A Threshold Autoregressive Model for Wholesale Electricity Prices

A Threshold Autoregressive Model for Wholesale
Electricity Prices
B. Ricky Rambharat, Department of Statistics, Carnegie Mellon University
Anthony Brockwell, Department of Statistics, Carnegie Mellon University
Duane Seppi, Graduate School of Industrial Administration, Carnegie Mellon University
July 16, 2003
Abstract
We introduce a discrete-time model for electricity prices, which accounts for both spikes
and temperature effects. The model allows for different mean-reversion rates, one around
spikes, and another for the remainder of the process. We demonstrate how a Markov chain
Monte Carlo approach can be used for parameter estimation, using data recorded in Allegheny
County, Pennsylvania. For this data set, we also demonstrate that the model outperforms
existing stochastic jump-diffusion models. Results also demonstrate the importance of model
parameters corresponding to both the temperature effect and the two-level mean-reversion rate.
1 Introduction
The study of electricity price dynamics has attracted significant attention from researchers and industry
practitioners in recent years, in part due to the deregulation of the electricity market. In
modern financial markets, there exist derivative securities whose value depends on the price of
electricity, which warrants the study of these price dynamics. In addition, a sound model for electricity
is relevant for prediction purposes, and this is pertinent to day-ahead markets. However,
electricity prices exhibit features making them difficult to analyze and/or predict. Specifically,
occasional jumps (or “spikes”) in electricity prices merit careful scrutiny, and the temperature is
well-known to have a significant effect on the prices. The situation in the Midwest United States
during the Summer of 1998, when the price of wholesale electricity soared to an unprecedented
level of $7,500 per MwH (see [14]), illustrates just how important it is to understand the spiky behavior
in electricity prices. The importance of these spikes can be recognized from the perspective
of an investor who might own a financial derivative (for example, a put or call option) on electricity.
In this context, price spikes can result in favorable or unfavorable outcomes for the investor.
1

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

1200
Dollars per MwH
1000
800
600
400
200
0
Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01
Season of Year
Figure 1: Maximum daily wholesale prices of electricity, from Jan 1999 to Dec 2001, for Allegheny
County, PA.
Figure 1 clearly illustrates the spikes, where the price jumps for a short period of time to values
as high as per megawatt-hour. The portion of the price process without spikes is known to
exhibit mean-reversion (see Kellerhals [11]). In order to avoid problems associated with modeling
¢¡¤£¦¥§¥
of highly-skewed distributions, in our analysis we work with the logarithm of the maximum daily
prices. In Figure 2, we also include a non-parametric estimate of the mean of the log data.
Figure 3 shows the sample partial autocorrelation function (PACF) of the logarithm of the maximum
daily prices. This particular plot suggests that a low-order autoregressive model might
be appropriate for the data, supporting the use of a lower-order autoregressive model such as
a continuous-time AR(1) model (also referred to as an Ornstein-Uhlenbeck model or a Vasicek
model, Vasicek [17]), a discrete-time version of such a model, or some variant of one of these.
2.2 Temperature Data
As a general rule, temperatures which deviate substantially from some “optimal value” are associated
with high electricity prices. To investigate this phenomenon, we obtained temperature data for
Allegheny County, PA from the National Weather Services website (www.nws.noaa.gov), and
extracted measurements corresponding to the times at which our electricity prices were recorded.
These temperatures are plotted, for the time period from January 1999 to December 2001, in Figure
4.
3

8
Log Dollars per MwH
7
6
5
4
3
2
Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01
Season of Year
Figure 2: The logarithm of the maximum daily wholesale price of electricity, Jan 1999 to Dec
2001, for Allegheny County, PA, along with a non-parametric estimate of the mean.
4

Partial ACF
0.0 0.2 0.4
0 5 10 15 20 25 30
Lag
Figure 3: Sample partial autocorrelation function of the log of the maximum of daily electricity
prices.
5

Temperature (deg F)
80
60
40
20
0
Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01
Season of Year
Figure 4: Temperature data corresponding to price data over the period Jan 1999 - Dec 2001.
Log of Daily Max Prices ($/MwH)
3 4 5 6 7
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0 20 40 60 80
Temperature (deg F)
Figure 5: Log maximum prices vs. corresponding temperature.
6

