H. D. VINOD AND R. R. GEDDESlegacy.fordham.edu/economics/vinod/jlegee.pdf · H. D. VINOD AND R. R....

CHAPTER 1

GENERALIZED ESTIMATING

EQUATIONS FOR PANEL DATA

AND MANAGERIAL

MONITORING IN ELECTRIC

UTILITIES

H. D. VINOD AND R. R. GEDDES

Economics Department, Fordham University, Bronx, NY 10458

Abstract: Vinod (1997, 1998) discuss the Godambe-Durbin theory of estimatingfunctions (EFs) and its potential in econometrics. Here we consider a popularapplication of EFs called generalized estimating equations (GEE). It is typicallyapplied to panel data, where the heteroscedasticity is analytically related to �,the regression parameter, and where the dependent variable is binary. Geddes(1997) studies panel data on regulated electric utilities with exclusive geographicfranchises, and the turnover of the chief executive o�cer (CEO) on the job. OurGEE estimates reverse his somewhat counterintuitive result that �rm performancevariables do not a�ect the turnover of the CEO. We test the empirical validityof predictions of (i) regulatory slack, (ii) rent seeking, and (iii) political pressurehypotheses, and reject the �rst.

Keywords and phrases: Estimating Functions, Panel Data Logits, Economet-rics, GLM, Managerial Turnover.

1.1 THE INTRODUCTION AND MOTIVATION

Vinod (1997, 1998) discuss applications of Godambe-Durbin EFs in econo-metrics. This paper provides a new panel data application of EFs called

1

2 H. D. VINOD AND R. R. GEDDES

GEE, which is popular in biostatistics (Dunlop, 1994, Diggle et al, 1994,Liang and Zeger, 1995). We provide an explanation of why GEE is popularby showing that it is simpler and theoretically superior to its competition:least squares (LS) and maximum likelihood (ML). Since econometriciansrarely use anything other than LS or ML, this explanation is novel. Al-though the underlying results are known in the EF literature (Godambeand Kale, 1991, and Heyde, 1997), their application to the panel data caseclari�es and highlights the advantages of EFs.

We consider a typical logit-type speci�cation and apply GEE to paneldata with limited (binary) dependent variables. This application of EFtheory will explain why EFs yield simpler and superior estimators here.Consider T real variables yi(i = 1; 2; � � � ; T ):

yi � IND(�i(�); �2�i(�)); where � is p� 1 (1)

where IND suggests an independently (not necessarily identically) distributedrandom variable (r.v.) with mean �i(�), variance �

2�i(�) and �2 does notdepend on �. Let y = yi be a T �1 vector and V = V ar(y) = �2 V ar(�(�))denote the T�T covariance matrix. The IND assumption implies that V (�)is a diagonal matrix depending only on the i-th component �i of the T � 1vector �.

The common parameter vector of interest is � that measures how �depends on covariates x. The heteroscedastic variances �i(�) are somewhatunusual. We emphasize that �i(�) and �i(�) in (1) are functionally relatedto each other through �, implying a \special" kind of heteroscedasticity. Ifyi are discrete stochastic processes, (time series data) then �i and �i areconditional on past data. The usual log-likelihood is:

LnL = �(T=2)ln2� � (T=2)(ln�2)� S1 � S2 (2)

where S1 = (1=2)�Ti=1ln�i, and S2 = �T

i=1[yi � �i(�)]2=[2�2�i(�)]. The�rst order condition (FOC) for generalized LS (or GLS) is @(S2)=@� = 0.The FOC for maximizing the LnL (ML estimator) is

@(S1 + S2)=@� = [@S2=@�i][@�i=@�] + [@(S1 + S2)=@�i][@�i=@�] = 0,using @S1=@�i = 0.

Thus @LnL=@� = �Ti=1(yi � �i)(@�i=@�)=(�2�i)�

�Ti=1(@�i=@�)=(2�i)

+�Ti=1(yi � �i)2(@�i=@�)=(2�2�2i ).

In our context, the quasi score function (QSF) equals the �rst term

[@S2=@�i] [@�i=@� ].Its expectation, E(QSF ) = �T

i=1(yi � �i)(@�i=@�)=(�2�i) = 0;

GEE PANEL ESTIMATOR FOR CEO MONITORING 3

since �i > 0 and Eyi = �i are assumed. Thus the QSF de�ned this wayalone yields an unbiased EF.Since (@�i=@�) 6= 0 is assumed, the inclusion of the remaining two terms ofthe FOC for ML would obviously lead to a biased EF.

Wedderburn (1974) was motivated by applications to the generalizedlinear model (GLM), where one is unwilling to specify any more than meanand variance properties. His quasi-likelihood function (QLF) is a hypothet-ical integral of the QSF. The true integral ( i.e., the likelihood function)can fail to exist when the \integrability condition" of symmetric partialsis violated, McCullagh and Nelder (1989, p. 333). The EFs are de�ned asfunctions of data and parameters, g(y; �). Unbiased EFs satisfy E(g) = 0.Godambe's (1960) optimal EFs minimize [V ar(g)]=(E@g=@�)2.

Godambe (1985) proved that the optimal EF is the quasi-score function(QSF). The optimal EFs (QSFs) are computed from the means and vari-ances, without assuming further knowledge of higher moments (skewness,kurtosis) or the form of the density. The methods based on QLFs are gen-erally regarded as \more robust." For example, Liang et al (1992, p.11)show that the traditional likelihood requires additional restrictions.

