I. INTRODUCTION

Starting from seminal research works of Iwata (1974), Gollop and Roberts (1979), and Appelbaum (1982), measuring the degree of competition in oligopolistic markets has become one of the key activities in empirical industrial organization. A large and growing economic literature in New Empirical Industrial Organization (NEIO) relies on structural models to infer what types of firm behavior ("conduct") are associated with prices that exceed marginal costs. (1) A typical structural model based on the conduct parameter approach for homogenous product markets starts with specifying a demand function and writing down the first-order condition (FOC) of the firm's static profit-maximization problem:

(1) P ([Q.sub.t]) - mc ([q.sub.it]) + P' ([Q.sub.t]) x [q.sub.it] = 0,

where P([Q.sub.t]) is inverse demand, [Q.sub.t] = [[SIGMA].sup.N.sub.i] [q.sub.it] is total industry's output, [q.sub.it] is the firm's output in period t, mc([q.sub.it]) is the firm's marginal cost, and [[theta].sub.it] is a "conduct" parameter that parameterizes the firm's profit maximization condition. Under perfect competition, [[theta].sub.it] = 0 and price equals marginal cost. In static equilibrium, when [[theta].sub.it] = 1/[S.sub.it] (where [s.sub.it] denotes firm's market share of output) we face a perfect cartel, and when 0 < [[theta].sub.it] < 1/[s.sub.it] various oligopoly regimes apply. (2) As the interpretation of estimated conduct parameter 0 becomes more complicated in a dynamic setting, (3) we interpret this parameter as a descriptive measure of the firm's degree of market power based on a structural economic model of the firm's behavior. (4)

This paper is concerned about econometric estimation of the conduct parameter [[theta].sub.it] along with other cost and demand parameters. The conduct parameter may vary across time as market conditions change, and firms change their own pricing strategies. (5) As collusive arrangements are unstable and might only be successful for short periods, looking for evidence of collusion using averages over long time periods might miss a lot of what is going on. Moreover, the conduct parameter may also vary across firms as "there is nothing in the logic of oligopoly theory to force all firms to have the same conduct" (Bresnahan 1989, p. 1030). (6)

Obviously, allowing the conduct parameter to vary both by firms and time series results in an overparameterized model. To avoid this problem, empirical studies in structural econometric literature always impose some restrictions on the way the value of conduct parameter varies across firms and time. The overparameterization is typically solved by estimating the average of the conduct parameters of the firms in the industry (Appelbaum 1982; Wolfram 1999), reducing the time variation into a period of successful cartel cooperation and a period of price wars or similar breakdowns in cooperation (Porter 1983a), allowing for different conduct parameters between two or more groups of firms (Gollop and Roberts 1979), or assuming firm specific, but time invariant, conduct parameters in a panel data framework (Puller 2007).

This study contributes to the NEIO literature by proposing a novel econometric approach that deals with the overparameterization problem and helps obtain the values of firms' conduct that vary across both time and market participants. Instead of estimating the firm's conduct as a common parameter together with other parameters defining cost and demand, we propose treating firms' conduct [[theta].sub.it] as a random variable. Our approach is based on a composed error model, where the stochastic part is formed by two random variables--a traditional error term, which captures random shocks, and a random conduct term, which measures market power. The model is estimated in three stages. (7) In the first stage, all parameters describing the structure of the pricing Equation (1) are estimated using appropriate econometric techniques. In the second stage, distributional assumptions on random conduct term are invoked to obtain consistent estimates of the parameters describing the structure of the two error components. In the third stage, market power scores are obtained for each firm by decomposing the estimated residual into a noise component and a market-power component.

The distinctive feature of our approach is that it takes advantage of the fact that the unobserved conduct term is assumed to be non-negative, that is, [[theta].sub.it] is identified based on its one-sided structural restriction imposed by the economic theory. This allows us to obtain firm-specific market power estimates that vary across both time and firms without any parametric constraints, except for the distribution of the conduct parameter 0,;. (8) The flexibility of our approach thus permits the estimated market-power scores to represent many underlying, but unobserved, oligopoly equilibria. Though the idea of identification of structural econometric models through asymmetries in variance of error term is not new in applied econometric literature, (9) to our knowledge one-sided nature (skewness) of conduct parameter in oligopolistic industry settings is not examined explicitly in most (if any) of the previous studies.

The proposed approach can also be viewed as belonging to the same family as Porter (1983b), Brander and Zhang (1993), and Gallet and Schroeter (1995) who estimate a regime-switching model where market power enters in the model as a supply shock. As in our model, the identification of market power in these studies relies on making assumptions about the structure of unobservable error term. However, while previous papers estimated the pricing relationship (1) assuming [[theta].sub.it] = [[theta].sub.t] , to be a discrete random variable that follows a bimodal distribution ("price wars" vs. "collusion"), here [[theta].sub.it] varies both across firms and over time and is treated as a continuous random term. Therefore, while the switching regression models can only be estimated when there are discrete "collusive" and "punishment" phases that are either observable or could be inferred from the data, our model can be estimated in absence of regime switches. (10) The continuous nature of our conduct random term thus allows us to capture gradual changes in firm's degree of market power." In this sense, our model can be viewed as the continuous counterpart of the discrete regime-switching models.

Another feature that distinguishes our paper from previous studies is the attempt to estimate a double-bounded distribution that imposes both lower and upper theoretical bounds (i.e., 0 [less than or equal to] [[theta].sub.it] [less than or equal to] 1/[S.sub.it]) to a continuous random conduct term. To achieve this objective we have explored the stochastic frontier literature, (12) and adapted the doubly truncated normal distribution recently introduced by Almanidis, Qian, and Sickles (2010) to our framework. (13) To our knowledge, this study is among the first applications of stochastic frontier models for estimating market power. The only other known papers to do so are Huang, Chiang, and Chao (2017b) and Huang, Liu, and Kumbhakar (2017a), which use a copula-based stochastic frontier method to model the correlation between cost efficiency and market power. (14)

Our paper differs from the above two papers in two aspects. First, our paper uses a double-truncated distribution to impose a theoretical upper bound to the market power error term, while Huang, Liu, and Kumbhakar (2017a) and Huang, Chiang, and Chao (2017b) do not impose any upper bound to 0it. Second, we are using nonparametric (engineering) estimates of firms' marginal costs based on the unit costs associated to different production technologies, and hence our empirical strategy does not rely on estimating a cost function aiming to measure firms' marginal costs. Both approaches have advantages and disadvantages. For instance, Genesove and Mullin (1994), Clay and Troesken (2003), and Kim and Knittel (2006) show that market power estimates might be strongly biased due to inaccurate estimates of marginal costs. When firms' marginal costs are not observed, they must be inferred from previous estimates of the utilities' cost function. The traditional Translog function provides a second-order approximation to the underlying cost function, but only a first-order approximation to the underlying marginal costs (see Jamasb, Orea, and Pollitt 2012), which is highly inaccurate. On the contrary, the use of nonparametric estimates of firms' marginal costs based on engineering approach assumes that firms are fully efficient. Thus, our present estimation framework ignores the potential correlation between cost/profit inefficiency and market power. (15)

We illustrate the model with an application to the California electricity generating market between April 1998 and December 2000. This industry is an ideal setting to apply our model because there were high concerns regarding market power levels in California restructured electricity markets during that period, and detailed price, cost, and output data are available as a result of the long history of regulation and the transparency of the production technology. This data set allows us to compute directly hourly marginal cost and residual demand elasticities for each firm. We can therefore avoid complications from estimating demand and cost parameters and focus our research on market power, avoiding biases due inaccurate estimates of marginal cost and residual demand. Hence, this data set provides a proper framework to discuss methodological issues and to apply the empirical approach proposed in the present paper. In addition, the robustness of our data has been extensively tested in previous studies focusing on market power in the California electricity market (e.g., Joskow and Kahn 2002; Borenstein, Bushnell, and Wolak 2002; Wolak 1999; Wolak 2003; and Puller 2007).

Our empirical results are compared to a well-known study of Puller (2007), which employs fixed-effect regression approach to obtain firm-specific conduct parameter averages over three periods between July 1, 1998 and November 30, 2000. In the first stage we estimate firm-specific conduct parameter averages, which are very similar to findings of Puller (2007). This result demonstrates that both approaches are, in practice, equivalent or interchangeable for estimating time-invariant firm-specific market power scores. In a panel data setting, our methodology has a significant advantage over fixed-effects regression approach in that we can analyze changes in firm's market conduct over time. The analysis of firm-specific conduct parameters suggests that realization of market power varies over both time and firms, and rejects the assumption of a common conduct parameter for all firms. Estimated firm-specific conduct parameters generally tend to move in the same direction across time, suggesting that the potential for exercising market power across all firms varies as market conditions change.

The rest of the paper is structured as follows. In Section II we describe the empirical specification of the model. In Section III we discuss the three-stage procedure to estimate the model. The empirical illustration of the model using California electricity data is described in Section IV and Section V concludes.

II. EMPIRICAL SPECIFICATION

The traditional structural econometric model of market power is formed by a demand function and a pricing equation. Because we are interested in the estimation of industry or firm-specific market power scores, we only discuss here the estimation of the pricing Equation (1), conditional on observed realization of residual demand. (16) If the demand function parameters are not known, they should be estimated jointly with cost and market power parameters.

In this section, we develop a simple model where firms sell homogenous products (e.g., kilowatt-hours of electricity) and choose individual quantities each period so as to maximize their profits. Our model is static as we assume that firms maximize their profits each period without explicit consideration of the competitive environment in other periods. (17) Firm i's profit function in period t can be written as:

(2) [[pi].sub.it] = P([Q.sub.t]) x [q.sub.it] - C ([q.sub.it],[alpha]),

where [alpha] is a vector of cost parameters to be estimated. We assume that firms choose different quantities each period and their marginal cost varies across firms and over time.

In a static setting, the firm's profit maximization problem is.

(3) [mathematical expression not reproducible]

The FOCs of the static model are captured by Equation (1), that is:

[P.sub.t] = mc ([q.sub.it], [alpha]) + [q.sub.it] x [[theta].sub.it],

where mc([q.sub.it] [alpha]) stands for firm's marginal cost, [g.sub.it] = [P.sub.t] [q.sub.it]/[Q.sub.t][[eta].sup.D.sub.it] and [[eta].sup.D.sub.it] = P' ([Q.sub.t]) [P.sub.t]/[Q.sub.t] is the (observed) elasticity of product demand. The stochastic specification of the above FOCs can be obtained by adding the error term, capturing measurement and optimization errors:

(4) [P.sub.t] = mc [[q.sub.it], [alpha]) + [g.sub.it] x [Q.sub.it] + [v.sub.it].

Instead of viewing firms' behavior as a structural parameter to be estimated we here treat firms' behavior as a random variable. While retaining standard assumption that the error term [v.sub.it] is i.i.d. and symmetric with zero mean, we also assume that [[theta].sub.it] follows a truncated distribution that incorporates the theoretical restriction that 0 [less than or equal to] [q.sub.it] [less than or equal to] 1/[S.sub.it]. The distinctive feature of our model is that the stochastic part is formed by two random variables--the traditional symmetric error term, [v.sub.it], and a one-sided random conduct term, [g.sub.it], x [[theta].sub.it] that reflects the market power. The restriction that the composed error term is asymmetric (as [[theta].sub.it] (is one-sided distributed) allows us to obtain separate estimates of [[theta].sub.it] and [v.sub.it] from an estimate of the composed error term.

III. ESTIMATION STRATEGY

We now turn to explaining how to estimate the pricing relationships presented in the previous section. Two estimation methods are possible: a method-of-moments (MM) approach and maximum likelihood (ML). The MM approach involves three stages. In the first stage, all parameters describing the structure of the pricing equation (i.e., cost, demand, and dynamic parameters) are estimated using appropriate econometric techniques. In particular, because some regressors are endogenous, a generalized method of moments (GMM) method should be employed to get consistent estimates in this stage. (18) This stage is independent of distributional assumptions on either error component. In the second stage of the estimation procedure, distributional assumptions are invoked to obtain consistent estimates of the parameter(s) describing the structure of the two error components, conditional on the first-stage estimated parameters. In the third stage, market power scores are estimated for each firm by decomposing the estimated residual into an error-term component and a market-power component.

The ML approach uses ML techniques to obtain second-stage estimates of the parameters) describing the structure of the two error components, conditional on the first-stage estimated parameters. It can be also used to estimate simultaneously both types of parameters, if the regressors in the pricing equation are exogenous. In this case, the ML approach combines the two first stages of the MM approach into one.

