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Essays in identification and estimation of duration models and varying coefficient models He, Xiaoqi 2016

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Essays in Identification andEstimation of Duration Models andVarying Coefficient ModelsbyXiaoqi HeB.S., Wuhan University, 2006B.A., Wuhan University, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2016c© Xiaoqi He 2016AbstractChapter 1 studies the identification of a preemption game where thetiming decisions are expressed as mixed hitting time (MHT). It considersa preemption game with private information, where agents choose optimaltime to invest, with payoffs driven by Geometric Brownian Motion. Thegame delivers the optimal timing of investment based on a threshold rulethat depends on both the observed covariates and the unobserved hetero-geneity. The timing decision rules specify durations before the irreversibleinvestment as the first time the Geometric Brownian Motion hits a heteroge-neous threshold, which fits the MHT framework. As a result, identificationstrategies for MHT can be used for a first stage identification analysis ofthe model primitives. Estimation results are performed in a Monte Carlosimulation study.Chapter 2 studies the identification of a real options game similar tochapter 1, but with complete information. Because of the multiple equi-libria problems associated with the complete information game, the pointidentification is only achieved for a duopoly case. This simple complete in-formation game delivers two possible different kinds of equilibria, and wecan separate the parameter space of unobserved investment cost accord-ingly for different equilibria. We also show the non-identification result fora three-player case in appendix B.4.Chapter 3 studies the estimation of a varying coefficient model withouta complete data set. We use a nearest-matching method to combine twoincomplete samples to get a complete data set. We demonstrate that thesimple local linear estimator of the varying coefficient model using the com-bined sample is inconsistent and in general the convergence rate is slowerthan the parametric rate to its probability limit. We propose the bias-corrected estimator and investigate the asymptotic properties. In particu-lar, the bias-corrected estimator attains the parametric convergence rate ifthe number of matching variables is one. Monte Carlo simulation results areconsistent with our findings.iiPrefaceChapter 3 is based on work collaborated with Zhu,Yajing, a Ph.D studentfrom Concordia University. I am very grateful to Zhu for allowing me to putthis joint work into my thesis. All errors are my own.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Identification of Mixed Hitting Time in a Preemption Game 11.1 The Economic Model . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Setting and Notation . . . . . . . . . . . . . . . . . . 51.1.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Parametric Estimation . . . . . . . . . . . . . . . . . . . . . 191.3.1 Estimation Strategies . . . . . . . . . . . . . . . . . . 191.3.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . 221.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Identification of Optimal Hitting Time in a Real OptionsGame with Complete Information . . . . . . . . . . . . . . . 252.1 The Economic Model . . . . . . . . . . . . . . . . . . . . . . 272.1.1 A Duopoly with Complete Information . . . . . . . . 272.1.2 Value Functions and Investment Thresholds . . . . . 282.1.3 Preemptive and Sequential Equilibrium . . . . . . . . 302.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36ivTable of Contents3 Estimation of Varying Coefficient Models with MatchingData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Two-Sample Matching Estimator . . . . . . . . . . . . . . . 403.1.1 Setting and Notation . . . . . . . . . . . . . . . . . . 403.1.2 Identification of the Two-sample Estimator . . . . . . 413.1.3 Two-Sample Naive Local Linear Estimator . . . . . . 433.1.4 Large Sample Properties of the Two-Sample Naive Lo-cal Linear Estimator . . . . . . . . . . . . . . . . . . 443.2 Bias Correction and the Consistent Estimator . . . . . . . . 503.3 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . 523.3.1 Case with One Complete Sample . . . . . . . . . . . 533.3.2 Case with Two Missing-data Samples . . . . . . . . . 53Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60AppendicesA Appendix to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 65A.1 Expected Value Functions for the Stochastic Control Problem 65A.2 Equilibrium Strategies . . . . . . . . . . . . . . . . . . . . . . 68A.3 A Duopoly Example . . . . . . . . . . . . . . . . . . . . . . . 71A.4 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . 75B Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 81B.1 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . 81B.2 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . 82B.3 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . 83B.4 A Three-player Real Options Game with Complete Informa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.4.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 84B.4.2 The Two-Firm Subgame . . . . . . . . . . . . . . . . 85B.4.3 Multiple Equilibria in the Three-Player Game . . . . 86C Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 90C.1 Convergence of the Denominator . . . . . . . . . . . . . . . . 90C.2 Convergence of the Numerator . . . . . . . . . . . . . . . . . 94C.3 Convergence of V ar(I10) . . . . . . . . . . . . . . . . . . . . 97C.4 Boundedness of V (z) . . . . . . . . . . . . . . . . . . . . . . 99vList of Tables1.1 Parametric Estimation . . . . . . . . . . . . . . . . . . . . . . 233.1 Finite Sample Comparison of the Local Linear (LL) Estimatorand the Bias-Corrected Local Linear (BCLL) Estimator . . . 57viList of Figures3.1 Local linear estimator in one complete sample case: intercept,β0(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Local linear estimator in one complete sample case: coefficientof X1, β1(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Local linear estimator in one complete sample case: coefficientof X2, β2(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Estimators in two-sample case: intercept, β0(Z) . . . . . . . . 573.5 Estimators in two-sample case: coefficient of X1, β1(Z) . . . . 583.6 Estimators in two-sample case: coefficient of X2, β2(Z) . . . . 59viiAcknowledgementsI am especially grateful to my supervisor Kevin Song for his invalu-able advice, unfailing encouragement and extreme patience over my years ofstudy at UBC. I also would like to thank my thesis committee Hiro Kasa-hara, Vadim Marmer and Paul Schrimpf for their insightful comments andfeedback which improved my thesis greatly.viiiDedicationTo my parents and my husband.ixChapter 1Identification of MixedHitting Time in aPreemption GameReal options theory tells us that an investment opportunity can be seenas an option to invest and the value of this option can be very large inthe case of irreversible investment. For example, to make a real estateinvestment, a company may wait until the market price is very high toseize this investment opportunity. Therefore, irreversible investments underuncertainty will be delayed compared with the traditional net present valuerule. The empirical industrial organization literature also suggests that thereis often first mover’s advantage in an entry game. For example, the investormay enjoy a period of monopoly benefit by introducing a new product firstinto the market. However the first investor may also want to delay hisinvestment due to the market uncertainty. Therefore, there is a tradeoffbetween pre-emption and delay in a real options game. This paper studiessuch situations theoretically and provides a methodology to identify thegame primitives nonparametrically.Let us consider a game of timing in continuous time. Suppose thatthere is an investment opportunity available to all the investors, and theinvestment is irreversible. The payoffs before the investment is normalized tozero while the payoffs after the investment are driven by Geometric BrownianMotion. Each investor observes the value of the payoffs at each instant timeand knows his own investment cost as private information at the beginningof the game, and then chooses his optimal time to invest. The optimaldecision rule to this game involves a threshold: each player will invest assoon as the Geometric Brownian Motion hits the threshold. In other words,the optimal investment timing is a duration determined by the first timea Geometric Brownian Motion hits a threshold. Hitting time models havebeen used in the statistic literature (Lee and Whitmore (2006)) to investigatedurations determined by optimal threshold decision rules. To study the1Chapter 1. Identification of Mixed Hitting Time in a Preemption Gameinteractive investment decisions under uncertainty, we will further explorethe interactive durations in a preemption game.This paper studies the identification of a preemption game where thetiming decisions are expressed as mixed hitting time (MHT). The MHTmodels (Abbring (2012)) are mixture duration models where durations arespecified as the first time a latent stochastic process crosses a heterogeneousthreshold. Our model builds on both the literature of MHT models and theliterature investigating preemption games with incomplete information forirreversible investments with real options values. In this context, a sym-metric Bayesian-Nash equilibrium is derived and the solution characterizesthe optimal timing of investment based on a threshold rule that depends onboth the observed covariates and the unobserved heterogeneity. The timingdecision rules specify durations before the investment as the first time theGeometric Brownian Motion hitting a heterogeneous threshold, which fitsthe MHT framework. Therefore, the identification strategies for the MHTmodel (Abbring (2012)) can be used for the identification of the last player’spayoff function in the first stage. In the second stage analysis, we utilizethe unique symmetric Bayesian-Nash equilibrium solution to this game andachieve point identification of the payoff function for the next to the lastplayer. Repeating the procedure and we will achieve point identification ofall the model primitives. We also present some parametric estimation resultsfrom a Monte Carlo simulation. The results are related to the literature onthe threshold regression for first-hitting time models, in which the thresh-old is dependent on covariates. Because the duration before investment inour game follows a mixed inverse gaussian distribution, given the observedcovariates and the distribution of the unobserved covariates, the estimationanalysis is also connected to the research on the generalized linear mixedregression models.The paper contributes to the joint framework of real options and preemp-tion games. Unlike most of the real options game in the literature, investorsin our model have incomplete information about their competitors’ invest-ment costs and we are considering an oligopoly framework in a preemptiongame. In the real options game literature, a duopoly game has been studiedintensively under complete information (see, for instance, Pawlina and Kort(2006); Kong and Kwok (2007)). Recently, a duopoly game under incom-plete information has also been presented (Lambrecht and William (2003);Janssens and Kort (2012)). Anderson, Friedman and Oprea (2010) inves-tigate an N players’ preemption game under incomplete information aboutinvestors’ costs to study a stochastic investment opportunity and provide alaboratory experiment to test some hypotheses implied by the theory. Our2Chapter 1. Identification of Mixed Hitting Time in a Preemption Gamestructural model extends theirs further by considering a more general payoffstructure. In our model, all the players make some positive profits from theinvestment, while the winner taking all of the returns is assumed in theirgame. Therefore, it is a one-shot move in Anderson, Friedman and Oprea(2010)’s setup, while our model delivers a sequential move of the players.The paper also adds to the research on the estimation of timing games.Honore´ and de Paula (2010) formulate a game theoretic model with com-plete information where the durations are endogenously determined. In theirmodel, the timing decision by the pioneer will have a positive effect on theother player’s utility and both the sequential timing equilibrium and simul-taneous timing equilibria could occur under different parameter realizationsof the game. Our model shows the existence of a unique symmetric BayesianNash equilibrium under incomplete information and allows for point iden-tification and a full solution approach (nested fixed point approach) in theestimation. In another example of timing game, Takahashi (2013) exam-ines the strategic exit decisions in a declining demand period in the U.S.movie theater industry. In his setting, competition resembles a war of attri-tion and generates a unique sequential exit under incomplete information.In our model, a sequential entry occurs due to the effect of the first-moveradvantage in the preemption game and also players’ beliefs about their com-petitors’ costs which is revealed over time. Schmidt-Dengler (2006) investi-gates the timing of technology adoption with a preemption motive. In hismodel, it is a discrete time dynamic game in which firms decide every periodwhether to adopt the new technology or not and subgame perfect equilibriaare considered. Our model is a continuous time model with a stochastic pay-off structure and we investigate the effect of both the preemption incentiveand the incomplete information on the investment timing under uncertainty.Our model provides a game theoretic MHT model. Abbring and Yu(2015) provide another approach and study a two-player optimal stoppinggame with complete information. Abbring and Yu (2015) use their setupto investigate the peer effect of quit smoking in a couple. Our model studyan N -player optimal stopping game with incomplete information and canbe used to study an entry competition between oligopolies in an industry.However the estimation method of the game theoretic MHT model we pro-pose relates to threshold regression models. Lee and Whitmore (2006) hasreviewed research on first-hitting-time models with regression structures,referred to as threshold regression models for survival data. The thresh-old regression models make the threshold and sometimes the parameters ofthe latent process dependent on the covariates through link functions. Thereduced-form model from our game structure resembles the threshold regres-3Chapter 1. Identification of Mixed Hitting Time in