In matrix notation write (1) as: y = � + �, E� = 0, E��0 = �2V =�2diag(�i). If D = f@�i=@�jg is a T � p matrix, McCullagh and Nelder(1989, p.327) show that the QSF (�; �) is:

QSF (�; �) = D0V �1(y � �)=�2 = �Ti=1(yi � �i)(@�i=@�)=(�

2�i): (3)

The optimal EF estimator of � is obtained by solving the (nonlinear)equation QSF = 0 for �. The following three key properties of the QSFslead to optimality of EF estimators:

(i) Since E(y��) = 0; E(QSF ) = 0; implying that QSF is an unbiasedEF.

(ii)Cov(QSF ) = D0�1D=�2 = IF , the Fisher information matrix.(iii) Since �E(@QSF=@�) = Cov(QSF ) = IF , its variance reaches the

Cramer-Rao lower bound.These statements do not require V to be diagonal as in (3), only that V

be symmetric positive de�nite having known functions of �. Vinod (1997)gives examples where the EF estimator coincides with the least squares (LS)and maximum likelihood (ML) estimators. Our panel data logit examplehere is more interesting, because the EF estimator is distinct from both LSand ML estimators. While the LS and ML solve FOCs similar to (4) below,the EF estimator solves QSF = 0: The chain rule on the FOC requiresa second term involving (@�i=@�), which is nonzero from (1) due to thespecial heteroscedasticity. Hence the FOC's of LS and ML are unnecessarilycomplicated. Our arguments in favor of EFs are (a) that FOCs can be


biased EFs and (b) that LS and ML can fail to reach the Cramer-Raobound, i.e., property (iii) above. We summarize this as:

Result 1: The �rst order conditions for ML imply a super uous secondterm in:

@(S1 + S2)=@� = QSF (�; �) + [@(S1 + S2)=@�i][@�i=@�] (4)

where QSF (�; � ) is from (3). The FOCs for GLS are similar to (4),except that the S1 term is absent. For both ML and GLS, the secondterm of (4) is nonzero under special heteroscedasticity conditions. Onlywhen (@�i=@�) = 0, i.e., when the heteroscedasticity does not depend on�, FOCs lead to unbiased EFs, proved to be desirable in EF theory, Heyde(1997).

Depending on how complicated (@�i=@�) is, the second term in (4) obvi-ously complicates the derivation of ML (normal) equations. Our discussionsurrounding Result 2 of the following section will explain why similarlycomplicated second terms are present in the so-called `panel logit/probit'models in econometrics. Econometric literature surveyed in Hajivassiliouand Ruud (1994) uses ingenuity and simulations to surmount the compli-cations. Unfortunately, these attempts ignore the deeper fault of the �rstorder conditions causing biased and/or ine�cient equations. We empha-size that QSF = 0 is always unbiased and its variance always reaches theCramer-Rao bound. By contrast the �rst order conditions de�ning the ML(or GLS) estimators can be biased equations and their variance may notreach the lower bound whenever @�i=@� values are nonzero.

A lesson of the EF-theory is that biased estimators can be acceptable butbiased and ine�cient EFs should be avoided. This is why the EF estimatorobtained by solving QSF = 0 cannot be worse than the full-blown MLestimator. Although counter-intuitive, the simpler QSF = 0 is actuallysuperior to ML whenever the heteroscedastic variance �i(�) depends on�, in light of (4) above. The following section provides further detailsregarding the GEE model for panel data logits and probits.

1.2 GLM, GEE & PANEL LOGIT/PROBIT (LDV) MODELS

This section provides an introduction to general linear model (GLM) andto the EF literature leading to GEE models. It may be skipped by statis-ticians familiar with the GLM and the GEE. We include it because evenrecent econometric literature dealing with logit, probit and limited depen-dent variable (LDV) models continues to ignore GLM and GEE models.For example, Baltagi (1995), Hajivassiliou and Ruud (1994) and Bertschekand Lechner (1998)and their references do not even mention GLM or GEE.


The econometric context of this paper is a limited dependent variablemodel for panel data (time series of cross sections) typically estimated bythe logit or probit. These LDV models are well known in biostatistics sincethe 1930s. GEE models generalize the LDV models by incorporating timedependence among repeated measurements for an individual subject. Whenthe biometric panel is of laboratory animals having a common heritage, thetime dependence is sometimes called the \litter e�ect." The GEE modelsincorporate di�erent kinds of litter e�ects characterized by serial correla-tion matrices R(�) de�ned later as functions of a parameter vector �. Thissection ends with a statement of the formulas for the GEE estimator andits variance. We have included a limited discussion of the economic is-sues regarding our application to CEO turnover, although the details arepostponed till the next section.

In light of Result 1 above, to show that a quasi-ML (or GEE) estimatorfor panel data logit models is superior to the ML and LS, we have toestablish that it has a special kind of heteroscedasticity. This is done inResult 2 of this section. In preparation for that result and for a betterunderstanding of GEE, we include some discussion of the generalized linearmodel (GLM) literature, McCullagh and Nelder (1989). This literatureshows that the logit is not merely convenient, but implies a \canonical link"for which a \su�cient" statistic exists. Since EF theory and GLMmodelingterminology is not well known in econometrics, we begin by placing thismaterial in the familiar context of a regression model with T (t = 1; � � � ; T )observations and p regressors:

y = X� + �; E(�) = 0; E��0 = �2: (5)

The generalized least squares estimator (GLS) minimizes the error sumof squares. If E(��0) = �2 is a known diagonal matrix which is not a func-tion of �, GLS is obtained by solving the normal equations: g(y;X;) =X0�1X� � X 0�1y = 0; for �. Under normality of errors, the GLS co-incides with the maximum likelihood estimator. Here the EF estimatorde�ned by QSF = 0 leads to the same normal equations. If the diagonalsof are functions of � as in panel logit models, the QSF = 0 equationsare called GEE and are developed as (10), (15) and (16) after we explainthe link functions of GLM.