While the first stage is standard in the NEIO literature, the second and third stages take advantage of the fact that the conduct term is likely positively or negatively skewed, depending on the oligopolistic equilibrium that is behind the data generating process. Models with both symmetric and asymmetric random terms of the form in Section II have been proposed and estimated in the stochastic frontier analysis literature (see Fried, Knox Lovell, and Schmidt 2008; Kumbhakar and Lovell 2000).

While economic theory imposes both lower and upper theoretical bounds to the random conduct term, the skewness of its distribution is an empirical issue. Oligopolistic equilibrium outcomes often yield skewed conduct random terms where larger (collusive) conduct parameter values are either less or more probable than smaller (competitive) conduct values. For instance, the dominant firm theory assumes that one (few) firm(s) has enough market power to fix prices over marginal cost. This market power is, however, attenuated by a fringe of (small) firms that do not behave strategically. (19) The most important characteristic of this equilibrium is that the modal value of the conduct random term (i.e., the most frequent value) is close to zero, and higher values of [[theta].sub.it] are increasingly less likely (frequent). In other markets all firms might be involved in a perfect cartel scheme. In such a cartel equilibrium, firms usually agree to sell "target" quantities, and the resulting market price is the monopoly price, which is associated with the maximum conduct value, for example, [[theta].sub.it] = 1/[S.sub.it] Smaller values of [[theta].sub.it] are possible due, for instance, to cheating behavior. (20) This means that the modal value of the conduct random term in this equilibrium is one, with smaller values of [[theta].sub.it] increasingly less likely. That is, firm conduct is negatively skewed. In general, similar equilibria that yield asymmetric distributions for the firm-conduct parameter with modal values close to zero or to the number of colluding firms may also arise. While the above examples of asymmetry are across firms, the firm-conduct can be also negatively skewed over time if a particular degree of competition (collusion) in the market occurs more frequently. This could happen, for example, due to different length of high and low demand periods that are found to affect the degree of market competition (e.g., Green and Porter 1984 and Rotemberg and Saloner 1986).

A. First Stage: Pricing Equation Estimates

Let us denote the average of the conduct parameters of the firms in the industry as [theta] = E([[theta].sub.it]). If we now add and subtract [g.sub.it][theta] from Equation (4), the pricing equation to be estimated can be rewritten as:

(5) [P.sub.t] = mc ([q.sub.it], [alpha]) + [g.sub.it] * [theta] + [g.sub.it] * {[[theta].sub.it] - [theta]} + [v.sub.it] [equivalent to] mc ([q.sub.it], [alpha]) + [g.sub.it] * [theta] + [[epsilon].sub.it]

where [alpha] is the vector of cost parameters and [[epsilon].sub.it] a composed error term:

(6) [[epsilon].sub.it] = [v.sub.it] + [g.sub.it] x {[[theta].sub.it] - [theta]}.

As we explain above, in Equation (6) the composed error term, [[epsilon].sub.it], comprises of a traditional symmetric error term, [v.sub.it], and a one-sided random conduct term, [g.sub.it]. {[[theta].sub.it] - [theta]}, which reflects the market power.

The possible endogeneity of some regressors will lead to least squares being biased and inconsistent. This source of inconsistency can be dealt with by using GMM. Note that the parameter estimates can still be inconsistent as in our second-stage we assume that E([[theta].sub.it]) is firm and time specific. This is because the upper truncation of [[theta].sub.it] depends on the firm's market share, and [[theta].sub.it] is heteroscedastic itself. To achieve consistent estimates it is critical to ensure that chosen instruments do not include determinants of [[theta].sub.it]. (21)

Though first-step GMM parameter estimates are consistent, they are not efficient by construction because the [[epsilon].sub.it]s' are not identically distributed. Indeed, assuming that [[theta].sub.it] and [v.sub.it] are distributed independently of each other, the second moment of the composed error term can be written as:

(7) E([[epsilon].sup.2.sub.it])= [[sigma].sup.2.sub.v] + [g.sup.2,sub.it] * [[sigma].sup.2.sub.[theta]],

where [mathematical expression not reproducible]. Equation (7) shows that the error in the regression indicated by Equation (5) is heteroscedastic. Therefore an efficient GMM estimator is needed. (22) Suppose that we can find a vector of m instruments [M.sub.it] that satisfy the following moment condition:

(8) [mathematical expression not reproducible]

The efficient two-step GMM estimator is then the parameter vector that solves:

(9) ([??], [??]) = arg min [[[SIGMA].sub.i][[SIGMA].sub.t][m.sub.it]([alpha],[theta])]'[W.sup.-1] [[[SIGMA].sub.i][[SIGMA].sub.t][m.sub.it]([alpha],[theta])],

where W is an optimal weighting matrix obtained from a consistent preliminary GMM estimator. This optimal weighting matrix can take into account both heteroscedasticity and autocorrelation of the error term.

B. Second Stage: Variance Decomposition

The pricing Equation (5) estimated in the first stage is equivalent to standard specification of a structural market power econometric model, where an industry-average conduct is estimated (jointly with other demand and cost parameters in most applications). As we mentioned earlier in this section, our paper aims to exploit the asymmetry of the random conduct term [[theta].sub.it] to get firm-specific market power estimates in the second and third stages. Because, as it is customary, we are going to assume that the noise term [v.sub.it] follows a normal distribution, this implies that the composed error term in Equation (6) is also asymmetrically distributed.

In the second stage of the estimation procedure, distributional assumptions are invoked to obtain consistent estimates of the parameter(s) describing the variance of [theta] and [v.sub.it] (i.e., [sigma][theta] and [[sigma].sub.v]), conditional on the first-stage estimated parameters. This stage is critical as it allows us to distinguish variation in market conduct, measured by [[sigma].sub.[theta]], from variation in demand and costs, measured by [[sigma].sub.v]. Given that we are going to assume a particular distribution for the conduct term, both variances can be estimated using ML. The ML estimators are obtained by maximizing the likelihood function associated to the error term [[epsilon].sub.it] = [v.sub.it] + [g.sub.it] * {[[theta].sub.it] - [theta]} that can be obtained from an estimate of the first-stage pricing Equation (5).

The second-stage model can be also estimated by MM that relies on the second and third moments of the error term [[epsilon].sub.it] in Equation (5). This approach takes advantage of the fact that, while the second moment provides information about both and [[sigma].sub.v], the third moment only provides information about the asymmetric random conduct term. In the empirical application (Section IV) we only report the results using the ML approach for both theoretical and practical reasons. First, Olson, Schmidt, and Waldman (1980) showed using simulation exercises that the choice of the estimator (ML vs. MM) depends on the relative values of the variance of both random terms and the sample size. When the sample size is large (as in our application) and the variance of the one-sided error component is small, compared to the variance of the noise term, ML outperforms MM. Second, the MM approach has some practical problems. As it is well known in the stochastic frontier literature, neglected heteroscedasticity in either or both of the two random terms causes estimates (here, the market power scores) to be biased. Kumbhakar and Lovell (2000) pointed out that only the ML approach can be used to address this problem. Another practical problem arises when, in homoscedastic specifications of the model, the implied ce becomes sufficiently large to cause [[sigma].sub.v] < 0, which violates the assumptions of the econometric theory.

Whatever the approach we choose in the present stage, we need to choose a distribution for [[theta].sub.it] The chosen distribution for the random conduct term reflects the researcher's beliefs about the underlying oligopolistic equilibrium that generates the data. Therefore, different distributions for the conduct random term can be estimated to test for different types of oligopolistic equilibrium. The pool of distribution functions is, however, limited as we need to choose a simple distribution for the asymmetric term to be able to estimate the empirical model, while satisfying the restrictions of the economic theory. The need for tractability prevents us from using more sophisticated distributions that, for instance, would allow us to model industries formed by two groups of firms with two different types of behavior, that is, an industry with two modes of the conduct term.

The distribution for the asymmetric term adopted in this study is the double-bounded distribution that imposes both lower and upper theoretical bounds on the values of the random conduct term, that is, 0 [less than or equal to] [[theta].sub.it] [less than or equal to] 1/[S.sub.it]. In doing so, we follow Almanidis, Qian, and Sickles (2010), who propose a model where the distribution of the inefficiency (here, the conduct) term is a normal distribution N([mu],[[sigma].sub.v]) that is truncated at zero on the left tail and at 1/[S.sub.it] on the right tail. (23,24) Regarding the noise term, we will assume that [v.sub.it] follows a normal distribution with zero mean and standard deviation (SD) [[sigma].sub.v]. The model can then be estimated by maximizing a well-defined likelihood function associated to the error term that can be obtained from an estimate of the first-stage pricing equation:

(10) [mathematical expression not reproducible].

Note that this residual term is the observed counterpart of:

(11) [[epsilon].sub.it] = [v.sub.it] + [g.sub.it] * ([[theta].sub.it] E([[theta].sub.it])).

When [[theta].sub.it] follows the doubly truncated normal distribution introduced by Almanidis, Qian, and Sickles (2010), the likelihood function associated to Equation (11) can be written as:

(12) [mathematical expression not reproducible]

where [mathematical expression not reproducible]

It should be noted that [[??].sub.it] depends on the expected value of the conduct term, that is, E([[theta].sub.it]). In principle, this expected value is unknown. To deal with this issue we can follow two alternative empirical strategies. The first strategy relies on the assumed distribution for the conduct term. Indeed, if [[theta].sub.it]is assumed to follow a doubly truncated normal distribution, the expected value of the conduct term can be written as:

(13) [mathematical expression not reproducible].

The above likelihood function in Equation (11) can then be maximized once we have replaced the expected value of the conduct term with the mathematical expression of E([[theta].sub.it]) in Equation (13). It should be noted that there are neither new parameters to be estimated nor first-stage parameters in this equation. This equation also indicates that, regardless whether [[sigma].sub.[theta]] is homoscedastic or heteroscedastic, the expected value of the conduct term is observation specific as it depends on firms' market shares.

The second strategy to replace the expected value of the conduct term in Equation (11) relies on the estimated parameters of the first-stage pricing equation. In this case, the expected value E([[theta].sub.it]) is simply replaced with our first-stage estimate of the conduct parameter, that is, [??]. Note that in this strategy the expected value of the conduct term is restricted to be common to all firms, or time invariant if firm-specific conduct parameters are estimated. The advantage of this strategy is that the estimated expectation does not depend on distributional assumptions on [v.sub.it] and [[theta].sub.it].

As we have mentioned earlier, neglected heteroscedasticity in either or both of the two random terms produces biased estimates of the market power scores. To address this problem, we propose estimating our model allowing for firm-specific and/or heteroscedastic random terms. In particular, we extend the classical homoscedastic model by assuming that variation in the error term is an exponential function of an intercept term, the day-ahead forecast of total demand and its square (i.e., FQ, [FQ.sup.2]), that are included in the model in order to capture possible demand-size effects, and a vector of days-of-the-week dummies (DAY). These variables allow for time-varying heteroscedasticity in the error term. In addition, firm-specific dummy variables (FIRM) are included to test whether variation of the error term is correlated with (unobservable) characteristics of firms/observations. Therefore, the variation in the noise term can be written in logs as (25):

(14) [mathematical expression not reproducible]

Regarding the conduct random term, we assume that both its mean and its variance are firm and time specific. This is achieved by modeling [[sigma].sub.[theta]] as an exponential function of several covariates. (26) Because the upper bounds are firm-specific, we should expect a higher variation in [[theta].sub.it] for those firms with lower market share, and vice versa. For this reason, we include sit as a determinant of variation in market conduct and we expect a negative coefficient for this variable. Since Porter (1983a), who estimates a regime-switching model, there is a large tradition in the empirical industrial organization literature that extended Porter's model by adding a Markov structure to the state (i.e., discrete) random variable capturing periods of either price wars or collusion (see, for instance, Ellison 1994, and Fabra and Toro 2005). Under this structure, the regimes are not independent and they are correlated over time, so that a collusion state today can be likely to lead to another collusion state next day.

