A Threshold Autoregressive Model for Wholesale Electricity Prices

The extensive literature on the study of electricity prices includes varied attempts to capture the
spiky behavior in electricity price series. Kellerhals [11] discusses some stochastic volatility models,
but does not offer a specific treatment of the electricity spikes. Moreover, there exist stochastic
jump-diffusion models such as the one set forth by Ethier and Dorris [8] (henceforth, the E&D
model) that combine a mean-reverting component and a positive jump term modeled according to
a Poisson process (see also [2]). However, this assumes that the rate of mean reversion is the same
throughout the price series, regardless of whether spikes are present or not. Additionally, there
exist models that separate the mean-reversion in electricity prices from the spikes (Huisman and
Mahieu [10]) via a regime jump model. In this case, if the prices spike up to a “jump regime”, they
must immediately return to the “mean-reversion regime.” However, there is no study of the rate
of mean-reversion in the presence of spikes, and how that might differ when there are no spikes.
Deng [7] details various stochastic models for electricity prices, which include the jump-diffusion
and regime switching types mentioned above, and also a stochastic volatility model. There is also
discussion on how these models are applicable in pricing derivative securities. However, procedures
to estimate parameters in these models would not be straightforward to implement.
In this article, we propose a variant of a first-order threshold autoregressive model (TAR(1)) for
wholesale electricity prices. (For discussion of TAR models, see, e.g., Tong [16], or, for the
continuous-time versions of these models, Stramer, Brockwell, and Tweedie [15].) In addition to
accounting for spikes and allowing for a temperature-driven effect, the model allows the threshold
to be time-varying. Effectively, it allows the mean-reversion parameter to change in the presence
of spikes. (Alvaredo and Rajaraman [1] noted this phenomenon, but they did not make a specific
proposal for modeling it. In fact, to the authors’ knowledge, the literature does not contain analyses
focusing on how the rate of mean-reversion might depend on the price level.) We estimate model
parameters using a Markov chain Monte Carlo (MCMC) procedure, and compare our model to the
E&D model, showing that our model provides a better fit to wholesale electricity prices collected
in Allegheny County, Pennsylvania over a three-year period.
The paper is organized as follows. Section 2 describes the data set, and discusses several important
features. In Section 3, we construct the model for the data, and describe the fitting procedure. A
discussion of the fitted model is given in Section 4, while Section 5 gives concluding remarks.
2 Description of Data
2.1 The Wholesale Electricity Price Data
We consider the maximum daily wholesale electricity prices in Allegheny County, Pennsylvania,
collected over the three-year period from January 1999 to December 2001. The data were taken
from www.pjm.com, a website that publishes wholesale electricity prices for regions in the Mid-
Atlantic. We choose to use the maximum daily price in order to preserve the effects of the spikes
in the price process (typically the spikes only last for a few hours). The maximum daily prices and
their logs are shown, respectively, in Figures 1 and 2.
2