# A Latent-Factor System Model for Real-Time Electricity Prices in Texas
## 5 Conclusions  
This paper demonstrated the relevance of latent factors on real-time energy prices using a system-wide approach. The ERCOT system served as the case study. Using energy prices from eight inter-connected zones as endogenous variables, we found that hidden factors significantly impact the merit-order effects of baseload and renewable energy generation. The latent-factor approach developed here can be improved and extended. [PERSON] et al. [29] use a hierarchical Bayesian approach to compare the impact of day-ahead and real-time prices on wholesale demand in ERCOT. However, they do not model latent factors. This paper clearly shows the importance of accounting for such factors. A Bayesian latent factor system-wide model for prices and/or demand is possible in principle; see [30]. However, the challenges are formidable. First, since the parameter space is very large, convergence issues will be a difficult problem to overcome. Concurrently, while studying energy prices or demand, the attendant datasets tend to be very large, as in this paper. This too will add to convergence issues since the likelihood function will have to be evaluated many-fold in any Markov chain Monte Carlo scheme that is required to obtain posterior distributions. Another future topic for research that this paper proposes is to model the system of equations in Equation (1) via non-normal errors. For example, [PERSON] et al. [1] use a nonparametric error distribution--the Indian Buffet Process--to develop a new class of latent factor models. But with large datasets, such nonparametric approaches are even more computationally involved compared to parametric formulations. Why should a non-normal error structure matter in the context of energy prices, and in the estimation of merit-order effects? Recent studies [21; 22] have shown that prices have asymmetric distributions with large kurtosis. Subsequently, error distributions from normal linear models tend to be non-normal heteroscedastic and autocorrelated. Hence, quantile regressions have been proposed and exemplified in the energy literature. However, there is a trade-off. Because of the mathematics underlying them, quantile regressions are essentially single-equation models. Thus, the prices of each ERCOT's eight zones can be modeled separately using quantile regressions; see [31]. But the results in this paper clearly demonstrate the importance of treating the eight zones as part of an interconnected system so that we can better understand how latent factors influence prices jointly. This leads to an open question: how should one construct a system-wide, latent-factor quantile regression model that is equivalent to Equation (1) in this paper? This is a very challenging problem for multiple reasons. For example, consider a bivariate time-series that represent prices from, say two of ERCOT's eight zones. Further, suppose the error term in the observation model in Equation (1) follows a bivariate skew-t distribution since this distribution allows for varying degrees of skewness. How should one jointly model the quantiles of this bivariate distribution as functions of latent factors and exogenous variables? The answer is not at all evident even in this simple bivariate setup. Therefore, instead of multivariate quantile regression systems, we believe, as a first step, it may be easier to recast Equation (1) using nonparametric prior distributions. Indeed, this could also lead to stronger correlations between the factor and price series since nonparametric priors can better treat outliers. The resulting estimation of the merit-order effects in energy markets would be a useful advancement. **Author Contributions:** Conceptualization, P.D. and J.Z. ; methodology, P.D. ; software, K.H.C. ; validation, K.H.C., P.D. and J.Z. ; formal analysis, K.H.C. and P.D. ; investigation, P.D. and J.Z. ; resources, J.Z. ; data curation, K.H.C. and J.Z. ; writing--original draft preparation, K.H.C., P.D. and J.Z. ; writing--review and editing, K.H.C., P.D. and J.Z. ; visualization, K.H.C. ; supervision, P.D.