Although imposing an autoregressive structure on the conduct term [[theta].sub.it] might be a more realistic assumption, in this study we still assume that [[theta].sub.it] is independent over time. There are two reasons for doing so. First, in our model, random conduct parameter [[theta].sub.it] varies across both firms and over time, and is treated as a continuous random term that, in addition, it is truncated twice. This makes it difficult to allow for correlation over time in the random conduct term. In a finite-state framework, the model can be estimated by maximizing the joint likelihood function of [v.sub.it] and [[theta].sub.it] if a Markov structure is not imposed. When this structure is added, the computation of the likelihood function of the model is much more complicated because it necessities to integrate out [[theta].sub.il], ... , [[theta].sub.ilt]. Several filtering methods have been proposed (e.g., Hamilton 1989) to make tractable the likelihood function, and to jointly estimate the hidden states and the parameters of the model. As pointed out by Emvalomatis, Stefanou, and Lansink (2011), these filtering methods cannot be easily adapted to a continuous and non-negative random variable. For instance, the traditional Kalman filtering techniques cannot be used in our framework when the latent variable (here [[theta].sub.it]) is not normally distributed, and a one-to-one, nonlinear transformation of [[theta].sub.it] should be used before putting 0,r in an autoregressive form. It is clearly out of scope of the present paper to extend the proposed approach to double truncated random variables. Second, Alvarez et al. (2006) pointed out that we can still get consistent parameter estimates if the correlation of unobserved conduct term over time is ignored. The justification is based on a quasi-ML argument, where the density of a firm's efficiency score at time t, could still be correctly specified, marginally with respect to the efficiency score in previous periods.

Although we do not explicitly incorporate autoregressive specification of unobserved conduct term [[theta].sub.it], we do attempt to control for observed past behavior in some target variables. (27) In particular, and following Fabra and Toro's (2005) application to the Spanish electricity market, we include the lagged first-difference of market shares, that is, [DELTA][S.sub.it - 1] = [s.sub.it-i] - [S.sub.it-2], as a target variable. A negative value of [DELTA][S.sub.it-1], indicates that other strategic rivals have got yesterday a higher market share than the day before. If the increase in rivals' market share is taken as a signal of weakness of a potential tacit collusion arrangement among firms, it might encourage firm i to behave more aggressive next day. If this is the case, we should expect a positive sign of the coefficient associated to this variable. (28)

Hence, our final specification of the conduct variation is:

(15) ln [[sigma].sub.[theta],it] = [[upsilon].sub.0] + [[upsilon].sub.1][S.sub.it] +[[upsilon].sub.2] + [DELTA][S.sub.it-1]+ [N.summation over (i=2)][[??].sub.i] x [FIRM.sub.i].

C. Third Stage: Obtaining Firm-Specific Market Power Estimates

In the third stage, we obtain the estimates of market power for each firm. From previous stages we have estimates of [[epsilon].sub.it] = [v.sub.it] + [g.sub.it] x ([[theta].sub.it] - E([[theta].sub.it])) or, in other words, of [[??].sub.it] = [[epsilon].sub.it] + [g.sub.it]E([[epsilon].sub.it]) = [v.sub.it] + [g.sub.it] x [[theta].sub.it] which obviously contain information on [[theta].sub.it]. The problem is to extract the information that [[??].sub.it] contains on [[theta].sub.it]. Jondrow et al. (1982) face the same problem in the frontier production function literature and propose using the conditional distribution of the asymmetric random term (here [??][[theta].sub.it] = [g.sub.it] x [[theta].sub.it]) given the composed error term (here [[??].sub.it]). The best predictor of the conduct term is the conditional expectation E ([[??].sub.it][parallel][[??].sub.it]) (see Kumbhakar and Lovell 2000). (29) Given our distributional assumptions, Almanidis, Qian, and Sickles (2010) show that the analytical form for E ([[??].sub.it][parallel][[??].sub.it]) can be written as follows (30):

(16) [mathematical expression not reproducible]

where [[??].sub.it] = [[mu] x [[sigma].sup.2.sub.v] + [[??].sub.it] [([g.sub.it][[sigma].sub.[theta]]).sup.2]]/[[sigma].sup.2.sub.it] and [[bar.[sigma]].sub.it] = ([g.sub.it][[sigma].sub.[theta]]) [[sigma].sub.v]/[[sigma].sub.it]

Once we have a point estimator for [[??].sup.2.sub.it], a point estimator for the conduct parameter [[theta].sub.it] (hereafter [[??].sub.it]) can be obtained using the identity [[theta].sub.it] [equivalent to] [[??].sub.it]/[g.sub.it] That is, [[??].sub.it] = E([[??].sub.it] [parallel][[??].sub.it]) /[g.sub.it]. (31,32)

IV. EMPIRICAL APPLICATION TO CALIFORNIA ELECTRICITY MARKET

In this section, we illustrate the proposed approach with an application to the California electricity generating market. This market was opened to competition in 1998 allowing firms to compete to supply electricity to the network. The wholesale prices stayed at "normal" levels from 1998 to May 2000, and then skyrocketed during summer and fall 2000, resulting in the breakdown of the liberalized electricity market by the end of 2000. While the California electricity crisis was a complex situation affected by a number of factors, such as poor wholesale market design, absence of long-term contracting, unexpected increase in generation input costs, and hike in end-use electricity demand due to unusually hot weather, a number of studies pointed to the evidence of significant market power in this restructured market. (33)

Our empirical application analyzes the competitive behavior of five strategic large firms from Puller's (2007) study of monopoly power in California restructured electricity markets using the same sample period (from April 1998 to November 2000). Following Borenstein, Bushnell, and Wolak (2002), Kim and Knittel (2006), and Puller (2007), we define five large firms that owned fossil-fueled generators (AES, DST/Dynegy, Duke, Reliant, and Southern) as "strategic" firms, that is, pricing according to Equation (5). The competitive fringe includes generation from nuclear, hydroelectric, and small independent producers, and imports from outside California. Puller (2007, p. 77) argues that these suppliers were either relatively small or did not face strong incentives to influence the price.34 Other studies (Borenstein et al. 2008; Bushnell and Wolak 1999), however, find that competitive fringe occasionally did have incentives to act strategically and bid elastic supply and demand schedules to counter exercise of market power by the strategic firms. Because electricity storage is prohibitively costly, [[upsilon].sub.0] both strategic and non-strategic firms had to produce a quantity equal to demand at all times. (36) The five large firms and a competitive fringe interacted daily in a market where rivals' costs were nearly common knowledge, which created strong incentives for tacit collusion (Puller 2007). And the residual demand for electricity was highly inelastic, which, given institutional weaknesses of California Power Exchange (PX), allowed individual firms to raise prices unilaterally (Wolak 2003).

We first carry out a standard econometric exercise and estimate consistently by GMM the parameters of the pricing Equation (5). In particular, and in order to be sure that our first stage is sound, we try to reproduce Puller's (2007) results, using the same data set, and the same specification for the pricing Equation (5), and the same set of dependent and explanatory variables. (37)

After estimating the parameters of the pricing equation, we carry out the second and third stages assuming particular distributions for the conduct random term, all of them imposing the conduct term to be positive and less than the inverse of firms' market shares.

A Pricing Equation and Data

Following Puller (2007, eq. 3) the pricing equation to be estimated in the first stage of our procedure is:

(17) [(P - mc).sub.it] = [alpha] x [CAPBIND.sub.it] + [theta]. ([P.sub.t] [q.sub.it]/[Q.sup.S.sub.strat,t])/[[eta].sup.D.sub.strat,t] + [[epsilon].sub.it],

where [alpha] and [theta] = E([[theta].sub.it]) are parameters to be estimated, P, is market price, [mc.sub.it] is firm's marginal costs, [q.sub.it] is firm's output, [CAPBIND.sub.it] is a dummy variable that is equal to 1 if capacity constraints are binding and equal to 0 otherwise, [Q.sup.S.sub.strat] is total electricity supply by the strategic firms, and [[eta].sup.D.sub.strat,t] is the elasticity of residual hourly demand function of the five strategic firms.

We use hourly firm-level data on output and marginal cost. As in Puller (2007), we focus on an hour of sustained peak demand from 5 to 6 p.m. (hour 18) each day, when intertemporal adjustment constraints on the rate at which power plants can increase or decrease output are unlikely to bind. Following Borenstein, Bushnell, and Wolak (2002), we calculate the hourly marginal cost of fossil-fuel electricity plants as the sum of marginal fuel, emission permit, and variable operating and maintenance costs.38 We assume the marginal cost function to be constant up to the capacity of the generator. A firm's marginal cost of producing one more megawatt hour of electricity is defined as the marginal cost of the most expensive unit that it is operating and that has excess capacity.

Our measure of output is the total production by each firm's generating units as reported in the Continuous Emissions Monitoring System (CEMS), that contains data on the hourly operation status and power output of fossil-fueled generation units in California. We use the PX day-ahead electricity price, because 80%-90% of all transactions occurred in the PX. Prices vary by location when transmission constraints between the north and south bind. (39) Most firms own power plants in a single transmission zone, so we use a PX zonal price.

For comparison purposes we also replicate Puller's (2007) residual demand elasticity estimates to compute the expected value of the random conduct term. Puller (2007) computes residual demand elasticity as

(18) [[eta].sup.D.sub.strat,t] = [??][Q.sup.S.sub.fringe]/[Q.sup.S.sub.strat,t],

where [Q.sup.S.sub.fringe] is electric power supply by the competitive fringe, and [??] = [P.sub.t]/[P'.sub.t] [Q.sup.S.sub.fringe,t] is the price elasticity of the fringe supply. obtain the estimates of [??] from Puller (2007, Table 3, p. 83). Table 1 reports the summary statistics for all these variables.

Figure 1 shows calculated price-cost margins. This figure is almost identical to Figure 1 in Puller (2007), and shows that margins vary considerably over sample period. They are also higher during the third and fourth quarters of each year, when total demand for electricity is high.

B. Pricing Equation Estimates

This section describes estimation results of pricing Equation (5), which result in the first-stage parameter estimates. We consider different specifications, estimation methods, and time periods. First, we estimate Equation (5) using elasticities of residual demand, calculated based on PX data and based on Puller's (2007) estimates. Second, we allow for output to be an endogenous variable as the error term [[epsilon].sub.it] in Equation (5) could include marginal cost shocks that are observed by the utility. (40) To account for endogeneity of output we estimate Equation (5) by the ordinary least squares (OLS), treating [P.sub.t] x [q.sub.it]/[Q.sup.S.sub.strat,t] (hereafter [x.sub.it]) as exogenous variable, and by GMM using instruments for [x.sub.it]. We use three

instruments for [x.sub.it]: the inverse of the day-ahead forecast of total electricity output, 1/F[Q.sub.t], the dummy variable for binding capacity constraints, [CAPBIND.sub.it] and firm's nameplate generation capacity, [k.sub.it]. (41) The first two instruments are from Puller (2007). (42) We assume that firm's generation capacity is orthogonal to the error term because it can be viewed as a quasi-fixed variable, independent of current levels of operation. We then perform Hansen's (1982) J test, F-test for weak instruments (Staiger and Stock 1997) and Hausman's (1978) specification test to test for over-identifying restrictions, instruments' strength, and consistency of the OLS estimates. Finally, we estimate Equation (5) over two periods described in Puller (2007). The first period from July 1998 to April 1999 covers four strategic firms (AES, DST/Dynegy, Duke, and Reliant). The second period from May 1999 to November 2000 covers five strategic firms following Southern entry. (43)

Table 2 summarizes the specification, estimation, and fit of the pricing Equation (5) over the periods analyzed in Puller (2007). The results of Hansen's J test and F-test for weak instruments indicate that the chosen instruments are generally valid, (44) whereas Hausman's (1978) specification test indicates that the OLS results are biased and inconsistent. The size of this OLS bias (measured by the difference between OLS and GMM estimates) is large indicating a significant correlation between the term [x.sub.it] and unobserved error term. All estimated values of the conduct parameter are statistically significant from zero. The GMM estimates of the conduct parameter, the GMM estimated values of the conduct parameter are a bit smaller in both periods to Puller's (2007) estimate of 0.97. However, the difference from Puller's (2007) estimate is not statistically significant, in the first period. It should be finally noted that Coelli's (1995) tests indicate that the estimated residuals in both periods do not follow a normal distribution and, in particular, that the distribution of the estimated residual are positive skewed. This result suggests the presence of a one-sided error term in Equation (5), as expected given our specification of the composed error term in Equation (6).

C. Variance Decomposition

Once all parameters of the pricing Equation (5) are estimated, we can get estimates of the parameters describing the structure of the two error components included in the composed random term [[epsilon].sub.it] (second stage). Conditional on these parameter estimates, market power scores can be then estimated for each firm by decomposing the estimated residual into a noise component and a market-power component (third stage). Following the discussion in Section III.B, to obtain the estimates of the parameters describing the structure of error components we first need to specify the distribution of the unobserved random conduct term. We must also impose both lower and upper theoretical bounds on the values of the random conduct term, that is, 0 [less than or equal to] [[theta].sub.it] [less than or equal to] 1/[S.sub.it] To achieve this objective, we use the truncated half normal distribution model introduced by Almanidis, Qian, and Sickles (2010) that allows us to impose both theoretical restrictions. (45)

In the empirical application we thus use two empirical strategies to deal with the expected value of the conduct random term, which in principle is unknown. The first strategy uses the mathematical expression in Equation (13), and the second strategy relies on the first-stage estimate of the conduct parameter [??]. The results based on both of these strategies are summarized in the tables and figures below.

