The Impact of Coskewness and Cokurtosis as Augmentation Factors in Modeling Colombian Electricity Price Returns

Abstract

This paper explores the empirical validity of an augmented volume model for Colombian electricity price returns (in the present study, the definition of returns is simply the “rate of change” of observed prices for different periods). Of particular interest is the impact of coskewness and cokurtosis when modeling Colombian electricity price returns. We found that coskewness as an augmentation factor is highly significant and should be considered when modeling Colombian electricity price returns. The results obtained for coskewness as an augmentation factor in a volume model are consistent when using either an Ordinary Least Square (OLS) and Generalized Method of Moments (GMM) specification for the data employed. On the other hand, the effect of cokurtosis is highly irrelevant and not significant in most cases under the proposed specification.

1. Introduction

Electricity spot price returns present seasonality, extreme volatility (price peaks), mean reversion, and generally deviate from the expected properties of a normal distribution [1]. Therefore, simple time series linear models would not be the most adequate since electricity spot price returns result from a unit root process. This is a problem because prices can rise to infinity from a theoretical point of view [2]. However, since mean reversion is present in most electricity price returns time series, the most common models employed in forecasting electricity prices are autoregressive. Studies have been made from the simple first autoregressive model AR(1) to more complex autoregressive moving average models (ARMAs) and generalized autoregressive conditional heteroscedasticity (GARCH) with all sorts of innovations such as jump diffusions and time-varying intercepts [3,4]. Ref. [5] argue that for electricity prices, extreme price movements account for the large standard deviations of electricity prices, which of course have a direct effect on the skewness and kurtosis of the observed distribution that deviates from what is expected in a normal distribution. Ref. [6] used a GARCH model which incorporated volume and found a statistically significant relationship between price and volume. But whether this relationship was positive or negative depended on the nature of each market. Ref. [4] argue that the stylized facts of energy prices (stationarity, seasonality, and extreme price swings) tend to depart from the foundations of traditional asset pricing research, and that other kinds of models should be considered when trying to model electricity prices. The present study attempts to fill this gap by proposing an augmented volume–price asset pricing model that incorporates the effect of higher distributional moments commonly present in the observed distribution of electricity prices.

The effects of higher moments, such as skewness and kurtosis, have been a widely discussed topic in asset pricing. The basic premise is that investors are willing to pay a premium for individual assets with positive skewness and for those assets that exhibit positive coskewness with the market since there is a higher probability of obtaining higher abnormal positive returns. For example, ref. [7] extended the traditional capital asset pricing model (CAPM) into a three-moment capital asset pricing model, which incorporates the risk premium attributable to skewness and argues that ignoring the effect of systematic skewness can lead to misspecification errors when testing the CAPM in its traditional form. [8] empirically tested the effectiveness of the three-moment capital asset model and found that coskewness is as essential as covariance in predicting stock returns. In the case of kurtosis for individual assets and cokurtosis with the market, the premise is that higher kurtosis implies that the asset has a higher probability of extreme losses. Therefore, investors will demand a premium for holding the individual asset or those assets with high cokurtosis with the market. Ref. [9] proposed a four-moment capital asset pricing model that incorporated the effects of cokurtosis and coskewness and found that the systematic risks attributable to skewness and kurtosis contributed significantly to explaining the variance in individual asset returns. In electricity markets, there is evidence that skewness is caused by the variability in supply (volume) and demand: when demand is high relative to volume, the spot prices tend to be positively skewed, and that skewness is a factor that should be included in electricity pricing models [10]. The use of asset pricing in electricity markets has been applied to thermoelectric power plants in Brazil [11]. As happens with skewness, electricity returns also exhibit high kurtosis due to “peak” prices in times of short supply, so the observed distributions tend to have fat tails due to extreme values, and this is a stylized fact among electricity markets around the world [12]. Ref. [13] used a multivariate GARCH framework with a generalized error distribution to incorporate kurtosis and skewness when modeling the variance in electricity prices in Germany. They highlighted that modeling kurtosis in electricity prices is essential, especially when dealing with extreme returns values. There are complex alternatives for addressing extreme price movements in electricity markets. Recent research proposes dynamic-based market models, linear optimization interactive models, and machine learning as alternatives to forecasting electricity prices [14,15,16,17].

