Using Nested Logit Model in the Study of Proxy Contest

The nested logit model has been extensively used in studies focused on transportation. This model is based on a choice modeling framework and is also applicable to the study of a proxy contest. In this study, we illustrate how the nested logit model can be effectively used to evaluate a case of a proxy contest. We also present the specific model that can be used to predict the possibility of the occurrence of proxy contests.


Introduction
The nested logit model has been commonly used in various studies of transportation and commercial behaviors ([1] [2]). The nested logit model is specifically based on a "choice" modeling framework. It considers human beings as highly rational as we tend to make the best choices that we think has more value in terms of utility. From this perspective, when faced with the assessing the utility function of different choices, the respondents are observed to make the choices that best suit their needs. Considering the expected value of the available choices, with the logit model, the optimum probability for each choice can be evaluated.
For studies in finance, the use of the nested logit model as a statistical tool is not very common. But there are several circumstances in which the nested logit model can be involved in finance because of the need to make choices regarding money/cash holdings and expenditures, which is a common situation in the financial world.
In this study, we illustrate how to build the nested logit model using the proxy contest as the framework. From a theoretical point of view, the person who controls the board has an absolute authority over the entire firm. This can be achieved by shareholder activists, who will leverage on the proxy contest to achieve this. There will be a formidable resistance in a situation where the firm is publicly traded, having as many as many as ten or more board members. In the corporate world, there is no shortcut to getting a board seat. As a candidate vying for a board seat, one would have to win the support of a sufficient number of shareholders (a majority) during the annual shareholder meeting in order to get a board seat over time. The shareholder population is a widely dispersed one; this means that a candidate would need to use undertake a convincing campaign to win the needed votes during an election. This can be a very expensive plan to execute. For larger firms, the total expenses incurred from an election process can reach as high as $1 million and about $100,000 for smaller firms. On the average, the candidate should expect expenses anywhere between $200,000 and $1

million ([3] [4]). A good example of a proxy test contest was the case between
Sam Wyly and Computer Associates Inc. As reported in the Los Angeles Times, the tussle for victory cost both contenders as much as $20 million each. In the end, the management team of Computer Associates Inc. emerged as the winner.
Proxy contests are a rather unlikely choice for many investors because of the high-cost implications. Many stakeholders have found alternatives to the expensive proxy contests as ways to prevent any potential uprising that could lead to a takeover of the company. A good example is a situation where in a staggered board restricts one-third of the board seats to be reelected every year; hence, it might take as long as three years before activists could potentially replace the entire board.
Shareholder activists can achieve an early end for a proxy contest before the annual shareholders meeting when shareholders come together to vote for the next set of board representatives. Once the shareholder activist makes a public announcement indicating the intention of shareholders to initiate a proxy fight at some time in the future, the start of a proxy contest is initiated. The end of a proxy contest is not concluded until the two parties have reached agreement in private and have amicably settled the disputes. Settlements can be made when the representative activist shareholder is invited to become a board member. In a situation where it is not possible to arrive at an agreement, the proxy contest would proceed to the next stage where the shareholder activist would make public declarations that the two parties have irresolvable disputes at this stage, the proxy contest will become a fierce tussle for authority. Even at this stage of the proxy contest, when the SEC has been notified of the disputes, it is still possible for the two parties to arrive at a private agreement before the annual shareholders meeting. The finalization of the negotiations signifies the end of a proxy contest. If the negotiations fail, then the final resolution will be undertaken at the annual shareholder meeting, when the shareholders will be expected to vote.
The new board members would be announced to the public, and the winning In summary, the entire process of a proxy contest can be broken down into three stages, with the end of each stage either resulting in an agreement or with the parties involved proceeding to the next stage. This is why a three-stage nested logit model is the most suitable statistical tool for the study of a proxy contest.
The remainder of the study is organized as follows. Section 2 provides the review of proxy contest and introduces the model. Section 3 describes the model assumptions and specification. Section 4 reveals the empirical results using nested logit model. Section 5 provides conclusions from the study. Firms that fail to prevent moves for a proxy fight will appear vulnerable at this stage. This implies that the top management team will potentially lose their positions at the helm of affairs. Faced with this threat, a proxy contest could be averted if the management team gives in to the requests of the shareholder activ-