Table 3 describes the parameter estimates of the model describing the structure of [[theta].sub.it] (and [v.sub.it] (i.e., [[sigma].sub.[theta]] and [[sigma].sub.v]) based on different identification strategies and across different time periods, conditional on the first-stage estimated parameters. Corrected standard errors have been computed using the "sandwich form" expression suggested by Alvarez et al. (2006) to allow for the fact that [[theta].sub.it] is likely not independent over time, as it is actually assumed in our specification. These authors point out that the estimated standard errors, calculated under the assumption of independence observations, will not be correct if independence does not hold.

In all cases, the variance of the conduct term is lower than the variance of the traditional error term. This outcome indicates that both demand and cost random shocks, which are captured by the traditional error term, explains most of the overall variance of the composed error term, [[sigma].sub.[epsilon]]. In all models we reject the hypothesis of homoscedastic variation in both the noise term and the conduct term. Many of the day-of-the-week dummy variables are statistically significant in most periods. As expected, variation in conduct decreases with firms' market shares, [S.sub.it]. The coefficient of the target variable [DELTA][S.sub.it-1] is not significant in all periods and using either the Equation (13) or the first-stage estimate of the conduct parameter. This result is robust to the inclusion of other alternative variables to capture the influence of the past behavior on the present market conduct, such as week-differences and other lags of the first-differences of market shares. The coefficient of dummy for DST in the conduct term part of the model has a large positive and significant coefficient in the first period. This result and the fact that the average market share of DST in the first period is much less than the average market share of its rivals explain our subsequent finding that DST market power scores are much higher than those obtained for the other strategic firms.

D. Firm-Specific Market Power Scores

Based on the previous estimates, the third stage allows us to obtain firm-specific market power scores. Table 4 provides the arithmetic average scores of each firm obtained using ML estimates of the doubly truncated normal model. For comparison purposes we also report the firm-specific estimates of Puller (2007).

Table 4 illustrates several interesting points that are worth mentioning. First, like in Puller (2007), the estimated firm-level values of the conduct parameter are closer to Cournot ([[theta].sub.it] = 1) than to static collusion ([[theta].sub.it] = 1/[S.sub.it]) across all specifications. A notable exception is DST, whose average market power score is much larger than the other averages during this period. Puller (2007, p. 84) finds similar result and argues that from these high conduct parameter estimates may result from incomplete quantity data for some of Dynegy's small peaker units. We do not, however, find an increase in market power if we compare the average values in the first period with those obtained in the second, regardless of which empirical strategy we have used to deal with the expected value of the conduct term.

Second, we find notable differences among utilities in terms of market power. This suggests that assuming a common conduct parameter for all firms is not appropriate. For instance, firms with smaller market shares (e.g., DST) have consistently higher market power scores, whereas firms with larger market shares (e.g., Duke) have consistently lower market power scores, compared to other firms. This result is somehow expected as the upper bound of the firm-market power scores is inversely related to firms' market power. On the other hand, the estimated differences in market power simply indicate that the traditional first-stage parameter estimates can be interpreted as the industry average market power, and hence it tends, as any average variable, to overweight the market power of larger firms and underweight the market power of smaller firms.

Third, as illustrated in Figure 2, our approach based on the estimated distribution of the random conduct yields similar time-invariant firm-specific market power scores to those estimated using a fixed-effect regression approach over subperiods analyzed in Puller (2007). This result demonstrates that both approaches are, in practice, equivalent or interchangeable for estimating time-invariant firm-specific market power scores.

Fourth, Figures 3A and 3B show the histograms of all the estimated market power scores. (46) Analyzing the skewness of the estimated market power scores is not easy because, as Wang and Schmidt (2009) pointed out, a conditional expectation is a shrinkage estimator that tends to attenuate the asymmetry of the observed distribution. Despite this, some of the distributions in these figures are highly skewed. To examine this issue in detail, we provide in Table 5 several tests of normality of the estimated market power scores for the whole industry (including DST) and for each firm separately. The numbers in this table suggest that we can reject the null hypothesis of normality. Moreover, the positive values of these tests indicate that all distributions are positively skewed, regardless we have modeled the expected value of [[theta].sub.it] using Equation (13) or the expected value of the conduct term is replaced with the first-stage estimate of E([[theta].sub.it]).

Fifth, in a panel data setting the most important advantage of our methodology over fixed-effects regression approach employed in Puller (2007) is that we can analyze changes in market conduct over time. Because our approach does not impose the restrictions on the temporal path of these scores they are allowed to change from one day to another. In Figures 4A, 4B, 5A, and 5B we show the temporal evolution of the average market power scores of the four/five strategic firms during the periods analyzed in the present paper. (47) Our results indicate that the estimated firm-specific conduct parameters do vary significantly across time. Notwithstanding these differences, firm-specific conduct parameters generally tend to move in the same direction across time. The notable exception is Duke, whose market strategies are occasionally different from other firms. Puller (2007) notes that there is a widespread belief that Duke violated California electricity market rules and forward-contracted some of its production, which in part explains observed Duke's conduct. The results are very robust to the choice of the empirical strategy to identify the expected value of [[theta].sub.it] (see Section IV.C).

Figures 4A and 4B show the intertemporal variation in estimated conduct parameters over the period from July 1, 1998 to April 15, 1999. These figures show that during this period firms electricity pricing were at or slightly below Cournot levels. The most notable exception is DST/Dynegy, whose conduct was well above Cournot level during summer 1998 and close to full collusion in winter 1998/1999. As explained above, high estimates of the conduct parameter for DST during these periods may reflect the bias from incomplete generation asset data for this firm. Another notable observation is rapid increase in the conduct term for Reliant and DST in winter 1998/1999.

Figures 5A and 5B show the intertemporal variation in estimated conduct parameters over the period from April 16, 1999 to May 30, 2000 following the entry of Southern. These figures demonstrate that firms' pricing strategies are still close to Cournot levels for most of this period. On average, over this period, the new entrant Southern tends to have a higher value of the estimated conduct parameter, whereas Duke tends to have a lower value of the estimated conduct parameter. Firms' pricing strategies exhibit a larger variation during this period. For example, the market conduct of Southern increases above Cournot levels in summer 1999, and the market conduct of Southern, Reliant, and AES increases above Cournot levels in winter 1999. The results show that the conduct parameter of all firms (and most notably DST) increases above Cournot levels during the notorious price run-up period of summer 2000. These findings are consistent with earlier studies of market power in California electricity market. Joskow and Kahn (2002), for example, find evidence of the strategic withholding of capacity by some generating firms during summer 2000. As regards Southern, though pricing strategy is above Cournot levels, it is not different from its strategy in summer 1999.

The important advantage of having time-varying and firm-specific conduct parameters is that they contribute to better understanding of firm-specific effects of a variety of external factors that impacted the market during California electricity crisis. These include, among other factors, changes of FERC price caps, variation in natural gas prices resulting from disruptions in gas supply, and the costs of NOx permits affected by the number of available pollution credits (Sweeney 2002).

Figures 5A and 5B illustrate the regulatory changes in FERC price caps, which have effectively acted as a cap on PX prices. As of October 1999 this cap was set to $750 per MWh and was not binding until the summer of 2000. After the ISO lowered the cap twice in 2000Q3 it began to play a significant role in the firms' ability to exercise market power (Puller 2007). Figures 5A and 5B show that the market power scores of AES and DST have considerably increased over the period of higher price caps in October 1999 to August 2000, whereas the market power scores of other firms were less affected. Tightening the price caps in August 2000 has lowered market power scores for all market participants.

Figures 6 and 7 demonstrate correlations between market power scores and NOx permit prices and natural gas prices, respectively. Figure 6 shows that Duke and Southern market power scores are positively correlated with NOx permit prices, whereas other three firms show little (if any) correlation between market power scores and NOx permit prices. Figure 7 shows that Duke and Southern market power scores are positively correlated with natural gas prices, DST market power scores are negatively correlated with natural gas prices, whereas Reliant and AES market power scores are not correlated with natural gas prices. These results indicate that variations in regulatory policies and input prices have indeed affected strategic behavior of at least some market participants.

E. Robustness Analyses

In order to examine the robustness of our results, we summarize in this section the results that have been obtained using alternative specifications of the basic model discussed in previous sections. In particular, in addition to the firm-specific market power estimates of Puller (2007) and our previous third-stage market power scores, Table 6 reports the firm-specific market power scores when: (a) we exclude those hours in which capacity binds; (b) we allow for firm-specific conduct parameters in the first-stage pricing equation; (c) the upper bound of the market power random term is time invariant; and (d) the demand elasticities are computed using PX data. Table 6 provides all these scores either using the mathematical expression in Equation (13) or the first-stage estimate of 9.48

The estimated coefficient of the capacity binding variable in Table 2 is very noisy. Conditional on capacity being binding, the shadow value associated to this restriction might be quite different as it likely depends on observed and unobserved market conditions. For this reason, it is interesting to examine the results excluding those hours in which capacity binds and therefore the FOC does not hold. While the GMM estimate of 9 is 9.96 for the first period, it is 9.81 for the second period. As the first-stage estimates of the conduct parameter in both periods are quite similar to that obtained using our previous model that include the capacity binding variable, the computed market power scores are also comparable to the previous ones. Therefore, our results are robust to the existence of heterogeneous shadow values associated to the capacity binding restriction.

The second robustness analysis involves estimating the conduct parameter at the firm level in the first-stage pricing equation, in order to account for heterogeneity across firms. This model reproduces the fixed-effect strategy used by Puller (2997). In this case, both first- and third-stage market power scores are provided in Table 6. This model allows us to examine whether those firms with larger conduct terms every period (e.g., DST) might appear "by surprise" as being above the unconditional mean because in the first stage we use a common conduct parameter for all firms. That could generate biases on how the market power scores are being estimated (conditional expectation) in the second and third stages of our procedure. Again our firm-specific first-stage market power estimates are consistent with those obtained by Puller using also a fixed-effect treatment of the conduct parameters, the average market power scores are akin to those obtained using the basic (common conduct parameter) model. Just the average market power score for DST in the first period is a bit less than the previous ones.

The next issue that is examined in this section is the potential endogeneity of the upper bound of the doubly truncated random term. Our upper bound is determined by the economic theory, but it not clear from an econometric point of view whether this bound is endogenous or exogenous as it endogenously changes over time. To examine this issue we have re-estimated the basic model using a time-invariant upper bound. Instead of using the inverse of the contemporaneous market shares, that is, 1/[S.sub.it], in this model we use the inverse of the firm-specific averages of their contemporaneous market shares, that is, 1/[[bar.S].sub.i] where [[bar.S].sub.i] = 1/T[[SIGMA].sup.T.sub.t=1][s.sub.it]. The computed firm-specific averages are likely exogenous variables because the effect of a single contemporaneous market share on [[bar.S].sub.i] is negligible due to the large number of observations of each firm, and because firms' size is mostly predetermined in the onset of the liberalization process. Overall, the average market power scores of this model are quite similar to those obtained using the basic model, except for DST in the first period of the sample that once again decreases. (49)

Finally, in Table 6 we show the average market power scores that are obtained using a different approach to compute the elasticity of the residual demand function of the five strategic firms. This is because Puller (2007) does not observe actual residual demand schedules. Instead, he estimates the supply function of competitive fringe, and calculates the slope of the fringe supply, "which has the same magnitude but opposite sign of the slope of the residual demand faced by the five strategic firms" (Puller 2007, p. 78). This is problematic because Puller's (2007) estimates are correct if and only if all fringe firms bid competitive schedules. This assumption is questioned by a number of studies. Instead we use the estimates of residual demand elasticities based on actual bids from PX as suggested by Wolak (2003). (50)

Similar to our basic model our first-stage conduct parameter estimates are closer to Cournot than to static collusion. While the GMM estimate of [??] is 0.71 for the first period, it is 1.11 for the second period. Therefore, the first-stage estimate of the conduct parameter in the first (second) period is a bit smaller (larger) than that obtained using our previous model based on Puller's demand elasticity. Regarding the computed market power scores in Table 6, the results are highly consistent with our previous model based on Puller's demand elasticity when the GMM estimate of [??] is used to deal with the expected value of the conduct term when estimating the structure of both [v.sub.it] and [S.sub.it] [[theta].sub.it] random terms. They are also comparable to the previous ones when the mathematical expression in Equation (13) is used to model in the second stage the expected value of the conduct term, except for DST in the first period. Indeed, while the other market power levels in this period are highly consistent with our previous scores, we find a zero market power score for DST. (51) Frutos and Fabra (2012) show in a residual demand framework that the FOC of profit maximization in Equation (1) only holds for those firms setting the price. They also show that non price-setters behave "as if' they were price-takers. The zero market power score for DST can then be justified from a theoretical point of view if DST has normally behaved as a price-taker firm and bid at marginal cost.