Like its European counterparts, the Colombian electricity market is auction oriented, in which the spot price at time (t) depends on the electricity volume set by the different power suppliers of the previous day (t − 1). One characteristic that makes the Colombian electricity market an interesting case study is that there are no negative prices because most of the electricity in the market is hydroelectric. This is important, because without negative prices, we can assume the lognormality of the prices. This fact does not affect the usual theoretical framework for asset pricing that is based on Gaussian assumptions [18]. Therefore, in the context of asset pricing, the theoretical relationship between volume and price returns is explained by the Kyle model, commonly referred to as the ($\lambda$). Ref. [19] postulates that in a continuous auction market (like the Colombian electricity market), the market makers (power suppliers) have no way of knowing the exact quantity of energy demanded by traders the next day. Therefore, according to Kyle, the spot price at any given time is a function of the volume demanded during the day by competing traders. In asset pricing, the Kyle ($\lambda$) is also known as the pricing factor for illiquidity [20]. The modified version of the Kyle ($\lambda$) used in this study does not use high-frequency order volume and prices returns, but instead daily volume and average daily price returns as proxies for obtaining ($\lambda$) as proposed by [21]. The paper is organized as follows: Section 2 focuses on the data employed in the study, Section 3 explains the volume model specification and augmentation, Section 4 discusses the results obtained, and finally, Section 5 concludes.

2. Data

As mentioned before, the data employed for this study are the Colombian daily volume measured in GW at (t − 1) and the average daily price at (t) since the Colombian electricity market is a day-ahead market. In a day-ahead market, the quantity of electricity bought today is dispatched on the following day, allowing clients to lock in today’s price in order to hedge the volatility in the next day real-time market. The data include 8145 observations for volume and price, respectively, and range from the period 1 January 2000 to 14 February 2022. The historical data were extracted from XM (https://sinergox.xm.com.co/Paginas/Home.aspx, accessed on 1 April 2022), the Colombian electricity market operator. In Figure 1, we can see the observed distributions for volume and price returns that are calculated as the daily and monthly change rate for the observations in the sample:

We can observe that in both the daily and monthly returns that both their distributions exhibit positive skewness and high kurtosis. Although electricity cannot be stored like financial assets, we use returns because the electricity bought today will be “dispatched” at a future date. In this way, the different agents on the market speculate on the price differences between periods. The Jarque–Bera test rejects the null hypothesis of normality for the observed distributions. Therefore, we know that neither kurtosis nor skewness has the theoretical values expected from a normal distribution. The next step was to calculate the series for coskewness and cokurtosis between the volume and price returns, which are defined as follows [22]:

$$Cos(P_{i,t},V_{i,t-1})=E\{[P_{i,t}-E(P_i)]{[V_{i,t-1}-E(V_{i,t-1})]}^2\}$$

(1)

$$Cok(P_{i,t},V_{i,t-1})=E\{[P_{i,t}-E(P_i)]{[V_{i,t-1}-E(V_{i,t-1})]}^3\}$$

(2)

where $Cos(P_{i,t},V_{i,t})$ and $Cok(P_{i,t},V_{i,t})$ are the coskewness and cokurtosis between the change in volume in observation (i) and price returns in observation (i), respectively, $P_i$ and $V_i$ are the returns of prices and volume at time (t), and $E(P_i)$ and $E(V_i)$ are the expected average returns for the time series of price and volume, respectively. From Equations (1) and (2), we obtain a time series for the same length of the sample of the returns $P_i$ and $V_i$ of cokurtosis and coskewness for each individual observation (i) at time (t).

3. Model

The results are obtained using the following models and their respective augmentations in Equation (3):

$$\begin{array}{l}{P}_{i,t}={\alpha }_t+{\lambda }_{v,t}{V}_{i,t-1}+{\epsilon }_t\to \mathrm{Volume}\mathrm{model}\\ P_{i,t}={\alpha }_t+{\lambda }_{v,t}{V}_{i,t}-1+{\lambda }_{cok,t}Cok(P_{i,t}{V}_{i,t-1})+{\epsilon }_t\to \mathrm{Volume}\mathrm{and}\mathrm{cokurtosis}\mathrm{model}\\ P_{i,t}={\alpha }_t+{\lambda }_{v,t}{V}_{i,t-1}+{\lambda }_{cos,t}Cos(P_{i,t}{V}_{i,t-1})+{\epsilon }_t\to \mathrm{Volume}\mathrm{and}\mathrm{coskewness}\mathrm{model}\\ P_{i,t}={\alpha }_t+{\lambda }_{v,t}{V}_{i,t-1}+{\lambda }_{cok,t}Cok(P_{i,t}{V}_{i,t-1})+{\lambda }_{cos,t}Cos(P_{i,t}{V}_{i,t-1})+{\epsilon }_t\to \mathrm{Volume},\mathrm{cokurtosis}\mathrm{and}\mathrm{coskewness}\mathrm{model}\end{array}$$

