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Citation: Cayon, E.; Sarmiento, J. The

Impact of Coskewness and

Cokurtosis as Augmentation Factors

in Modeling Colombian Electricity

Price Returns. Energies 2022,15, 6930.

https://doi.org/10.3390/en15196930

Academic Editor: Raymond Li

Received: 18 July 2022

Accepted: 17 September 2022

Published: 22 September 2022

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energies

Article

The Impact of Coskewness and Cokurtosis as Augmentation

Factors in Modeling Colombian Electricity Price Returns

Edgardo Cayon 1, * and Julio Sarmiento 2

1Finance Department, CESA Business School, Bogotá111311, Colombia

2Business Department, Pontificia Universidad Javeriana, Bogotá110231, Colombia; sarmien@javeriana.edu.co

*Correspondence: ecayon@cesa.edu.co

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.

Keywords: electricity markets; asset pricing; higher moments

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 autore-

gressive 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

Energies 2022,15, 6930. https://doi.org/10.3390/en15196930 https://www.mdpi.com/journal/energies

Energies 2022,15, 6930 2 of 8

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 [14–17].

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 (

λ

). 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 (

λ

) is also known as the pricing factor for illiquidity [

20

]. The modiﬁed

version of the Kyle (

λ

) 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 (

λ

) as proposed by [

21

]. The paper is organized as follows: Section 2focuses

on the data employed in the study, Section 3explains the volume model speciﬁcation and

augmentation, Section 4discusses the results obtained, and ﬁnally, Section 5concludes.

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

Energies 2022,15, 6930 3 of 8

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:

Energies 2022, 15, x FOR PEER REVIEW 3 of 8

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://siner-

gox.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:

0

500

1,000

1,500

2,000

2,500

3,000

3,500

-0.04 -0.02 0.00 0.02 0.04 0.06

Series : RETURNS VOLUME

Sa mpl e 1/02/ 2000 4/18/2022

Obs ervation s 8143

Mean -1.11 x 10(-5)

Medi an -0.000691

Maximum 0.073613

Mini mum -0.050974

Std. Dev. 0.005616

Ske wnes s 1.528966

Kurtosi s 13.87518

Jarq ue-Be ra 43300.57

Probability 0.000000

Series : RETURNS VOLUME

Sa mpl e 1/02/ 2000 4/18/2022

Obs ervation s 8143

Mean -1.11 x 10(-5)

Medi an -0.000691

Maximum 0.073613

Mini mum -0.050974

Std. Dev. 0.005616

Ske wnes s 1.528966

Kurtosi s 13.87518

Jarq ue-Be ra 43300.57

Probability 0.000000

0

400

800

1,200

1,600

2,000

2,400

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0. 6

Series : RETURNS PRI CE

Sa mpl e 1/02/2000 4/ 18/2022

Obs ervat ion s 8143

Mean 0.000172

Med ia n -0.000778

Maximum 0.742374

Min imu m - 0.831320

Std. Dev. 0.11 8740

Ske wne ss 0.071258

Kurtos is 7.734689

Jarq ue -Bera 7612. 889

Proba bi li ty 0.000000

Series : RETURNS PRI CE

Sa mpl e 1/02/2000 4/ 18/2022

Obs ervat ion s 8143

Mean 0.000172

Med ia n -0.000778

Maximum 0.742374

Min imu m - 0.831320

Std. Dev. 0.11 8740

Ske wne ss 0.071258

Kurtos is 7.734689

Jarq ue -Bera 7612. 889

Proba bi li ty 0.000000

Figure 1. Observed distribution of daily volume and prices returns in the Colombian electricity mar-

ket (1 January 2000 to 14 February 2022).

We can observe that in both the daily and monthly returns that both their distribu-

tions exhibit positive skewness and high kurtosis. Although electricity cannot be stored

like financial assets, we use returns because the electricity bought today will be “dis-

patched” 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 nor-

mality for the observed distributions. Therefore, we know that neither kurtosis nor skew-

ness 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]:

2

, ,1 , ,1 ,1

(, ) {[ ()][ ( )]}

it it it i it it

Cos P V E P E P V E V

−−−

=− − (1)

3

,,1 , ,1 ,1

(, ) {[ ()][ ( )]}

it it it i it it

Cok P V E P E P V E V

−−−

=− − (2)

where ,,

(,)

it it

Cos P V and ,,

(, )

it it

Cok P V are the coskewness and cokurtosis between the

change in volume in observation (i) and price returns in observation (i), respectively, i

P

and i

V are the returns of prices and volume at time (t), and ()

i

EP and ()

i

EV are the

expected average returns for the time series of price and volume, respectively. From Equa-

tions (1) and (2), we obtain a time series for the same length of the sample of the returns

i

Pand i

V 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 augmenta-

tions in Equation (3):

,,,1

,,,co,,,1

,,,1cos,,,1

,,,1c

Volume model

1 ( ) Volume and cokurtosis model

Cos( ) Volume and coskewness model

it t vt it t

it t vtit kt itit t

it t vt it t it it t

it t vt it

PV

PV CokPV

PV PV

PV

αλ ε

αλ λ ε

αλ λ ε

αλ λ

−

−

−−

−

=+ +→

=+ −+ +→

=+ + +→

=+ +

o, , , 1 cos, , , 1

( ) Cos( ) Volume, cokurtosis and coskewness model

kt it it t it it t

Cok P V P V

λε

−−

++→

(3)

Figure 1.

Observed distribution of daily volume and prices returns in the Colombian electricity

market (1 January 2000 to 14 February 2022).