a Preemption Gamesion models and differs from theirs by incorporating agents’ heterogeneityinto the regressors. Our investment timing model arises naturally from thereal options literature, which has been applied in real estate markets, R&Dinvestment, and many other irreversible investment decisions(Clapp, Bardosand Wong (2012), Bulan, Mayer and Somerville (2009),Weeds (2002) andthe others).There are two general strands of literature that are related to this paper’smodel setup. The literature on irreversible investments under uncertaintystates that there is an option value of waiting and it delays investmentcompared to the traditional net present value (NPV) rule. The NPV rule iscalculated as the deterministic discounted cash flow of an investment projectand is considered generally incorrect (Dixit and Pindyck (1994)), because itfails to accommodate the ability to delay the irreversible investment whenfuture return is uncertain. The real options approach values the opportunityto invest as a real option when determining the optimal investment timing.And real options theory has been applied both theoretically and empiricallyin many areas to characterize this negative relationship between investmentand uncertainty. Since the option value of waiting to invest in the real worldis often strategically interdependent, recent research has incorporated thegame theoretic setting into the real options literature and argued that com-petition erodes the real options value and the fear to be preempted reducesthe investment delay(e.g.,Grenadier (1996); Mason and Weeds (2010) etc.).Azevedo and Paxson (2010) review the recent literature on real optionsgame and its applications. Our model studies the real options game withincomplete information and therefore we can further investigate the effectof asymmetric information along with the effect of preemptive advantage onthe investment timing.Our model is a multiple duration model in continuous time. For a givenstochastic process, we have multiple observations of the investment timingdurations determined by different thresholds. This is different from thecompeting risk models in survival analysis. Competing risk models studythe multiple causes of failure. For an observed duration, it is induced bymultiple latent processes competing with each other. While the durationin our model is determined by players’ interactive decisions following thesame latent process. However, we develop most of our results similar to theidentification ideas in the duration analysis and obtain point identification.The remainder of the chapter is organized as follows. Section 1 presentsthe economic model. Section 2 shows the main identification results. Section3 first illustrates the estimation strategies and then conducts small MonteCarlo simulations and examines the cost of incomplete information on the41.1. The Economic Modelinvestment timing. Section 4 concludes and shows possible applications ofour structural model. All the mathematical derivations and main proofsdetails are in the appendix.1.1 The Economic ModelIn this section, we analyze a real options game of oligopoly competitionwith incomplete information. We consider each firm has private informationconcerning its own investment cost, and has the opportunity to invest in thesame market. The investment opportunity can be a development project foreach firm to gain entry into a real estate market. We assume the investmentopportunity is perpetual and the investment decision is irreversible. Hencethe firms choose their optimal investment timing once with the sunk costs.There are no variable costs after investment. Further, we assume that thefirms are value maximizing and risk-neutral, that is, the time discount rateis set to be the risk-free rate r.1.1.1 Setting and NotationSuppose that there are N firms. Firm i observes its own investment costCi, i ∈ {1, ..., N}, at the beginning of the game but is uncertain about itsrivals’ costs. The game starts at t = 0 and the time is continuous. At eachinstant, each firm decides whether to make the investment or not. Beforethe investment, the firms have an option to wait for their optimal timing ofinvestment and receive no profits. Once it chooses to invest at time t, firmi pays an amount of Ci, i ∈ {1, ..., N}, and obtains an instantaneous profitflow ofpi(yt, Dj(x)) = ytDj(x), j ∈ {1, ..., N},where j stands for the total number of firms that have entered the marketand the state variable yt follows a Geometric Brownian Motion with driftparameter µ and standard deviation parameter σ:dyt = µytdt+ σytdWt,where yt can be interpreted as a demand shock and parameter µ can beinterpreted as the industry growth rate and parameter σ as the industryvolatility. We assume the stochastic process starts from initial state y0 lowenough such that the immediate investment is not optimal for any firm.The deterministic multiplier Dj(x), j ∈ {1, ..., N}, reflects the effect ofthe competition on profits and the state variables x are covariates to describe51.1. The Economic Modelmarket conditions and also used for the identification of function Dj(·). Herewe want to impose some assumptions on Dj(x) to characterize the negativemarket externality, that is, when j increases, the market has to be sharedby more firms, which means each active firm in the market has lower profits.The following assumptions are made:Assumption 1. If n firms invest at the same instant t, then only one firm ischosen randomly with probability 1n , n ∈ {2, ..., N}, and invests successfully.Assumption 2. D1(x) > D2(x) > ... > DN (x).Assumption 3. The values of Ci, i ∈ {1, ..., N}, are drawn independentlyfrom the distribution function G, with support on [CL, CU ].Assumption 1 implies if more than one firm choose to enter at the sameinstant t, a randomization device is introduced to avoid the problem ofthe non-existence of a pure strategy equilibrium1. The game proceeds asfollows: conditional on how many firms have already invested, each of theremaining firms decides when to invest. Once one of the firms invests, theother remaining firms revise their planned entry time based on the currentinformation, which reflects how many firms are active in the market. Hence afirm’s investment timing strategy depends on both the number of the activefirms in the market, and his investment cost rank among the remainingfirms. The equilibrium policy function suggests that all the firms will investsequentially according to their investment costs order.Assumption 2 imposes negative externalities in the model, which meansmonopoly profits are highest and oligopoly profits are decreasing in thenumber of the firms in the market.Assumption 3 implies that all the firms have the identical priors abouteach other’s investment costs. Note in this incomplete information game,each firm has the same information about the state variable, which repre-sents the public market information. While each firm knows its own invest-ment cost, which represents the private firm specific information, it onlyknows about its rivals’ costs up to a prior G.1.1.2 SolutionWe will look for a symmetric Bayesian Nash equilibrium (BNE) of thisgame. At each instant, suppose the state variables of each player i consist of1This randomization device rules out the possibility that simultaneous investmentshappen because of coordination failures. See the discussion about how it works in aN-player preemption game on page 374 in Argenziano and Schmidt-Dengler (2014).61.1. The Economic Modelits private investment cost Ci and the common state variables ωt = (l, t, ht),where l is the number of firms entered the market, t is the current time, andht is the history of the game up to time t, which involves the history of thestochastic state variable yt up to time t. Given this information, the firm idecides when to enter the market. The optimal entering time Ti is definedas:Ti = inf{t ≥ 0 : (ωt, Ci) ∈ S × [CL, CU ]},where S is the state space of all the common state variables ω. Ti is alsocalled the optimal stopping time such that firm i maximizes its expecteddiscounted value to enter the market, due to the stochastic nature of itsexpected discounted value. As a result, the pure timing strategy Ti weconsider here is solved as the best response to a stochastic control problem,which is the first time the stochastic process yt hits a threshold.Let Vi(τ, T−i, ω, Ci) be the present expected discounted value of firmi when i decides to enter at time τ , and the other firms’ strategies arerepresented by T−i, given the state variables are (ω,Ci). Let h(Ci|ht) be thefirm i’s belief about its competitors’ investment costs when firm i and allits competitors have survived until time t. The Bayesian Nash equilibriumis defined asDEFINITION: A set of strategies (Tˆi(ω,Ci))Ni=1 with posterior beliefsh(Ci|ht) is a perfect Bayesian Nash equilibrium if for all ω ∈ S and Ci ∈[CL, CU ],• if for all i and any strategy Ti,Vi(Tˆi, Tˆ−i, ω, Ci) ≥ Vi(Ti, T−i, ω, Ci),and• if for any i’s competitor j, h(Cj |ht) is given by Bayes’ rule.Since the optimal decision rule to this game involves a threshold: eachplayer will invest as soon as the Geometric Brownian Motion hits the thresh-old, we will use the timing strategy and the corresponding threshold inter-changeably from now on. The optimal threshold function is a function ofits own investment cost, above which a firm immediately invests. Based onthe investment timing order of the N firms’ decisions, we will classify theentry order of the N firms as the first, ..., the kth, ..., the Nth active playerin the market. We will also distinguish the identity of the N firms in thesubscript accordingly as firm i or its rival j in the context without confusion,∀ i, j ∈ {1, 2, ..., N}.71.1. The Economic ModelNow suppose that firm i enters as the kth active firm in the market,followed by subsequent N − k firms who seize the investment opportunityat time t∗j , with expected investment threshold y∗j accordingly, j ∈ {k +1, ..., N}, k ∈ {1, ..., N − 1}. Appendix A.1 shows the derivation of theexpected value functions in details. When firm i chooses to invest at thethreshold yi, the expected value function of the kth active firm at time t isE[V ki (yt | yi, y∗k+1, ..., y∗N )]=(yiDk(x)r − µ − Ci)(ytyi)β+N∑j=k+1(y∗j (Dj(x)−Dj−1(x))r − µ)(yty∗j)β,(1.1)where β is the positive solution to the fundamental quadratic, i.e. 12σ2β2 +(µ − 12σ2)β − r = 0. Note that the expected value function of the kth ac-tive firm, E[V ki ], consists of the net present value from its own investment,yiDk(x)r−µ − Ci, discounting back from the time when yt hits yi, plus the netpresent value of a group of dividends caused by its rivals’ eventual invest-ments,y∗j (Dj(x)−Dj−1(x))r−µ , discounting back from the time when yt hits y∗jsequentially, j ∈ {k + 1, ..., N}.Clearly, the threshold functions depend on firms’ entry order in this gameand each firm has to consider all the possibilities of its entry order in hisstrategy. Since the Nth entrant in the market has complete information,his investment trigger is given by fNi =ββ−1(r−µ)CiDN (x)with investment costCi. The derivation of fNi is in Appendix A.1. Hence the strategy of firm iis denoted as ({bk(Ci)}N−1k=1 , fNi ), where the investment threshold functionsbk(Ci) map his own investment cost to the real line, given his entry orderk, k ∈ {1, ..., N − 1}. Note that we make the dependence of the thresholdfunctions {bk(Ci)}N−1k=1 on the other state variables implicit. Now let usspecify the state variables available for each firm thoroughly. At the time t,a firm i will know the number of active firms in the market is l, the currentstate value is yt, the highest possible cost for the rivals to invest is Cˆ, whichwill be specified later and his own investment cost is Ci. Now after the linvestments, firm i has to compete against N − l− 1 firms, and firm i couldinvest as the kth active firm in the market with investment threshold bk(Ci),k ∈ {l+ 1, ..., N − 1}, or the Nth active firm with investment threshold fNi .If firm i wants to invest as the kth firm, then he must invest before all hisrivals and this implies all his rivals’ investment costs are higher than firmi’s. Suppose that the probability for firm i to invest as the kth active firm is81.1. The Economic Model1−Hk(Ci), which will be specified later. Then firm i’s total expected valuefunction after l investments isV (yi, y∗−i, N − l, yt, Cˆ, Ci)=N−1∑k=l+1(1−Hk(Ci))E[V ki (yt | bk(Ci), y∗k+1, ..., y∗N )]+(1−N−1∑k=l+1(1−Hk(Ci)))V Ni (yt | fNi ),where yi = {bk(Ci)|k = l + 1, ..., N − 1} and y∗−i = {y∗k+1, ..., y∗N |k = l +1, ..., N−1}. The expected value function as the kth firm is given in equation(1.1). Now we have to specify 1−Hk(Ci), the probability for firm i to investas the kth active firm, k ∈ {l + 1, ..., N − 1}.Firstly, let us see what can be inferred from the information set at timet. Since each firm has observed a history of state values of yt until the timet and obtained the information that none of the remaining N − l firms haveinvested yet. Then we can infer the probability of the kth entry firm aftertime t, ∀ k ∈ {l + 1, ..., N − 1}. Let yˆ be the highest value of state variableobserved until the time t, yˆ = max0≤τ≤t{yτ} and Cˆ = (bk)−1(yˆ) be thehighest cost draw that would lead a rival of the kth entry firm to invest atyˆ. Suppose that firm i is the kth entry firm, ∀ k ∈ {l + 1, ..., N − 1}. Thenthe fact that none of firm i’s rivals have invested until yˆ is reached meanstheir thresholds are larger than yˆ, then firm i would learn that Cj > Cˆ,∀ j ∈ {k + 1, ..., N}.The probability that firm i’s rival j, has cost Cj higher than Ci, condi-tional on Cj > Cˆ is1−G(Ci)1−G(Cˆ) . When firm i enters as the kth firm, implyingfirm i has N − k rivals to consider, the probability that none of the N − krivals will preempt isPr(ti < tj , ∀ j 6= i) =(1−G(Ci)1−G(Cˆ))N−k≡ 1−Hk(Ci).Denote b(m) = {bk(m)|k = l + 1, ..., N − 1}, m ∈ [Cˆ, CU ]. Given invest-ment cost Ci, firm i’s problem is reduced to find a threshold function b(m)91.1. The Economic Modelthat solves:maxm∈[Cˆ,CU ]V (b(m), y−i, N − l, yt, Cˆ, Ci)= maxm∈[Cˆ,CU ](N−1∑k=l+1(1−Hk(m))E[V ki (yt | bk(m), y∗k+1, ..., y∗N )]+(1−N−1∑k=l+1(1−Hk(m)))V Ni (yt | fNi )),(1.2)where y−i = {y∗k+1, ..., y∗N |k = l + 1, ..., N − 1}.In equilibrium, the investors choose their true type to obtain the BNEstrategy function, such that investor i maximizes equation (1.2) when m =Ci. The first order condition (FOC) for bk(Ci) satisfies the following firstorder differential equation,hk(Ci)1−Hk(Ci) =E[V k′i (yt | bk(Ci), y∗k+1, ..., y∗N )]E[V ki (yt | bk(Ci), y∗k+1, ..., y∗N )]− V Ni (yt | fNi ), (1.3)where hk(Ci) ≡ Hk′(Ci). The intuition behind the above FOC reflects thatfirm i’s expected marginal benefits of winning as the kth entrant is equal toits expected marginal costs of waiting to enter the market