(3)

where ${\alpha }_t$ = is the intercept for each model, $P_{i,t}$ = the daily or monthly returns of the Colombian electricity spot prices from the period under observation, $V_{i,t-1}$ = the daily or monthly returns of the volume of electricity negotiated the previous day in the Colombian electricity market, ${\lambda }_{v,t}$ = the coefficient obtained for volume in each model, $Cok(P_{i,t}{V}_{i,t-1})$ = the daily or monthly cokurtosis values obtained using the procedure in Equation (2), ${\lambda }_{cok,t}$ = the coefficient obtained for cokurtosis in each model, $Cos(P_{i,t}{V}_{i,t-1})$ = the daily or monthly coskewness values obtained using the procedure in Equation (3), ${\lambda }_{cos,t}$ = the coefficient obtained for coskewness in each model, and ${\epsilon }_t$ = the error term for each model. We adapted the method proposed by [22] to electricity markets for testing extensions of the CAPM models, which are simply different extensions of the traditional market model proposed by Markowitz [23]. The basic postulate of the market model is that an underlying factor explains the changes in prices, and that can vary depending on the market under analysis. The different volume models described in Equation (3) are simply extensions of the traditional market model. In our specific market model, the volume of energy traded in the Colombian electricity market is the underlying factor that proxies the evolution of electricity prices. We can test the validity of the models in Equation (3) by testing a different set of hypotheses for each model. We expect to accept the null hypothesis that the intercept is zero (if the intercept is different from zero and statistically significant, in the context of asset pricing, this is evidence of omitted information) and that the lambda of the volume model is negative and statistically significant (in electricity markets, higher production volumes are negatively correlated to spot prices). Therefore, the hypotheses are:

$$\begin{array}{l}{\alpha }_t=0\to H_0:\mathrm{Accept}\mathrm{the}\mathrm{null}\mathrm{that}\mathrm{alpha}\mathrm{is}\mathrm{zero}\\ {\lambda }_{v,t}<0\to H_1:\mathrm{Accept}\mathrm{the}\mathrm{alternative}\mathrm{that}\mathrm{lambda}\mathrm{is}\mathrm{negative}\end{array}$$

(4)

In the case of the other models, cokurtosis and coskewness are tested as individual and additional augmentations of the volume model. As augmentation factors, we expect the lambdas of coskewness for each model to be positive and statistically significant since in the empirical distribution (see Figure 1 and Figure 2) of the Colombian electricity spot prices and volume returns, there is evidence of positive skewness. The positive skewness can be interpreted as a sign that positive returns are more frequent than negative returns in the observed distributions. In the case of cokurtosis, the observed kurtosis for the empirical distributions in Figure 1 and Figure 2 is high, which can be evidence of extreme positive and negative returns variations. Therefore, the sign of the lambdas for cokurtosis for each model can be either negative or positive. In summary, the hypotheses are:

$$\begin{array}{l}{\alpha }_t=0\to H_0:\mathrm{Accept}\mathrm{the}\mathrm{null}\mathrm{that}\mathrm{alpha}\mathrm{is}\mathrm{zero}\\ {\lambda }_{v,t}<0\to H_1:\mathrm{Accept}\mathrm{the}\mathrm{alternative}\mathrm{that}\mathrm{lambda}\mathrm{is}\mathrm{negative}\\ {\lambda }_{cos,t}>0\to H_1:\mathrm{Accept}\mathrm{the}\mathrm{alternative}\mathrm{that}\mathrm{lambda}\mathrm{is}\mathrm{positive}\\ {\lambda }_{cok,t}<0,{\lambda }_{cok,t}>0\to H_1:\mathrm{Accept}\mathrm{the}\mathrm{alternative}\mathrm{that}\mathrm{lambda}\mathrm{is}\mathrm{either}\mathrm{positive}\mathrm{or}\mathrm{negative}\end{array}$$

(5)

To test the consistency of the results, we also ran the models of Equation (3) using the generalized method of moments (GMM), which is widely used in asset pricing. Since our data deviate from what is expected from a normal distribution, GMM addresses the problems of non-normality in our data by correcting for serial correlation, heteroscedasticity, and leptokurtosis [24]. Additionally, we tested for the consistency of the instruments employed in the GMM regression by using the J-statistic in which the null is that the instruments (usually lagged terms of the independent variables) are adequate for the proposed models.