Remark 1: The GLS is extended into the general linear model (GLM)in three steps, McCullagh and Nelder (1989).

(i) Instead of y � N (�; �2) we allow non-normal distributions withvarious relations between mean and variance functions. Non-normality per-mits the expectation E(y) = � to take on values only in a meaningfulrestricted range (e.g., nonnegative integer counts or binary outcomes).


(ii) De�ne the systematic component � = X� = �pj=1xj�j ; where

� 2 (�1;1); is a linear predictor.

(iii) A monotonic di�erentiable link function � = h(�) relates E(y) tothe systematic component X�. The t-th observation satis�es �t = h(�t):For GLS, the link function is identity, or � = �, since y 2 (�1;1). Wheny data are counts of something, we need a link function which makes surethat X� = � > 0: Similarly, for y as binary (dummy variable) outcomes,y 2 [0; 1], we need a link function h(�) which maps the interval [0; 1] for yon (�1;1) for X�: In the CEO example below, we use a binary dummydependent variable.

Remark 2: To obtain generality, the normal distribution is often re-placed by a member of the exponential family of distributions, which in-cludes Poisson, binomial, gamma, inverse-Gaussian, etc. It is well knownthat \su�cient statistics" are available for the exponential family. In ourcontext, X0y which is a p � 1 vector similar to �, is a su�cient statistic.A \canonical" link function is one for which a su�cient statistic of p � 1dimension exists. Some well known canonical link functions for distribu-tions in the exponential family are: h(�) = � for Normal, h(�) = log� forPoisson, h(�) = log[�=(1� �)] for Binomial, and h(�) = �1=� is negativefor gamma distributions.

Remark 3: Since h(�) = �1=�, based on the gamma distribution, israrely used in econometrics, it is useful to remark on the special featuresof this link function. The gamma density is:

f(x) = (1=�(�))e�x�x��1��; where; x � 0; � > 0; � > 0: (6)

Its mean is �=�, variance is �=�2, and the coe�cient of variation de�nedas the (standard deviation) / mean is ��0:5, which is a constant, since �is a constant parameter. Thus in applications where the variance increaseswith the mean, keeping the coe��cient of variation constant, the gammadistribution with a �xed � is attractive. Since the support of the gammadensity is [0;1), rather than the (�1;1), this is restrictive. However,for many economic variables, including our CEO example in the followingsection, this may be a desirable restriction. A competitor of the gammamodel is the Log-Normal. Firth (1988) supports the gamma over the Log-Normal under mutually reciprocal misspeci�cations.

Now we state and prove the known result that when y is binary, het-eroscedasticity measured by V ar(�t), the variance of �t, depends on theregression coe�cients �. This dependence result also holds true for themore general case, where y is a categorical variable (e.g., poor, good andexcellent as three categories) and to panel data where we have a time series


of cross sections. The general cases tend to be tedious and are discussed inthe EF literature.

Result 2: The heteroscedastic V ar(�t) is a function of the regressioncoe�cients � for a special case where yt is a binary (dummy) variablefrom time series (or cross sectional) data (up to a possibly unknown scaleparameter).

Proof: Let Pt denote the probability that yt = 1. Our interest is inrelating this probability to various regressors at time t, or Xt. If the binarydependent variable yt in (5) can assume only two values (1 or 0), thenregression errors �t also can and must assume only two values: 1�Xt� or�Xt�. The corresponding probabilities are: Pt and (1 � Pt) respectively,which can be viewed as realizations of a binomial process. Note that

E(�t) = Pt(1�Xt�) + (1� Pt)(�Xt�) = Pt �Xt�: (7)

Hence the assumption that E(�t) = 0 itself implies that Pt = Xt�. Thuswe have the result that Pt is a function of the regression parameters �.Since E(�t) = 0, the V ar(�t) is simply the square of the two values of �tweighted by the corresponding probabilities. After some algebra, thanks tocertain cancellations, we have V ar(�t) = Pt(1� Pt) = Xt�(1�Xt�). Thisproves the key result that both the mean and variance depend on �, whereEFs have superior properties.

We can extend the above result to other situations with limited depen-dent variables. In econometrics, the canonical link function terminology ofRemark 2 is rarely used. Econometricians typically replace yt by unobserv-able (latent) variables and write the regression model as:

y�t = Xt� + �t;

where the observable

yt = 1 if y�t > 0; and yt = 0 if y�t � 0: (8)

Now write Pt = Pr(yt = 1) = Pr(y�t > 0) = Pr(Xt� + �t > 0), whichimplies that:

Pt = Pr(�t > �Xt�) = 1� Pr(�t � Xt�) = 1�

Z Xt�

�1

f(�t)d�t; (9)

where we have used the fact that �t is a symmetric random variable de�nedover an in�nite range with density f(�t). In terms of cumulative distributionfunctions (CDF) we can write the last integral in (9) as F (Xt�) 2 [0; 1].Hence Pt 2 [0; 1] is guaranteed. It is obvious that if we choose a density


which has an analytic CDF, the Pt expressions will be convenient. Forexample,

F (Xt�)=[1+exp(�Xt�)]�1 is the analytic CDF of the standard logistic

distribution. From this, econometric texts derive the logit link functionh(Pt) = log[Pt=(1�Pt)] somewhat arduously. Since Pt=(1�Pt) is the ratioof the odds of yt = 1 to the odds of yt = 0, the practical implication ofthe logit link function is to regress the log odds ratio on Xt. Clearly, asPt in [0; 1], the logit is de�ned by h(P ) 2 (�1;1). The probit model issimilar and also popular in econometrics. It was �rst used for bioassay in1935 and uses the inverse of the CDF of the unit normal distribution as thelink function: h(Pt) = ��1(Pt):