The Proxy Contest and Setting up the Model
ists. If this happens, the shareholders can either back down from a proxy contest or choose to proceed with an election. This is the final stage of a proxy contest. In summary, shareholder activists become aggravated by firms with huge potential for growth which will benefit the shareholder but performing poorly due to poor management. These firms may be targeted for a proxy contest, which will often start with the launch of a media accusation. The shareholder activists announce their decision to proceed with a proxy contest with the firms' management in the media. The firms that make the best targets are the one that has made few or no provisions to prevent or frustrate a takeover. These firms run with an independent board who are not deeply involved in decision making hence they can be swayed by external influence. The impact of a media threat can cause the management to begin negotiations with the activists which can end the proxy contest when an amicable agreement is reached. Failure to come to an agreement will propel the activists to proceed with informing the SEC of its plans. Even at this stage, agreements can still be arrived before the annual shareholder meeting when a vote will be held. Firms that are usually targeted by the shareholder activists are those firms that have a larger percentage of external supporters than internal supporters. From the analysis above, an ideal nested structure in three distinctive levels clearly reveals the provisions for a proxy contest. This is illustrated in Figure 1. Specifically, firms at the lower levels have certain firm characteristics which are similar to those firms at the upper levels due to the nested structure representing the data. An example is a move by the shareholder activists against poorly performing firms by propagating media threats. This means firms involved in proxy contests that progress to the stage where the SEC is notified are poorly performing firms. This eliminates the assessment of performance for firms when a SEC filing has been made because they had already been targeted by the media threat at the initial stage of the proxy contest due to poor performance. This generalizes poor performance as a factor that affects both the firms at the upper and lower levels. In view of this, firm attributes included in the upper levels should be shared by firms in the lower levels and there should be different considerations for the different stages of a proxy contest.

Assumption of the Nested Logit Model
As a result of the structure illustrated in Figure 1, the nested logit model is adopted in this study. The model requires loads of the assumptions explained below.
First, the nested logit model is applicable under the assumption that the various alternatives could be separated into nests. From Figure 1, the nested structure for the proxy contests is compellingly demonstrated. This means the assumptions have been convincingly satisfied in this study.
Second, the nested logit model relaxes assumption of Independent & Identically Distributed (IID) and recognizes the possibility that each alternative may have information in the unobserved influences of that alternative, which in turn has a role to play in determining an outcome that is different across the alternative branches ( [8]). This difference implies that the variances of error might be different but the information content could be similar among subsets of alternatives generating some amount of correlation among these subsets ( [8]). Under the context of the proxy contests, IID indeed does not hold. In fact, the utilities of the third and the fourth outcome are highly correlated in that the occurrence of both events depends on the contingency of the SEC Filing.
Lastly, the application of a nested logit model relaxes the restriction that the choice between any two pairs of alternatives is simply a binary logit model ( [9]).
In other words, this Independence of Irrelevant Alternatives (IIA) can be too restrictive under the nested logit model. Indeed, if the 'No further action after a Media Threat' in Figure 1 is not an option anymore, then more cases of the 'No further action after SEC Filing' will occur. This will change the ratio of the occurrence of the two mutually exclusive alternatives nested with SEC Filing. It should be noted that if the IIA is not applicable, then it can be assumed that IID will not be applicable because the IIA is a consequence of the assumption that the errors are IID ([2] [10]).
In accordance with the standard procedure, we test the assumption of IIA by applying the Hausman ( [11]) specification test to re-estimate the model on a subset of the alternatives since this test is not sensitive to the tree structure that we specify for a nested logit model ([2] [10]). Hence, if IIA is valid, there will be a minimal difference between the parameters arrived at for the subset of alterna-

Specifications of the Nested Logit Model
The basic nested logit model is dependent on a "choice" modeling framework. It is based on the fact that people act rationally to make choices among options that have the highest utility values to them. This indicates that for a utility function for the different choices, respondents would be expected to make the optimum choice. This will depend on the estimated utility value expected from the available choices, as obtainable from the logit model. An attempt to the optimum probability for the various choices will be possible.    outcome is the probability of a "No further action after SEC filing" or a proxy fight; this will be conditional depending on if the firm is classified under the SEC filing category. However, there are three nodes, Limb 1, Branch 1 and Branch 2; these three nodes are subject to degeneration with a single extension.
In summary, the nested logit models begin with the calculation of the expected utilities for the lower level and the utilities at the higher levels which include the expected utilities from the lower levels. This relies on the expected utilities achievable at the different levels. The maximum likelihood function (ML function) is equal to the log of the summation for the different utilities multiplied by the corresponding proxy contest results (indicator variable). This ML function is concave and it is characterized by efficient and consistent solutions. The utility models at each level are, as follows:

The Application of the Nested Logit Model in the Study of Proxy Fight
This section will cover the reports for the theoretical analysis of the nested logit model in the context of the proxy contests. For this study, the choice behavior is described by applying a random utility framework. Also, the alternative chosen should provide the highest utility.
According to Hensher, Rose, and Greene ( [12]), the scale parameters for the different limbs, branch and elemental alternatives for the three nested structure above is illustrated as follows: where: γ(i) is the scale parameter for the i th limb, λ(j,i) is the scale parameter for the j th branch of limb i. Then, μ(j,i) represents the scale parameter for the elemental alternatives of branch j and limb i. It is identical for each alternative present within any given branch, because the variance of each elemental alternative present within branch j is equal.
The utility functions in the higher levels also include IV (inclusive value) which is calculated as the natural logarithm of the sum of the exponentials of the utility expressions for the level directly below which the IV is calculated. For simplicity of model presentation, we normalize on μ(j,i) following random utility model 1 (RU 1).

Level 1 (elemental alternatives):
As discussed in Section 2, we focus on variables describing shareholder structures for the bottom level: 1) Aggregated ownership by insiders.

4) Number of institutional funds.
5) The indicator variable for filing of schedule 13-d.
For the elemental alternatives at Level 1, the utility functions can be expressed as follows.
No further actions after SEC filing Using RU1 1 as suggested by Hensher, Ross and Greene (2005), we normalize Since No further actions after SEC Filing and proxy fight are nested under the branch of SEC Filing, the probabilities of the two alternatives are defined as follows: , No further actions after SEC Filinĝ The utility of Branch 3 can be expressed as follows.
V a and V b are known from Level 1.

For Branch 2, it is assumed that the utility for No further actions after Media
Threat is defined at this level. Hence, the expected utilities, the IV, from Level 1 is zero.
In comparison, Branch 1 linked with Nothing at Level 1 is not defined at this level, but at Level 3. Hence, V branch 1 or V (3,2) is still zero.
Next, similar to the conditional logit model, the probability that a branch is chosen is calculated only by considering other branches partitioned within the same limb.
3) Industry adjusted log market to book ratio. 4) Industry adjusted free cash flow.
5) Industry adjusted square root leverage ratio.
6) Industry adjusted log firm size.
There are two limbs stemming from the same trunk. The utility function of Limb 2 regarding Media Threat with two branches can be written as follows.
Limb 1 is degenerate and it is assumed that the utility for Nothing is defined at this level. Hence, the expected utility, the IV, from Level 2 is zero. Since, The probability of the first limb being selected under RU1 is as follows. Probability summary: The above probabilities calculated in each level are conditional probabilities.
The following summarizes the conditional probabilities.
For the indicator variables, we use the following method to get the ME. N is the number of observations and i represents each observation. Interpreting the estimated average marginal effects is that 1 unit increase in x is related to ME% increase in y, while all other variables remain fixed.
Explaining the nest logit model using the AME is straightforward. It should be noted that the standard errors would not be addressed when the AME is used. Cameron and Trivedi ( [13]) pointed out this challenge, however no solution was proffered for it. Also, Jones and Hensher ([8]) attempted to introduce the nested logit model into accounting research but failed to proffer solutions to this challenge. The important fact is that they used p-values derived directly from the nested logit models in the discussion of the elasticity. This also implies that the elasticity (marginal effects) and the coefficient estimation have similar p-values. From this review, we follow Jones and Hensher ([8]) and Bethel, Liebeskind and Opler ( [14]) and use the p-values obtained directly from the nested logit models in the discussions of marginal effects in Section 4. performance. This will be achieved by using the three-level nested logit models.

Nested Logit Model-Examination of Assumptions
As seen in Figure  form of (β a − β b )'(Ω a − Ω b ) −1 (β a − β b ) has a chi-square distribution. Figure 3 shows the new tree structure after eliminating Nothing from the sample: Next, we use total assets, free cash flow, market to book ratio, leverage, prior shareholder proposals, poison pills, board independence, institutional ownership, and block-holder ownership in the Hausman test ( [11]), rather than all variables in the nested logit model. This strategy is appropriate due to the small sample size of the subgroup. The purpose of the Hausman test is to demonstrate IIA assumption for some variables; hence, using certain variables while excluding the rest will not affect the conclusion.  A different approach that can be used to test the IIA assumption is to apply the likelihood ratio test (LRT), which tests the "goodness of fit" between the nested logit model and the conditional nested model. The two models differ only in the dissimilarity parameters, which are set to equal to 1 in the conditional logit model while they are free for estimation in the nested logit model. The LRT uses the following equation to compare the likelihood scores of the two models: This LRT statistic follows a chi-square distribution. The degree of freedom is equal to the number of additional parameters in the more complete model, which is 2, coming from the dissimilarity parameters for Media Threat and SEC Filing. The dissimilarity parameters for the two degenerate branches are set to one in the nested logit model; this also applies to the conditional logit model. The result of LRT is reported in Table 1 as part of the procedures of a nested logit model. The chi-square is 87.24, with 2 degrees of freedom. Thus, the IIA assumption is strongly rejected at the 1% level again, indicating that nested logit model is suitable. In summary, the IIA assumption does not hold by applying both the Hausman ( [11]) test and the likelihood ratio test. Thus, the nested logit model is more appropriate 2 . The detailed utility and probability functions derived from the nested logit model are reported in the Appendix.