4. Results

The results for the models in Equation (3) for daily and monthly returns based on the sample are given in

Table 1. Different augmented volume models with cokurtosis and coskewness for daily returns of Colombian electricity spot prices.

Panel A-OLS Regression-Augmented Volume Models
	Volume Model	Volume with Cokurtosis	Volume with Coskewness	Volume with Coskewness and Cokurtosis
αt	0.0001	0.0002	0.0008	0.0015 *
	(0.1648)	(0.1880)	(0.8936)	(1.7959)
λv,t−1	−2.4937 ***	−2.7753 ***	−2.3919 ***	−0.9888
	−(10.8580)	−(14.0247)	−(5.5231)	−(1.4454)
λcok,t		23,769.2500 ***		−108,425.1000 ***
		(3.3498)		−(3.2712)
λcos,t			2984.7110 ***	6469.6310 ***
			(4.2303)	−(1.4454)
Number of Observations	8143	8143	8143	8143
R2	1.39%	2.37%	11.24%	18.26%
Panel B-GMM Regression-Augmented Volume Models
	Volume Model	Volume with Cokurtosis	Volume with Coskewness	Volume with Coskewness and Cokurtosis
αt	0.1910	0.3204	0.1128	0.1014 *
	(0.4532)	(0.1863)	(1.6223)	(1.7489)
λv,t−1	−2.3870 **	−4.0577 *	−1.6265 *	−2.2640 **
	−(1.9906)	−(1.8855)	−(1.8456)	−(2.1043)
λcok,t		116,200.0000		155,753.6000
		(1.1829)		(1.1829)
λcos,t			2368.5840 **	1408.9870
			(2.4344)	(0.7511)
J-statistic	286.0240	127.9153	0.0034	0.3821
Prob(J-statistic)	0.0000	0.0000	0.9533	0.8261
Number of observations	7983	8105	8140	8141

Notes: This table reports the results of daily Colombian electricity prices returns from the period 1 January 2000 to 14 February 2022 and regresses it on the electricity volume for the same period and augments it for cokurtosis, coskewness and cokurtosis and coskewness. The coefficients reported are the lambdas of each regression. The significance of the coefficients is at the 1 (*), 5 (**), and 10% (***) levels. The coskewness and cokurtosis are obtained using Equations (1) and (2) and the four models detailed in Equation (3). The results in panel A are obtained using a simple OLS model, and the results in panel B using with GMM for robustness purposes.

and

Table 2. Different augmented volume models with cokurtosis and coskewness for monthly returns of Colombian electricity spot prices.

Panel A-OLS Regression-Augmented Volume Models
	Volume Model	Volume with Cokurtosis	Volume with Coskewness	Volume with Coskewness and Cokurtosis
αt	0.0049	0.0116	0.0250	0.0238
	(0.2903)	(0.6038)	(1.5969)	−(4.5916)
λv,t	−1.1228 ***	−1.0365 ***	−0.3161 **	−1.7855 ***
	−(7.2201)	−(6.4252)	−(2.3306)	−(4.5916)
λcok,t		14.1173		−59.4133 **
		(0.9771)		−(2.1511)
λcos,t			21.2504 ***	24.1067 **
			(4.0302)	(2.1066)
Number of Observations	268	268	268	268
R2	9.88%	10.54%	29.88%	11.39%
Panel B-GMM Regression-Augmented Volume Models
	Volume Model	Volume with Cokurtosis	Volume with Coskewness	Volume with Coskewness and Cokurtosis
αt	0.0002	0.0454 ***	0.0717 **	−0.0090
	(0.0154)	(3.2133)	(2.1178)	−(0.2326)
λv,t	−0.6205 ***	−0.4345 **	1.8413 *	1.7868 *
	−(4.4899)	−(2.4081)	(1.6634)	(1.6634)
λcok,t		86.7519 ***		−266.7234 **
		(4.0549)		−(2.1524)
λcos,t			93.2383 ***	127.6382 ***
			(2.6517)	(0.0000)
J-statistic	23.5120	27.4642	0.7926	1.9356
Prob(J-statistic)	0.1333	0.1560	0.3733	0.7476
Number of observations	250	247	266	265

Notes: This table reports the results of monthly Colombian electricity prices returns from the period 1 January 2000 to 14 February 2022 and regresses it on the electricity volume for the same period and augments it for cokurtosis, coskewness and cokurtosis and coskewness. The coefficients reported are the lambdas of each regression. The significance of the coefficients is at the 1 (*), 5 (**), and 10% (***) levels. The coskewness and cokurtosis are obtained using Equations (1) and (2) and the four models detailed in Equation (3). The results in panel A are obtained using a simple OLS model, and the results in panel B using with GMM for robustness purposes.