later. That is, firmi’s expected payoff gain when it wins against its rival to enter the kth marketfirst is E[V ki (yt | bk(Ci), y∗k+1, ..., y∗N )]−V Ni (yt | fNi ), with probability hk(Ci)that firm i wins at trigger bk(Ci). When it loses the competition, firm i’sexpected payoff loss is E[V k′i (yt | bk(Ci), y∗k+1, ..., y∗N )], with probability 1−Hk(Ci) that one of i’s competitors invests first, conditional on not investinguntil bk(Ci) is reached.In the symmetric BNE, all the firms use the same threshold functions{bk(·)|k ∈ {1, ..., N − 1}}. Therefore, the expected investment trigger y∗j ,j ∈ {k + 1, ..., N} in the total expected value function (1.2) is given byy∗j = E[bj(C) | C > Ci, Ci]=∫ CUCibj(C)g(C)dC1−G(Ci) ≡ Bj(Ci), j ∈ {k + 1, ..., N − 1}(1.4)y∗N = E[fNC | C > Ci, Ci]=ββ − 1(r − µ)DN (x)∫ CUCiCg(C)dC1−G(Ci) ≡ BN (Ci),(1.5)101.1. The Economic ModelwherefNC =ββ − 1(r − µ)CDN (x),and fNC is the investment trigger of the Nth active firm. From firm i’sperspective, when it invests as the kth active firm, the expected trigger y∗j ,j ∈ {k+1, ..., N} is the same across all of i’s rivals, which depends on firm i’sinformation set {∀j ∈ {k + 1, ..., N} : Cj > Ci} and firm i’s prior for all itscompetitors. Now, we substitute y∗j = Bj(Ci) into (1.3), ∀j ∈ {k+ 1, ..., N},and let yt = bk(Ci). Then we will get(N − k)g(Ci)1−G(Ci) =I1I2, (1.6)whereI1 = (1− β)bk′(Ci)Dk(x)r − µ +N∑j=k+1(B′j(Ci)(bk(Ci)Bj(Ci))β ((Dj(x)−Dj−1(x))r − µ))+ βbk′(Ci)bk(Ci)CiandI2 =(bk(Ci)Dk(x)r − µ − Ci)+N∑j=k+1(Bj(Ci)(Dj(x)−Dj−1(x))r − µ)(bk(Ci)Bj(Ci))β−(fNCiDN (x)r − µ − Ci)(bk(Ci)fNCi)β ,with boundary condition,bk(CU )DN (x)r − µ = CU +(fNCUDN (x)r − µ − CU)(bk(CU )fNCU)β, (1.7)whereg(C) ≡ G′(C),fNy =ββ − 1(r − µ)yDN (x), y ∈ {Ci, CU}.To compute the equilibrium of the model, we note that the game pro-111.1. The Economic Modelceeds sequentially as if an (N − 1)-players game is nested in an N -playergame. Therefore, we solve the game sequentially starting from the Nthplayer. Since the Nth active firm enters the market with complete informa-tion, we first solve the (N − 1)th active firm’s investment trigger bN−1(·).The derivation of bN−1(·) is illustrated in Appendix A.3. Repeating theprocedure, the bk(·) can be expressed in the following recursive formula,bk(Ci)Dk(x)r − µ= Ci −Rk(Ci)+∫ CUCi((bk(Ci)bk(y))β−(bk(Ci)fNy)β)(1−G(y)1−G(Ci))N−kdy,(1.8)whereRk(Ci) =N∑j=k+1(Bj(Ci)(Dj(x)−Dj−1(x))r − µ)(bk(Ci)Bj(Ci))β−(bk(Ci)fNCi)β (fNCiDN (x)r − µ − Ci),andfNy =ββ − 1(r − µ)yDN (x), fNCi =ββ − 1(r − µ)CiDN (x),given the expected investment triggers of firm i’s rivals {Bj(·)}Nj=k+1 definedin equation (1.4 − 1.5).As proved in Theorem 1, the above boundary value problem has aunique solution bk(·), given the expected triggers {Bj(·)}Nj=k+1. And theinvestment threshold functions {bk(·)|k ∈ {1, ..., N − 1}} characterize oursymmetric BNE.Theorem 1. Under Assumptions 1−3, we have the following conclusions:1. A function b∗k(·) satisfies the recursive formula (1.8) if and only if itsolves the boundary problem (1.6 − 1.7), k ∈ {1, ..., N − 1}.2. Given β > 0, boundary value problem (1.6 − 1.7) has a unique solu-tion: b∗k(·) : [CL, CU ] −→ R, k ∈ {1, ..., N − 1}.3. The solution to the boundary problem (1.6 − 1.7), k ∈ {1, ..., N − 1}constitutes a symmetric Bayesian Nash Equilibrium in which each121.1. The Economic Modelplayer invests according to the thresholds ({b∗k(Ci)}N−1k=1 , fNCi), with itsown investment cost Ci. Furthermore, {b∗k(Ci)}N−1k=1 are solved se-quentially, starting from k = N − 1 by the recursive formula (1.8).4. The symmetric equilibrium is unique, if it exists.Proof. See Appendix A.4 for details.To obtain b∗k(·) numerically, we iterate (1.8) from an initial approxima-tion using the upper boundary value (1.7), substitute it for b∗k(·) on theright hand side of equation (1.8) to obtain a better approximation, and it-erate. The symmetric BNE threshold b∗k(·) is a fixed point of (1.8). In fact,the recursive formula (1.8) is a nonhomogeneous nonlinear Volterra integralequation. M. Eshaghi Gordji and Baghani (2011) provides the existence anduniqueness conditions for the solution to this type of integral equation andtheir conditions are applied in the proof of Theorem 1. Note the existenceof a symmetric Bayesian Nash equilibrium is not proved in the general case.The argument is similarly to the discussions as in Proposition 5 of Taka-hashi (2013). However, if a symmetric equilibrium of the game exists, thenit is the unique symmetric equilibrium as discussed in the Theorem 1 of ourpaper. For a duopoly case, the existence and uniqueness of the symmetricBNE is proved in Janssens and Kort (2012).Anderson, Friedman and Oprea (2010) investigate an N players’ pre-emption game under incomplete information with a payoff structure thatonly the first firm can make a profit. Our structural model considers amore general payoff structure. In our model, all the followers make somepositive profits from the investment, while the winner takes all of the re-turns in their game. As a result, the symmetric BNE threshold functionin Anderson, Friedman and Oprea (2010)’s model is consistent with theboundary problem (1.6 − 1.7) for the special case that the payoff of allthe followers’ after the first entrant is equal to zero. Therefore, Anderson,Friedman and Oprea (2010) use a different method to prove the existenceand uniqueness of the symmetric BNE threshold. The boundary problem inAnderson, Friedman and Oprea (2010) is equivalent to an ODE of b∗k(·) andthe Picard-Lindelof theorem is applied accordingly to prove the symmetricBNE exists and is unique. In our model, the followers capture a positiveprofit, and affect the first entrant’s expected profit through the expected in-vestment triggers {Bj(·)}Nj=k+1, which is reflected in the boundary problem(1.6 − 1.7). Naturally, we obtain an equivalent recursive integral equationsystem of (b∗k(·), {Bj(·)}Nj=k+1) and adopt an alternative way to prove theexistence and uniqueness of the symmetric BNE threshold.131.2. Identification1.2 IdentificationFor the identification and estimation of the preemption game, we observea random sample of M markets. In each market m, the game is playedamong N players and we will classify the entry order of the N firms asthe first,...,the kth,..., the Nth active player in the market. We will alsodistinguish the identity of the N firms in the subscript accordingly as firm ior its rival j, ∀ i, j ∈ {1, 2, ..., N}. The identification proceeds in sequentialstages. In the first stage, we determine both the parameters of the payoffstructure for the Nth player and the distribution function of the unobservedinvestment costs. Then we apply the first-stage identification results to thesecond-stage identification analysis of the remaining parameters, which arefrom the (N − 1)th player’s payoff structure. As shown in Theorem 3,the identification strategy for the (N − 1)th player’s payoff structure can beapplied to any players j’s payoff structure, j ∈ {1, ..., N − 2}. Therefore, wewill focus on the identification of a duopoly game in the following analysis.Suppose that we observe a random sample {(Tm, xm)}Mm=1, where sub-script m ∈ {1, ...,M} indicates different markets. Let Tm = {T (N−1)m , T (N)m }be the investment timing of the players according to the entry order in themarket m and xm = {x(N−1)m , x(N)m } are the market-specific covariates ac-cording to the entry order in the market m.2 The superscript (N − 1) and(N) indicate the entry order of the players in each game. For example,(N − 1) indicates the first entrant in the duopoly game and also named asthe leader in the game and (N) is called the follower. For the simplicity ofthe notation, we omit the subscript m in the following analysis without con-fusion and use subscript i to indicate the player i in the market m. Whenfirm i is the Nth entrant in the market m, its investment timing is T(N)iwith investment threshold fNi . Define T(N)i asT(N)i = inf{t ≥ 0 : yt ≥ fNi } = inf{t ≥ 0 : ln(yt) ≥ ln(fNi )}.When firm i is the (N − 1)th entrant, its investment timing is T (N−1)i withinvestment trigger bN−1(Ci),T(N−1)i = inf{t ≥ 0 : ln(yt) ≥ ln(bN−1(Ci))}.2The function Dj(x) in the profit flow, j ∈ {N − 1, N} can still be identified evenwhen x(N−1) = x(N) = x. For the identification of Dj(·) we only need some variation inthe covariates of x.141.2. IdentificationSince yt is a Geometric Brownian Motion with instantaneous drift µ andinstantaneous standard deviation σ, ln(yt) is a standard Brownian Motionwith instantaneous drift µˆ = µ− 12σ2 and instantaneous standard deviationσˆ = σ. Therefore, T(N)i is the first time that the standard Brownian Motionln(yt) crosses a threshold ln(fNi ). Abbring (2012) shows that although thedistribution of T(N)i cannot be explicitly expressed, the Laplace transformof the distribution of T(N)i satisfiesLT(N)i(s) ≡ E[exp(−sT (N)i 1(T(N)i <∞))]= E[exp(−Λ(s) ln (fNi ))] , (1.9)whereΛ(s) =√µˆ2 + 2σˆ2s− µˆσˆ2,and Λ(s) is the inverse function of the Laplace exponent of the standardBrownian Motion given by the Le´vy-Khintchine formula3.Suppose that we observe market covariates x(N) when the (N)th playerinvests in the market and market covariates x(N−1) when the (N − 1)thinvestment happens in the market. Note the market covariates only dependson the order of the investment and not depend on the individual firm i.Therefore, we can write the investment trigger of fNi asfNi =ββ − 1(r − µ)C(N)iDN (x(N)), (1.10)where C(N)i = maxi∈{1,...,N}{C1, ..., CN}.Assumption 4. Dj(x) : χ −→ (0,∞) is such that Dj(x1) 6= Dj(x2) forsome x1, x2 ∈ χ, ∀j ∈ {1, ..., N}.Assumption 5. The time discount factor r is known to the econometrician.Assumption 6. Investment costs Ci and Cj are drawn independently fromthe same distribution function G(·). G(·) is a continuous function withfull support on R2+. And the marginal density function g(·) satisfies that0 < g(·) <∞ on R+4.3See Abbring (2012) for an illustration of the Le´vy-Khintchine formula in the generalcase of a Le´vy process in Section 4.14Note the unobservables are the sunk costs of each player, so it is in the economicsense that Ci > 0, i ∈ {1, 2, ..., N}. In Abbring and Yu (2015)’s paper, they consider aform of unobservables of the form Ci = e−vi and for vi point mass at zero is allowed.151.2. IdentificationTheorem 2. Identification of DN (·). Under Assumptions 1−6, thefunction DN (·) and parameters µ and σ are identified up to scale, and thedistribution of investment cost G(·) is identified up to scale.Proof. Define φN (x) = ln(ββ−1(r−µ)DN (x)). Then from equation (1.10), thelogarithm of the trigger value for T(N)i is ln fNi = φN (x(N)) + ln(C(N)i ). Byequation (1.9), we haveLT(N)i(s | x(N)) = E(exp[−Λ(s)(φN (x(N)) + ln(C(N)i ))] | x(N))= E(exp[−Λ(s) ln(C(N)i )]) exp[−Λ(s)φN (x(N))]= LlnC(N)i(Λ(s)) exp[−Λ(s)φN (x(N))],where LlnC(N)iis the Laplace transform of the distribution of lnC(N)i , andwe transform the Geometric Brownian Motion yt into a standard BrownianMotion ln(yt), so thatΛ(s) =√(µ− 12σ2)2 + 2σ2s− (µ− 12σ2)σ2.Given two sets of covariates x(N)1 and x(N)2 , the conditional distributionof T(N)i given x(N)1 and that of T(N)i given x(N)2 are characterized byLx(N)1(s) ≡ LT(N)i(s | x(N)1 ) = LlnC(N)i (Λ(s)) exp[−Λ(s)φN (x(N)1 )],andLx(N)2(s) ≡ LT(N)i(s | x(N)2 ) = LlnC(N)i (Λ(s)) exp[−Λ(s)φN (x(N)2 )].NoteLx(N)1(s)Lx(N)2(s)= exp[−Λ(s)(φN (x(N)1 )− φN (x(N)2 ))]= exp[−Λ(s) ln(DN (x(N)2 )DN (x(N)1 ))].161.2. IdentificationWhen Lx(N)1(s)/Lx(N)2(s) 6= 0, we can write(Lx(N)1(s)/Lx(N)2(s))′(Lx(N)1(s)/Lx(N)2(s)) = −Λ′(s) ln(DN (x(N)2 )DN (x(N)1 )). (1.11)Sincelims→0Λ′(s) =(µ− 12σ2)−1,solims→0(Lx(N)1(s)/Lx(N)2(s))′(Lx(N)1(s)/Lx(N)2(s)) = −(µ− 12σ2)−1ln(DN (x(N)2 )DN (x(N)1 )). (1.12)Note the LHS of equation (1.12) is not degenerate. To see this, we rewriteit using the definition of the Laplace transform,lims→0(Lx(N)1(s)/Lx(N)2(s))′(Lx(N)1(s)/Lx(N)2(s))= lims→0(E(exp(−sT (N)) | xN1 )/E(exp(−sT (N)) | xN2 ))′(E(exp(−sT (N)) | xN1 )/E(exp(−sT (N)) | xN2 ))= E(−T (N) | xN1 )− E(−T (N) | xN2 ).Next we substitute equation (1.12) intoLx(N)1(s)Lx(N)2(s)= exp[−Λ(s) ln DN (x(N)2 )DN (x(N)1 )],thenLx(N)1(s)Lx(N)2(s)= exp[Λ(s)(µ− 12σ2)lims→0(Lx(N)1(s)/Lx(N)2(s))′(Lx(N)1(s)/Lx(N)2(s)) ], (1.13)so µ and σ in Λ(s) are identified up to scale. Once Λ(s) is known, so isDN (·) identified up to scale from equation (1.11).Given µ and σ are identified up to scale and the time discount factor r171.2. Identificationis known, β is identified up to scale as well. So φN (x) is identified up toscale. Lastly,Lx(N)1(s) = LlnC(N)i(Λ(s)) exp[−Λ(s)φN (x(N)1 )],therefore, LlnC(N)i(s), is identified and so are the distributions of lnC(N)iand C(N)i . Suppose gC(N)i(x) is the density function of C(N)i . Since C(N)i =maxi∈{1,...,N}{C1, ..., CN} and C1, ..., CN are i.i.d., then we can identify thedistribution of Ci fromgC(N)i(x) = Ng(x)G(x)N−1.Because the investment costs are drawn from the same distribution G(·), wecan use the distribution of lnCi, i ∈ {1, 2}, to identify G(·).Theorem 3. Identification of DN−1(·). Under Assumptions 1−6, thefunction DN−1(·) is identified up to scale.Proof. Suppose that firm i enters the market as the (N − 1)th player. ThenT(N−1)i = inf{t ≥ 0 : ln(yt) ≥ ln(bN−1(Ci))}.The Laplace transform of the distribution of T(N−1)i , conditional on CisatisfiesLT(N−1)i |Ci(s) ≡ E[exp(−sT (N−1)i 1(T (N−1)i <∞)]= exp[−Λ(s) ln(bN−1(Ci))].The unconditional Laplace transform of the distribution of T(N−1)i is,LT(N−1)i(s) ≡ E[exp[−Λ(s) ln(bN−1(Ci))]]= Lln(bN−1(Ci))(Λ(s)).By Theorem 2, Λ is identified, so the density of ln(bN−1(Ci)) is known.Denote the conditional quantile function of ln(bN−1(Ci)) byQln(bN−1(Ci))|x(τ),given x at τ ∈ (0, 1). Then the policy function bN−1(·) is identified bybN−1(G−1Ci (τ) | x) = Qln(bN−1(Ci))|x(τ),181.3. Parametric Estimationwhere G−1Ci (·) is the inverse of the conditional distribution function GCi(·),which is identified by Theorem 2.Once the policy function bN−1(· | x) is known, we plug the policy functioninto the FOCs (1.6 − 1.7) and the only unknown part of the payoff functionDN−1 is solved.Finally, the identification for the oligopoly game can be proved just as theduopoly game. We first identify model primitives from the last player andthen identify the market externality multipliers Dj , j = N − 1, ..., 1 from(N − 1)th player to the first, each following the method from Theorem3.1.3 Parametric EstimationIn this section, we consider the parametric estimation of a duopoly game.Then we can extend the procedure to an N−players’ game. For the duopolygame, we first estimate the parameters in the threshold function of the lastentrant in the market, and then estimate the remaining parameters of thethreshold function of the first entrant in the same market. Since the firsthitting time in the case of a standard Brownian Motion process follows theinverse Gaussian distribution, we will parameterize the threshold functionsto apply the maximum likelihood estimation. The generalized moment con-dition estimation method may also be applicable but will be difficult toimplement due to the complicated moment conditions in our model. Themoment conditions of the durations depend on the threshold policy func-tions, which are solutions to a system of very complicated differential equa-tions. In the two-player complete information case studied by Abbring andYu (2015), they suggest that a likelihood-based approach is more promisingthan GMM and worth investigating in the future.1.3.1 Estimation StrategiesThe estimation strategies follow the arguments of the threshold regres-sion in the case of a standard Brownian Motion associated with its inverseGaussian first hitting time as in Lee and Whitmore (2006). Consider astandard Brownian Motion process yt with instantaneous drift µ and in-stantaneous standard deviation σ. The time T when the process hits thethreshold a0 for the first time has an inverse Gaussian distribution. The191.3. Parametric Estimationdensity function for the first hitting time is given by,f(t | µ, σ, a0) = a0σ√2pit3exp(−(a0 − µt)22σ2t).The latent process yt can be given an arbitrary measurement unit. There-fore one parameter need to be normalized. We set the variance parameterσ to unity. The threshold a0 is linked to regression covariates that are rep-resented by vector x = (1, x1, ..., xk). In the threshold regression analysis, alogarithmic functionln(a0) = βx = β0 + β1x1 + ...