# A Latent-Factor System Model for Real-Time Electricity Prices in Texas
## 4 Results  
Here, we report and discuss the results for three regions: Houston, Austin, and West; details on all other regions are available on request. Where appropriate, we highlight the empirics from the other regions as well. For the three regions, we report the results for 4:00 a.m., 12:00 p.m., and 4:00 p.m.; these are representative of the other off-peak and peak hours. Thus, we estimate Equation (1) nine times since we have nine models in total. We have the following major results. _Wald Test._ This test has a chi-square distribution. It tests the null hypothesis of whether or not all the unknown parameters in the observation and latent factor equations are jointly significant; this is similar to the F-test in multiple linear regression. For all nine models, the Wald Statistic soundly rejects the null hypothesis at any significance level (\(p<0.00001\)). _Actual versus predicted price series._ Consider Figure 3 which shows the actual and predicted series. As expected, there are some outliers in the data, especially during the 4:00 p.m. hour for all three zones. Also, again consider Figure 2. Note that the predicted time series, shown in grey, track the original price series in red quite well for the three different hours, barring the time points corresponding to the outliers. _Correlations between actual price series and estimated factor series \(\mathbf{f}_{t}\)._ Table 2 shows the correlations between each of the price series from all eight zones for the three hours. They are all positively correlated to the predicted latent factors. We highlighted the correlations for the regions Houston, Austin, and West in Table 2 in order to emphasize two points. First, note that the West zone has the weakest correlation during the peak hours of 12:00 p.m. and 4:00 p.m., compared to other regions. This is because of the larger impact of wind generation in the West during these hours, compared to other zones. Second, consider Figures 4-6. Each comprises four plots. For the sake of clarity, let us focus on just Figure 6 corresponding to the 4:00 p.m. hour. The top left plot is the latent factor one-step-ahead estimated time series. The other three plots in each of the panels are the actual price series for Houston, Austin, and West. The corresponding correlations between the latent factor series and these three price series from Table 2 are: 0.587, 0.591, and 0.457, respectively. It is evident that the latent factor series _structurally evolves_ like the three price series, which are representative of the price series for the entire ERCOT system. The presence of outliers in the price series is unavoidable in the ERCOT data. This would explain why some of the correlations are not as high as one might expect. We experimented with higher-order lags in the autoregressive error structure for the latent factor series in Equation (1). But such an increase in model dimensionality does not change the overall conclusions by much. Hence, we err on the side of parsimony. Figure 3: Scatter plots of actual and predicted prices, along with 45-degree lines, in Houston, Austin, and the West for the hours 4:00 a.m., 12:00 p.m., and 4:00 p.m.  
Figure 4: 4:00 a.m.—Latent Factor and Houston series (top left and right); Austin and West (Bottom left and right). Figure 5: 12:00 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 c} \hline \hline  & **Austin** & **Houston** & **LCRA** & **North** & **RAYB** & **CPS** & **South** & **West** \\ \hline
4:00 a.m. & **0.343** & **0.323** & 0.343 & 0.339 & 0.318 & 0.344 & 0.338 & **0.346** \\
12:00 p.m. & **0.532** & **0.473** & 0.531 & 0.541 & 0.517 & 0.504 & 0.428 & **0.365** \\
4:00 p.m. & **0.591** & **0.587** & 0.594 & 0.618 & 0.593 & 0.559 & 0.533 & **0.457** \\ \hline \hline \end{tabular} Note: Certain values are bold in order to better understand the Figure 6 discussion.  
\end{table} 
Table 2: Correlations between the actual price series and estimated factor series for all zones.