Remark 4: The normality assumption is obviously unrealistic whenthe variable assumes only a few values, or when the researcher is unwillingto assume precise knowledge about skewness, kurtosis, etc. Recall that QSFof (3) is the optimum EF and satis�es three key properties. Econometri-cians generally use a \feasible GLS" estimator, where the heteroscedasticityproblem is solved by simply replacing V ar(�t) by its sample estimates. Inthe present context of binary data, V ar(�t) is a function of � and minimiz-ing the S2 with respect to (wrt) � would have to allow for the dependenceof V ar(�t) on �. See (4) and Result 1 above. In the 1970's some biostatisti-cians simply ignored such dependence on � for computational convenience.The EF-theory proves the surprising result that it would be suboptimal toincorporate the dependence of V ar(�t) on � by including the extra term inthe FOCs of (4). An initial appeal of EF-theory in biostatistics was thatit provided a formal justi�cation for the quasi-ML estimator used since the1970's. We shall see that GEE goes beyond quasi-ML by o�ering more exible correlation structures for panel data.

As in McCullagh and Nelder (1989), we denote the log of the quasi-likelihood by Q(�; y) for � based on the data y. For the Normal distributionQ(�; y) = �0:5(y��)2, the variance function V ar(�) = 1 and the canonicallink is h(�) = �. For the binomial, Q(�; y) = ylog[�=(1 � �)] + log(1 ��); V ar(�) = �(1 � �); h(�) = log[�=(1 � �)]. For the gamma, Q(�; y) =�y=� � log�; V ar(�) = �2 and h(�) = �1=�. Since the link functionof the gamma has a negative sign, the signs of all regression coe�cientsare reversed if the gamma distribution is used. The quasi-score functions(QSFs) become our EFs as in (3):

@Q=@� = D0�1(y � �) = 0 (10)

where � = h�1(X�) and D = f@�t=@�jg is a T � p matrix of partials and is T �T diagonal matrix with entries V ar(�t) as noted above. The GLMestimate of � is given by solving (10) for �. Thus the complication arising


from a binary (or limited range) dependent variable is solved by using theGLM method.

1.2.1 GLM for Panel Data:

The panel data involve an additional complication from three possible sub-scripts i, j and t. There are (i = 1; � � � ; N ) individuals about which crosssectional data are available in addition to the time series over (t = 1; � � � ; T )on the dependent variable yit and p regressors xijt, with j = 1; � � �p. Weavoid subscript j by de�ning xit as a p�1 vector. Geddes (1997) estimatesa logit model for panel data from electric utilities focusing on the tenureof the chief executive o�cer (CEO) relating it to age, salary, job perfor-mance, price charged for electricity, etc. His logit model estimates suggestthe somewhat counterintuitive result that CEO job performance variablesdo not have a statistically signi�cant e�ect on the survival of the CEO.This paper reviews that result from the GEE perspective.

Let yit represent a binary choice variable such that yit = 1; if the CEO isremoved and yit = 0, otherwise. Let Pi;t denote the probability of turnoverof i-th CEO (i = 1; � � � ; N ) at time t (t = 1; � � � ; T ) and note that:

E(yit) = 1Pi;t + 0(1� Pi;t) = Pi;t: (11)

Now, we remove the time subscript by collecting elements of Pi;t and yitinto T�1 vectors and write E(yi) = Pi, as a vector of turnover probabilitiesfor the i-th individual CEO. Let Xi be a T �p matrix of data on regressorsfor i-th individual. As before, let � be a p�1 vector of regression parameters.If the method of latent variables is used, the decision to remove a CEO isassumed to be based on latent unobservable positive dissatisfaction y�it bythe board of directors with the CEO's performance. Thus

yit = 1; if y�it > 0 or

yit = 0; if y�it � 0 (12)

where yit = 1 if the board is dissatis�ed with the CEO and yit = 0 if theboard is satis�ed.

Following the GLM terminology of link functions, we may write thepanel data model as:

h(Pi) = Xi�i + �i; E(�i) = 0; E�i�0i = �2i for i = 1; � � �N: (13)

Now the logit link has h(Pi) = log[Pi=(1�Pi)], probit link has h(Pi) =��1(Pi) and the gamma density of (6) implies reciprocal link h(�) = �1=�.


1.2.2 Random E�ects Model from Econometrics:

Instead of N separate �i parameters for each CEO as in (13), econometri-cians often pool the data for all CEOs and split the errors as �it =Mi+�it,where �it represents \random e�ects" and Mi denotes the \individual ef-fects." Using the logit link, the log-odds ratio in a so-called random e�ectsmodel is written as:

log(Pi;t=(1� Pi;t)) = x0it� +Mi + �it; (14)

The random e�ects model also assumes that Mi � IID(0; �2M ) and�it � IID(0; �2�) are independent of each other and also independent of theregressors xit. It is explained in the panel data literature, Baltagi (1995,p.178), that these individual e�ects complicate matters signi�cantly. Notethat under the random e�ects assumptions in (13), covariance over time isnonzero, E(�it�is) = �2M . Hence independence is lost and the joint likeli-hood (probability) cannot be rewritten as a product of marginal likelihoods(probabilities). Since the only feasible maximum likelihood implementa-tion involves numerical integration, we may consider a less realistic \�xede�ects" model where the likelihood function is a product of marginals. Un-fortunately, the �xed e�ects model still faces the so-called \problem of inci-dental parameters" (the number of parameters Mi increases inde�nitely asN !1). Some other solutions from the econometrics literature referencedby Baltagi include Chamberlain's (1980) suggestion to maximize a condi-tional likelihood function. These ML or LS methods continue to su�er fromunnecessary complications arising from the extra term (See eq. 4), whichwould make their FOCs (See eq. 3) biased and ine�cient.