+ βkxk,is used to link a0 to the covariates. A sample log-likelihood function will beconstructed accordingly to define the maximum likelihood estimators for βand µ.In our model, the latent process yt is a Geometric Brownian Motionprocess with drift parameter µ˜ and standard deviation σ˜. To transform theprocess into a standard Brownian Motion, we obtain ln(yt) with instanta-neous drift µ = µ˜− 12 σ˜2 and instantaneous standard deviation σ = σ˜.Suppose that firm i is the leader and firm j is the follower. For theleader i, we observe the optimal investment timing T(1)i , threshold b1(Ci),and market covariates x(1). For the follower j, we observe the optimalinvestment timing T(2)j , threshold fj , and market covariates x(2). DefineT(1)i and T(2)j asT(1)i = inf{t ≥ 0 : yt ≥ b1(Ci)} = inf{t ≥ 0 : ln(yt) ≥ ln(b1(Ci))}, (1.14)andT(2)j = inf{t ≥ 0 : yt ≥ fj} = inf{t ≥ 0 : ln(yt) ≥ ln(fj)}, (1.15)whereln fj = ln(ββ − 1(r − µ)D2(x(2)))+ ln(Cj),and b1(Ci | β, µ, σ,D1(x(1)), D2(x(2))) is defined as the solution to the in-tegral equation (1.8). Note that the cost variables Ci and Cj are unob-servable in the threshold functions to the econometrician and are assumedto be independently drawn from the distribution function G(·). Followingthe threshold regression, we take D2(x) = exp (β20 + β21x), and D1(x) =201.3. Parametric Estimationexp (β10 + β11x).Let {(Tm, Xm)}Mm=1 be a complete random sample induced from ourmodel at the true parameter values, where Tm = {T (1)i,m, T (2)j,m} and Xm ={X(1)m , X(2)m }. The parameters (µ, σ, β10 , β11 , β20 , β21) can be uniquely estimatedfrom the conditional distribution of T1 given X and T(2) given X underproper normalizations (Abbring and Salimans (2012)).We first estimate structural parameters of the follower j’s threshold.Rewrite the threshold function ln fj for the followerln fj = ln(ββ − 1(r − µ)D2(x(2)))+ ln(Cj)= ln(β(r − µ)β − 1)+ β20 − β21x(2) + ln(Cj)= φ(µ)− β21x(2) + ln(Cj),where β20 is normalized to 0, and φ(µ) is a function of µ, given r and σ. Sincethe conditional distribution of T (2) given X(2) and Cj follows an inverseGaussian distribution with density functionf(t2 | x(2), Cj) = φ(µ)− β21x(2) + ln(Cj)√2pit32× exp(− (φ(µ)− β21x(2) + ln(Cj)− (µ− 12 )t2)22t2).(1.16)The likelihood is computed asl2(µ, β) =M∑m=1ln∫f(t2,m | x(2)m , Cj,m)dG(Cj),where t2,m is a realization of T(2).After we obtain the estimators for µˆ and βˆ21 , we will substitute them intothe conditional density function of T (1), given X(1) and Ci and estimate theremaining parameters β10 and β11 of the leader i’s threshold. The conditional211.3. Parametric Estimationdensity function for T (1), given X(1) and Ci is,f(t1 | x(1), x(2), Ci) = ln b(Ci, x(1), x(2), µˆ, βˆ21 , β10 , β11)√2pit31× exp(− (ln b(Ci, x(1), x(2), µˆ, βˆ21 , β10 , β11)− (µˆ− 12 )t1)22t1).(1.17)Following Abbring and Salimans (2012), we normalize β10 to 1.5 Here thepolicy function b(Ci, x(1), µˆ, βˆ21) is a fixed point for the integral equation (1.8)and the resulting MLE estimation will have a nested fixed point problemto solve, which increase the computational time significantly than the MLEestimation for parameters in the threshold of T (2).1.3.2 Monte Carlo SimulationThe data-generating process is defined as follows: we simulate a standardBrownian Motion yt with drift parameter µ = 1.5 and standard deviationσ = 1 and simulate the first hitting time according to (1.15) and (1.14).To simulate the thresholds in (1.15) and (1.14), we set the time discountr = 5 and taking D2(x) = exp (β20 + β21x) and D1(x) = exp (β10 + β11x).We set (β10 , β11 , β20 , β21) = (1, 0.3, 0, 0.6) and assume that the market specificcovariables x = [x(1), x(2)] are drawn independently from uniform distribu-tion on [−1, 1]2. We will assume Ci and Cj are independent and uniformlydistributed on [50, 80] in the parametrization following Anderson, Friedmanand Oprea (2010).To make a comparison, we also estimate a linear approximation to thepolicy function b(C1, x1, µˆ, βˆ) as the misspecification model. Supposeln b(Ci, x(1), µˆ, βˆ21) = φ(µˆ)− lnD1(x(1)) + ln(Ci), (1.18)5The normalization of the threshold is needed because of the existence of both thecovariates and the heterogeneity in the threshold. When the covariates are multipliedby a constant, the heterogeneity can also be multiplied by the reciprocal of the constantwithout changing the magnitude of the threshold. See the argument given in Abbring andSalimans (2012) on page 8− 9.221.4. ConclusionTable 1.1: Parametric EstimationTrue Model Linear ApproximationTrue value Bias Std. True value Bias Std.µ 1.5 0.0164 0.0183β21 0.6 0.2456 0.1935β11 0.3 0.03 0.1014 0.3 0.0707 0.2126Notes:σ is normalized to 1;β20 is normalized to 0 while β10 is normalizedto 1.The true estimation is based on 100 random samples of size 200observations each.The linear approximation estimation is based on 100random samples of size 200 observations each.thenf(t1 | x(1), x(2), Ci) = φ(µˆ)− β10 − β11x(1) + ln(C1)√2pit31× exp(−(φ(µˆ)− β10 − β11x(1) + ln(C1)− (µˆ− 12)t1)22t1).(1.19)All the estimation results reported in table 1.1 are based on 100 repli-cations of datasets of size 200. In table 1.1, the estimation of true model isbased on (1.16) and (1.17), while the estimation of linear approximation isbased on (1.18). Because the estimation proceeds in two stages according tothe identification, we first estimate µ and β21 from (1.16) in the true modeland then estimate β11 from (1.17). The difference between true model andlinear approximation is the estimation of the parameters in the second stage.The estimation of linear approximation uses the estimators of µˆ and βˆ21 fromthe first stage, and then applies a misspecified policy function to the density(1.19), which is implied by the leader’s strategy in a complete informationduopoly game. As expected, the estimation of β11 is more biased comparedto the result from the true model.1.4 ConclusionIn this paper, we study the interactive mixed duration models drivenby Brownian Motion. This paper relates to the continuous time durationanalysis and also contributes to the real options game literature. We use231.4. Conclusionthe identification strategies of MHT (Abbring (2012)) for the first-stageidentification analysis of the model primitives and we utilize the uniquesymmetric Bayesian-Nash equilibrium solution to this game for the second-stage identification analysis of the rest of the model primitives.There are many applications of real options game in the real estate mar-ket. The development of a new apartment is an irreversible investment andit takes time for the developers to start the construction. Therefore un-certainty of the market plays an important role in the investment timingdecision for the developers. Also the studies of pioneer survival can demon-strate an interesting application. For a market pioneer, both the uncertaintyof the market’s response to the new product and the future technology de-velopment delay the entry timing of firms. But the possible high profit fromthe monopoly also induces the preemptive competition among firms. Wangand Xie (2014) investigate both the newspaper industry and the high-techindustries to analyze the first-mover’s advantage. They estimate and com-pare the survival distributions by separating the monopoly period and thecompetition period. We could show the effect of preemptive incentive onthe investment timing directly by examining the survival distributions ofthe first mover and the second mover. And it would be interesting to fur-ther quantify the effect of asymmetric information on the investment timingand compare it with the effect of first-mover advantage.24Chapter 2Identification of OptimalHitting Time in a RealOptions Game withComplete InformationThis chapter studies the identification of a real options game with com-plete information. As in the first chapter, the investment opportunity con-sidered here is irreversible and the payoffs before the investment is normal-ized to zero while the payoffs after the investment are driven by GeometricBrownian Motion. Each player observes the value of the payoffs at eachinstant time and all the players’ investment costs are known as public infor-mation. The optimal timing decision rule to this game involves a threshold:each player will invest as soon as the Geometric Brownian Motion hits thethreshold. In other words, the optimal investment timing is a duration de-termined by the first time a Geometric Brownian Motion hits a threshold,which fits the mixed hitting time (MHT) model framework. The point iden-tification of the model primitives is achieved in a duopoly game by applyingthe identification strategies for the MHT model and exploring the equilib-ria’s specification. And non-identification results are provided when thereare three or more players.The analysis of investment decisions in the framework of a real optionsgame has been studied intensively for a duopoly game with complete infor-mation. Pawlina and Kort (2006) propose a duopoly game with asymmetriccosts and Kong and Kwok (2007) study a duopoly game that is asymmetricboth in the costs and in the profits for the duopolies. As is well known, theequilibrium for a complete information game is not unique in a more gen-eral set-up, and this complicates the identification problem. However, wecan achieve point identification of the primitives of an optimal stopping timegame in a duopoly case. Although there are two possible different kinds ofequilibria in the duopoly game studied in this chapter, the point identifica-25Chapter 2. Identification of Optimal Hitting Time in a Real Options Game with Complete Informationtion can still be achieved because the equilibria in this duopoly game havea unique characteristic that the entry order is according to the efficiencyorder, such that the more efficient firm with a lower entry cost always en-ters the market first, which helps to separate the different equilibria on theparameter space of the unobserved variables.However, when we consider a three-player game with complete infor-mation, the entry order is not necessarily the same as the efficiency order.In fact, even when the three players have only two different types of cost,namely, low-cost and high-cost, Argenziano and Schmidt-Dengler (2012)show that the high-cost firm could preempt the low-cost firm in the equilib-rium. In Appendix B.4, the equilibria of a three-player game with threedifferent cost types is derived, and there are three possible different typesof equilibria. In this three-player game, we cannot separate the parame-ter space of the investment cost monotonically according to the cost rankand therefore, we cannot achieve point identification of all the model prim-itives. And the identification of a complete information game that involvesmultiple equilibria typically relies on the equilibrium selection mechanism(Bresnahan and Reiss (1991), Berry (1992), Einav (2010) etc). Even underthe assumption of the same type of players, a multiple-player investmentgame with complete information may have an equilibrium that two or moreinvest simutaneously when the preemption race is intensive among the ac-tive players, as shown in Argenziano and Schmidt-Dengler (2014) and Bouis,Huisman and Kort (2009). There are some N -player real options games withcomplete information delivering a unique equilibrium. For example,Wangand Zhou (2006) have studied an N -player game incorporating both stochas-tic demand and stochastic construction costs in the real estate market andderived a unique Markov subgame perfect equilibrium. The firms behavein the same way as the symmetric firms as in Grenadier (1996) and all theplayers achieve the same level of utility in the equilibrium.The contribution of this chapter is point identification results for aduopoly game and non-identification results for a three-player game. Fora duopoly game, we observe two outcomes from the two different kind ofequilibria results. From the last entrant’s investment timing, since the twoplayers share the same payoff structure and the unobserved investment costare drawn independently from the same distribution, we can achieve pointidentification of the payoff structure and the distribution of the unobserv-ables as in the the identification in a single agent problem. And then fromthe first entrant’s ivestment timing, we just have to identify the remainingpayoff term only related to the first entrant. However, as discussed in athree-player game, the multiple equilibria results also depend on entry or-262.1. The Economic Modelders. Therefore, now we observe three outcomes from the game, but wehave to identify three sets of payoff structure terms and the three possibleentry orders as well. Even after normalization, we still cannot achieve pointidentification in this case.The remainder of this chapter is organized as follows. Section 1 presentsthe economic model. Section 2 shows the main identification results. Section3 concludes and shows possible applications of our structural model. All themathematical derivations and main proofs details are in the appendix.2.1 The Economic ModelThis section describes Kong and Kwok (2007)’s real options game ofa duopoly with asymmetric investment costs, which will be used to studythe identification. In Appendix B.4, we also study a real options game ofthree