, wherein Panel A shows the results obtained running single and augmented volume models with OLS, and Panel B shows the same results for the same models with GMM. The results show that the volume lambda (${\lambda }_{v,t}$) is significant at the daily and monthly level and with the expected negative sign. In hydroelectricity, lower volumes lead to higher prices, the only exception being the volume model with coskewness and cokurtosis in the OLS specification, but statistically significant in the GMM specification in which the J-stat accepts the null hypothesis that the instruments (lagged terms of the independent variables) are not mis-specified in the model. In the case of cokurtosis, the results from the OLS model at the daily level are significant. In the GMM specification, cokurtosis is statistically insignificant at the daily level, but the J-stat rejects the null hypothesis that the instruments are adequate; therefore, the results are inconclusive. For the monthly OLS model, cokurtosis is insignificant. However, under the GMM specification, cokurtosis is significant, and the J-stat leads us to accept the null hypothesis that the instruments (lagged terms of the independent variables) are not mis-specified in the model. For the augmented OLS and GMM models with only coskewness, the models are significant at both the daily and monthly level, and the J-stat for the GMM specification accepts the null hypothesis that the instruments (lagged terms of the independent variables) are not mis-specified in the model.

In the augmented model with both cokurtosis and coskewness, the results for the different specifications (OLS and GMM) have mixed results. In the OLS specification, both cokurtosis and coskewness are significant at the daily level. However, the volume becomes insignificant, and the intercept ($\alpha$) becomes significant, which can be a sign of omitted information. Additionally, when we analyze the results under the GMM specification, volume becomes significant, but both cokurtosis and coskewness become insignificant. The intercept ($\alpha$) continues to be significant, and the J-stat for the GMM specification accepts the null hypothesis that the instruments (lagged terms of the independent variables) are not mis-specified in the model. At the monthly level, all three variables (volume, cokurtosis, and coskewness) are significant, regardless of the specification. The J-stat for the GMM specification accepts the null hypothesis that the instruments (lagged terms of the independent variables) are not mis-specified in the model.

In summary, the results show that according to our hypothesis in Equation (4), both the volume model and the augmented volume model with coskewness are the most stable under the OLS and GMM specifications, and with the expected signs and an intercept ($\alpha$) that is not statistically different from zero. In the case of cokurtosis, there are conflicting results for the different models at both the daily and monthly level. Therefore, we can infer that the effect of cokurtosis as an augmentation factor for volume models is insignificant and, in most cases, irrelevant. Finally, an augmented volume model with cokurtosis and coskewness is not significant at the daily level but at the monthly level, regardless of the specification (OLS or GMM) employed. At the economic level, coskewness as an augmentation factor helps to improve a traditional volume model in the sense that the positive sign of the coefficient under different specifications is an indication that coskewness must be taken into account when modeling Colombian electricity spot price returns in order to correct for the non-normality of the data. For all the models tested, the coefficients of coskewness are positive, as expected from the hypothesis in Equation (5), which means that higher positive coskewness can lead to higher returns, and therefore, a lower price.

5. Conclusions

This paper proposes an augmented volume for modeling Colombian electricity prices. Our proposed model is based on the theoretical relationship between price and volume that in asset pricing is commonly referred to as the Kyle ($\lambda$) and on the augmentations proposed by Kraus and Litzenberger in their three-factor asset pricing model. Using cokurtosis and coskewness as augmentation factors for a simple volume model shows that coskewness is highly significant as an augmentation factor. Finally, the positive and robust relationship of the coskewness of volume with Colombian spot electricity price returns indicates that coskewness is a significant factor to be considered when modeling Colombian electricity spot price returns. This has an important market implication in the sense that the risk attributable to coskewness is a relevant pricing factor in day-ahead electricity markets. One limitation of the research is that it is limited to hydroelectric markets that are not affected heavily by the existence of negative prices. Future research on the subject should explore the effect of coskewness in markets with other sources of electricity power.