## 3 Regression Analysis of Price (USD/MWh) Data for West, North, and Houston Regions in ERCOT  
A common assumption in many statistical applications, including the current context, is that the error distribution in a linear regression model follows a normal distribution. But, for this assumption to hold, the data distribution has to be examined carefully; see Refs. [24; 25]. The assumption of normality for the error distribution might be justified if the third and fourth moments--skewness and kurtosis--of the data distribution were approximately symmetric and exhibited low kurtosis. Additionally, the dataset should be free from multiple outliers. Skewness is a measure of symmetry, and kurtosis measures how the probability at the peak of the data distribution varies due to the thickness of the tails. For a standard normal distribution, the skew and kurtosis are zero and three, respectively. In a regression context, it is important to understand the properties of the data distribution, since inferences critically depend on them [26]. Based on our examination of the pricing data distributions (see Table 2), we use a natural log transformation of the pricing data, since they are highly asymmetric, with excessive kurtosis. Moreover, as shown in the previous section, due to the LMP feature in electricity pricing data, our dataset also has many outliers. In order to address these problems, in this section, we will assume that the error term has a skew-t distribution, instead of a normal distribution, for all of the pricing data regressions. The use of skew-t distributions to model the error term in energy regression models and other contexts is well-documented; see, as examples, Refs. [9; 27; 19]. More generally, [PERSON] and [PERSON] (2008) state: \"In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew-t distribution is explored in more detail and reasons are given to adopt this option as a sensible, general-purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome\" [28]. The univariate skew-t distribution, denoted ST (\(\xi\), \(\omega^{2}\), \(\alpha\), \(\
u\)), is defined by introducing a multiplier to the Student-\(t\) density, which is a heavier-tailed distribution than the normal distribution. For a random variable \(x\), the probability density function is given through:  
\[f_{ST}\Big{(}x;\ \xi,\omega^{2},\ \alpha,\ \
u\Big{)}=2\ \omega^{-1}t(z;\ \
u)T\left\{ \frac{\alpha z\sqrt{\
u+1}}{\
u+\ z^{2}};\ \
u+1\right\},\ x\in\ \mathbb{R} \tag{1}\]  
where \(t\) (\(z\); \(\
u\)) is the density of a univariate Student-\(t\) distribution with degrees of freedom \(\
u\), and T (\(\ ;\
u+1\)) is the cumulative distribution function of a univariate Student-t distribution with \(\
u+1\) degrees of freedom. The parameter \(\xi\in R\) regulates the location of the distribution, \(\omega>0\) regulates the scale of the distribution, the shape parameter \(\alpha\in R\) controls the asymmetry of the distribution, and the degrees-of-freedom parameter \(\
u>0\) controls the tails of the distribution. When \(\alpha=0\), the density in Equation (1) reduces to the standard Student-\(t\) density. When \(\alpha=0\), and as \(\
u\) tends to \(\infty\), the skew-t density reduces toward the normal density. By introducing an extra parameter for regulating the tails, the skew-t distribution accommodates outlying observations. Thus, the skew-t distribution explicitly allows for varying levels of asymmetry and kurtosis in the data distributions, while also coping with outliers. All three of these features are prevalent in the pricing data we use in this research. Hence, the skew-t distribution will prove useful in estimating the regression coefficients. We construct regression models for the hours ending at 3 AM, 11 AM, and 4 PM within each of the three regions--West, North and Houston.