1.2.3 Derivation of GEE, the Estimator for � and Standard Er-rors:

Next, we describe how panel data GEE methods can avoid the di�cultand ine�cient LS or ML solutions in the econometrics literature. We shallwrite a quasi score function justi�ed by the EF-theory as our GEE. Weachieve a fully exible choice of error covariance structures by using linkfunctions of the GLM. Since GEE is based on the QSF (See eq.3), only themean and variance are assumed to be known. The distribution itself canbe any member of the exponential family with almost arbitrary skewnessand kurtosis - not just the normal distribution assumed in the literature.Denoting the log likelihood for i-th individual by Li we construct a T � 1vector @Li=@�. Similarly, we construct yi and �i = h�1(X0

i�) as T �1 vectors and suppress the time subscripts. Denote a T � p matrix of


partial derivatives by Di = f@�i=@�jg for j = 1; � � �p. When there isheteroscedasticity but no autocorrelation, V ar(yi) = i = diag(t) is T�Tdiagonal matrix of variances of yi over time. Using these notations, the i-thQSF similar to (10) above is:

@Li=@� = D0i

�1i (yi � �i) = 0: (15)

When panel data are available with repeated N measurements over Ttime units, GEE methods view this as an opportunity to allow for bothautocorrelation and heteroscedasticity. The pooling over i leads to an ag-gregate QSF from (15) called generalized estimating equation (GEE):

�Ni=1D

0iV

�1i (yi � �i) = 0; where Vi = 0:5

i R(�) 0:5i ; (16)

where R(�) is a T � T matrix of serial correlations viewed as a functionof a vector of parameters �. The sandwiching of R(�) autocorrelationsbetween two matrices of (heteroscedasticity) standard deviations in (16)makes Vi a proper covariance matrix. The GEE user can simply specify thegeneral nature of the autocorrelations by choosing R(�) from the followinglist, stated in increasing order of exibility. The list contains commonabbreviations used by authors of software.

(i) Ìndependence' means R(�) is the identity matrix.

(ii) Èxchangeable' R(�) means that all intertemporal correlations de-�ned by corr(yit; yis)=�, are constant.

(iii) ÀR(1)' or �rst order autoregressive model implies that R(�) orcorr(yit; yis) simply equals �jt�sj:

(iv) Ùnstructured' correlations in R(�) means that corr(yit; yis) = �tswith T (T � 1)=2 distinct values for all pairwise correlations.

Finally, solving (16) for � gives the GEE estimator, which is usuallyiteratively estimated. Liang and Zeger (1986) suggest a \modi�ed Fisherscoring" algorithm for these iterations. The initial choice of R(�) is usuallythe identity matrix and standard GLM is �rst estimated. The GEE algo-rithm then estimates R(�) from the residuals of the GLM and iterates untilconvergence. We use Smith (1996) software in S-PLUS language on an IBMcompatible computer. The theoretical justi�cation for iterations exploitsthe property that a QML estimate is consistent even if R(�) is misspeci�ed,(Zeger and Liang, 1986, McCullagh and Nelder, 1989, p.333). Denoting byR̂ the estimates of R, the asymptotic covariance matrix of GEE estimatoris:

V ar(�̂gee) = �2A�1BA�1; with A = �Ni=1Di0 V̂

�1i Di


andB = �N

i=1Di0R̂�1i RiR̂

�1i Di: (17)

This expression yields the robust standard errors reported in our nu-merical work in the next section. See Zeger and Liang (1986) and Dunlop(1994) for further discussion and references.

In the following section we use the gamma family to �x the relation be-tween the mean and variance, instead of the traditional binomial or Poissonfamily. This is mainly because the gamma family gives better �ts as mea-sured by the lowest residual sum of squares (RSS) than other families ofdistributions. Remark 3 above notes other reasons why the gamma familyand its canonical link may be appropriate. McCullagh and Nelder (1989,p. 290) suggest a deviance function as the di�erence between two log like-lihoods instead of RSS. We do not use deviances, since they need arti�cialtruncation to avoid computing log of a near zero number for the gammafamily. A proof of consistency of the GEE estimator is given by B. Li(1997). Heyde (1997, p.89) gives the necessary and su�cient conditionsunder which GEE are fully e�cient asymptotically. Lipsitz et al (1994) re-port simulations showing that GEE are more e�cient than ordinary logisticregressions. In conclusion, this section has shown that the GEE estimatoris practical with attractive properties for the type of data studied here.

1.3 GEE ESTIMATION OF CEO TURNOVER AND THREEHYPOTHESES

In this section, we discuss the motivation for examining managerial turnoverusing GEE. An important way to align the interests of managers with thoseof owners is by linking managerial turnover to �rm performance. Removalof a manager by a board of directors for performance reasons is a negativesignal to the managerial labor market. If boards remove managers when a�rm performs poorly, an inverse relationship between managerial turnoverand performance will result. Using a variety of data sets, performance mea-sures and empirical techniques, researchers have con�rmed this relationshipin many industries. Salancik and Pfe�er (1980) report that when outsidersown stock there is a positive and signi�cant correlation between pro�t mar-gin and managerial tenure. Warner, Watts, and Wruck (1988) and Cough-lan and Schmidt (1985) �nd that the probability of managerial turnoveris inversely related to abnormal stock price performance. Weisbach (1988)�nds that, given the behavior of stock returns, the probability of manage-rial turnover is negatively related to accounting performance. Barro andBarro (1990) report a negative relationship between turnover and perfor-mance for bank CEOs. The evidence is thus strongly supportive of the


hypothesis that performance and managerial turnover are negatively re-lated, consistent with an incentive-alignment view of CEO turnover. Thereare a number of reasons to believe that the performance-turnover relation-ship described above may be a�ected by utility regulation. Investor-ownedutilities are typically regulated by state public utility commissions, whichadminister rate-of-return regulation under exclusive geographic franchises.Two fundamental managerial functions, investment and �nancing, are de-termined through the regulatory process. Hence regulation is crucial formanagerial decisions.