players with three different cost types and show that under completeinformation, there are three different kinds of equilibria based on the entryorder.2.1.1 A Duopoly with Complete InformationTwo risk-neutral firms compete for the optimal development timing ofreal estate projects. The two developers are assumed to have a perpetualinvestment opportunity on a development site. The investment decisionis irreversible and the sunk costs of investment are asymmetric betweenthe two firms. Before the investment, both firms have the option to waitfor their optimal timing of investment and receive a zero profit until theinvestment. At any point in time, a firm i can choose to invest the amountof Ki, i ∈ {1, 2}, and obtain an instantaneous profit flow ofpi(yt, DN (x)) = ytDN (x).The state variable yt represents demand shock of real estate market andfollows a Geometric Brownian Motion:dyt = µytdt+ σytdWt,where µ and σ are instantaneous drift parameter and instantaneous stan-dard deviation parameter. dt is the time increment and dWt is the Wienerincrement. Parameter µ can be interpreted as the industry growth rate andparameter σ as the industry volatility. We assume the stochastic process272.1. The Economic Modelstarts from y0 low enough such that immediate investment is not optimalfor either firm.The deterministic multiplier DN (x), N ∈ {1, 2}, stands for the marketexternality for each firm i, i ∈ {1, 2}, where N indicates the number ofactive firms in the market. N = 1 represents a monopoly market and N = 2a duopoly market. Note that x are the covariates to describe the marketconditions and are also useful for the identification. We define the profitadjusted cost for firm i as K˜iN = Ki/DN (·), i ∈ {1, 2}. We also denote firmi’s rival firm as firm j in the context without confusion. Here we considerthe asymmetry between the two firms in the sunk costs.The following assumptions are made:Assumption 7. D1(·) > D2(·).Assumption 8. If Ki < Kj, then K˜iN < K˜jN , N ∈ {1, 2}, i ∈ {1, 2}.Assumption 7 imposes negative externalities in the model, which meansmonopoly profits are higher than duopoly profits for both firms. Assumption8 specifies that if firm i is a lower investment cost firm, then firm i also hasthe lower profit adjusted cost. Under Assumption 7 and Assumption 8, wewill show that the low cost firm is always a leader in the equilibrium of aduopoly game and achieve the identification and estimation results usingdata on firms’ timing decisions T1 and T2 and their respective covariates x1and x2. Kong and Kwok (2007) provide an equilibrium analysis of a generalcase of our model. However, our special case shows an equilibrium scenarioof keen competition between the two firms upon their entry into a marketand the identification and estimation results can be adapted into other setsof strategic equilibriums accordingly.2.1.2 Value Functions and Investment ThresholdsIn this section, we define value functions of the firms given their possi-ble different strategies. Each firm has three possible choices of investmenttiming. It could choose to invest first, as the leader or invest after its rival,as the follower, or invest simultaneously with the other firm. As a standardapproach to analyze complete information games, we use backward induc-tion to solve for the firms’ optimal strategies. We first solve the follower’soptimal development timing, given the leader has started the investment.We derive the optimal value functions and investment thresholds similarlyas in Appendix A.1 and the optimal value function for follower i, i ∈ {1, 2},282.1. The Economic Modelis given asV Fi (yt | xi) =(fiD2(x)r−µ −Ki)(ytfi)βif yt ≤ fi (wait),ytD2(x)r−µ −Ki if yt > fi (invest).(2.1)When yt ≤ fi, the value function of the follower is the value of the optionto wait. It reflects the present value of the follower’s investment discountedback by a stochastic discount factor ytfi from the random time of reachingthe trigger value fi. When yt > fi, the value function of the follower is thenet present value of the project when immediate investment is optimal.Follower i’s the investment trigger is given byfi =ββ − 1(r − µ)KiD2(x), i ∈ {1, 2},where β > 1 is the positive root of the associated fundamental quadratic(see Dixit and Pindyck (1994) and Stokey (2008))Q(β) =12σ2β(β − 1) + µβ − r = 0.When the leader faces a preemptive threat from its rival, it may chooseto invest immediately. Then the value function for firm i when the otherfirm j chooses to invest at fj is given byV Li (yt | fj) = ytD1(x)r−µ −Ki + fj(D2(x)−D1(x))r−µ(ytfj)βif yt ≤ fj ,ytD2(x)r−µ −Ki if yt > fj .(2.2)Here the leader’s value function is divided into two parts by the follower’sinvestment trigger fj . When yt ≤ fj , the value function of the leader is thenet present value from immediate investment plus the decrease in the ex-pected present value caused by its rival’s eventual investment. By comparingthe value of investing immediately as the leader with the value of waiting toinvest as the follower, we could consider the incentive of a firm to preemptits competitor to be the leader in the game.However, if the leader i chooses its optimal investment timing without a292.1. The Economic Modelpreemptive threat, then the value function becomesV LLi (yt | fj)=(liD1(x)r−µ −Ki)(ytli)β+fj(D2(x)−D1(x))r−µ(ytfj)βif yt ≤ li,ytD1(x)r−µ −Ki +fj(D2(x)−D1(x))r−µ(ytfj)βif li < yt ≤ fj ,ytD2(x)r−µ −Ki if yt > fj ,(2.3)where the leader’s the investment trigger without preemption is given byli =ββ − 1(r − µ)KiD1(x), i ∈ {1, 2}. (2.4)2.1.3 Preemptive and Sequential EquilibriumThere are two types of equilibria played by the two competing firms,namely the preemptive and sequential equilibrium. The different outcomesof strategic equilibria come from the properties of different investment thresh-olds. Since a firm in the leader position may face the preemption by its rival,we have to consider the existence of firms’ preemptive thresholds first.As in Pawlina and Kort (2006), we construct the preemptive trigger (orbreak-even points as in Janssens and Kort (2012)) of each firm as bi and bjrespectively, at which the firms are indifferent between being a leader andbeing a follower. Like Kong and Kwok (2007), we define the break-evenfunction ξi(y), i ∈ {1, 2}, asξi(y) ≡ VLi (y)− V Fi (y)Ki.After substituting the value functions of V Li (y) and VFi (y), we obtainξi(y) = − 1β − 1[β(1li− 1fi)(1fj)β−1+(1fi)β]yβ +ββ − 11liy−1. (2.5)The preemptive threshold bi is defined as the lowest value of y such thatξi(y) = 0, i ∈ {1, 2}. Therefore, the preemptive trigger is the first timea firm’s value function in a leader position exceeds in a follower position.In the following analysis of strategic equilibria, we will conclude that thelow-cost firm i is the leader and the high-cost firm j is the follower under302.1. The Economic Modelthe additional assumption that both preemptive thresholds bi and bj exist.6Theorem 4. Under Assumptions 7−8 and the assumptions that bothpreemptive thresholds exist, the preemptive trigger of the low-cost firm bi,i ∈ {1, 2}, is always lower than the preemptive trigger of the high-cost firmbj.Proof. See Appendix B.1.Under the assumption of negative externalities, the low-cost firm alwayshas a lower preemptive threshold than the high-cost firm, which means theequilibrium of simultaneous entry does not exist. Therefore, the high-costfirm is a natural follower in the duopoly game.Theorem 5. Under Assumptions 7−8 and the assumptions that bothpreemptive thresholds exist, the optimal strategy for the low-cost firm i,i ∈ {1, 2}, is to invest at the trigger of min(bj , li), where bj is the pre-emptive trigger of the high-cost firm j and li is the stand-alone trigger inthe leader position for firm i.Proof. See Appendix B.2.When li < bj , the low-cost firm i ∈ {1, 2}, invests at the monopolist’sthreshold li and the high-cost firm j invests at the follower’s threshold fj ,which we call the sequential equilibrium. On the other hand, the preemptiveequilibrium is played when the low-cost firm i ∈ {1, 2}, chooses to invest atbj .Theorem 6. Suppose that firm i is the low-cost firm and firm j is thehigh-cost firm. Under Assumptions 7−8 and the assumptions that bothpreemptive thresholds exist.1. When we haveβ(lilj− β − 1β)<[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi <lilj, (2.6)then the sequential equilibrium (li, fj) is played.6Kong and Kwok (2007) discuss different cases of the existence conditions of preemp-tive thresholds in the Proposition 1 of their paper. To apply their results in our model, biexists if fifj< 1; both bi and bj exist if1q(fi/fj)< fifj< 1, where q(x) = ( xβ−1β(x−1) )1β−1 .312.2. Identification2. When we haveββ − 1lilj− 1 > 1β − 1[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi , (2.7)then the preemptive equilibrium (bj , fj) is played.Proof. See Appendix B.3.Therefore, we can separate the sequential equilibrium from the preemp-tive equilibrium in terms of the parameters when both the break-even thresh-olds bi and bj exist.2.2 IdentificationIn this section, we will investigate how to identify the duopoly modelby the duration data. Suppose that we have data on both the optimaltiming {(Ti1, Ti2)}Ni=1 and covariates {(xi1, xi2)}Ni=1, where subscripts i1 andi2 refer to firm 1 and firm 2 respectively in the market i. The cost pairs{(Ki1,Ki2)}Ni=1 are unobservable to the econometrician. The optimal stop-ping time Ti1 and Ti2 can be seen as the mixed hitting time defined inAbbring (2012) and therefore we can achieve the identification by applyingthe identification strategies in Abbring (2012) and utilizing the probabilityof the events Ti1 < Ti2 and Ti1 > Ti2 as in Honore´ and de Paula (2010).Assumption 9. Ki1 and Ki2 are jointly distributed with cdf G(·, ·). G(·, ·)is a continuous function with full support on R2+.Assumption 10. For each i ∈ {1, ..., N} and each j ∈ {1, 2}, at least onecomponent of xij, say xij,k, has a support that contains a nonempty opensubset of R.Assumption 11. The time discount factor r is known.Assumption 12. Investment costs Ki1 and Ki2 are independent and iden-tically distributed, for each i ∈ {1, ..., N}.Take the event Ti1 < Ti2 for example, which means firm i1 is the leaderand firm i2 is the follower, i ∈ {1, ..., N}. From the results in Theorem5, the optimal trigger values for firm i1 are defined as li1 in the sequentialequilibrium and bi2 in the preemption equilibrium, while the optimal trigger322.2. Identificationfor firm i2 is denoted as fi2, i ∈ {1, ..., N}. The optimal timing of thefollower, Ti2, i ∈ {1, ..., N}, is given asTi2 = inf{t ≥ 0 : yt ≥ fi2} = inf{t ≥ 0 : ln(yt) ≥ ln(fi2)}.And the optimal timing of the leader, Ti1, i ∈ {1, ..., N}, is defined as,Ti1 = inf{t ≥ 0 : yt ≥ min(li1, bi2)} = inf{t ≥ 0 : ln(yt) ≥ min(ln(li1), ln(bi2))}.The identification is divided into two parts. We first apply the identifi-cation strategies in Abbring (2012) to the follower’s timing duration Ti2,i ∈ {1, ..., N}. Since yt is a Geometric Brownian Motion with instantaneousdrift µ and instantaneous standard deviation σ, ln(yt) is a standard Brown-ian Motion with instantaneous drift µˆ = µ− 12σ2 and instantaneous standarddeviation σˆ = σ. Therefore, Ti2 is the first time that the standard Brow-nian Motion ln(yt) crosses a threshold ln(fi2). Abbring (2012) shows thatalthough the distribution of Ti2 cannot be explicitly expressed, the Laplacetransform of the distribution of Ti2 satisfies,LTi2(s) ≡ E[exp(−sTi21(Ti2 <∞)]= exp[−Λ(s) ln(fi2)],whereΛ(s) =√µˆ2 + 2σˆ2s− µˆσˆ2,and Λ(s) is an inverse function of the Laplace exponent of the standardBrownian Motion given by the Le´vy-Khintchine formula.Theorem 7. Identification of D2(·). Under Assumptions 7−11, thefunction D2(·) and parameters µ and σ are identified up to scale. Further-more, with additional Assumption 12, both the marginal distribution func-tions G1 and G2 are identified up to scale.Proof. When Ti1 < Ti2, the logarithm of the trigger value for Ti2 is ln fi2 =ln(ββ−1(r−µ)D2(xi2))+ ln(Ki2). Let φ2(x) = ln(ββ−1(r−µ)D2(x)). ThenLTi2(s | xi2, Ti1 < Ti2) = E(exp[−Λ(s)(φ2(xi2) + ln(Ki2))] | xi2, Ti1 < Ti2)= E(exp[−Λ(s) ln(Ki2)] | Ti1 < Ti2) exp[−Λ(s)φ2(xi2)]= LlnKi2|Ti1<Ti2(Λ(s)) exp[−Λ(s)φ2(xi2)],where LlnKi2|Ti1<Ti2 is the Laplace transform of the distribution lnKi2 con-ditional on the event Ti1 < Ti2, and we transform the Geometric Brownian332.2. IdentificationMotion yt into a standard Brownian Motion ln(yt), so thatΛ(s) =√(µ− 12σ2)2 + 2σ2s− (µ− 12σ2)σ2.Given two sets of covariates xi2,1 and xi2,2 for firm i2, the conditionaldistribution of Ti2 given xi2,1 and xi2,2 respectively are characterized byLxi2,1|Ti1<Ti2(s) ≡ LTi2(s | xi2,1, Ti1 < Ti2)= LlnKi2|Ti1<Ti2(Λ(s)) exp[Λ(s)φ2(xi2,1)],andLxi2,2|Ti1<Ti2(s) ≡ LTi2(s | xi2,2, Ti1 < Ti2)= LlnKi2,Ti1<Ti2(Λ(s)) exp[−Λ(s)φ2(xi2,2)].NoteLxi2,1|Ti1<Ti2(s)Lxi2,2|Ti1<Ti2(s)= exp [−Λ(s)(φ2(xi2,1)− φ2(xi2,2))]= exp[−Λ(s) ln D2(xi2,2)D2(xi2,1)].When Lxi2,1|Ti1<Ti2(s)/Lxi2,2|Ti1<Ti2(s) 6= 0, we can write(Lxi2,1|Ti1<Ti2(s)/Lxi2,2|Ti1<Ti2(s))′(Lxi2,1|Ti1<Ti2(s)/Lxi2,2|Ti1<Ti2(s)) = −Λ′(s) ln(D2(xi2,2)D2(xi2,1)). (2.8)Notelims→0Λ′(s) =(µ− 12σ2)−1.Solims→0(Lxi2,1|Ti1<Ti2(s)/Lxi2,2|Ti1<Ti2(s))′(Lxi2,1|Ti1<Ti2(s)/Lxi2,2|Ti1<Ti2(s)) = −(µ− 12σ2)−1ln(D2(xi2,2)D2(xi2,1)).(2.9)Note the LHS of equation (2.9) is not degenerate as discussed in Chapter 1.342.2. IdentificationNext we substitute equation (2.9) intoLxi2,1|Ti1<Ti2(s)Lxi2,2|Ti1<Ti2(s)= exp(−Λ(s) ln(D2(xi2,2)D2(xi2,1))).Hence µ and σ in Λ(s) are identified up to scale. Once Λ(s) is known, so isD2(·) identified up to scale from equation (2.8).Given µ and σ are identified up to scale and time discount factor r isknown, β is identified up to scale as well. Hence φ2(x) is identified up toscale. Lastly,Lxi2,1|Ti1<Ti2(s) = LlnKi2|Ti1<Ti2(Λ(s)) exp (−Λ(s)φ2(xi2,1)) .therefore LlnKi2|Ti1<Ti2(s) is identified and so is the distribution of lnKi2,i ∈ {1, ..., N}.Theorem 8. Identification of D1(·). Under Assumptions 7−12, thefunction D1(·) is identified up to scale.Proof. We consider the event Ti1 < Ti2 and apply the equilibrium separa-tion conditions we discussed in section 2.1 to achieve the identification ofD1(·). From Theorem 5, we know i1 would choose a strategy min{li1, bi2}.let KSi2 be the investment cost of firm i2 such that ξi2(li1;KSi2,Ki1) = 0.By Proposition 5.2 in Janssens and Kort (2012), we know that bi2 is anincreasing function of Ki2. Therefore, when Ki1 < Ki2 < KSi2, firm i1 wouldface an preemption threat from firm i2, and therefore choose to invest atbi2. When Ki2 > KSi2, either bi2 does not exist as the solution to firm i2’sbreak-even function ξi2(·) or li1 < bi2 holds, therefore, firm i1 always choosesli1 to invest. We apply this property to complete the identification.Firstly,Pr(Ti1 < Ti2 | xi1, xi2)= Pr(Ki1 < Ki2 < KSi2 | xi1, xi2) + Pr(Ki1 < KSi2 < Ki2 | xi1, xi2)= Pr(Ki1 < min(Ki2,KSi2) | xi1, xi2).352.3. ConclusionThen, we consider the following conditional probabilityPr(Ti1 < Ti2 | xi1, xi2,Ki1 < Ki2)= Pr(Ki1 < KSi2 | xi1, xi2,Ki1 < Ki2)= Pr(Ki1 < (1− (D2(xi2)Ki1D1(xi1)Ki2)β)−1(ββ − 1)(Ki1+(D2(xi2)−D1(xi2)D2(xi1))(D2(xi1)D1(xi1))βKi1 −Ki2(D2(xi2)Ki1D1(xi1)Ki2)β) | xi1, xi2,Ki1 < Ki2)= Pr(1− (D2(xi2)Ki1D1(xi1)Ki2)β < (ββ − 1)(1 + (D2(xi2)−D1(xi2)D2(xi1))(D2(xi1)D1(xi1))β)− ( ββ − 1)(D2(xi2)D1(xi1))β(Ki1Ki2)β−1 | xi1, xi2,Ki1 < Ki2)= Pr((ββ − 1)(D2(xi2)D1(xi1))β(Ki1Ki2)β−1 − (D2(xi2)Ki1D1(xi1)Ki2)β < (ββ − 1)(1+ (D2(xi2)−D1(xi2)D2(xi1))(D2(xi1)D1(xi1))β)− 1 | xi1, xi2,Ki1 < Ki2)= Pr((ββ − 1)(Ki1Ki2)β−1 − (Ki1Ki2)β < (D2(xi2)D1(xi1))−β(ββ − 1)(1+ (D2(xi2)−D1(xi2)D2(xi1))(D2(xi1)D1(xi1))β)− 1) | xi1, xi2,Ki1 < Ki2).