¥
£
¡
£
¥
I
0
The temperature time series exhibits a clear periodic component, with higher temperatures associated
with summer months. In order to illustrate the relationship between temperatures and
log-electricity prices, we give a scatter-plot of log-price versus temperature in Figure 5. The figure
shows a parabolic relationship between the log of the maximum prices and temperature. We use
the loess function in S-Plus (see [18]) in order to obtain a smooth fit to the plot (the smoothing
parameter was chosen to be 2/3). This technique fits polynomials locally so that nearby data points
are most influential in the fit (see Cleveland et al. [5] for details on LOESS). Using this method of
smoothing, we find that the minimizing temperature is about 57.0 degrees Fahrenheit.
In light of this, we assume that the log-daily maximum price ¢¡¤£¦¥ includes a component of the
§©¨£
form , for some positive § constant , £ where denotes the temperature on the -th day,
and represents the minimizing temperature. ¨¡¤£¡£ Hence should include a difference of
this component, that is,
§©¨%©£&' (¨£!'©£ *) (1)
£$#
§©¨£!
"§©¨£
3 The Proposed Model
The sample PACF plot in Figure 3 suggests that a low-order autoregressive model might be appropriate
for the data. We use the discrete-time version of the model, that is an AR(1) model, but
we generalize it in several ways. We introduce a time-varying mean, along with a time-varying
threshold. The mean-reversion parameter is then allowed to take two different values - one when
the process is above the threshold, and another when it is below. In a difference equation for logdaily
price, we also include the term (1) to model the effect of temperature, as well as a term which
mixes spikes into the process.
3.1 The Model
Our discrete-time model for the daily log-electricity price series ,+£¦¥ is given by
/ 01¨%¡£ 324¡£657§©¨%©£! 98:£;5=£;5@?;A£B (2)
¡£.-
E£65"¡£FB (3)
+C£D-
where
(4)
represents the volatility, §RP
01¨%GH-
0J*BKGMLONB
BKGMPQN©B
is a constant which determines the magnitude of the
?.P
S temperature-effect, is the temperature at which price is typically £ minimal, is the temperature
that corresponds to the maximum price recorded on the -th 8T£U-V£W©£ day, XA£¦¥ , is
a sequence of independent and identically distributed Gaussian random variables with zero mean
and unit variance, YE£¦¥ and represents the time-varying mean of the process, which we introduced
7

¡
¨%¡£ ¡£3B© - ¨%¡£ ¡£FB© B£&-
5
¡
¡
¨¡ £ ¡£3B© B

0
¡
¥
)
)
)
)
That is, we assign a prior distribution ¨ ©
¡
3.3 Estimating Model Parameters
We adopt a Bayesian approach to model fitting.
to the parameter © vector , then the posterior distribution of the parameters ¨ ©
¡ B)()) B ¡
£¢
¨ © ¨ ¡ B))) B ¡
© effectively specifies the fitted model. We take the prior for © to be the product
of a truncated gamma densities 0 for 0 and , a normal density § for , a normal density N for , a
uniform(0,1) density for , and a gamma density for ? . All priors are relatively diffuse (that is,
they have high variance and have virtually no influence on the results).
¡
Rather than attempting to compute the posterior directly, we implement a block-Metropolis-Hastings
Markov chain Monte Carlo (MCMC) routine (see, e.g., Gilks et. al. [9], Chapter 1), which allows
us to draw samples from the posterior distribution. The block-Metropolis-Hastings algorithms
performs a sequence of five updates, ¨ 0 B 0 B §CBNB
for ? and , respectively. Each update uses
a random-walk proposal with zero-mean (multivariate) normal steps, whose variances are, corresponding
to the aforementioned sequence of blocks,
¤ ¥
¥¦¥ ¥
¥§¥ ¡
¥§¥ ¡ ¥
¥¦¥¨§ B
0.0001, 0.25, 0.0001, and 0.01. Using this approach, we generate a Markov chain of length 25,000,
and discard the first 5,000 iterations as burn-in.
4 Results
Table 1 shows posterior means and 95% credible intervals obtained for the parameters using the
MCMC algorithm. The 95% credible intervals are estimated by the range from the 0.025 to 0.975
sample quantiles of the last 20,000 iterations of the Markov chain.
0J
Parameter Posterior Mean 95% Credible Interval
0.650 (0.572, 0.712)
1.010 (0.811, 1.130)
¡¤¥
N
0.234 (0.122, 0.380)
2.352 (-0.029, 2.739)
0.019 (0.011, 0.027)
"0J 0.360 (0.198, 0.501)
¨%0
§©
? 0.412 (0.394, 0.430)
Table 1: Proposed model: posterior means and 95% credible intervals.
The results presented in Table 1 yield some important insights. First, observe that the meanreversion
0 rate above the threshold N level is higher than the mean reversion 01 rate below
the threshold, meaning that reversion to the mean is faster at higher levels than at normal levels.
Also, the posterior mean for the difference in the mean-reversion (0 0J rates ) is approximately
9