Regulation supplants decisions normally made by �rm owners and theirmanagers with the administrative process. The signi�cance of this controlhas been the subject of considerable debate. Some researchers suggest thatrate-of-return regulation allows managers to incur high \agency costs" withlittle fear of removal by owners. After all, the return to managerial e�-ciency, given that the maximum rate-of-return is achieved, is zero. In theirclassic article, Alchian and Kessel (1962) state: \If regulated monopolistsare able to earn more than the permissible pecuniary rate of return, then\ine�ciency" is a free good, because the alternative to ine�ciency is thesame pecuniary income and no \ine�ciency." This is essentially a \reg-ulatory slack" view of regulation, which takes regulation as exogenous: ifthe �rm is earning at least the allowed rate-of-return then managers neednot be concerned with operating in owners' interests, and managerial inef-�ciency has zero opportunity cost in the alternative use of maximizing �rmvalue. Regulation leads to a situation where monitoring by owners has alow return, and owners engage in little of it.

Others suggest that regulatory \rent-seeking" behavior will lead to man-agerial monitoring by owners. For example, Crain and Zardkoohi (1978,1980) rely on a rent-seeking hypothesis to arrive at the conclusion thatmanagerial control mechanisms operate in regulated �rms. They submitthat there are potential monopoly rents available through regulation, whichcan be obtained via rent-seeking activity by managers, and that privateowners monitor on this basis. Firm resources, rather than being devotedto pro�t-enhancing activities such as product development and marketing,are directed at in uencing the allowed rate-of-return and other regulatedvariables that a�ect economic pro�ts, making regulatory outcomes endoge-nous to managerial behavior. Such rent-seeking activities include politicalcontributions and public relations programs, as well as payments to lawyersand consultants. Here managerial ine�ciency carries a non-zero opportu-nity cost, providing an incentive for owners to monitor managers.

In a third view, commentators suggest that the regulatory process ex-poses managerial behavior to political forces. In Stigler's (1971) immortal


words, the regulatory process \automatically admits powerful outsiders toindustry's councils," or as Joskow, Rose and Shepard (1993) state, \Eco-nomic regulation imposes political outcomes in place of some private de-cisions or market outcomes." This view implies that the regulatory pro-cess provides organized pressure groups with a mechanism for translatingtheir interests into outcomes. One important party likely to obtain greatercontrol over the regulatory process is the consumer. In many regulatoryprocesses, consumers are granted avenues by which they can organize andare given a special say in proceedings. The political power of consumersrelative to shareholders is expected to increase under a \political pressure"hypothesis. The variable that best measures consumer wealth is the realprice of electricity, and we expect consumers to monitor managers on thatbasis. As applied to turnover, this implies that the political forum in whichregulated CEOs operate results in turnover that responds positively to in-creases in electricity prices.

Our data set, described below, allows us to compare the predictions of(i) regulatory-slack, (ii) rent-seeking, and (iii) political pressure hypothesesin the case of US electric utilities. In testing these alternative hypotheses,we are able to show how the GEE approach represents an improvementover standard limited dependent variable techniques. Below, we describeour data sources, variables, and present summary statistics. We show howour data allow tests of these hypotheses. We then present estimates ofmanagerial turnover using GEE.

1.3.1 Description of Data

A sample of 95 investor-owned electric utilities (IOUs) was taken from thoselisted in the Statistics of Privately Owned Electric Utilities in the UnitedStates, for which �nancial and managerial data were available. The testyears run from 1966 through 1988. We used numerous data sources tocompile a large data set on managerial turnover and variables potentiallya�ecting turnover; see Geddes (1997) for details. The variables in the dataset are brie y described below. Due to our need for data on CEO age,salary, and performance measures, the ultimate number of observations inthe data set was 790 in variable groups discussed below.

Turnover measure:

The focus of this study is on the probability of a change in the seniormanager of an electric utility. Senior managers are de�ned as the presidentor CEO of an IOU. If a change in the president or CEO was observed butactually the individual moved into the chairman's position this was not


counted as a turnover. Our measure of managerial turnover, TURN=1 ifthe CEO left the �rm, and zero otherwise. Data on reasons for departure(�ring, quits or illness) are unavailable.

Managerial characteristics:

We control for three important managerial characteristics. AGE, the age ofthe senior manager, is expected to a�ect turnover positively since managersare more likely to change as they approach retirement age. This e�ect ismore explicitly studied by a dummy variable: RETIRE=1, if the manageris aged 63 through 66. A non-trivial number of managerial changes occuraround normal retirement age, and are likely to be unrelated to perfor-mance. TENURE is the number of years served as the CEO, and is alsoexpected to positively a�ect turnover. SALARY is the annual real compen-sation including bonus of the CEO and we expect that it will be negativelyrelated to turnover, if higher paid CEOs are less likely to leave.

