## 3 Regression Analysis of Price (USD/MWh) Data for West, North, and Houston Regions in ERCOT
### Analysis of Hourly Models for West Region Price Data
* For the variable Wprice32, when prices are at least USD 32, real-time prices in the West region increase by about 380%, compared to when prices are below USD 32. In this regard, recall the summary statistics of the price distributions from Section 2. The median and mean prices at 3 AM in the West region are USD 16.93 and USD 17.16, respectively. The minimum and maximum prices are USD 0.01 and USD 161.02. Indeed, a percentile analysis revealed that roughly 5% of the prices are extreme at both tails of the price distribution. As described previously, extreme prices could be the result of seasonality and/or the LMP effect. Lower prices are typically the result of low demand, as at 3 AM. On the other hand, when demand and prices are high, if the MOE from renewable resource generation is sizeable, that would dampen the price, since the cost of delivering energy to the grid will decline. It is clear from the above \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Skew-t Regression** & **Hour 3 AM** & **Hour 11 AM** & **Hour 4 PM** \\
**Independent Variables** & **(1)** & **(2)** & **(3)** \\ \hline WSapp18 & \(-0.0492\)* & \(-0.0664\)* & \(-0.0744\)* \\  & (0.0038) & (0.0058) & (0.0073) \\ Wprice1 & \(-2.7683\)* & - & - \\  & (0.0059) & - & - \\  & 1.5691 * & & \\  & (0.0070) & - & - \\  & & 1.0229 * & \\  & & (0.0189) & - \\  & & & 1.3520 * \\  & - & - & (0.0169) \\ Ln(NGprice) & 0.5133 * & 0.4399 * & 0.4198 * \\  & (0.0124) & (0.0126) & (0.0153) \\ Ln(Wgen) & \(-0.0851\)* & \(-0.1032\)* & \(-0.1100\)* \\  & (0.0026) & (0.0026) & (0.0037) \\ Ln(Sgen) & - & \(-0.0170\)* & \(0.0157\)* \\  & - & (0.0035) & (0.0032) \\ Ln(Ngen) & \(-0.1119\)* & \(-0.1118\)* & \(-0.1253\)* \\  & (0.0081) & (0.0107) & (0.0136) \\ Ln(Wload) & 0.2554 * & 0.6642 * & 0.3194 * \\  & (0.0185) & (0.0331) & (0.0351) \\ Ln(REload) & 0.1273 * & 0.2897 * & 0.6883 * \\  & (0.0117) & (0.0182) & (0.0194) \\ Constant & 0.9502 * & \(-2.9376\)* & \(-4.2033\)* \\  & (0.1076) & (0.1364) & (0.1547) \\ \hline \hline \end{tabular} Legend: * \(p\) < 0.0001.  
\end{table} 
Table 3: West region skew-t regression estimates. parameter estimates discussion of the MOE that more gains from wind generation could be had, if the number of active wind farms were to grow in Texas. * Natural Gas Price: The coefficient estimate for LN(NGprice) is 0.5133 (\(p<0.0001\)). For a 1% increase in natural gas price, the wholesale electricity price increases by 0.5133%. * West Load: The coefficient estimate for LN(Wload) is 0.2544 (\(p<0.0001\)). For a 1% increase in demand in the West, the wholesale electricity price in the West at 3 AM increases by 0.2544%. * Rest of ERCOT Load: The coefficient estimate for LN(REload) is 0.1273 (\(p<0.0001\)). For a 1% increase in demand in the rest of the ERCOT, the wholesale electricity price in the West at 3 AM increases by 0.1273%. * Normal Regression Comparison: Finally, consider the normal regression parameter estimates in Table 4. When compared to the skew-t estimates, a most striking feature here is that the WSapp18 variable is not significant, and its marginal effect is almost zero. A second counterintuitive conclusion is that the impact of demand in the West (Ln(Wload)) is not significant. A third key difference is that the standard errors under the normal regression model tend to be higher than those from the skew-t model. This is because the standard error (i.e., the standard deviation of the estimated errors) in the normal regression fails to account for the asymmetry, kurtosis, and outliers in the data distribution. Thus, these regression results validate the importance of using a skew-t distribution for the error term in the regression to better account for idiosyncrasies in the data distributions. Next, consider Table 3 and focus on the Hour 11 AM estimates. All of the regressor variables are statistically significant (\(p<0.0001\)), and the coefficient signs are as expected. We focus on the most important variables here, since the inferences for the others are similar to those from the 3 am description, given above.