V. CONCLUSIONS

This study contributes to the literature on estimating market power in homogenous product markets. Our econometric approach allows for the value of estimated conduct parameter to vary across both firms and time. We estimate a composed error model, where the stochastic part of the firm's pricing equation is formed by two random variables: the traditional error term, capturing random shocks, and a random conduct term, which measures the degree of market power. The model can be estimated in three stages. While the first stage of our model is similar to the previous literature, the second and the third stages allow us to distinguish variation in market power from volatility in demand and cost, and get firm-specific market power scores, conditional on the first-stage parameter estimates.

Treating firms' conduct as a random parameter helps solving the over-parameterization problem in the continuous time. The second and third stages of our procedure allow us to identify groups of suspected cartel members, "maverick" firms, or changes in mark-ups which cannot be explained by "normal" random shocks. In this sense, the proposed procedure can be viewed as a collusive screening procedure. Other advantages of our approach are its applicability to cross-sectional or short data sets. Moreover, firm-specific market power estimates can be obtained just using cross-sectional data sets because our approach relies on distributional assumptions. Our approach is also useful in a panel data setting when the assumption of time-invariant conduct is not reasonable. In addition, by imposing upper bound on the value of estimated conduct parameter we ensure that estimated market power scores are always consistent with the economic theory.

The main distinctive feature of our approach is that model identification is based on the assumption that the conduct term follows a one-sided distribution, which, to our best knowledge, has not been previously used in the empirical industrial organization literature. Another feature that distinguishes our paper from previous studies is the attempt to estimate a double-bounded distribution that imposes both lower and upper theoretical bounds to a continuous random conduct term. To achieve this objective, we adapt one of the most recent stochastic frontier models in production economics to our framework. To our knowledge, this is the first time the stochastic frontier models are used to measure market power.

We illustrate the proposed approach with an application to the California wholesale electricity market using a well-known data set from Puller (2007). After estimating the parameters of the pricing equation, we implement the second and third stages based on the truncated normal distribution, which imposes both lower and upper theoretical bounds on the values of the random conduct term. Similar to the findings of Puller (2007) our estimated average firm-level values of the conduct parameter are closer to Cournot than to static collusion across all samples and specifications. This result demonstrates that both approaches are, in practice, equivalent or interchangeable for estimating time-invariant firm-specific market power scores. The analysis of firm-specific conduct parameters suggests that realization of market power varies over both time and firms, and rejects the assumption of a common conduct parameter for all firms.

There are several interesting extensions of the proposed model than can be explored in the future. (52) First, if firms are not fully efficient, our price equation should include a cost inefficiency term in addition to the noise and conduct terms. Estimating a model with 2 one-sided error terms is not trivial. Promising strategies could be using copula-based methods, or using different distributions for the cost and conduct terms (e.g., homoscedastic vs. heteroscedastic). Second, it would be interesting to examine whether the model could be estimated without distributional assumptions on the conduct term, using similar semiparametric specifications to Tran and Tsionas (2009) and Parmeter, Wang and Kumbhakar (2017). Third, our model prevents zeroconduct values from occurring. In this case, a zero inefficiency stochastic frontier (ZISF) model of Kumbhakar, Parmeter, and Tsionas (2013) could be estimated. More details on this model can be found in Rho and Schmidt (2015). The socalled ZISF model would allow us to distinguish between firms that tend to behave as perfectly competitive firms (i.e., tend to be "fully efficient") and firms that, for some reasons, do have market power. A ZISF-market power specification of our model would also permit to identify the determinants of being a perfectly competitive firm. Regulators or policy makers might find the latter information very useful to design measures aiming to promote competitive behavior in the homogenous products market.

ABBREVIATIONS

CEMS: Continuous Emissions Monitoring System

FOC: First-Order Condition

GMM: Generalized Method of Moments

ML: Maximum Likelihood

MM: Method of Moment

NEIO: New Empirical Industrial Organization

OLS: Ordinary Least Squares

PX: California Power Exchange

ZISF: Zero Inefficiency Stochastic Frontier

doi: 10.1111/ecin.12539

Online Early Publication December 19, 2017

REFERENCES

Abreu, D., D. Pearce, and E. Stacchetti. "Optimal Cartel Equilibria with Imperfect Monitoring." Journal of Economic Theory, 39(1), 1986, 251-69.

Almanidis, P., J. Qian, and R. Sickles. "Bounded Stochastic Frontiers with an Application to the US Banking Industry: 1984-2009." Unpublished manuscript, Rice University. 2010. Accessed November 28, 2017. http:// www.uh.edu/~cmurray/TCE15/Papers/Almanidis.pdf

Alvarez, A., C. Amsler, L. Orea, and P. Schmidt. "Interpreting and Testing the Scaling Property in Models Where Inefficiency Depends on Firm Characteristics." Journal of Productivity Analysis, 25(3), 2006, 201-12.

Appelbaum, E. "The Estimation of the Degree of Oligopoly Power." Journal of Econometrics, 19(2-3), 1982, 287-99.

Borenstein, S. "The Trouble with Electricity Markets: Understanding California's Restructuring Disaster." Journal of Economic Perspectives, 16(1), 2002, 191-211.

52. We thank an anonymous reviewer for suggesting some of these extensions.

Borenstein, S., and N. L. Rose. "Competition and Price Dispersion in the US Airline Industry." Journal of Political Economy, 102(4), 1994,653-83.

Borenstein, S., J. B. Bushnell, and F. A. Wolak. "Measuring Market Inefficiencies in California's Restructured Wholesale Electricity Market." American Economic Review, 92(5), 2002, 1376-405.

Borenstein, S., J. B. Bushnell, C. R. Knittel, and C. Wolfram. "Inefficiencies and Market Power in Financial Arbitrage: A Study of California's Electricity Markets." Journal of Industrial Economics, 56(2), 2008, 347-78.

Brander, A. J., and A. Zhang. "Dynamic Oligopoly Behavior in the Airline Industry." International Journal of Industrial Organization, 11(3), 1993, 407-35.

Bresnahan, T. "Empirical Studies of Industries with Market Power," in Handbook of Industrial Organization, Vol. 2, Chapter 17, edited by R. Schmalensee and R. Willig. Amsterdam, The Netherlands: North-Holland, 1989, 1011-57.

Bushnell, J. B., and F. A. Wolak, "Regulation and the Leverage of Local Market Power: Reliability Must-Run Contracts in the California Electricity Market." POWER Working Paper No. PWP-070, University of California Energy Institute, 1999.

Clay, K., and W. Troesken. "Further Tests of Static Oligopoly ' Models: Whiskey, 1882-1898." Journal of Industrial Economics, 51(2), 2003, 151-66.

Coelli, T. "Estimators and Hypothesis Tests for a Stochastic Frontier Function: A Monte Carlo Analysis." Journal of Productivity Analysis, 6(3), 1995, 247-68.

Corts, K. "Conduct Parameters and the Measurement of Market Power." Journal of Econometrics, 88(2), 1999, 227-50.

Das, A., and S. C. Kumbhakar. "Markup and Efficiency of Indian Banks: An Input Distance Function Approach." Empirical Economics, 51(4), 2016, 1689-719.

Delis, M. D.. and E. G. Tsionas. "The Joint Estimation of Bank-Level Market Power and Efficiency." Journal of Banking and Finance, 33(10), 2009, 1842-50.

Ellison, G. "Theories of Cartel Stability and the Joint Executive Committee." RAND Journal of Economics, 25(1), 1994, 37-57.

Emvalomatis, G., S. E. Stefanou, and A. O. Lansink. "A Reduced-Form Model for Dynamic Efficiency Measurement: Application to Dairy Farms in Germany and the Netherlands." American Journal of Agricultural Economics, 93(1), 2011, 161-74.

Fabra, N., and J. Toro. "Price Wars and Collusion in the Spanish Electricity Market." International Journal of Industrial Organization, 23(3-4), 2005, 155-81.

Figuieres, C., A. Jean-Marie, N. Querou, and M. Tidball. Theory of Conjectural Variations. Singapore: World Scientific Publishing, 2004.

Fried, H., C. A. Knox Lovell, and S. S. Schmidt. The Measurement of Productive Efficiency and Productivity Growth. Oxford: Oxford University Press, 2008.

Frutos, M.-A., and N. Fabra. "How to Allocate Forward Contracts: The Case of Electricity Markets." European Economic Review, 56(3), 2012,451 -69.

Gallet, A. G., and J. R. Schroeter. "The Effects of the Business Cycle on Oligopoly Coordination: Evidence from the U.S. Rayon Industry." Review of Industrial Organization, 10(2), 1995, 181-96.

Genesove, D., and W. Mullin. "Testing Static Oligopoly Models: Conduct and Cost in the Sugar Industry, 1890-1914." RAND Journal of Economics, 29(2), 1994, 355-77.

Gollop, D., and M. Roberts. "Firm Interdependence in Oligopolistic Markets." Journal of Econometrics, 10(3), 1979,313-31.

Green, E. J., and R. H. Porter. "Noncooperative Collusion under Imperfect Price Information." Econometrica, 52(1), 1984, 87-100.

Hamilton, J. D. "Analysis of Time Series Subject to Changes in Regime." Journal of Econometrics, 45(1-2), 1989, 39-70.

Hansen, L. P. "Large Sample Properties of Generalized Method of Moments Estimators." Econometrica, 50(4), 1982, 1029-54.

Hausman, J. A. "Specification Tests in Econometrics." Econometrica, 46(6), 1978, 1251-71.

Huang, T.-H., N.-H. Liu, and S. C. Kumbhakar. "Joint Estimation of the Lerner Index and Cost Efficiency Using Copula Methods." Empirical Economics, 2017a, https:// doi .org/10.1007/s00181 - 016-1216- z.

Huang, T.-H., D.-L. Chiang, and S.-W. Chao. "A New Approach to Jointly Estimating the Lerner Index and Cost Efficiency for Multi-Output Banks under a Stochastic Meta-Frontier Framework." Quarterly Review of Economics and Finance, 65, 2017b, 212-26.

Itaya, J.-I., and M. Okamura. "Conjectural Variations and Voluntary Public Good Provision in a Repeated Game Setting." Journal of Public Economic Theory, 5(1), 2003,51-66.

Itaya, J.-I., and K. Shimomura. "A Dynamic Conjectural Variations Model in the Private Provision of Public Goods: A Differential Game Approach." Journal of Public Economics, 81(1), 2001, 153-72.

Iwata, G. "Measurement of Conjectural Variations in Oligopoly." Econometrica, 42, 1974, 949-66.

Jamasb, T., L. Orea, and M. Pollitt. "Estimating Marginal Cost of Quality Improvements: The Case of the UK Electricity Distribution Companies." Energy Economics, 34(5), 2012, 1498-506.

Jaumandreu, J., and J. Lorences. "Modelling Price Competition across Many Markets (An Application to the Spanish Loans Market)." European Economic Review, 46(1), 2002, 93-115.

Jondrow, J., C. A. K. Lovell, S. Materov, and P. Schmidt. "On the Estimation of Technical Efficiency in the Stochastic Frontier Production Function Model." Journal of Econometrics, 19(2/3), 1982, 233-38.

Joskow, P. L., and E. Kahn. "A Quantitative Analysis of Pricing Behavior in California's Wholesale Electricity Market during Summer 2000." Energy Journal, 23, 2002, 1-35.

Kim, D. W., and C. R. Knittel. "Biased in Static Oligopoly Models? Evidence from the California Electricity Market." Journal of Industrial Economics, 54(4), 2006, 451-70.

Koetter, M., and T. Poghosyan. "The Identification of Technology Regimes in Banking: Implications for the Market Power-Fragility Nexus." Journal of Banking and Finance, 33(8), 2009, 1413-22.

Koetter, M., J. W. Kolari, and L. Spierdijk. "Enjoying the Quiet Life under Deregulation? Evidence from Adjusted Lerner Indices for U.S. Banks." Review of Economics and Statistics, 94(2), 2012,462-80.

Kole, S., and K. Lehn. "Deregulation and the Adaptation of Governance Structure: The Case of the U.S. Airline Industry." Journal of Financial Economics, 52(1), 1999, 79-117.

Kumbhakar, S. C., and C. A. K. Lovell. Stochastic Frontier Analysis. Cambridge: Cambridge University Press, 2000.