Firm characteristics:

We include SIZE as measured by the annual real sales of the �rm in dollars.It is often alleged that larger �rms have a higher rate of CEO turnover(Warner, Watts, and Wruck (1988). We also examine the responsivenessof CEO turnover at regulated �rms to shareholder wealth, as measuredby accounting returns. Despite the problems with using earnings data tomeasure economic pro�ts (Fisher and McGowan 1983), accounting returnsdo measure short-term pro�ts, rather than the discounted present value ofthe expected future cash ows of the �rm, as measured by the stock price.Since stock prices are forward-looking, they incorporate the possibility thatthe board will remove the CEO after poor performance (Weisbach 1988).The use of stock prices may therefore understate the e�ects of managerialmonitoring. Also, Joskow, Rose and Shepard (1996) note that accountingreturns are likely to be relatively more important in electric utilities, whichare regulated. Changes in accounting returns are thus the best availablemeasure of changes in owner wealth when examining managerial turnoverin electricals. The two measures used here are ROA and ROE. ROA is the�rm's realized return-on-assets = (gross income) / (total assets). ROE isthe realized return on equity = (gross income) / (total stockholder's equity).

Regional variables:

These dummy variables control for the area or region of the country in whichthe �rm operates. They are a proxy for such diverse factors as di�erent fed-


eral air pollution standards, availability of cheap hydroelectric power, ageof the capital stock, di�erent population densities, weather, etc. To cre-ate these variables, the country was divided into seven regions. Northeast(NEDUM), Mid-Atlantic (MADUM), Southeast (SEDUM), South-central(SCDUM), Northwest (NWDUM), Midwest (MWDUM), and the South-west (SWDUM). The su�x DUM is for dummy variable, i.e., they equalunity if the electric utility operated in that region and zero otherwise. Theomitted category was the Southwest. Table 1 summarizes the predictionsof the three hypotheses and the variables used to test them. Table 2 reportsthe GEE estimation results using the gamma density and assuming thatthe correlation structure over time is unrestricted (compare to Table 2 ofGeddes, 1997). Since the likely sign of the coe�cient is known from thetheories discussed above, we use one-sided tests.

1.3.2 Shareholder and Consumer Wealth Variables for Hypoth-esis Testing

Shareholder wealth:

Sharholder wealth comes from accounting returns. Since it is unlikely thatmanagers would respond to the level of shareholder wealth, the year-to-yearchanges in ROA and ROE were used to construct two new variables, �ROAand �ROE. These variables measure the change in shareholder wealth priorto a particular observed �rm-year. The regulatory slack hypothesis predictsthat managerial turnover will be unrelated to changes in returns, while therent-seeking hypothesis predicts that turnover will be negatively relatedto changes in returns. An alternative test of the rent-seeking hypothesisinvolves \allowed" returns. Since regulatory rent-seeking activity includesattempts by managers to in uence the return allowed by the regulatorycommission it may result in a greater deviation of the realized rate-of-return from that allowed by the commission. That is, managers may devoteresources to maximizing the deviation between the realized and allowedreturn, and may be monitored by owners on that basis.

The rent-seeking hypothesis is further tested by relating managerialturnover in IOUs to di�erences between the actual and allowed returns, andto changes in allowed returns. Two new variables were created, �DEVROAand �DEVROE, which are the year-to-year deviations between the returnallowed by the regulatory commission and the realized return for ROA andROE respectively. The rent-seeking hypothesis predicts that the probabilityof turnover will decrease as �DEVROA and �DEVROE increase.


Customer wealth:

To test the political pressure hypothesis, a measure of the change in cus-tomer wealth was developed. A variable called �PRICE was created,which is the year-to-year change in the real price of electricity sold by the�rm. If consumers exercise power via the political process, then managerialturnover will be positively related to changes in price. That is, managerswill be removed when the real price of electricity rises, and conversely.

TABLE 1 Tested Hypotheses and Predictions.

Hypothesis Variable Predicted E�ecton Turnover

Regulatory slack �ROA, �ROE No e�ectRent-seeking �ROA, �ROE, Negative

�DEVROA, �DEVROEPolitical pressure �PRICE Positive

1.3.3 Empirical Results

Firm performance measures

Table 2 presents GEE estimates of the e�ects of �rm performance on man-agerial turnover. It is important to note that the gamma function reversesthe signs of the coe�cients relative to the logit. In Table 2, most variableshave the expected e�ect on turnover. For example, RETIRE, TENUREand ln(SIZE) signi�cantly increase the probability of managerial change.With retirement e�ects controlled for by RETIRE, it does not appear thatAGE a�ects CEO turnover. SALARY signi�cantly decreases turnover, aspredicted. Most regional variables have little e�ect, with the exceptionof the Southeast dummy, which decreases the turnover probability. Im-portantly, both �ROA and �ROE decrease the probability of managerialturnover, with �ROE signi�cant at the 2 percent level. This is stronglyat odds with logit estimates reported in Table 2 of Geddes (1997, p.275).These GEE estimates provide support for the rent-seeking hypothesis overthe regulatory slack hypothesis, as discussed above.

Turnover and Allowed Returns:

The rent-seeking hypothesis can also be tested by examining the e�ect man-agers have on di�erences between allowed and realized rates of return. Thatis, managers may be monitored by owners on the basis of their ability toachieve a return higher than that allowed by the regulatory commission.


Here, the rent-seeking hypothesis predicts that the probability of turnoverwill be negatively related to the deviation between actual and allowed re-turns, �DEVROA and �DEVROE. GEE parameter estimates including�DEVROA and �DEVROE are reported in Columns 2 and 3 of Table 3.

TABLE 2Generalized Estimating Equations Estimates of the E�ects of FirmPerformance on Managerial Turnover in Investor-Owned Utilities.