Graphs Based on Estimated Probabilities
In this section of the study, we provide the graphs based on the estimated probabilities with the independent variable of interest for Media Threat, SEC Filing, and Proxy Fight, respectively, while holding all the other variables fixed at 2 We use Stata 10 to run the nested logit model. Stata 10 produces a nested logit model that is consistent with random utility maximization (RUM) ([1] [10]). We use the default settings for nested logit model under Stata 10. For the numerical root finding method, the default setting is Newton-Raphson algorithm. For the type of variance-covariance matrix, the default is observed information matrix (OIM), the inverse of the negative Hessian matrix ( [15]). In addition, all models have no constant variables since all independent variables are non-variant among alternatives, which require keeping one alternative as base in each level and using the interaction between the alternative and the independent variables to create variations in independent variables. This is the last step in testing IIA assumption. This last step is to calculate (β a − β b )'(Ω a − Ω b ) −1 (β a − β b ). β a and β b are from step one and step two respectively and Ω a and Ω b are covariance matrix. (β a − β b )'(Ω a − Ω b ) −1 (β a − β b ) has a chi-square distribution when IIA holds, with two degree of freedom in this test. sample mean, except for indicator variables whose values equal to one for all observations. Figure 4 plots the marginal probabilities of Media Threat for holding period return. Figure 5 and Figure 6 plot the marginal probabilities of SEC Filing for board independence level and the proxy fight for insider ownership, respectively. The estimated probabilities seem a little low. This might be because proxy contests are rare corporate events. Figure 4 shows that when holding period return increases, the estimated probability of a media threat tends to drop. Figure 5 shows that when board independence level increases, the estimated probability of a SEC filing tends to drop. However, when board independence level is more than 80%, the possibility of a SEC filing might increase. Figure 6 shows that when board insiders' ownership increases, the estimated probability of a formal proxy fight tends to fall.

Conclusions
In this study, we have illustrated how the nested logit model can be effectively

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.

Appendix-The Functions of the Nested Logit Model
The coefficients and average marginal effects estimated from the nested logit model are reported in Panel (a) of Table A1. Based on the coefficients, we obtain the estimated utility functions and derive the estimated probability functions.
They are, as follows: Utility Functions: In general, nj nj nj U V ε = + , where V nj is observed by the researcher and ε nj is a random variable whose value is not observable. Additionally, n represents the n th observations and j represents the j th alternatives.  The Table shows  Free cash flows are calculated by subtracting capital expenditures (Schedule V) of property, plant and equipment from operating income before depreciation and scaled by total assets. Leverage equals the sum of long term debt and short term debt and then divided by total assets. Market to book ratio equals the sum of total assets and the product of stock price at the end of the fiscal year and outstanding common shares, minus total equity and then divided by total asset. Firm size equals total assets. Firm size and market to book ratio is transformed by log and leverage by square root to reduce kurtosis and skewness. Additionally, to control for the industry effect, all of the above accounting variables are adjusted by industry means. We collected the status of classified boards and poison pills, percentage of outstanding shares required to amend bylaws, percentage of outstanding shares required to amend company charter and percentage of outstanding shares required to approve a merger or sale from TCL. Where the percentage of the above three variables is greater than 66.67%, the thresholds for super majority, it is recorded as 1. Otherwise, it is recorded as 0. Additionally, board size and number of outside directors from TCL is recorded to ascertain the level of board independence. Quarterly ownership by institutional investors from the Thompson CDA/Spectrum 13F institutional ownership data is collected for the sample. We also divided the sample into five groups by manager type code as defined in Thompson. We then calculated the aggregate ownership and counted the number of funds by all institutional investors and by each manager type. Ownership by insiders and block-holders as listed in TCL, is also collected. Shareholder proposal data is collected from Investor Responsibility Research Center (IRRC).