# A Latent-Factor System Model for Real-Time Electricity Prices in Texas
## 4 Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Coefficient** & \(p\)**-Value** & **95\% Confidence Intervals** \\ \hline \multicolumn{5}{c}{Latent Factor Equation} \\ \(\mathbf{f_{f-1}}\) & \(0.2240\) & \(0.00001\) & \(0.1412\) & \(0.3067\) \\ SystemLoad & \(2.0997\) & \(0.00001\) & \(1.8646\) & \(2.3348\) \\ Lag(WtPr) & \(-0.2542\) & \(0.0430\) & \(-0.4999\) & \(-0.0085\) \\ \multicolumn{5}{c}{Houston Equation} \\ \(\mathbf{f_{f}}\) & \(0.2845\) & \(0.00001\) & \(0.2713\) & \(0.2977\) \\ Wind & \(-0.0943\) & \(0.00001\) & \(-0.1151\) & \(-0.0736\) \\ Nuclear & \(-0.5532\) & \(0.00001\) & \(-0.6576\) & \(-0.4487\) \\ Solar & \(0.0243\) & \(0.0630\) & \(-0.0013\) & \(0.0500\) \\ Henry Hub & \(0.6154\) & \(0.00001\) & \(0.4786\) & \(0.7522\) \\ Dummy & \(2.4006\) & \(0.00001\) & \(2.1030\) & \(2.6982\) \\ \multicolumn{5}{c}{Austin Equation} \\ \(\mathbf{f_{f}}\) & \(0.2661\) & \(0.00001\) & \(0.2559\) & \(0.2763\) \\ Wind & \(-0.1432\) & \(0.00001\) & \(-0.1603\) & \(-0.1261\) \\ Nuclear & \(-0.4445\) & \(0.00001\) & \(-0.5392\) & \(-0.3497\) \\ Solar & \(0.0090\) & \(0.4140\) & \(-0.0126\) & \(0.0307\) \\ [PERSON] & \(0.6454\) & \(0.00001\) & \(0.5272\) & \(0.7636\) \\ Dummy & \(1.2841\) & \(0.00001\) & \(1.0444\) & \(1.5237\) \\ \multicolumn{5}{c}{West Equation} \\ \(\mathbf{f_{f}}\) & \(0.2916\) & \(0.00001\) & \(0.2684\) & \(0.3148\) \\ Wind & \(-0.2826\) & \(0.00001\) & \(-0.3155\) & \(-0.2497\) \\ Nuclear & \(-0.3945\) & \(0.00001\) & \(-0.5201\) & \(-0.2689\) \\ Solar & \(0.0093\) & \(0.6340\) & \(-0.0292\) & \(0.0478\) \\ Henry Hub & \(0.5949\) & \(0.00001\) & \(0.4013\) & \(0.7885\) \\ Dummy & \(0.9767\) & \(0.00001\) & \(0.4842\) & \(1.4692\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: ML coefficients for the 12:00 p.m. sample. Since we are dealing with the natural logs of all the variables, the MLEs represent elasticities. We first describe some overarching conclusions from all three tables here, saving for later the discussion of the merit-order effects. First, from the latent factor equations for all three hours and zones, system-wide load (SystemLoad) positively and significantly impacts the hidden factors. Second, lagged weighted price (LagWtPr) is not significant in the off-peak hour but is significant during the peak hours. Interestingly, it impacts the hidden factors negatively at the noon hour and positively at the 4:00 p.m. hour. Third, the lagged latent factor variable is positive and statistically significant at all three hours for all three zones in the latent factor equation. In conjunction with the plots shown in Figures 4-6, this further confirms the importance of the latent factor dynamics on energy prices in all eight zones. Fourth, from the observation equation for the three zones, during all three hours, as expected, wind generation and nuclear generation have negative elasticities, and Henry Hub gas has positive elasticity. Fifth, solar generation is a mixed bag, largely because this resource is still growing in Texas, and as such its data are non-stationary. Thus, solar generation is not significant at 12:00 p.m. and its elasticities are positive and weak at 4:00 p.m. Finally, the impact of extreme spikes in real-time prices (the dummy variable) at 12:00 p.m. and 4:00 p.m. is highly significant in all three zones. _System-wide merit-order effects._ To best understand the merit-order effects shown as elasticities in Tables 4-6, consider the price boxplots shown in Figure 7. The top, middle and bottom panels, corresponding to hours 4:00 a.m. 12:00 p.m., and 4:00 p.m., respectively, have three boxplots in each panel. On the \(X\)-axis, the box titled \"Before Price\" is the group of mean prices in the eight ERCOT zones before accounting for any merit-order effect. The second and third boxes are the change in mean prices after accounting for merit-order effects in wind and nuclear generation, respectively.