Kumbhakar, S. C., C. F. Parmeter, and E. G. Tsionas. "A Zero Inefficiency Stochastic Frontier Model." Journal of Econometrics, 172(1), 2013, 66-76.

Lewbel, A. "Using Heteroscedasticity to Identify and Estimate Mismeasured and Endogenous Regressor

Models." Journal of Business & Economic Statistics, 30(1), 2012, 67-80.

Li, Q. "Estimating a Stochastic Production Frontier When the Adjusted Error Is Symmetric." Economics letters, 52(3), 1996, 221-28.

Nevo, A. "Measuring Market Power in the Ready-to-Eat Cereal Industry." Econometrica, 69(2), 2001, 307-42.

Newbery, D. "Predicting Market Power in Wholesale Electricity Markets." Working Paper No. 2009/03, European University Institute, 2009.

Olson, J. A., P. Schmidt, and D. M. Waldman. "A Monte Carlo Study of Estimators of Stochastic Frontier Production Functions." Journal of Econometrics, 13(1), 1980, 67-82.

Parmeter, C. F., and S. C. Kumbhakar. "Efficiency Analysis: A Primer on Recent Advances." Foundations and Trends in Econometrics, 7(3-4), 2014, 191-385.

Parmeter, C. F., H.-J. Wang, and S. C. Kumbhakar. "Nonparametric Estimation of the Determinants of Inefficiency." Journal of Productivity Analysis, 47(3), 2017, 205-21.

Perloff, J. M., L. Karp, and A. Golan. Estimating Market Power and Strategies. Cambridge: Cambridge University Press, 2007.

Porter, R. H. "A Study of Cartel Stability: The Joint Executive Committee, 1880-1886." Bell Journal of Economics, 14(2), 1983a, 301-14.

--. "Optimal Cartel Trigger Price Strategies." Journal of Economic Theory, 29(2), 1983b, 313-38.

Puller, S. "Pricing and Firm Conduct in California's Deregulated Electricity Market." Review of Economics and Statistics, 89(1), 2007, 75-87.

--. "Estimation of Competitive Conduct When Firms Are Efficiently Colluding: Addressing the Corts Critique." Applied Economics Letters, 16(15), 2009, 1497-500.

Reiss, P. C., and F. A. Wolak. "Structural Econometric Modeling: Rationales and Examples from Industrial Organization." Handbook of Econometrics, 6, 2007, 4280-370.

Rho, S., and P. Schmidt. "Are All Firms Inefficient?" Journal of Productivity Analysis, 43(3), 2015, 327-49.

Rigobon, R. "Identification through Heteroskedasticity." Review of Economics and Statistics, 85(4), 2003, 777-92.

Rotemberg, J. J., and G. A. Saloner. "Supergame-Theoretic Model of Price Wars during Booms." American Economic Review, 76, 1986, 390-407.

Staiger, D., and J. H. Stock. "Instrumental Variables Regression with Weak Instruments." Econometrica, 65(3), 1997, 557-86.

Stigler, G. J. "A Theory of Oligopoly." Journal of Political Economy, 72(1), 1964, 44-61.

Sweeney, J. The California Electricity Crisis. Stanford, CA: Hoover Institution Press, 2002.

Tran, K. C., and E. G. Tsionas. "Estimation of Nonparametric Inefficiency Effects Stochastic Frontier Models with an Application to British Manufacturing." Economic Modelling, 26(5), 2009, 904-9.

Verbeek, M. A Guide to Modern Econometrics. 1st ed. Chichester, UK: John Wiley and Sons, 2000.

Von der Fehr, N.-H. M., E. S. Amundsen, and L. Bergman. "The Nordic Market: Signs of Stress?" The Energy Journal, 26, 2005,71-98.

Wang, W. S., and P. Schmidt. "On the Distribution of Estimated Technical Efficiency in Stochastic Frontier Models." Journal of Econometrics, 148(1), 2009, 36-45.

Wolak, F. A. "Measuring Unilateral Market Power in Wholesale Electricity Markets: The California Market, 1998-2000." American Economic Review, 93(2), 2003, 425-30.

--. "Lessons from the California Electricity Crisis," in Electricity Deregulation Choices and Challenges, edited by J. M. Griffin and S. L. Puller. Chicago: University of Chicago Press, 2005.

Wolfram, C. D. "Measuring Duopoly Power in the British Electricity Spot Market." American Economic Review, 89(4), 1999, 805-26.

Woo, C. K, A. Olson, I. Horowitzc, and S. Lu. "BiDirectional Causality in California's Electricity and Natural-Gas Markets." Energy Policy, 34(15), 2006, 2060-207.

LUIS OREA and JEVGENIJS STEINBUKS *

* The authors would like to express their sincere gratitude to Ben Hobbs for his invaluable help and support. They are extremely grateful to Steve Puller and Carolyn Berry for helping them with getting CEMS and California PX bidding data, and to Chiara Lo Prete for assisting them with computation of residual demand elasticities based on PX bidding data. They also thank William Greene, David Newbery, Jacob LaRiviere, Shaun McRae, Mar Reguant, Peter Schmidt, two anonymous reviewers, and the participants of the 3rd International Workshop on Empirical Methods in Energy Economics, the 9th Annual International Industrial Organization Conference, the Spanish Economic Association Annual Congress, and the American Economic Association Annual Meetings for their helpful comments and suggestions. Responsibility for the content of the paper is the authors' alone and does not necessarily reflect the views of their institutions, or member countries of the World Bank. J.S. appreciates the financial support from the UK Engineering and Physical Sciences Research Council, grant "Supergen FlexNet." L.O. is grateful for financial assistance for this work provided by the Regional Government of the Principality of Asturias in conjunction with FEDER (Grant FC-15-GRUPIN14-048). Orea: Professor, Department of Economics, School of Economics and Business, University of Oviedo, 33006, Oviedo, Spain. Phone +34 985106243, Fax +34 985236670, E-mail lorea@uniovi.es Steinbuks: Economist, Development Research Group, The World Bank, Washington, DC 20433. Phone 202 473-9345, Fax 202 522-2714, E-mail jsteinbuks@worldbank.org

(1.) For an excellent survey of other approaches to estimating market power in industrial organization literature, see Perloff, Karp, and Golan (2007).

(2.) In a symmetric equilibrium, the upper bound of inequality 0 < [[theta].sub.it] < 1/[S.sub.it] would be equal to the number of firms, N.

(3.) Some studies interpret conduct parameter as a "conjectural variation," that is, how rivals' output changes in response to an increase in firm i's output. It is also sometimes argued that the conjectural variation parameter results from the reduced form of a more complex dynamic game, such as a tacit collusion game (e.g., Itaya and Shimomura 2001; Itaya and Okamura 2003; Figuieres et al. 2004, and references therein). Other studies (Bresnahan 1989; Reiss and Wolak 2007) argue that with the exception of a limited number of special cases (e.g., perfect competition, Cournot-Nash, and monopoly) there is no satisfactory economic interpretation of this parameter as a measure of firm behavior. Sorting out between these theoretical complications is beyond the scope of this study.

(4.) Newbery (2009) argues that more theoretically attractive models, such as, for example, supply function equilibrium models, pose formidable practical and conceptual problems if they are to be used for market monitoring, and even more so in quasi-judicial investigations of the kind conducted by competition authorities. On the other hand more simple descriptive measures based on the Cournot model of oligopolistic competition, such as, for example, price-cost margins or Herfindahl Hirshman Index are inconsistent with empirical evidence. High market shares and low elasticities may lead to very high price-cost mark-ups that are considerably higher than observed (e.g., Von der Fehr, Amundsen, and Bergman 2005).

(5.) As the problem of repeated oligopoly interaction has received greater attention, the estimation of time-varying conduct parameters that are truly dynamic has become an issue. Indeed, Stigler's (1964) theory of collusive oligopoly implies that, in an uncertain environment, both collusive and price-war periods will be seen in the data (see for instance Green and Porter 1984 and Rotemberg and Saloner 1986, who predict opposite relationships between prices/mark-ups and demand evolution). Moreover, Abreu, Pearce, and Stacchetti (1986) find that in complex cartels the length of price wars (i.e., changes in conduct parameter) is random because there are "triggers" for both beginning a price war and for ending one. It is therefore difficult to impose plausible structural conditions and estimate firms' conduct over time.

(6.) In many treatments of oligopoly as a repeated game, firms expect deviations from the collusive outcome. Firms expect that if they deviate from the collusive arrangement, others will too. This expectation deters them from departing from their share of the collusive output. Because these deviations are unobserved in an uncertain environment, each firm might have its own expectation about what would happen if it deviates from collusive output.

(7.) As in Porter (1983b), Brander and Zhang (1993), and Gallet and Schroeter (1995), Maximum Likelihood techniques can be used to estimate all parameters of the model in a unique stage. However this does not allow us to address the endogeneity issues that appear when estimating the pricing equation (1).

(8.) Although the model does not require asymmetry of the one-sided distribution to be identified (see Li 1996), our specification of the conduct term allows for different degrees of asymmetries in the distribution of [[theta].sub.it].

(9.) See Rigobon (2003), Lewbel (2012), and references therein.

(10.) The regime switches only occur when a firm's quantity is never observed by another firm and, hence, deviations cannot be directly observed. This is not the case in the electricity generating industry analyzed in the empirical section as market participants had access to accurate data on rivals' real-time generation.

(11.) Kole and Lehn (1999) argue that for many firms the decision-making apparatus is slow to react to changes in the market environment within which it operates, due to the costs to reorient decision-makers to a new "game plan." In particular, the existing culture or the limited experience of the firm in newly restructured markets may be such that enhancing market power may not be immediately possible. In addition, we would also expect gradual changes in firms' conduct in a dynamic framework if firms are engaging in efficient tacit collusion and are pricing below the static monopoly level, and when there is a high persistence in regimes (Ellison 1994).

(12.) For a comprehensive survey of this literature, see Kumbhakar and Lovell (2000), Fried, Knox Lovell, and Schmidt (2008), and Parmeter and Kumbhakar (2014).

(13.) As pointed out by an anonymous reviewer, an alternative distribution would be to use the uniform distribution, as proposed by Li (1996). We cannot use this specification in our setting because the upper limit of the uniform distribution is set by the economic theory, and it is not a parameter to be estimated as in Li (1996). As the variance of the inefficiency (market power) term is attached to the upper limit of the uniform distribution, this distribution does not allow for heroscedastic specifications. From an economic view, it is also worth mentioning that the uniform distribution assumes that all market power values are equally probable, and this might prevent obtaining industry outcomes where most market power scores are similar, such as in all-inclusive collusion outcomes or when all firms are competing, say, a la Cournot.

(14.) Delis and Tsionas (2009), Koetter and Poghosyan (2009), Koetter, Kolari, and Spierdijk (2012), and Das and Kumbhakar (2016) also estimate market power within a model that uses stochastic frontier analysis, but in these papers, the use of stochastic frontier analysis is only to recover cost inefficiency first, and then determine its impact on market power.

(15.) Ignoring that firms might be pricing to reflect their inefficiencies is likely more important in applications where the products are heterogeneous and firms set their own prices (see Huang, Chiang, and Chao 2017b, p. 4).

(16.) This is the strategy followed, for instance, by Brander and Zhang (1993), Nevo (2001), and Jaumandreu and Lorences (2002).

(17.) Corts (1999) argues that traditional approaches to estimating the conduct parameter from static pricing equations yield inconsistent estimates of the conduct parameter if firms are engaged in an effective tacit collusion. The robustness of the conduct parameter approach depends, in addition, on the discount factor and the persistency of the demand. Puller (2009) derives and estimates a more general model that addresses the Corts critique. The results from estimating the more general model for the California market yielded estimates very similar to the static model. This similarity comes from the fact that "California market [can be] viewed as an infinitely repeated game with a discount factor between days very close to 1" Puller (2007, p. 84). Our empirical application to California electricity market as a static model is therefore sufficient for estimating market power consistently.

(18.) The GMM estimator has the additional advantage over ML in that it does not require a specific distributional assumption for the errors, which makes the approach robust to non-normality and heteroscedasticity of unknown term (Verbeek 2000, p. 143).

(19.) This partial collusion equilibrium is reasonable in markets with many firms, where coordination among all firms is extremely difficult to maintain as the number of firms in the collusive scheme is too high or other market characteristics (e.g., markets with differentiated products) make coordination too costly.

(20.) It is well known that secret price cuts (or secret sales) by cartel members are almost always a problem in cartels. For instance, Ellison (1994) finds that secret price cuts occurred during 25% of the cartel period and that the price discounts averaged about 20%. Also see Borenstein and Rose (1994).