(2.10)Since the marginal distribution of Ki1 and Ki2, i ∈ {1, ..., N}, are identi-fied in Theorem 7, and under the independence assumption between Ki1and Ki2, distribution of Ki1/Ki2 is identified, i ∈ {1, ..., N}. Therefore,the expression of the conditional probability function (2.10) can be used toidentify function D1(·).2.3 ConclusionThis chapter studies a game theoretic MHT model and contributes tothe literature of continuous time duration analysis. We use the identificationstrategies for MHT (Abbring (2012)) for the first stage identification analysisof the model primitives and we achieve point identification for the rest ofthe model primitives by separating the two different equilibria accordingto the efficiency order of the two firms. The three-player or N -player realoptions game with complete information is more difficult to analyze the pointidentification of all the model primitives because the multiple equilibria ofthe game cannot be separated in general. See the discussion for detailsin Appendix B.4. Therefore a further investigation about the multiple362.3. Conclusionequilibria selection would be interesting.37Chapter 3Estimation of VaryingCoefficient Models withMatching DataThe purpose of this chapter is to investigate the estimation of the varyingcoefficient model that involves matching estimators. Suppose that we areinterested in estimating the varying coefficient model:Y = X · β(Z) + u. (3.1)Unlike a linear parametric regression model with constant coefficients, westudy a case of nonlinear parametric regression models with coefficients vary-ing with the models’ specification. For example, the marginal propensity toconsume would be different between generations and the rate of return toschooling would be different for individuals with different working experi-ence. And when we investigate issues between generations, we sometimesface the problem that (Y,X,Z) can not be obtained from a single data setand we have to combine information from two or more samples drawn fromthe same population. For example, there is a large literature that stud-ies the intergenerational income mobility. Let Y be son’s income, and X1be control variables of son’s characteristics, such as son’s education and/oryears of working experience. Let X2 be father’s income or family income atthe time of son’s childhood and Z be father’s education. It is quite likelythat X2 and Y can not be observed in the same sample set. Usually wecan only observe (Y,X1, Z) in one sample and (X2, Z) in another sample,where Z are some common variables (not necessarily common observations).Combining different data sets is quite common when a complete data is notavailable. Arellano and Meghir (1992) estimate the female labor supplyequation using two survey data sets, the UK Labour Force Survey (LFS)and the Family Expenditure Survey (FES). The labour survey contains theinformation about the labour supply, which is the dependent variable, andthe information on job-search activity, which is one of the explanatory vari-38Chapter 3. Estimation of Varying Coefficient Models with Matching Dataables, while the budget survey contains information on the wage rate, otherincome and consumption. Common variables are education, age of husbandand regional labor market conditions. In Arellano and Meghir (1992), com-mon variables are excluded from the supply equation and only used for theimputation of wage and other income in the combined data set.There is a large literature studying how to identify and estimate the jointdensity of (Y,X,Z) based on the data combination, see the literature reviewby Ridder and Moffitt (2007). However, as noted in Ridder and Moffitt(2007), what can be recovered from the combined data is largely dependenton the nature of the available samples and the additional assumptions wewant to make. For example, when the population moment conditions areadditively separable into two samples and the available samples are richenough such that we can construct required moment conditions from thetwo samples respectively, then a two-sample generalized method of moments(GMM) estimation is enough and no additional assumptions are needed, seeAngrist and Krueger (1995) and Liu, Murtazashvili and Prokhorov (2015).However, when the two-sample GMM is not feasible due to the nature ofthe available samples, we need more assumptions. Suppose that Y and Xare only available in two different data sets. Usually, either the assumptionof conditional independence between Y and X given common variables Z,or the exclusion conditions are added for the full inference. Obviously, theconditional independence assumption is not very attractive when we areinterested in the estimation of E(Y | X,Z). On the other hand, exclusionconditions are similar to the instrumental variables (IV) approach. We needto find variables that are excluded from the regression we are interestedin, but also highly correlated to the missing data we want to impute intothe combined data. This approach is also known as two-sample IV. Insteadof making more assumptions or requiring rich samples, we can also simplycombine the missing data samples by applying some matching method andinvestigate the properties of the estimators involving matching, which havebeen rigorously studied in the average treatment effects models, see Abadieand Imbens (2006) and Abadie and Imbens (2011).We consider a general case where (Y,X1, Z) and (X2, Z) are collectedfrom two samples with one sample size potentially greater than the otherbut of the same order. And we will apply a nearest matching method tomatch these two samples based on the covariates Z. The first intuition, asin the literature of evaluation research on the average treatment effect, isto approximate the missing X2 in (Y,X1, Z) by reasonable X2 in the largersample determined through nearest matching over the corresponding Z. Ourinvestigation shows that the simple local linear estimator based on matching393.1. Two-Sample Matching Estimatoris inconsistent, which is due to the “matching discrepancy” termed in Abadieand Imbens (2006). Moreover, it is shown that the rate of convergence ofthe simple local linear estimator is dominated by the rate of the error terminvolving matching discrepancy, which in turn depends on the number ofmatching variables. In particular, the simple local linear estimator reachesthe parametric convergence rate only if matching is conducted over onevariable, instead of high dimensional Z. In addition to the above results, wediscuss the possible bias-corrected estimators.The rest of this chapter is organized as follows. Section 3.1 shows theinconsistency of the simple local linear estimation of the regression model(1) using matched samples. Section 3.2 proposes the methods for bias-corrected estimators and examines their convergence properties. Section 3.3conducts Monte Carlo simulations and examines the performance of the biascorrection in finite samples. Section 3.4 concludes. All proofs are given inthe Appendix. MATLAB codes implementing the estimators are availableupon request.3.1 Two-Sample Matching Estimator3.1.1 Setting and NotationConsider the following varying coefficient modelYi = XTi β(Zi) + ui= XT1iβ1(Zi) +XT2iβ2(Zi) + ui,E(ui | Xi, Zi) = 0E(ui2 | Xi, Zi) = σ2 a.s., i ∈ {1, ...n}(3.2)where the dependent variable Yi is a scalar random variable, and β(·) =(β1(·)T , β2(·)T )T is a d1 + d2 dimensional vector of unknown functions. LetXi = (XT1i, XT2i)T ∈ Rd1+d2 , where X1i and X2i denote d1 and d2 dimen-sional vectors of exogenous regressors respectively. Zi ∈ R1 are continuouscovariates with compact support.Suppose that we observe two independent random samples from the samepopulation, namely, {Yi, X1i, Z1i}ni=1 and {X2j , Z2j}mj=1. And we constructa matching data set of n observations {(Yi, X1i, X2j(i), Z1i, Z2j(i))}ni=1, whereZ2j(i) is denoted as the nearest match to Z1i and X2j(i) is the observation403.1. Two-Sample Matching Estimatorpaired with Z2j(i) in the sample {X2j , Z2j}mj=1. Definej(i) := argminj∈{1,...,m}|Z2j − Z1i|. (3.3)In other words, we will match the missing values X2i in {X1i, X2i}ni=1, withX2j(i) in {X2j , Z2j}mj=1, such that j(i) is the index of the unit that is thenearest match for unit i in terms of the matching variables Z7.Define C(j), the number of times that unit j in the sample {X2j , Z2j}mj=1is used as a match to unit i in the sample {Yi, X1i, Z1i}ni=1,C(j) =n∑i=11(j = j(i)), j ∈ {1, ...,m}where 1(·) is the indicator function, equal to one if j = j(i) is true and zerootherwise.We also define A(j) as the subset of the indices i, i ∈ {1, ..., n}, such thatj is used as a match to each observation indexed from A(j); for instance, ifi ∈ A(j), then j = j(i). Clearly, the number of the elements in the set A(j)is C(j).3.1.2 Identification of the Two-sample EstimatorIf we can observe a complete sample, the moment condition is E(Xiui |Zi) = 0d1+d2 , or(E(X1iui | Zi)E(X2iui | Zi))= 0d1+d2 . If we now have two samples ofmissing data {Yi, X1i, Z1i}ni=1 and {X2j , Z2j}mj=1 as well as the constructedsample {(Yi, X1i, X2j(i), Z1i, Z2j(i))}ni=1, the previous moment condition cannot be used directly to identify parameters. Instead, the applicable momentcondition in this case would be(E(X1iui | Zi)E(X2j(i)ui | Z1,2))= 0d1+d2 . Thereforewith the above moment condition, a straight forward calculation gives(E(X1iyi | Zi)E(X2j(i)yi | Z1,2))=(E(X1iXT1i | Zi), E(X1iXT2i | Zi)E(X2i | Zi)E(XT1i | Zi), E(X2i | Zi)E(XT2i | Zi))(β1(Zi)β2(Zi)).(3.4)7As discussed in Abadie and Imbens (2006), we can also apply the nearest kth matchmethod such that we firstly find k nearest matches to each i and then average the kmatches to impute into the combined sample set. This will improve the performance ofour corrected estimator using a single match and further proof will be explored in thefuture work413.1. Two-Sample Matching EstimatorTo further reduce equation 3.4, let us define some notations. SupposeΩ(z) = E(XX ′ | Z = z) is positive definite for each z and uniformly con-tinuous in z. And let Ω(z)(ij) be the (i, j)th block element of matrix Ω(z).Further denote the conditional expectations of X1 and X2, given Z, asg1(Z) = E(X1 | Z), g2(Z) = E(X2 | Z).And the conditional expectation errors arev1 = X1 − g1(Z), v2 = X2 − g2(Z).Let g(Z) = (g1(Z)T , g2(Z)T )T and v = (vT1 , vT2 )T . Denote the conditionalvariance of v as E(vvT |Z) = Σ, the conditional variance of v1 as E(v1vT1 |Z) =Σ11, the conditional variance of v2 as E(v2vT2 | Z) = Σ22, and the conditionalcross-covariance as E(v1vT2 |Z) = Σ12 and E(v2vT1 |Z) = Σ21.Recall that the model is identifiable if the matrix(E(X1iXT1i | Zi) E(X1iXT2i | Zi)E(X2i | Zi)E(XT1i | Zi) E(X2i | Zi)E(XT2i | Zi))(3.5)is invertible from equation 3.4. Also matrix 3.5 is equivalent to(Ω(z)(11) Ω(z)(12)Ω(z)(21) Ω(z)(22) − Σ22)is invertible when Σ12 = 0, which means that X1 and X2 are uncorrelatedconditional on Z.Firstly, we note matrix 3.5 is symmetric positive definite when Σ12 = 0,and matrix 3.5 is invertible only if diagonal elements are invertible. There-fore we assume both E(X1iXT1i | Zi) and E(X2i | Zi)E(XT2i | Zi) are invert-ible, which means X2i’s dimension is d2 = 1.Secondly, it is interesting to notice that the conditional distribution ofX | Z is not needed to obtain the GMM estimator in this two-sample VCMmodel. Similarly to the discussions in Ridder and Moffitt (2007) aboutstatistical matching methods in the case of a simple OLS regression, thisis due to the moment conditions are quadratic in X = (X1, X2)T and onlyE(X | Z) and E(XXT | Z) are needed. In the special case parallel to thesimple OLS in Ridder and Moffitt (2007)(Page 5499, equation 63), whenwe have two independent samples (Y,Z) and (X,Z), now the matrix 3.5 isreduced to E(X | Z)E(XT | Z), which is only invertible when X’s dimensionis d = 1.423.1. Two-Sample Matching Estimator3.1.3 Two-Sample Naive Local Linear EstimatorSuppose that {Yi, X1i, X2j(i), Z1i, Z2j(i)}ni=1 is a matched sample fromtwo samples {Yi, X1i, Z1i}ni=1 and {X2j , Z2j}mj=1. DenoteXj(i) = (XT1i, X2j(i))Tas the matching pair for Xi = (XT1i, X2i)T , i ∈ {1, ..., n}. We can approxi-mate β(Z1i) in a small neighbourhood of z by a linear functionβ(Z1i) ≈ θ0 + 1h(Z1i − z)θ1,where θ0 = β(z), and θ1 = hβ′(z). Then we consider a naive local linearestimator as minimizers with respect to (θ0, θ1) of the following weightedlocal least-squares problem:n∑i=1(Yi −XTj(i)θ0 −Z2j(i) − zhXTj(i)θ1)2Kh(Z2j(i) − z), (3.6)where K(·) is a kernel function, h is a bandwidth and Kh(·) = K(·/h)h .The local linear estimator βˆ(z) is given by the solution for θ0 to the prob-lem of minimizing (3.6). And the solution admits the following expression:(βˆ(z)T , hβˆ′(z)T)T=((DXm)TWDXm)−1 (DXm)TWY, (3.7)whereDXm =XTj(1) XTj(1)Z2j(1)−zh... ...XTj(n) XTj(n)Z2j(n)−zh ,W = diag(Kh(Z2j(1) − z), ...,Kh(Z2j(n) − z)),andY = (Y1, ..., Yn)T .Let Θ(z) =(β(z)T , β′(z)T)T, and define its estimator to beΘˆ(z) = H−1((DXm)TWDXm)−1 (DXm)TWY,where H = diag(1, ..., 1, h, ..., h) is a 2(d1 + 1) × 2(d1 + 1) matrix with thefirst d1 + 1 diagonal elements being 1 and the remaining diagonal elements433.1. Two-Sample Matching Estimatorh. We can write the estimator of β(z) asβˆ(z)T = eH−1((DXm)TWDXm)−1(DXm)TWY,where e is a 1 × 2(d1 + 1) matrix with the first d1 + 1 elements being 1and the rest diagonal elements 0. We call βˆ(z) the two-sample naive locallinear estimator for varying coefficient models. For the following analysis,we modify the expression for the naive local linear estimator in a conciseway,HΘˆ(z) = Dn(z)−1Nn(z),where Dn(z) =(Dn,0 Dn,1Dn,1 Dn,2)and Nn(z) =(Nn,0Nn,1),Dn,0(z) =1nn∑i=1Kh(Z2j(i) − z)Xj(i)XTj(i),Dn,1(z) =1nhn∑i=1Kh(Z2j(i) − z)(Z2j(i) − z)Xj(i)XTj(i),Dn,2(z) =1nh2n∑i=1Kh(Z2j(i) − z)(Z2j(i) − z)2Xj(i)XTj(i),Nn,0(z) =1nn∑i=1Kh(Z2j(i) − z)Xj(i)Yi,Nn,1(z) =1nhn∑i=1Kh(Z2j(i) − z)(Z2j(i) − z)Xj(i)Yi.3.1.4 Large Sample Properties of the Two-Sample NaiveLocal Linear EstimatorAssumption 13. {Yi, X1i, Z1i}ni=1 and {X2j , Z2j}mj=1 are two independentsamples from the same population {Y,X1, X2, Z} with missing data.Assumption 14. 