Regressor (1) (2)Constant 73.207 (1.797)** 83.243 (2.147)**RETIRE -.806 (-.172) -.173 (-.044)AGE .032 (.076) -.077 (-.235)TENURE -.984 (-3.527)** -.968 (-3.523)**SALARY .042 (1.704)** .0475 (2.028)**Ln(SIZE) -4.672 (-1.911)** -5.065 (-2.083)**�ROA 21.378 (1.452)* {�ROE { .478 (2.086)**NEDUM 2.378 (.502) 2.156 (.491)SEDUM 28.340 (1.304)* 28.339 (1.298)*MADUM -.046 (-.010) .585 (.128)NWDUM 24.628 (.570) 24.869 (.578)MWDUM .422 (.127) 1.072 (.376)SCDUM -.068 (-.019) .199 (.061)RSS 47.158 46.928N 790 790

Note: Robust Z-statistics are in parenthesis. * Signi�cant at the .10level; ** Signi�cant at the .05 level, one-tailed test. The Variance-to-meanrelation is gamma. The link function is reciprocal. Signs are reversed whenusing gamma.

The reduced number of observations available here is due to lack of dataon allowed returns for certain �rm-years. Here, �DEVROA is signi�cantat approximately the 6 percent level, with greater deviations of the realizedreturn from the allowed decreasing the probability of managerial turnover.These tests do not provide support for a regulatory slack hypothesis, butare consistent with a rent-seeking hypothesis. They suggest that owners domonitor managers on the basis of deviations from allowed returns.

Managerial Turnover and Electricity Price Changes:

GEE estimates incorporating �PRICE are reported in column 1 of Table3. These estimates support an important conclusion: managerial turnover


in IOUs is sensitive to increases in price. This is consistent with estimatesreported in Geddes (1997), but levels of signi�cance are higher here. Itappears that managers in electric utilities are also monitored on the basisof price changes, consistent with a \political pressure" hypothesis.

TABLE 3GEE Estimates of the E�ects of Output Price and Allowed Returns

on Managerial Turnover in Investor-Owned Utilities.

Regressor (1) (2) (3)Constant 59.122 (1.312)* 76.509 (1.676)** 68.388 (1.583)*RETIRE 2.694 (-0.547) 1.805 (0.418) -1.46 (-0.301)AGE 0.113 (0.253) -0.045 (-0.09) 0.0003 (0.001)TENURE -0.994 (-3.07)** -1.362(3.401)** -1.053 (-2.906)**SALARY 0.0298 (1.131) 0.042 (1.285) 0.0334 (1.251)ln(SIZE) -3.566(-1.372)* -4.244(-1.372)* -3.846 (-1.594)*DeltaPRICE -4.065 (-2.51)** || ||DeltaDEVROA || 0.623 (1.522)* ||DeltaDEVROE || || 0.247 (0.635)NEDUM 1.783 (0.379) 3.336 (0.577) 0.142 (0.027)SEDUM 25.03 (1.213) 41.015 (1.143) 40.646 (1.112)MADUM -1.955 (-0.458) 0.250 (0.049) -0.76 (-0.15)NWDUM 23.397 (0.541) 19.496 (0.498) 20.242 (0.515)MWDUM -0.802 (-0.201) -0.388 (-0.079) -0.564 (-0.125)SCDUM -0.36 (-0.091) -0.829 (-0.176) -1.239 (-0.288)RSS 48.067 61.923 40.506N 776 685 692

Note: Robust Z-statistics are in parenthesis. * Signi�cant at the .10level; ** Signi�cant at the .05 level, one-tailed test. The Variance-to-meanrelation is gamma. The link function is reciprocal. Signs are reversed whenusing gamma.

1.4 CONCLUDING REMARKS

This paper reviews recent developments in the estimating function liter-ature and explains why estimation problems involving limited dependentvariables are particularly promising for applications of the EF theory. OurResult 1 shows that whenever heteroscedasticity is related to � the tra-ditional LS or ML estimators have an unnecessary extra term leading tobiased and ine�cient EFs. We note that recent econometric literature, sur-veyed in Hajivassiliou and Ruud (1994) or Baltagi (1995) while ignoring


the simpler GEE methods, is suggesting computer intensive nonparamet-ric and simulation based method of moments estimators. These estimatorsare obviously suboptimal, since they fail to remove the extra term men-tioned above. Our Result 2 shows why binary dependent variables havesuch heteroscedasticity.

The panel data GEE estimator in (16) is implemented by Liang andZeger's (1986) \modi�ed Fisher scoring" algorithm with variances given in(17). The exibility of the GEE estimator arises from its ability to specifythe matrix of autocorrelations R(�) as a function of a set of parameters�. We use unstructured R(�) with minimum prior assumptions to achieverobustness and use a \canonical link" function satisfying \su�ciency" prop-erties available for all distributions from the exponential family. It is wellknown that this family includes many of the familiar distributions includ-ing Normal, binomial, Poisson, exponential, gamma, etc. Our Remark 3explains the advantages of the gamma family with its canonical link usedhere, which is almost never used in econometrics.

The regression results reported here are consistent with the \rent-seeking"hypothesis regarding the intensity of managerial monitoring of �rm man-agers by owners in the electric utility industry. The GEE estimates implyrejection of the \monopoly slack" view. This reverses the conclusions sug-gested by logit tests in Geddes (1997). There is also evidence that managersare monitored by consumers on the basis of price changes, which is consis-tent with a \political pressure" hypothesis. Overall, this suggests that theGEE estimation technique represents an improvement over standard binarydependent variable techniques, especially for panel data.

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