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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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