# A Latent-Factor System Model for Real-Time Electricity Prices in Texas
## 4 Results
_Significance of the latent factor coefficient._ From Table 3, the endogenous latent factor variable, \(\mathbf{f}_{t}\), when it appears as an exogenous variable in the observation equation is statistically significant for all the nine models (\(p<0.00001\)). This result confirms one of the principal assertions in this paper, namely that there are hidden, _unobserved_ factors that influence the distribution of real-time prices throughout a 24-h cycle across all zones. [PERSON] et al. (2019) do not use latent factors in their system-wide price and demand ERCOT model. It is evident from this research that latent factors play a significant role in ERCOT's pricing structure. _The marginal effects of the exogenous variables._ Consider Tables 4-6 which show the maximum likelihood estimates (MLEs) for coefficients that appear in the observation and latent factor equations in (1); the corresponding \(p\)-values; and the 95% confidence intervals for Hours 4:00 a.m., 12:00 p.m., and 4:00 p.m., respectively, for the three zones. Figure 6: 4:00 p.m.—Latent Factor and Houston price series (top left and right); Austin and West (Bottom left and right). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Austin** & **Houston** & **LCRA** & **North** & **RAYB** & **CPS** & **South** & **West** \\ \hline
4:00 a.m. & **0.571** & **0.532** & 0.581 & 0.533 & 0.510 & 0.550 & 0.541 & **0.621** \\
12:00 p.m. & **0.266** & **0.284** & 0.273 & 0.223 & 0.204 & 0.272 & 0.268 & **0.292** \\
4:00 p.m. & **0.433** & **0.430** & 0.441 & 0.427 & 0.425 & 0.423 & 0.412 & **0.461** \\ \hline \hline \end{tabular} Note: All coefficients have \(p\)-values \(<0.00001\). The latent factor is a vector quantity; hence it appears in bold font to be consistent with the notation in Section 3.  
\end{table} 
Table 3: Coefficients for the Latent Factor \(\mathbf{f}_{t}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Coefficient** & \(p\)**-Value** & **95\% Confidence Intervals** \\ \hline \multicolumn{5}{c}{Latent Factor Equation} \\ \(\mathbf{f_{f-1}}\) & \(0.1681\) & \(0.0030\) & \(0.0579\) & \(0.2783\) \\ SystemLoad & \(1.6092\) & \(0.00001\) & \(1.3573\) & \(1.8610\) \\ Lag(WtPr) & \(-0.1248\) & \(0.1660\) & \(-0.3015\) & \(0.0518\) \\ \multicolumn{5}{c}{Houston Equation} \\ \(\mathbf{f_{f}}\) & \(0.5331\) & \(0.00001\) & \(0.5117\) & \(0.5545\) \\ Wind & \(-0.3570\) & \(0.00001\) & \(-0.4027\) & \(-0.3112\) \\ Nuclear & \(-0.7034\) & \(0.00001\) & \(-0.9073\) & \(-0.4995\) \\ Henry Hub & \(1.1530\) & \(0.00001\) & \(0.9328\) & \(1.3733\) \\ \multicolumn{5}{c}{Austin Equation} \\ \(\mathbf{f_{f}}\) & \(0.5709\) & \(0.00001\) & \(0.5502\) & \(0.5917\) \\ Wind & \(-0.3674\) & \(0.00001\) & \(-0.4143\) & \(-0.3204\) \\ Nuclear & \(-0.7780\) & \(0.00001\) & \(-0.9948\) & \(-0.5612\) \\ Henry Hub & \(1.1212\) & \(0.00001\) & \(0.8919\) & \(1.3505\) \\ \multicolumn{5}{c}{West Equation} \\ \(\mathbf{f_{f}}\) & \(0.6207\) & \(0.00001\) & \(0.5831\) & \(0.6583\) \\ Wind & \(-0.5517\) & \(0.00001\) & \(-0.6190\) & \(-0.4845\) \\ Nuclear & \(-0.7153\) & \(0.00001\) & \(-0.9645\) & \(-0.4662\) \\ Henry Hub & \(1.1317\) & \(0.00001\) & \(0.8288\) & \(1.4345\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ML coefficients for the 4:00 a.m. hour.