£
£
0.360, and 0 does not lie in the 95% credible interval. Hence a formal test would reject the null
hypothesis that mean-reversion rates are the same around spikes as at other times. We also see that
the estimate for is about 0.019, which is consistent with the proportion of spikes present in the
log data.
¡
Our results also support the hypothesis that temperature has a significant effect on electricity prices.
The estimate for § (the parameter controlling the effect of temperature), as expected, is positive.
The 95% credible interval does not contain 0, indicating the effect to be significant. The results for
the threshold parameter N indicate a mean threshold level of about 2.352 for the log price data.
We are able to compare the performance of our model (2, 3) with that of the E&D model. We fit the
E&D model and compute posterior means of parameters. We then use the posterior means from
both models and compute log-likelihoods, obtaining a log-likelihood of approximately -692.832
for our model, and -772.285 for the E&D model. Penalizing for the number of parameters, we
compute the Akaike Information Criterion (AIC) of the two models ( - ¨ log-likelihood 5
AIC
number of parameters ) as 1397.664 for our model, and 1552.570 for the E&D model. (Note
¨
that our model has 6 parameters, while the E&D model has 4.) By this measure, our proposed
model clearly out-performs the E&D model. Additional diagnostics and comparisons are given in
the appendix.
5 Concluding Remarks
In this article, we have proposed a first-order threshold-autoregressive process with a temperature
effect and a spiking component, for the logarithm of daily wholesale electricity price data. For
the Allegheny County data we have considered, the model appears to fit noticeably better than the
E&D model. However, we believe there are potential refinements to the model which could yield
further improvement. One possibility would be to allow the spike probability ¡ to vary with time,
and to depend on the price ¡ £ and temperature £ . In addition to allowing for seasonal effects
in spiking probabilities and improving the fit of the model, this could yield further insight into
factors which lead to high spiking probabilities. Another natural way to extend our model would
be to allow for time-varying or stochastic volatility instead of constant volatility (specified by the
parameter ?T ). The literature offers several alternatives along this direction (see for example, Cox
et al. [6]). Also, while our non-parametric smoothing approach to estimating E £ is effective for
short-range forecasting, other methods could be used. For instance, a stochastic process, such as a
random walk, could be specified for YET£¦¥ itself. Additional refinements could include estimation
of the jump process parameters ¢ and £ .
Clearly, a useful application of this model would be in generating forecasts. For short-range forecasting,
the mean process E£ can easily be projected into the future, although for longer range
forecasting, it would be desirable to use a stochastic process of some sort to characterize YE £¦¥ ,
since this would remove the implicit assumption that the mean is known, and thus yield safer
(higher variance) predictive distributions. In addition, temperature forecasts would be required,
and a separate model could be used to generate this, although in most parts of the world, fairly reli-
10

¥
¥
B
B
¡
¡
able forecasts are publicly available. A related problem is that of pricing energy derivatives, which
are becoming prominent in modern financial markets (see [14]). These instruments are important
for hedging purposes, and their prices obviously depend on the underlying price processes. The
use of the model proposed in this paper for pricing such derivatives is thus a potentially interesting
area for future study.
6 Acknowledgements
The authors are grateful to Peter Brockwell for comments and suggestions. This work was supported
in part by National Science Foundation Grant IIS-0083148.
A
Additional Diagnostics
One method to assess goodness-of-fit of our model is based on the following fact, derived from the
results in Rosenblatt [13]. £ Let denote the one-step predictive cumulative density ¡¤£ of , given
B))()(B ¡£ , evaluated at the observed value G©£ , written as £ -¢¡ ¢¡ £L GC£ GC£ *B9GC£ B))) B G¥ .
¡
If the model fits, £ £¦¥ then forms a sequence of independent uniform random variables on the
interval /
2 .
Thus we can assess goodness of fit for our model by computing the £ ,£¦¥ sequence , and performing
a standard Kolmogorov-Smirnov test to check whether they are plausibly iid uniform on the
/ 2 interval .
We compute these values for both our model (2, 3) and the E&D model, using the respective posterior
parameter estimates. Figures 6 shows plots of the empirical cdf of the £ Y£¦¥ sequence versus
the desired uniform cdf, along with 95% confidence bands for the empirical cdf of a uniform(0,1)
sample, using, respectively, our model and the E&D model for the maximum deviation from the
desired cdf. These plots were generated using the function cpgram from the library(MASS)
module in S-Plus (see Venables and Ripley [18]). In this case, the empirical CDF for the E&D
model comes slightly out of the confidence bands, suggesting that model (2,3) fits the data better.
As a further illustration of goodness-of-fit, we simulate data directly from the two models. The
simulations, shown in Figure 7, when compared with the actual data shown in Figure 2 show
that our model does indeed capture many of the visible features of the data. A visual inspection
suggests that both models are certainly reasonable, although arguably the variance of the nonspiking
portions simulated with the E&D model is a little high. This could be explained by the
fact that the E&D model uses only one set of dynamics to explain both spiking and non-spiking
behavior.
11

Proposed Model
E&D Model
0.0 0.2 0.4 0.6 0.8
0.0 0.1 0.2 0.3 0.4
frequency
0.0 0.2 0.4 0.6 0.8
0.0 0.1 0.2 0.3 0.4
frequency
Figure 6: Plots of the empirical CDFs of both models with 95% confidence bands for the empirical
CDF of a uniform(0,1) sample, obtained using the cpgram command in S-Plus.
12

8
8
7
7
Log Dollars per MwH
6
5
Log Dollars per MwH
6
5
4
4
3
3
2
Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01
Season of Year
Wi99 Su99 Fa99 Wi00 Sp00 Su00 Fa00 Wi01 Sp01 Su01 Fa01
Season of Year
Figure 7: Simulations from the proposed model (left) and the E&D model (right).
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