3.82 (>0.999) 4.26 (>0.999) 0.13 (0.551) 6.39 (>0.999) 4.07 (>0.999) −5.16 (<0.001) 3.82 (>0.999) 3.48 (>0.999) 4.76 (>0.999) 3.59 (>0.999) 1.44 (0.926) 3.84 (>0.999) 4.23 (>0.999) Logistic2−0.28 (0.389) −0.2 (0.419) 1.82 (0.965) −5.11 (<0.001) 3.46 (>0.999) 0.74 (0.769) −7.12 (<0.001) 1.92 (0.972) 1.59 (0.944) 1.63 (0.949) 1.34 (0.91) −0.66 (0.254) −1.39 (0.083) 1.02 (0.845) −6.57 (<0.001) −0.05 (0.479) −0.44 (0.331) −7.58 (<0.001) 1.19 (0.883) 1.07 (0.859) 1 (0.842) 0.77 (0.78) −0.71 (0.239) −1.35 (0.088) 1.01 (0.843) −6.37 (<0.001) Asinh1−0.1 (0.462) 0.19 (0.575) 1.67 (0.953) −3.53 (<0.001) 2.04 (0.979) 3.98 (>0.999) −7.61 (<0.001) 3.58 (>0.999) 2.71 (0.997) 2.42 (0.992) 2.02 (0.978) −0.55 (0.29) −0.49 (0.313) 1.07 (0.858) −4.24 (<0.001) 0.57 (0.717) 0.76 (0.775) −8.11 (<0.001) 2.56 (0.995) 2.01 (0.978) 1.65 (0.95) 1.32 (0.907) −0.61 (0.271) −0.47 (0.319) 1.07 (0.858) −4.16 (<0.001) 0.57 (0.717) Asinh29.18 (>0.999) 6.8 (>0.999) 8.5 (>0.999) 5.51 (>0.999) 8.16 (>0.999) 8.71 (>0.999) 2.2 (0.986) 10.28 (>0.999) 10.4 (>0.999) 8.6 (>0.999) 9.25 (>0.999) 8.73 (>0.999) 6.45 (>0.999) 8.16 (>0.999) 5.22 (>0.999) 7.68 (>0.999) 8.27 (>0.999) −0.97 (0.165) 9.82 (>0.999) 10.05 (>0.999) 8.25 (>0.999) 8.86 (>0.999) 8.71 (>0.999) 6.45 (>0.999) 8.17 (>0.999) 5.18 (>0.999) 7.64 (>0.999) 8.17 (>0.999) Mlog1−1.75 (0.04) −0.93 (0.177) 0.13 (0.55) −3.45 (<0.001) −0.28 (0.391) −1.11 (0.133) −9.16 (<0.001) 3.45 (>0.999) 2.1 (0.982) 0.16 (0.562) 0.38 (0.649) −2.52 (0.006) −1.49 (0.068) −0.48 (0.315) −3.9 (<0.001) −1.2 (0.114) −2.64 (0.004) −9.72 (<0.001) 0.42 (0.662) 0.27 (0.606) −0.49 (0.313) −0.44 (0.33) −2.62 (0.004) −1.47 (0.07) −0.48 (0.315) −3.86 (<0.001) −1.19 (0.117) −2.6 (0.005) −9.76 (<0.001) Mlog2−1.97 (0.024) −0.78 (0.216) 0.23 (0.59) −3.03 (0.001) −0.13 (0.447) −0.69 (0.244) −9.29 (<0.001) 3.56 (>0.999) 3.47 (>0.999) 0.25 (0.6) 0.49 (0.686) −2.96 (0.002) −1.3 (0.096) −0.34 (0.368) −3.44 (<0.001) −0.94 (0.173) −1.82 (0.034) −9.85 (<0.001) 0.71 (0.76) 2.43 (0.992) −0.3 (0.383) −0.26 (0.396) −3.09 (<0.001) −1.29 (0.098) −0.34 (0.368) −3.4 (<0.001) −0.93 (0.176) −1.82 (0.034) −9.87 (<0.001) 0.51 (0.693) N-PIT −0.71 (0.24) −0.74 (0.23) 0.78 (0.783) −4.13 (<0.001) 0.37 (0.643) −0.13 (0.447) −7.65 (<0.001) 1.5 (0.933) 1.17 (0.879) 3 (0.999) 1.51 (0.934) −1.14 (0.128) −1.51 (0.066) 0.07 (0.528) −4.7 (<0.001) −0.85 (0.198) −1.31 (0.095) −8.11 (<0.001) 0.64 (0.74) 0.59 (0.721) 1.82 (0.965) 0.38 (0.648) −1.19 (0.117) −1.49 (0.068) 0.07 (0.528) −4.68 (<0.001) −0.85 (0.199) −1.47 (0.071) −8.18 (<0.001) 0.62 (0.734) 0.42 (0.661) T-PIT −0.97 (0.166) −0.75 (0.228) 0.51 (0.693) −3.23 (<0.001) 0.07 (0.53) −0.32 (0.374) −8.23 (<0.001) 1.4 (0.92) 1.11 (0.867) 0.97 (0.834) 2.47 (0.993) −1.49 (0.069) −1.36 (0.087) −0.15 (0.439) −3.64 (<0.001) −0.79 (0.214) −1.24 (0.107) −8.72 (<0.001) 0.46 (0.676) 0.45 (0.673) −0.16 (0.436) −0.09 (0.466) −1.55 (0.06) −1.34 (0.09) −0.15 (0.439) −3.63 (<0.001) −0.79 (0.215) −1.34 (0.091) −8.79 (<0.001) 0.43 (0.665) 0.25 (0.599) −0.41 (0.341) Multivariate DM test statistic using MAE criterion for forecast horizon 1. 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