1. The density function f(·) of Z is bounded, and havecontinuous second derivatives on a compact set.2. The matrix f(z)Ω(z) is invertible, and so is the matrixf(z)(Ω(z)(11), Ω(z)(12)Ω(z)(21), Ω(z)(22) − Σ22)443.1. Two-Sample Matching Estimatorover the domain of z.3. βj(·), with j ∈ {1, ..., d1 + d2} have continuous second derivatives ateach point z in the support of ZAssumption 15. The kernel function K(·) is symmetric and a boundedsecond order kernel function with compact support. K(·) is Lipschitz con-tinuous. The bandwidth h satisfies nh→∞ and h→ 0 as n→∞, nh8 → 0and nh2/(log n)2 →∞ as n→∞.Assumption 16. 1. Functions g1(·) and g1(·) have continuous secondderivatives at each point z on the support of Z,2. the fourth moment of the conditional distribution of Y given Z = zexists and is bounded uniformly in z,3. σ2 is bounded away from zero,4. mn → κ ∈ (0,∞) as n, m→∞ jointly.Assumption 13 specifies our two-sample setup. As both sample sizesn→∞ and m→∞, we cannot apply the law of large numbers or the centrallimit theorem to the combined sample constructed by nearest matching,because the i.i.d. property of the combined sample is destroyed by replacingthe fixed index i with a random index j(i). Assumptions 14−15 are standardassumptions for the consistency and asymptotic normality of the local linearestimators of the varying coefficient models. Assumption 16 adds additionalassumptions needed to reestablish the consistency and asymptotic normalityresults for our two-sample nearest matching estimators.We first show that the denominator in our two-sample matching estima-tor is consistent for its expectation. And then we show that the numeratoris also consistent but the resulting two-sample matching estimator is biased.Then we will prove that without the conditional bias term, the matchingestimator is N1/2 consistent and asymptotically normal. In the followinganalysis, we denoteµj =∫ujK(u)du,andνj =∫ujK2(u)du, j = 1, 2, 3.And we use⊗to denote kronecker product.453.1. Two-Sample Matching EstimatorLemma 9. Convergence of Denominator. Under Assumptions 13−16,Dn(z)p−→ f(z)Ω(z)⊗(1 00 µ2),Proof. See Appendix C.1.Lemma 10. Convergence of Numerator. Under Assumptions 13−16,Nn(z)− 1n(DXm)TWDXm(β(z)hβ′(z))p−→f(z)(0 Σ120 −Σ22)β(z)⊗(10)+ f(z)(0 Σ120 −Σ22)hβ′(z)⊗( 0µ2).Proof. See Appendix C.2.Theorem 11. Inconsistency of the Naive Estimator. Suppose thatAssumptions 13−16 hold. Thenβˆ(z)− β(z) = bias(z)β(z) +Op(h2 +1√nh), (3.8)wherebias(z) = Ω(z)−1(0 Σ120 −Σ22).Proof. Theorem 11 holds by combining the results from Lemma 9 and10.Theorem 12. Asymptotic Normality for the Naive Estimator. Sup-pose that Assumptions 13−16 hold. Then√nhV (z)−1/2D(z)(H(Θˆ(z)−Θ(z))− bias(z)⊗( β(z)hβ′(z)))d−→ N (0, I),where D(z) = f(z)Ω(z)⊗(1 00 µ2)andV (z) =m∑j=1σ2(C(j)Kh(Z2j − z))2Ω(Z2j)⊗( 1 (Z2j − z)/h(Z2j − z)/h ((Z2j − z)/h)2).463.1. Two-Sample Matching EstimatorProof. From the argument in Lemma 10,√nh(H(Θˆ(z)−Θ(z))− bias(z)⊗( β(z)hβ′(z)))=√nhD−1n (z)I4,where I4 =(1n∑ni=1Xj(i)Kh(Z2j(i) − z)ui1n∑ni=1Xj(i)Kh(Z2j(i) − z)ui(Z2j(i) − z)/h). By Lemma 9,Dn(z)p−→ f(z)Ω(z)⊗(1 00 µ2).Firstly, we will show I4 is a random vector made up with sums of condi-tionally independent random variables, given all the Z from the two samples.Denote I4 = (IT4,0, IT4,1)T , whereI4,0 =1nn∑i=1Xj(i)Kh(Z2j(i) − z)ui,andI4,1 =1nn∑i=1Xj(i)Kh(Z2j(i) − z)ui(Z2j(i) − z)/h.Since A(j) indicates the subset of the index i, i ∈ {1, ..., n}, such that jis used as a match to each observation indexed by i, i ∈ {1, ..., n}, we canrewrite I4,0 and I4,1 asI4,0 =1nm∑j=1Ij4,0=1nm∑j=1∑i∈A(j)X(i,j)Kh(Z2j − z)ui,andI4,1 =1nm∑j=1Ij4,1=1nm∑j=1∑i∈A(j)X(i,j)Kh(Z2j − z)ui(Z2j − z)/h,where X(i,j) = (XT1i, XT2j)T .Recall that Z1,2 represents all the Z from the two samples. Then con-473.1. Two-Sample Matching Estimatorditional on Z1,2, the unit-level terms Ij4,0 =∑i∈A(j)X(i,j)Kh(Z2j − z)ui areindependent with zero means and nonidentical distributions. To see this,first, note the number of elements in the index set A(j) is C(j), which is thenumber of times Z2j is used as a match, j ∈ {1, ...,m}. And conditional onZ1,2, C(j) is nonstochastic. Therefore conditional on Z1,2, the sum in I4,0are made up with all i.i.d. terms for all j, j ∈ {1, ...,m}. As a result, Ij4,0are independent for all j, j ∈ {1, ...,m}. The conditional variance of Ij4,0 is(C(j)Kh(Z2j − z))2σ2Ω(Z2j) + o(1).Likewise, conditional on all the Z1,2, the unit-level terms Ij4,1 are also in-dependent with zero means and nonidentical distributions. The conditionalvariance of Ij4,1 is (C(j)Kh(Z2j − z)(Z2j − z)/h)2σ2Ω(Z2j) + o(1).Next, we will use the Crame`r-Wold device and the Lindeberg-Feller cen-tral limit theorem to derive the asymptotic distribution of√nI4. Denotethe dimension of X = (X1, X2) as p = d1 +d2. For any 2p×1 nonzero vectorτ = (τ1, ..., τ2p)T , we have√nhτT I4 =√mn√h√mm∑j=1p∑k=1∑i∈A(j)τkX(i,j)kKh(Z2j − z)ui+p∑k=1∑i∈A(j)τk+pX(i,j)kKh(Z2j − z) ((Z2j − z)/h)uiSimilarly to the argument for the conditional independence of Ij4,0 and Ij4,1given Z1,2,√hIj4 =√hp∑k=1∑i∈A(j)τkX(i,j)kKh(Z2j − z)ui+p∑k=1∑i∈A(j)τk+pX(i,j)kKh(Z2j − z) ((Z2j − z)/h)uiare independent and we can apply the Lindeberg-Feller central limit theo-rem if the Lindeberg-Feller condition are satisfied. Denote the variance of∑mj=1 Ij4 is Vτ .483.1. Two-Sample Matching EstimatorFor given Z1,2, the Lindeberg-Feller condition requires that1mVτm∑j=1E[(Ij4)21{|Ij4 | ≥ η√mVτ} | Z1,2]→ 0for all η > 0. By applying Ho¨lder’s and Markov’s inequalities we have1mVτm∑j=1E[(Ij4)21{|Ij4 | ≥ η√mVτ} | Z1,2]≤ 1mVτm∑j=1(E[(Ij4)4 | Z1,2]) 12E[(Ij4)2 | Z1,2]η2mVτ≤ 1mVτm∑j=1(C(j)4Kh(Z2j − z)4E[u4j | Z1,2]Ψ(τ, Z2j)) 12C(j)2Kh(Z2j − z)2σ2Φ(τ, Z2j)η2mVτ≤ c¯12η2σ21m 1mm∑j=1C(j)4 ,where c¯ = supzE[u4j | Z = z] < ∞, both Ψ(τ, Z2j) and Φ(τ, Z2j) are func-tions composed with deterministic coefficients. Because E(C(j)4) is uni-formly bounded by Lemma 3 in Abadie and Imbens (2006), by Markov’sinequality, the last term is bounded in probability. Hence, the Lindeberg-Feller condition is satisfied for almost all Z. As a result,√nhV (z)−1/2I4d−→ N (0, I),whereV (z) =m∑j=1σ2(C(j)Kh(Z2j−z))2Ω(Z2j)⊗( 1 (Z2j − z)/h(Z2j − z)/h ((Z2j − z)/h)2)+o(1).The boundedness of V (z) are proved in Appendix C.4. Then√nhV (z)−1/2D(z)(H(Θˆ(z)−Θ(z))− bias(z)⊗( β(z)hβ′(z)))d−→ N (0, I),493.2. Bias Correction and the Consistent Estimatorwhere D(z) = f(z)Ω(z)⊗(1 00 µ2).3.2 Bias Correction and the Consistent EstimatorIn this section we analyze the asymptotic properties of the bias-correctedmatching estimator. From Theorem 11, the bias term in the two-samplematching estimator is given bybias(z)⊗( β(z)hβ′(z))= Ω(z)−1(0 Σ120 −Σ22)⊗( β(z)µ2hβ′(z)).This bias comes from the fact that the denominator of the two-sample match-ing estimator, Dn(z), converges to its expectation D(z), while the numera-torNn(z) converges to(D(z) + f(z)(0 Σ120 −Σ22)⊗(1 00 µ2))(β(z)hβ′(z)).Therefore, we can replace the denominatorDn(z) with a consistent estimatorof Dn(z) + f(z)(0 Σ120 −Σ22)⊗(1 00 µ2), and leave the numerator Nn(z)unchanged to eliminate the bias in the two-sample matching estimator.In order to establish the asymptotic properties of the bias-corrected es-timator, we need to estimate Σ12 and Σ22 consistently. For Σ22, we considerthe difference-based variance estimator proposed in Rice (1984). In partic-ular,Σˆ22 =12(m− 1)m∑j=2(X2(j) −X2(j−1))(X2(j) −X2(j−1))T ,where X2(j) and Z2(j) are from the ordered sample {(X2(j), Z2(j))}mj=1 basedon Z2(1) ≤ · · · ≤ Z2(m).However, the estimator of Σ12 is more complicate, because Σ12 reflectsthe population correlation between X1 and X2, which are not available ina single sample. And there is a large literature about this kind of two-sample combination problems and the solutions to recover the populationjoint density of X1 and X2 are either assuming X1 and X2 are conditionallyindependent or adding more exclusive variables, which are excluded fromthe regression of Y on X1 and X2, but highly correlated to both X1 and X2.As a result, here we assume Σ12 = 0, and we can also consistently estimatethe density function f(z) by any nonparametric method.503.2. Bias Correction and the Consistent EstimatorDefine bias-corrected two-sample matching estimator asHΘˆbcll(z) =(Dn(z) + fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))−1Nn(z).Then we can estimate β(z) consistently usingβˆbcll(z) = eH−1(Dn(z) + fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))−1Nn(z),(3.9)where e is a 1× 2(d1 + d2) matrix with the first (d1 + d2) elements being 1and the remaining diagonal elements 0.Theorem 13. Asymptotic Normality for the Bias-corrected Match-ing Estimator. Suppose that Assumptions 13−16 hold, and assumeΣ12 = 0, then√nhV (z)−1/2D(z)(H(Θˆbcll(z)−Θ(z)))d−→ N (0, I),where D(z) = f(z)Ω(z)⊗(1 00 µ2)andV (z) =m∑j=1σ2(C(j)Kh(Z2j − z))2Ω(Z2j)⊗( 1 (Z2j − z)/h(Z2j − z)/h ((Z2j − z)/h)2).Proof. Firstly, we will show that√n(HΘˆbcll(z)−HΘˆ(z) + bias(z)⊗( β(z)hβ′(z)))p−→ 0,513.3. Monte Carlo Simulationswhere bias(z) = Ω(z)−1(0 00 −Σ22). NoteHΘˆbcll(z)−HΘˆ(z)=((Dn(z) + fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))−1− (Dn(z))−1)Nn(z)= (Dn(z))−1(fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))(Dn(z) + fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))−1Nn(z)= (Dn(z))−1(fˆ(z)(0 00 −Σˆ22)⊗(1 00 µ2))HΘˆbcll(z).The consistency of HΘˆbcll(z) can be established in line with the proof ofTheorem 11. Therefore we have HΘˆbcll(z)p−→ HΘ. As a result,HΘˆbcll(z)−HΘˆ(z) p−→ Ω(z)−1(0 00 −Σ22)⊗(1 00 µ2)HΘ.Then the asymptotic normality results will be applied by Theorem 12.The result of Theorem 13 suggests that the bias-corrected matching es-timator has the same asymptotic variance as the naive local linear matchingestimator.3.3 Monte Carlo SimulationsTo evaluate the naive local linear and the bias-corrected local linearestimator, we consider the following data generating process (DGP):Y = β0(Z) + β1(Z)X1 + β2(Z)X2 + U, (3.10)where β0(Z) = (1 − eZ + Z) and β1(Z) = (0.5 + 0.5Z). We consider twofunctional forms of β2(·). We assume β(1)2 (z) = 0.5 + 0.5Z+ 0.25Z2 for DGP(1) and β(2)2 (z) = 1 + ez for DGP (2). We set Z ∼ N(0, 1) truncated at ±2,X1 = 2Z + ξ1, X2 = 3Z + ξ2, (ξ1, ξ2)′ ∼ N(0, I2), and U ⊥ (ξ1, ξ2)′.The optimal bandwidth h of all the estimators are determined by crossvalidation (CV) method throughout this section.523.3. Monte Carlo Simulations−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4.5−4−3.5−3−2.5−2−1.5−1−0.500.5Intercept  InterceptLLFigure 3.1: Local linear estimator in one complete sample case: intercept,β0(Z)3.3.1 Case with One Complete SampleIn this section we show that the naive local linear estimator performs wellwhen we have one complete sample generated from DPG (1). The samplesize n = 3000. Figure 3.1 - Figure 3.3 are based on 500 replications. (In factthe performance of the estimator based on one estimation, including theorder of bias and variance, is similar with that based on 500 replications.)3.3.2 Case with Two Missing-data SamplesIn this section we first study the behaviour of the naive local linear es-timator (LL) and the bias-corrected local linear estimator (BCLL) whenwe have two i.i.d samples with missing data generated from DPG (1),{Yi, X1i, Z1i}ni=1, {X2j , Z2j}mj=1, where n = 3000, m = 4000. Figure 3.4 -Figure 3.6 are based on 500 replications. (In fact the performance of theLL and BCLL estimators in one estimation, including the order of bias and533.3. Monte Carlo Simulations−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.500.511.52Coefficient of X1  Coefficient of X1LLFigure 3.2: Local linear estimator in one complete sample case: coefficientof X1, β1(Z)543.3. Monte Carlo Simulations−2 −1.5 −1 −0.5 0 0.5 1 1.5 200.511.522.5Coefficient of X2  Coefficient of X2LLFigure 3.3: Local linear estimator in one complete sample case: coefficientof X2, β2(Z)553.3. Monte Carlo Simulationsvariance, are similar with that based on 500 replications, respectively.) Fig-ure 3.4, Figure 3.5 and Figure 3.6 show the performance of the LL and BCLLestimators on the estimation of the intercept, the coefficient function for X1and the coefficient function for X2 respectively. In these figures, the solidlines represent the true function parameters, while the dashed line and thedotted line represent the LL and BCLL respectively. It can be easily seenfrom Figure 3.4 that, the BCLL estimator has less average bias than the LLestimator in most part of the support of Z despite the boundary effect. Inaddition, BCLL identifies the right shape of the intercept function while LLdoes not. Figure 3.6 shows the similar properties of two estimators as inFigure 3.4. For Figure 3.5, the two estimators both identifies the true shapeof the coefficient function while BCLL has less average bias.To evaluate the finite sample performance of the LL and BCLL estimatorof the functional coefficient, we calculate both the mean absolute deviation(MAD) and mean squared error (MSE) for each estimate evaluated at 100evenly spaced points between the support of Z, which is [−2, 2]. In thiscase we consider both DGP (1) and DGP(2) with sample size n = 300 andm = 400.Table 3.1 reports the results where the MSEs and MADs are averagesover 500 replications for each functional coefficient. As expected, the bias-corrected local linear (BCLL) estimators perform better than the local linear(LL) estimators both in DGP(1) and DGP(2), which are specified differentlyonly for the coefficient of X2, β2(Z), in equation 3.10. However, all theestimators of the coefficients β0(Z), β1(Z) and β2(Z) are very sensitive tothis change in the DGP. This reflects the fact that even with the additionalassumption that Σ12 = 0, the BCLL estimator is not just corrected for thecoefficient β2. All the estimators of coefficients are affected because of theinverse of the additive bias-corrected term in the denominator of the BCLLestimator.563.3. Monte Carlo SimulationsTable 3.1: Finite Sample Comparison of the Local Linear (LL) Estimatorand the Bias-Corrected Local Linear (BCLL) EstimatorDGP Estimators β1(Z) β2(Z) β3(Z)MSE MAD MSE MAD MSE MADDGP (1) LL 1.1625 1.3886 0.7254 0.6272 0.8537 0.6072BCLL 0.2233 0.3040 0.2784 0.2297 0.3082 0.2237DGP (2) LL 1.3908 1.0438 1.4761 1.1572 1.3490 0.9765BCLL 0.8634 0.7173 0.1902 0.3199 0.3856 0.5050Notes: samples are generated from DGP (1) and DGP (2) respectively.The pair of sample sizes are (n = 300,m = 400), which is the same forboth DGP (1) and DPG (2). MSEs and MADs are averages over 500replications.−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−60−50−40−30−20−1001020Intercept  InterceptLLBCLLFigure 3.4: Estimators in two-sample case: intercept, β0(Z)573.3. Monte Carlo Simulations−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5−1−0.500.511.522.5Coefficient of X1  Coefficient of X1LLBCLLFigure 3.5: Estimators in two-sample case: coefficient of X1, β1(Z)583.3. 