(21.) We thank William Green and Peter Schmidt for clarifying this point. An alternative approach to addressing endogeneity problem is replacing [theta] with a formula measuring the (conditional) expectation of [[theta].sub.it]. This empirical strategy has two shortcomings. First, it implies assuming a particular distribution for [[theta].sub.it]. This assumption is made in the second stage of our procedure, not in our current first stage. And second, as this expectation term is nonlinear, this empirical strategy prevents using standard GMM estimators as in this case the GMM estimation becomes a nonlinear optimization problem where an explicit matrix result does not exist.

(22.) In our application we also allow both [v.sub.it] and [[theta].sub.it] be heteroscedastic, which further reinforces the use of an efficient GMM estimator.

(23.) Note that, for notational ease, we use o9 to indicate hereafter the standard deviation of the pretruncated normal distribution, and not the standard deviation of the post-truncated variable [[theta].sub.it] as before.

(24.) An important caveat in estimating doubly truncated normal models is whether it is globally identifiable. Almanidis, Qian, and Sickles (2010) show that identification problems may arise when both the mean and the upper-bound of the pretruncated normal distribution are estimated simultaneously. Fortunately, these problems vanish in a structural model of market power because the upper-bound is fixed by the theory and it does not need to be estimated in practice.

(25.) In our empirical application we have scaled the day-ahead forecast of total demand dividing it by its sample mean in order to put all explanatory variables in a similar scale.

(26.) An anonymous reviewer has correctly noted that we are modeling the variance (not the mean) of the pretruncated normal distribution. But, it should be taken into account that at the end we are modeling the mean (and the variance) of [[theta].sub.it] as it is a function of the variance of the original pretruncated normal distribution.

(27.) Since these variables in a regime-switching framework mainly affect the probability of starting a price war, they are labeled as "trigger" variables or "triggers." We prefer using the term "target" because in our model we do not have collusion and price-war regimes, and hence we do not have to estimate transition probabilities from one discrete regime to another.

(28.) We have also included other variables in order to capture the influence of past observables on actual market conduct. In particular, we have also used week-differences and other lags of the first-differences of market shares. Following Ellison (1994) we have also created more sophisticated target variables, such as deviations with respect to predicted value, using the average of the same variable for the previous 7 days. The results were almost the same as those obtained using [DELTA][S.sub.it-1].

(29.) Both the mean and the mode of the conditional distribution can be used as a point estimator for the conduct term [[??].sub.it]. However, the mean is, by far, the most employed in the frontier literature.

(30.) For notational ease, we ignore in Equation (16) that both [[sigma].sub.v] and [[sigma].sub.[theta]] are heteroscedastic.

(31.) Although [[??].sub.it] is the minimum mean squared error estimate of [[theta].sub.it], and it is unbiased in the unconditional sense [E ([[??].sub.it] - [[theta].sub.it]) = 0], it is a shrinkage of [[theta].sub.it] toward its mean (Wang and Schmidt 2009). An implication of shrinkage is that on average we will overestimate 0it when it is small and underestimate 0it when it is large. This result, however, simply reflects the familiar principle that an optimal (conditional expectation) forecast is less variable than the term being forecasted.

(32.) As pointed out by an anonymous reviewer, it is worth mentioning that our estimates of market power may not be invariant to scale because we are using prices and marginal costs in levels.

(33.) For excellent surveys of the California electricity market restructuring disaster, see Borenstein (2002), Sweeney (2002), and Wolak (2005).

(34.) Specifically, Puller (2007) argues that independent and nuclear units were paid under regulatory side agreements, so their revenues were independent of the price in the energy market. The owners of hydroelectric assets were the same utilities that were also buyers of power and had very dulled incentives to influence the price. Finally, firms importing power into California were likely to behave competitively because most were utilities with the primary responsibility of serving their native demand and then simply exporting any excess generation.

(35.) One of the ways of storing electricity for load balancing is through pumped-storage hydroelectricity. The method stores energy in the form of water, pumped from a lower elevation reservoir to a higher elevation. Low-cost off-peak electric power is used to run the pumps. During periods of high electrical demand, the stored water is released through turbines to produce electric power. In California, there is a significant amount of hydropower including some pumped storage. Notwithstanding relative abundance pumped storage in California, its potential for load balancing is limited as hydropower schedules are relatively fixed in part due to environmental (low flow maintenance, etc.) rules.

(36.) Modeling of market power in wholesale electricity markets becomes more complex if firms forward-contract some of their output. As Puller (2007, p. 85) notes, in the presence of unobserved contract positions the estimate of conduct parameters would be biased. This was generally not an issue in California wholesale electricity market during sample period. As Borenstein (2002, p. 199) points out, "Although the investor owned utilities had by 2000 received permission to buy a limited amount of power under long-term contracts, they were [... ] still procuring about 90 percent of their "net short" position [...] in the Power Exchange's day-ahead or the system operator's real-time market." Puller (2007, p. 85) argues that "there is a widespread belief that in 2000 Duke forward-contracted some of its production." If data on contract positions were available, one could correct this bias by adjusting infra-marginal sales by the amount that was forward-contracted. Unfortunately, as in earlier studies on market power in the California wholesale electricity market the contract positions are not observable in our data set.

(37.) Careful description of the data set can be found in the technical appendix of Puller (2007, pp. 86-87).

(38.) We do not observe the spot prices for natural gas for California hubs in 1998 and 1999, and use prices from Henry Hub instead. The difference between natural gas prices between these hubs before 2000 (for which we have the data available) was relatively small (see Woo et al. 2006, p. 2062, Fig. 2).

(39.) An important implication of transmission congestions is that they cause the slope of residual demand to differ for firms in the north and south of California. Puller (2007) estimated his model based on a subsample of uncongested hours and found smaller conduct parameter estimates relative to full sample (though his qualitative conclusions did not change). Our choice of residual demand elasticities based on PX data (see below) captures the effect of transmission constraints.

(40.) Puller (2007) makes a similar point.

(41.) Time variation in nameplate capacity is mostly due to acquisitions of fossil-fuel electric plants from divested utilities. For more details, see Borenstein et al. (2002, p. 1381).

(42.) Puller (2007) adopts the day-ahead forecast of total electricity output, rather than its inverse. We do not use the day-ahead forecast of total electricity output here as an instrument because it failed Hansen's (1982) J test. Notwithstanding this difference, the economic interpretation of using this instrument is the same as in Puller (2007).

(43.) Puller (2007) also reports estimates for the period from June 2000 to November 2000, which covers the price run-up preceding collapse of California liberalized electricity market. We chose not report these estimates because though the incentives of some market participants changed during this period (Borenstein et al. 2008), the market structure itself was not fundamentally different.

(44.) Chosen instruments fail Hansen's J test at 5% level of significance over the period from July 1998-April 1999 using residual demand elasticities calculated based on Puller's (2007) estimates.

(45.) For robustness grounds, several specifications of the doubly truncated normal model were estimated in previous versions of this paper, corresponding to different levels of [mu], that is, the mean of the pretruncated random term. In practice, this implies moving the mass of the distribution to the right and closer to 1/s. The Akaike information criterion showed that the preferred level of truncation is 0 across all specifications, so the conduct random term is better modeled using a half normal distribution that assumes zero modal value of [[theta].sub.it]. It is important to note here that the distribution of estimated market power is not the same as the assumed half-normal distribution for [[theta].sub.it] because the market power estimator is a shrinkage estimator of [[theta].sub.it] (see Wang and Schmidt 2009). In other words, although the modal value of a half normal distribution is zero, the average (industry) market power level is not restricted to be close to zero in practice.

(46.) We have excluded DST in the histograms of the first period because the estimated market power scores for this firm were much larger than the scores of the remainder firms. The following normality analyses are robust to the inclusion or exclusion of this firm from the sample.

(47.) To smooth the variation across time, we report the rolling 30-day average of the estimated conduct parameters.

(48.) The full set of first- and second-stage parameter estimates of all estimated models is available upon request from the authors.

(49.) As the economic theory does not back imposing a time invariant upper bound, the observed differences in market power scores for DST are likely due to both empirical (i.e., endogeneity) and theoretical problems. In this sense, it should be pointed out that this result might also have to do with the theoretical issue that is to be mentioned later on in this section.

(50.) It is important to point out that because suppliers had the opportunity to sell their capacity in the CAISO ancillary services markets and the real-time energy market, calculated residual demand elasticities may differ from actual ones. Unfortunately, we do not have the data for these markets. However, given that PX market accounted for 85% of all electricity delivered in the CAISO control area, whereas CAISO's real-time market accounted for just 5% (Borenstein, Bushnell, and Wolak 2002), the ancillary services market was very small, and there was no substantial divergence between PX and ISO market clearing prices for the most of the time covered in this study (Borenstein et al. 2008) we believe our calculations provide a reasonable approximation of actual residual demand elasticities.

(51.) This somewhat awkward result is caused by the fact that the second-stage coefficient of the DST dummy variable determining the variance of [[theta].sub.it] is negative and extremely large. Another awkward result of the second stage of this model has to do with the coefficient of sit. Indeed, while we expect a negative value for this coefficient due to the upper bound of [[theta].sub.it] is inversely related with firms' market share, we have found a positive effect.

Caption: FIGURE 1 Price-Cost Margins in Hour 18 (July 3, 1998 to November 30, 2000)

Caption: FIGURE 2 Comparison of Market Power Scores

Caption: FIGURE 3 Histograms of the Market Power Scores. (A) Modeling the Expected Value of [[theta].sub.it]. (B) Using the First-Stage Estimate of E([[theta].sub.it])

Caption: FIGURE 4 Firm-Specific Conduct Parameter Estimates. (A) Modeling the Expected Value of 8.,. (B) Using the First-Stage Estimate of E([[theta].sub.it])

Caption: FIGURE 5 Firm-Specific Conduct Parameter Estimates. (A) Modeling the Expected Value of 9/r. (B) Using the First-Stage Estimate of E([[theta].sub.it])

Caption: FIGURE 6 Correlations of NOx Permit Price and Market Power Scores. Modeling the Expected Value of [[theta].sub.it]

Caption: FIGURE 7 Correlations of Natural Gas Price and Market Power Scores. Modeling the Expected Value of [[theta].sub.it]

TABLE 1
Summary Statistics (Hour 18)

                                          Mean    Standard
                                                  Deviation

July 1, 1998 to April 15, 1999
  Price ([P.sub.t])                       35.2      21.0
  Marginal cost ([mc.sub.it])             26.6       3.1
  Margin ([P.sub.t] - [mc.sub.it]])       8.6       21.0
  [CAPBIND.sub.it]                        0.05      0.22
  Capacity ([k.sub.it])                  2,463      1,054
  Output ([q.sub.it])                     813        844
  Market demand ([Q.sub.it])             30,395     4,146
  Elasticities based on Puller (2007)     2.12      1.33
April 16, 1999 to November 30, 2000
  Price ([P.sub.it])                      61.2      68.4
  Marginal cost ([mc.sub.it])             42.7      22.9
  Margin ([P.sub.t]-[mc.sub.it])          18.4      57.3
  [CAPBIND.sub.it]                        0.05      0.21
  Capacity ([k.sub.it])                  2,955       769
  Output ([q.sub.it])                    1,223       793
  Market demand ([Q.sub.it])             30,604     3,658
  Elasticities based on Puller (2007)     1.02      0.68

                                          Min      Max      Obs

July 1, 1998 to April 15, 1999
  Price ([P.sub.t])                       4.9     180.4     864
  Marginal cost ([mc.sub.it])             19.5     33.7     864
  Margin ([P.sub.t] - [mc.sub.it]])      -25.0    158.6     864
  [CAPBIND.sub.it]                        0.00     1.00     864
  Capacity ([k.sub.it])                   670     3,879     864
  Output ([q.sub.it])                      0      3,720     864
  Market demand ([Q.sub.it])             20,057   43,847    864
  Elasticities based on Puller (2007)     0.56    10.77     864
April 16, 1999 to November 30, 2000
  Price ([P.sub.it])                      9.5     750.0    2,300
  Marginal cost ([mc.sub.it])             22.3    214.5    2,300
  Margin ([P.sub.t]-[mc.sub.it])         -33.4    697.1    2,300
  [CAPBIND.sub.it]                        0.00     1.00    2,300
  Capacity ([k.sub.it])                  1,020    3,879    2,300
  Output ([q.sub.it])                      0      3,317    2,300
  Market demand ([Q.sub.it])             22,076   42,404   2,300
  Elasticities based on Puller (2007)     0.35     5.26    2,300

TABLE 2
Pricing Equation Estimates. Dependent Variable:
[(P-mc).sub.it]; Method: OLS and Two-step GMM (a)

                                               July 1, 1998 to
                                               April 15, 1999

                                             Number of Strategic
                                                   Firms: 4

Explanatory Variables          Coefficient     OLS       GMM (b)

[CAPBlND.sub.it]                 [alpha]     -4.98 *    11.28 ***
                                              (2.64)     (3.89)

[x.sub.it] = [P.sub.t]           [theta]     1.41 ***   0.93 ***
[q.sub.it]/[[eta].sup.D                       (0.05)     (0.10)
.sub.strat't] [Q.sup.S
.sub.strat,t]

Observations                                   864         864

Mean of the dependent                          8.56       8.56
variable

Standard error of residuals                   13.14       14.23

Normality test (c)                                      50.85 ***

Hausman test (c)                                        32.25 ***

Hansen test (c)                                           0.666

Test for weak                                           226.5 ***
instruments (c)

                                April 16, 1999 to
                                November 30, 2000

                               Number of Strategic
                                     Firms: 5

Explanatory Variables            OLS       GMM (b)

[CAPBlND.sub.it]                -5.16     28.36 ***
                                (4.13)     (4.95)

[x.sub.it] = [P.sub.t]         1.36 ***   0.82 ***
[q.sub.it]/[[eta].sup.D         (0.03)     (0.06)
.sub.strat't] [Q.sup.S
.sub.strat,t]

Observations                     2300       23t)u

Mean of the dependent           18.43       18.43
variable

Standard error of residuals     27.83       34.4

Normality test (c)                        135.6 ***

Hausman test (c)                          120.8 ***

Hansen test (c)                             0.712

Test for weak                             412.5 ***
instruments (c)

(a) HAC Standard errors robust to heteroscedasticity
and autocorrelation in parenthesis.