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(Pi,t,Vi,t−1) = En[Pi,t−E(Pi)][Vi,t−1−E(Vi,t−1)]2o(1)

Cok(Pi,t,Vi,t−1) = En[Pi,t−E(Pi)][Vi,t−1−E(Vi,t−1)]3o(2)

where

Cos(Pi,t

,

Vi,t)

and

Cok(Pi,t

,

Vi,t)

are the coskewness and cokurtosis between the

change in volume in observation (i) and price returns in observation (i), respectively,

Pi

and

Vi

are the returns of prices and volume at time (t), and

E(Pi)

and

E(Vi)

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 re-

turns

Pi

and

Vi

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 augmenta-

tions in Equation (3):

Pi,t=αt+λv,tVi,t−1+εt→Volumemodel

Pi,t=αt+λv,tVi,t−1+λcok,tCok(Pi,tVi,t−1) + εt→Volume and cokurtosis model

Pi,t=αt+λv,tVi,t−1+λcos,tCos(Pi,tVi,t−1) + εt→Volume and coskewness model

Pi,t=αt+λv,tVi,t−1+λcok,tCok(Pi,tVi,t−1) + λc os,tCos(Pi,tVi,t−1) + εt→Volume, cokurtosis and coskewness model

(3)

where

αt

= is the intercept for each model,

Pi,t

= the daily or monthly returns of the Colom-

bian electricity spot prices from the period under observation,

Vi,t−1

= the daily or monthly

returns of the volume of electricity negotiated the previous day in the Colombian electricity

market,

λv,t

= the coefﬁcient obtained for volume in each model,

Cok(Pi,tVi,t−1)= the

daily

or monthly cokurtosis values obtained using the procedure in Equation (2),

λcok,t

= the

coefﬁcient obtained for cokurtosis in each model,

Cos(Pi,tVi,t−1)

= the daily or monthly

coskewness values obtained using the procedure in Equation (3),

λcos,t

= the coefﬁcient

obtained for coskewness in each model, and

εt

= the error term for each model. We adapted

Energies 2022,15, 6930 4 of 8

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 analy-

sis. 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:

αt=0→H0: Accept the null that alpha is zero

λv,t<0→H1: Accept the alternative that lambda is negative (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 Figures 1and 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 Figures 1and 2is 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:

αt=0→H0: Accept the null that alpha is zero

λv,t<0→H1: Accept the alternative that lambda is negative

λcos,t>0→H1: Accept the alternative that lambda is positive

λcok,t<0, λco k,t>0→H1: Accept the alternative that lambda is either positive or negative

(5)

Energies 2022, 15, x FOR PEER REVIEW 5 of 8

0

5

10

15

20

25

30

35

40

-0.2 -0.1 0.0 0.1 0.2 0.3

Series : RETURNS VOLUME

Sampl e 1 268

Obs erva tion s 268

Mea n -0.000337

Medi a n 0.003788

Maxi mum 0. 350326

Min imu m -0.225545

Std. Dev. 0.112331

Ske wne ss 0.249164

Kurtos i s 2.943930

Jarq ue -Bera 2. 808140

Probability 0.245595

Series : RETURNS VOLUME

Sampl e 1 268

Obs erva tion s 268

Mea n -0.000337

Medi a n 0.003788

Maxi mum 0. 350326

Min imu m -0.225545

Std. Dev. 0.112331

Ske wne ss 0.249164

Kurtos i s 2.943930

Jarq ue -Bera 2. 808140

Probability 0.245595

0

10

20

30

4

0

50

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Series: RETURNS PRICE

Sample 1 268

Observations 268

Mean 0.005232

Median -0.020171

Maximum 1.585594

Minimum -1.623706

Std. Dev. 0.401333

Skewness 0.404891

Kurtosis 5.611520

Jarque-Bera 83.47960

Probability 0.000000

Series: RETURNS PRICE

Sample 1 268

Observations 268

Mean 0.005232

Median -0.020171

Maximum 1.585594

Minimum -1.623706

Std. Dev. 0.401333

Skewness 0.404891

Kurtosis 5.611520

Jarque-Bera 83.47960

Probability 0.000000

Figure 2. Observed distribution of monthly volume and prices returns in the Colombian electricity

market (January 2000 to February 2022).

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, heteroscedas-

ticity, and leptokurtosis [24]. Additionally, we tested for the consistency of the instru-

ments 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 Tables 1 and 2, 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 ( ,vt

λ

) 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 inconclu-

sive. 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 specifica-

tion accepts the null hypothesis that the instruments (lagged terms of the independent

variables) are not mis-specified in the model.

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 Coskew-

ness 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 23769.2500 *** −108425.1000 ***

Figure 2.

Observed distribution of monthly volume and prices returns in the Colombian electricity

market (January 2000 to February 2022).

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.

Energies 2022,15, 6930 5 of 8

4. Results

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

sample are given in Tables 1and 2, 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 (

λv,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.

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

αt0.0001 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

R21.39% 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

αt0.1910 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.

Energies 2022,15, 6930 6 of 8

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

αt0.0049 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

R29.88% 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

αt0.0002 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.

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 (

α

) 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 (

α

) 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 (

α

)

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

Energies 2022,15, 6930 7 of 8

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 (

λ

) 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.

Author Contributions:

Data curation, E.C.; Investigation, E.C.; Writing—original draft, E.C.; Writing—

review & editing, J.S. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Data Availability Statement:

Data was obtained from XM (https://sinergox.xm.com.co/Paginas/

Home.aspx, accessed on 1 April 2022).

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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