(b) Instruments: [CAPBIND.sub.it], [k.sub.it], 1/F[Q.sub.t],
where FQ is day-ahead forecast of total (perfectly inelastic)
demand and [k.sub.it] is capacity.

(c) While the normality test (Coelli 1995) follows a standard
normal distribution, both Hausman and Hansen tests follow a
[chi square] distribution with 1 degree of freedom.
The Hausman test is sometimes based in only one parameter
in order to provide a positive value. The test for weak
instruments follows F distribution with 2 and (obs-3)
degrees of freedom.

* Significant at 10%; ** significant at 5%; *** significant at 1%.

TABLE 3 Second-stage Parameter Estimates
(Robust Standard Errors in Parenthesis)

                                       Modeling the Expected
                                     Value of the Conduct Term

                                 July 1998 to        April 1999 to
Component/Parameter               April 1999         November 2000

  Intercept                   1.40 ***    (0.11)   1.97 ***    (0.05)
  F[Q.sub.t]                    -0.02     (0.67)   -1.15 ***   (0.24)
  0.5 * F[Q.sub.t.sup.2]      11.41 ***   (3.16)     2.44      (1.98)
  [D.sub.DST]                 0.29 ***    (0.07)   0.12 ***    (0.02)
  [D.sub.Duke]                -0.16 ***   (0.01)   0.51 ***    (0.02)
  [D.sub.Reliant]               0.06      (0.04)    0.03 *     (0.02)
  [D.sub.Southern]            -0.27 ***   (0.02)   -0.38 ***   (0.02)
  [D.sub.tuesday]               0.10      (0.07)   0.31 ***    (0.16)
  [D.sub.Wednesday]           0.64 ***    (0.04)    0.15 **    (0.07)
  [D.sub.thursday]            0.52 ***    (0.05)   0.20 ***    (0.03)
  [D.sub.friday]                0.02      (0.11)     -0.01     (0.09)
  [D.sub.Saturday]            0.25 ***    (0.05)    0.08 *     (0.04)
  [D.sub.Sunday]              0.10 ***    (0.03)   0.13 ***    (0.04)
Asymmetric component, [[sigma].sub.[theta]]
  Intercept                     0.08      (0.22)     0.05      (0.18)
  [s.sub.it]                    -0.19     (0.64)     0.14      (0.62)
  [D.sub.DST]                 1.76 ***    (0.17)   0.49 ***    (0.11)
  [D.sub.Duke]                 0.04 *     (0.02)   0.14 ***    (0.03)
  [D.sub.Reliant]               0.09      (0.08)   -0.09 ***   (0.02)
  [D.sub.Southern]            -0.31 ***   (0.07)   0.09 ***    (0.00)
  [s.sub.it-1-][s.sub.it]-2     0.74      (0.51)     0.28      (0.75)

  Mean log-likelihood          -3.386               -4.023
  Observations                   864                 2300

                                      Using the First-Stage
                                   Estimate of the Conduct Term

                                 July 1998 to       April 1999 to
Component/Parameter               April 1999        November 2000

  Intercept                   1.15 ***    (0.08)   1.88 ***    (0.04)
  F[Q.sub.t]                  -1.73 ***   (0.50)   -1.32 ***   (0.31)
  0.5 * F[Q.sub.t.sup.2]      11.92 ***   (3.49)   3.40 ***    (1.43)
  [D.sub.DST]                 0.03 ***    (0.01)   0.29 ***    (0.01)
  [D.sub.Duke]                -0.17 **    (0.04)   0.43 ***    (0.02)
  [D.sub.Reliant]             0.10 ***    (0.02)   0.05 ***    (0.01)
  [D.sub.Southern]
  [D.sub.tuesday]             0.23 ***    (0.05)    0.25 **    (0.11)
  [D.sub.Wednesday]           0.74 ***    (0.04)   0.18 ***    (0.07)
  [D.sub.thursday]            0.57 ***    (0.04)    0.23 **    (0.04)
  [D.sub.friday]                -0.23     (0.15)     -0.09     (0.07)
  [D.sub.Saturday]             0.32 **    (0.13)     0.07      (0.04)
  [D.sub.Sunday]                0.18      (0.13)   0.15 ***    (0.05)
Asymmetric component, [[sigma].sub.[theta]]
  Intercept                   0.78 ***    (0.23)   1.22 ***    (0.16)
  [s.sub.it]                  -2.59 ***   (0.76)   -6.47 ***   (0.82)
  [D.sub.DST]                 1.69 ***    (0.20)   -0.15 **    (0.06)
  [D.sub.Duke]                0.43 ***    (0.19)   -0.07 ***   (0.02)
  [D.sub.Reliant]             0.16 ***    (0.06)   -0.04 ***   (0.02)
  [D.sub.Southern]
  [s.sub.it-1-][s.sub.it]-2     0.16      (0.44)     -0.18     (0.26)

  Mean log-likelihood          -3.244               -3.927
  Observations                   864                 2300

* Significant at 10%; ** significant at 5%;
*** significant at 1%.

TABLE 4
Firm-Specific Conduct Parameter Estimates

                                        Market Share
                                        ([s.sub.it])

                                              Standard
Firm                                   Mean   Deviation

July 1, 1998 to April 15, 1999
  AES                                  0.28     0.15
  DST                                  0.07     0.08
  Duke                                 0.48     0.20
  Reliant                              0.19     0.10
Industry average
Industry average (excl. DST)
First-stage mean
April 16, 1999 to November 30, 2000
  AES                                  0.17     0.09
  DST                                  0.12     0.05
  Duke                                 0.31     0.12
  Reliant                              0.20     0.07
  Southern                             0.20     0.08
Industry average
First-stage mean

                                         Modeling the
                                       Expected Value of
                                       [[theta].sub.it]

                                       Mean    Standard
Firm                                           Deviation

July 1, 1998 to April 15, 1999
  AES                                  0.74      0.38
  DST                                  4.61      2.27
  Duke                                 0.67      0.41
  Reliant                              0.91      0.46
Industry average                       1.73
Industry average (excl. DST)           0.70
First-stage mean                       0.93
April 16, 1999 to November 30, 2000
  AES                                  0.78      0.37
  DST                                  1.18      0.61
  Duke                                 0.68      0.40
  Reliant                              0.70      0.34
  Southern                             0.75      0.38
Industry average                       0.82
First-stage mean                       0.82

                                        Using the FirstStage
                                       Estimate of
                                        [[theta].sub.it]

                                       Mean    Standard    Puller
Firm                                           Deviation   (2007)

July 1, 1998 to April 15, 1999
  AES                                  0.92      0.53       0.99
  DST                                  6.88      4.57       5.15
  Duke                                 0.83      0.56       1.02
  Reliant                              1.29      0.74       1.48
Industry average                       2.48                 2.16
Industry average (excl. DST)           0.91                 1.05
First-stage mean                       0.93                 0.97
April 16, 1999 to November 30, 2000
  AES                                  1.02      0.70       0.82
  DST                                  1.11      0.65       1.75
  Duke                                 0.49      0.53       0.81
  Reliant                              0.76      0.47       1.01
  Southern                             0.94      0.54       1.21
Industry average                       0.86                 1.12
First-stage mean                       0.82                 0.97

TABLE 5
Normality Tests of the Market Power Scores (a)

                Modeling the        Using the
                  Expected         First-Stage
                 Value of         Estimate of E
              [[theta].sub.it]   [[theta].sub.it]

July 1, 1998 to April 15, 1999
  All firms      32.01 ***          34.90 ***
  AES             9.56 ***           8.97 ***
  DST             8.99 ***           9.53 ***
  Duke            9.45 ***           9.86 ***
  Reliant        10.11 ***           9.44 ***

April 16, 1999 to May 30, 2000
  All firms      30.70 ***          32.99 ***
  AES            11.78 ***          14.31 ***
  DST            12.89 ***          13.51 ***
  Duke           13.96 ***          20.38 ***
  Reliant        12.06 ***          14.24 ***
  Southern       12.20 ***          12.27 ***

(a) The normality test introduced by Coelli (1995)
follows a standard normal distribution.

* Significant at 10%; ** significant at 5%;
*** significant at 1%.

TABLE 6
Robustness Analyses. Average Market Power Scores

                                           Second Stage
                                           Modeling the
                                           Expected Value
                        Firm-Specific      of [Q.sub.it]
                         Estimate of
              Puller   [[theta].sub.i]   Basic    Capacity
Firm          (2007)    (First Stage)    Model   No Binding

July 1, 1998 to April 15, 1999
  AES          0.99         0.88         0.74       0.81
  DST          5.15         4.69         4.61       4.61
  Duke         1.02         0.94         0.67       0.63
  Reliant      1.48         1.35         0.91       0.96

April 16, 1999 to November 30, 2000
  AES          0.82         0.96         0.78       0.87
  DST          1.75         1.45         1.18       1.19
  Duke         0.81         0.48         0.68       0.67
  Reliant      1.01         0.85         0.70       0.74
  Southern     1.21         0.95         0.75       0.80

                        Second Stage Modeling the
                       Expected Value of [Q.sub.it]

               Firm-Specific                     Elasticities
                 Estimate       Time-Invariant     Based on
Firm          [[theta].sub.i]    Upper Bound       PX Data

July 1, 1998 to April 15, 1999
  AES              0.75              0.67            0.80
  DST              3.62              2.99            0.02
  Duke             0.69              0.65            0.65
  Reliant          0.90              0.97            0.84

April 16, 1999 to November 30, 2000
  AES              0.77              0.76            1.39
  DST              1.01              1.18            1.42
  Duke             0.78              0.68            1.20
  Reliant          0.69              0.70            1.28
  Southern         0.71              0.76            1.30

              Second Stage Using the First-Stage
                   Estimate of [Q.sub.it]

                                    Firm-Specific
              Basic    Capacity       Estimate
Firm          Model   No Binding   [[theta].sub.i]

July 1, 1998 to April 15, 1999
  AES         0.92       0.97           0.92
  DST         6.88       7.01           6.92
  Duke        0.83       0.85           0.84
  Reliant     1.29       1.32           1.30

April 16, 1999 to November 30, 2000
  AES         1.02       1.02           1.01
  DST         1.11       1.10           1.11
  Duke        0.49       0.49           0.47
  Reliant     0.76       0.76           0.76
  Southern    0.94       0.94           0.92

                 Second Stage Using the
                  First-Stage Estimate
                      of [Q.sub.it]

                               Elasticities
              Time-Invariant     Based on
Firm           Upper Bound       PX Data

July 1, 1998 to April 15, 1999
  AES              0.97            0.83
  DST              4.41            4.28
  Duke             0.79            0.79
  Reliant          1.34            1.16

April 16, 1999 to November 30, 2000
  AES              1.02            0.98
  DST              1.11            0.73
  Duke             0.47            0.49
  Reliant          0.76            0.66
  Southern         0.94            1.39