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And we will distinguish the identity of the Nfirms in the subscript accordingly as firm i or its rival j, ∀i, j ∈ {1, 2, ..., N}.To solve the expected value functions of N firms, we start with the Nthplayer’s problem, which is equivalent to a single agent’s stochastic controlproblem. The solution method has been orderly explored in the book ofDixit and Pindyck (1994) and Stokey (2008). We then solve the expectedvalue function of the (N − 1)th player. Following the same method, we canderive the expected value functions of the kth active firm in the market,k ∈ {1, ..., N}.Suppose that firm i is the Nth player. Before the investment, firm iholds the opportunity to invest, the value of which can also be seen as anoption value. Firm i chooses optimal investment time tiN to maximizeV Ni (yt | x) = suptiNEt(∫ ∞tiNe−r(u−t)DN (x)yudu− e−r(tiN−t)Ci | yt). (A.1)To solve this stochastic control problem, we first define a reward functiongN (s, ys) = Es(∫ ∞se−r(u−s)DN (x)yudu | ys)=ysDN (x)r − µ ,(A.2)and gN (tiN , ytiN ) is denoted as the optimal reward function. Then the valuefunction (A.1) is rewritten asV Ni (yt | x) = suptiNEt((gN (tiN , ytiN )− Ci)e−r(tiN−t) | yt).65A.1. Expected Value Functions for the Stochastic Control ProblemThe solution to the optimal investment time tiN is characterized as whenthe stochastic process yt first crosses the threshold ytiN . Let yNi = ytiN . TheHamilton-Jacobi-Bellman equation in the continuation region(i.e. before theinvestment is made) is:rV Ni (y) = µy∂V Ni (y)∂y+12σ2y2∂2V Ni (y)∂y2. (A.3)The general solution to (A.3) has the following form:V Ni (y) = Ayβ +Byγ ,whereβ =12− µσ2+√(µσ2− 12)2 +2rσ2> 1,andγ =12− µσ2−√(µσ2− 12)2 +2rσ2< 0.Applying the following boundary conditions:V Ni (yNi ) =yNi DN (x)r − µ − Ci, (A.4)∂V Ni (yNi )∂y=DN (x)r − µ , (A.5)limy−→0V Ni (y) = 0. (A.6)(A.4) is the value-matching condition, (A.5) is the smooth-passing conditionand (A.6) is the no bubble condition for value function V Ni (y) at y = 0. Sothe Nth player’s optimal value function and investment trigger is given byV Ni (yt | x) =(yNi DN (x)r−µ − Ci)(ytyNi)βif yt ≤ yNi (wait)ytDN (x)r−µ − Ci if yt > yNi (invest),(A.7)yNi =ββ − 1(r − µ)CiDN (x).When yt ≤ yNi , the value function is the value of the option to wait. Itreflects the present value of the Nth player i’s investment discounted backby a stochastic discount factor yt/yNi from the random time of reaching the66A.1. Expected Value Functions for the Stochastic Control Problemtrigger value yNi . When yt > yNi , the value function is the net present valueof the project when immediate investment is optimal.Next, we will calculate the (N−1)th player’s expected option value. Nowwe assume firm i is the (N − 1)th player and his optimal control problemis to choose optimal investment time tiN−1, given the Nth player’s optimalinvestment time t∗N . As I will show in the next section A.2, although theentry order matters for firm i’s optimal investment timing decision, in anSBNE, the entry order of firm i’s rivals does not affect firm i’s decision. Infact, firm i only considers the expected investment time of its rivals. As aresult, we use t∗N as the expected investment time of the Nth player thathas not invested yet. Then the optimal stochastic problem of the (N − 1)thplayer is:E[V N−1i (yt | x)]= suptiN−1Et(∫ t∗NtiN−1e−r(u−t)DN−1(x)yudu− e−r(tiN−1−t)Ci+∫ ∞t∗Ne−r(u−t)DN (x)yudu | yt)= suptiN−1Et(∫ ∞tiN−1e−r(u−t)DN−1(x)yudu− e−r(tiN−1−t)Ci+∫ ∞t∗Ne−r(u−t) (DN (x)−DN−1(x)) yudu | yt).Applying the reward function definition as in (A.2), we can divide the ex-pected option value function E[V N−1i ] into two parts,E[V N−1i (yt | x)]= suptiN−1Et((gN−1(tiN−1, ytiN−1)− Ci)e−r(tiN−1−t) | yt)+ Et((gN (t∗N , yt∗N )− gN−1(t∗N , yt∗N ))e−r(t∗N−t) | yt).Therefore, the expected option value function E[V N−1i ] is the summationof a perpetual payment flow ytDN−1, minus a sunk cost Ci, starting fromtiN−1 and a perpetual dividend rate yt(DN −DN−1) starting from t∗N .Let y∗N = yt∗N be the expected value of the Nth player’s optimal thresh-old, and yN−1i = ytiN−1 be player i’s threshold as the (N−1)th player. Apply67A.2. Equilibrium Strategiesthe same solution method as in the Nth player’s optimal stochastic controlproblem, we will have the expected value function before the investment as,E[V N−1i (yt | x)]=yN−1i D1(x)r − µ − Ci +y∗N (DN (x)−DN−1(x))r − µ(yN−1iy∗N)β( ytyN−1i)β.When yt ≤ yN−1i , the expected value function of (N − 1)th player is thenet present value from its own investment plus the decrease in the expectedpresent value caused by its rival’s eventual investment.Now suppose that firm i invests as the kth active firm, followed by sub-sequent N − k firms who seize the investment opportunity at time t∗j , withexpected investment threshold y∗j accordingly, j ∈ {k+1, ..., N}. When firmi chooses investment threshold yi, the expected option value function isE[V ki (y | yi, y∗k+1, ...y∗N )]=yiDk(x)r − µ − Ci +N∑j=k+1(y∗j (Dj(x)−Dj−1(x))r − µ)(yiy∗j)β( yyi)β.(A.8)Therefore, the expected option value function E[V ki ] consists of the netpresent value from its own investment, discounting back from the timewhen yt hits yi, plus the net present dividend caused by its rivals’ eventualinvestment, discounting back from the time when yt hits y∗j sequentially,j ∈ {k + 1, ..., N}.A.2 Equilibrium StrategiesRecall that each firm makes its entry-time decision after observing thenumber of active firms in the market and current state value yt. Supposethat firm i chooses a threshold yi at time t, when there are l active firms inthe market, l ∈ {0, ..., N − 1}. Then firm i could enter as the kth firm intothe market and is followed by subsequent N − k investments of i’s rivals,where k ∈ {l + 1, ..., N − 1}. Suppose that firm i’s strategy as the kthactive firm is yi = bk(C), which is a function of investment cost, and firmi’s expected value as the kth active firm is given in (A.8).Now we consider the information set for firm i when firm i chooses to68A.2. Equilibrium Strategiesenter as the kth firm into the market. Each firm has observed a history ofstate values of yt until the time t and obtained the information that none ofthe remaining firms have invested yet. And this information can be used toinfer the probability of the next first-entry firm after time t. Let yˆ be thehighest value of state variable observed until the time t, yˆ = max0≤τ≤t{yτ}and Cˆ = (bk)−1(yˆ) be the highest cost draw that would lead a rival to investat yˆ. And the fact that none of i’s rivals has invested until yˆ is reachedmeans their thresholds are larger than yˆ, then i would learn that all of theN − k rivals have the investment costs Cj > Cˆ, ∀j ∈ {k + 1, ..., N}.The probability that any of i’s rivals j, has cost Cj higher than i’s costC, conditional on Cj > Cˆ is1−G(C)1−G(Cˆ) . To invest first against all of the N − krivals, the probability that none of the N − k rivals will preempt isPr(ti < tj , ∀ j 6= i) =(1−G(Ci)1−G(Cˆ))N−k≡ 1−Hk(Ci). (A.9)Given investment cost Ci, firm i’s problem to find a threshold yi ={bk(Ci), fNi |k = l+ 1, ..., N − 1} is equivalent to find its optimal investmenttype m, m ∈ [Cˆ, CU ], given the threshold function bk(m). Therefore wesolvemaxm∈[Cˆ,CU ]V (yi, y−i, N − l, yt, Cˆ, Ci) =N−1∑k=l+1(1−Hk(m))E[V ki (yt | bk(m), yk+1, ..., yN )]+(1−N−1∑k=l+1(1−Hk(m)))V Ni (yt | fNi ),(A.10)where y−i = {y∗k+1, ..., y∗N |k = l + 1, ..., N − 1}.In the symmetric BNE, all the firms use the same threshold functions{bk(C)|k ∈ {1, ..., N−1}, C ∈ [CL, CU ]}. Therefore, the expected investmenttrigger y∗j , j ∈ {k + 1, ..., N} in the total expected value function A.10 isgiven byy∗j = E[bj(C) | C > Ci, Ci]=∫ CUCibj(C)g(C)dC1−G(Ci) ≡ Bj(Ci), j ∈ {k + 1, ..., N − 1},69A.2. Equilibrium Strategiesy∗N = E[fNC | C > Ci, Ci]=ββ − 1(r − µ)DN (x)∫ CUCiCg(C)dC1−G(Ci) ≡ BN (Ci),wherefNC =ββ − 1(r − µ)CDN (x),fNC is the investment trigger of the Nth active firm. From firm i’s per-spective, when it invests as the kth active firm, the expected trigger y∗j ,j ∈ {k + 1, ..., N} is the same across all of i’s rivals, which depends on firmi’s information set {∀j ∈ {k + 1, ..., N} : Cj > Ci} and firm i’s prior forall its competitors. Therefore, when we substitute yj = Bj(m) into (A.8),∀j ∈ {k + 1, ..., N}, the expected utility E[V ki ] becomesE[V ki (yt | bk(m), Bk+1(m), ..., BN (m))]=(bk(m)Dk(xk)r − µ − Ci)(ytbk(m))β+N∑j=k+1(Bj(m)(DN (xj)−Dj−1(x))r − µ)(ytBj(m))β.As a result, the optimization problem (A.10) is rewritten asmaxm∈[Cˆ,CU ]V (b(m), B(m), N − l, yt, Cˆ, Ci)= maxm∈[Cˆ,CU ]N−1∑k=l+1(1−Hk(m))E[V ki (yt | bk(m), Bk+1(m), ..., BN (m))]+(1−N−1∑k=l+1(1−Hk(m)))V Ni (yt | fNi ).(A.11)70A.3. A Duopoly ExampleThe FOC to this optimization problem is0 =N−1∑k=l+1(1−Hk(m))E[V k′i (yt | bk(m), Bk+1(m), ..., BN (m))]−N−1∑k=l+1Hk′(m)E[V ki (yt | bk(m), Bk+1(m), ..., BN (m))]+N−1∑k=l+1Hk′(m)V Ni (yt | fNi ).When hk(m)1−Hk(m) is properly defined (i.e. 0 < G(m) < 1), the above FOC isrearranged ashk(m)1−Hk(m) =E[V k′i (yt | bk(m), Bk+1(m), ..., BN (m))]E[V ki (yt | bk(m), Bk+1(m), ..., BN (m))]− V Ni (yt | fNi ).where hk(m) ≡ Hk′(m).Substituting the expression for hk(m) and Hk(m) from (A.9), now theFOC for bk(m) becomes(N − k)g(m)1−G(m) =E[V k′i (yt | bk(m), Bk+1(m), ..., BN (m))]E[V ki (yt | bk(m), Bk+1(m), ..., BN (m))]− V Ni (yt | fNi ).(A.12)From the FOC (A.12) and equilibrium condition that m = Ci, we will derivethe optimal strategy functions bk(Ci), ∀k ∈ {l + 1, ..., N − 1} under propertechnical conditions. Before we prove that the threshold functions bk(·),∀k ∈ {l+ 1, ..., N − 1} indeed constitute a SBNE, we will show how to solvethe threshold function bk(·) from the FOC in a two players game.A.3 A Duopoly ExampleWhen k = N − 1, we have the duopoly case: firm i competes against itsrival to decide when to enter the (N − 1)th market. Let b(C) = bN−1(C)and B(C) = BN (C). And the FOC for b(C) isg(C)[E[V N−1i (yt | b(C), B(C))]− V Ni (yt | fNi )]= E[V N−1′i (yt | b(C), B(C))](1−G(C)).71A.3. A Duopoly ExampleThe intuition of the above FOC is that firm i’s expected payoff gainwhen it wins against its rival to enter the (N − 1)th market first is equalto its expected payoff loss when it loses the competition, where g(C) is theprobability that firm i wins at trigger b(C) and 1−G(C) is the probabilitythat one of i’s competitors invest first, conditional on not investing untilb(C) is reached.After substituting the expression for E[V N−1i ] and V Ni into the FOC,and evaluating at the equilibrium condition Ci = C, we obtain the followingdifferential equation for optimal strategy b(C):g(C)1−G(C) =I1I2,whereI1 = (1− β)[b′(C)DN−1(x)r − µ +B′(C)(b(C)B(C))β ((DN (x)−DN−1(x))r − µ)]+ βb′(C)b(C)C,I2 =(b(C)DN−1(x)r − µ − C)+B(C)(DN (x)−DN−1(x))r − µ(b(C)B(C))β−(fNC DN (x)r − µ − C)(b(C)fNC)β,andfNC =ββ − 1(r − µ)CDN (x).To obtain an explicit recursion formula for b(C), we first rewrite the objectivefunction (A.11) in the duopoly case:V (b(m), B(m), 2, yt, Cˆ, Ci)=(1−G(m)1−G(Cˆ))(ytb(m))β (b(m)DN−1(x)r − µ − Ci+(B(m)(DN (x)−DN−1(x))r − µ)(b(m)B(m))β)+(1− 1−G(m)1−G(Cˆ))(ytb(m))β (b(m)fNi)β (fNi DN (x)r − µ − Ci),(A.13)72A.3. A Duopoly ExamplewherefNi =ββ − 1(r − µ)CiDN (x).It is convenient to define the following functions:1−W (m) =(1−G(m)1−G(Cˆ))(ytb(m))β,R(m;Ci) =(B(m)(DN (x)−DN−1(x))r − µ)(b(m)B(m))β−(b(m)fNi)β (fNi DN (x)r − µ − Ci).As a result, the optimization problem (A.13) is reduced asmaxm∈[Cˆ,CU ]V (b(m), B(m), 2, yt, Cˆ, Ci)= maxm∈[Cˆ,CU ](1−W (m))(b(m)DN−1(x)r − µ − Ci +R(m;Ci)).(A.14)The FOC is0 = −w(m)(b(m)DN−1(x)r − µ − Ci +R(m;Ci))+ (1−W (m))(b′(m)DN−1(x)r − µ +R′(m;Ci)),(A.15)where w is W ’s positive and continuous derivative.Here we insert the equilibrium condition m = Ci ≡ C into (A.15) toobtain−Cw(C) = −w(C)(b(C)DN−1(x)r − µ +R(C;C))+ (1−W (C))(b′(C)DN−1(x)r − µ +R′(C;C)).(A.16)To simplify the notation, we define R(C) = R(C;C). However, R′(C) 6=R′(C;C) andR′(C) = R′(C;C) +(b(C)fNC)β,73A.3. A Duopoly ExamplewherefNC =ββ − 1(r − µ)CDN (x).Integrate both sides of (A.16) from C to CU−∫ CUCyw(y)dy =∫ CUC(ddy[(1−W (y)) (b(y)DN−1(x)r − µ +R(y))]− (1−W (y))(b(y)fNy)β)dy,wherefNy =ββ − 1(r − µ)yDN (x).Then use the fact that 1−W (CU ) = 0 since 1−G(CU ) = 0,∫ CUCyw(y)dy= (1−W (C))(b(C)DN−1(x)r − µ +R(C))+∫ CUC(1−W (y))(b(y)fNy)βdy.(A.17)Rearrange (A.17)b(C)DN−1(x)r − µ +R(C)=11−W (C)∫ CUCyw(y)dy −∫ CUC(1−W (y)1−W (C))(b(y)fNy)βdy.(A.18)To obtain an explicit recursion formula, we first integrate∫ CUC yw(y)dyby parts and then substitute the expression for W (y) and 1−W (C) to get,11−W (C)∫ CUCyw(y)dy =11−W (C)(CU − CW (C)−∫ CUCW (y)dy)=11−W (C)(CU − CW (C)−∫ CUC(1−(1−G(y)1−G(Cˆ))(ytb(y))β)dy)= C +(∫ CUC(1−G(y)1−G(C))(ytb(y))βdy).74A.4. Proof of Theorem 1Also,∫ CUC(1−W (y)1−W (C))(b(y)fNy)βdy =∫ CUC(1−G(y)1−G(C))(b(C)fNy)βdy.Therefore, (A.18) is rewritten asb(C)DN−1(x)r − µ = C −R(C) +∫ CUC((b(C)b(y))β−(b(C)fNy)β)(1−G(y)1−G(C))dy,(A.19)whereR(C) =(B(C)(DN (x)−DN−1(x))r − µ)(b(C)B(C))β−(b(C)fNC)β (fNC DN (x)r − µ − C),B(C) =ββ − 1(r − µ)DN (x)∫ CUCiCg(C)dC1−G(Ci) .(A.19) shows that when C approaches its upper endpoint C = CU , we haveb(CU )DN (x)r − µ → CU + VFCU,whereV FCU = (fNCUDN (x)r − µ − CU )(b(CU )fNCU)β.A.4 Proof of Theorem 1Proof of Theorem 1: For the first part of the theorem, suppose thatb∗k(C) satisfies the recursive equation (1.8). Define the last term of equation(1.8) as fk(C; y), wherefk(C; y) =∫ CUCi((bk(Ci)bk(y))β−(bk(Ci)fNy)β)(1−G(y)1−G(Ci))N−kdy.Note first that the integrand fk(C; y) is the product of two increasing func-tions of C. Differentiating the integrand fk(C; y) with C and using the75A.4. Proof of Theorem 1product rule,∂fk(C; y)∂C=∫ CUCβb∗k′(C)b∗k(C)((b∗k(C)b∗k(y))β−(b∗k(C)fNy)β)(1−G(y)1−G(C))N−kdy− (N − k) g(C)1−G(C)∫ CUC((b∗k(C)b∗k(y))β−(b∗k(C)fNy)β)(1−G(y)1−G(C))N−kdy−(1− (b∗k(C)fNC)β).Differentiating Rk(C) with C,Rk′(C)= (1− β)N∑j=k+1B′j(C)((Dj(x)−Dj−1(x))r − µ)(b∗k(C)Bj(C))β+N∑j=k+1βb∗k′(C)((Dj(x)−Dj−1(x))r − µ)(b∗(C)Bj(C))β−1− β b∗k′(C)b∗k(C)(b∗k(C)fNC)β (fNC DN (x)r − µ − C)+(b∗k(C)fNC)β.Now differentiating both sides of (1.8), we obtainb∗k′(C)Dk(x)r − µ = 1−Rk′(C) +∂fk(C; y)∂C.After substituting the expression for Rk′(C) and ∂fk(C;y)∂C into the aboveequation, it is equivalent to the boundary problem (1.6 − 1.7).Conversely, suppose that b∗k(C) satisfies the boundary problem (1.6 −1.7). To prove it is also the solution to the recursive equation (1.8), let ususe the duopoly case in Appendix A.3 for an example. In the duopoly case,the recursive equation (1.8) is the FOC, derived from (A.13). The boundaryproblem (1.6 − 1.7) is the FOC for the same problem with a different butequivalent expression for the value function (A.11). For a general case, wefollow the argument as in the duopoly case. Now the value function can be76A.4. Proof of Theorem 1rewritten asmaxm∈[Cˆ,CU ]N−1∑k=l+1V k(bk(m), Bk+1(m), ..., BN (m), N − l, yt, Cˆ, Ci)= maxm∈[Cˆ,CU ]N−1∑k=l+1(1−W k(m))(bk(m)Dk(x)r − µ − Ci +Rk(m;Ci)),where1−W k(m) =(1−G(m)1−G(Cˆ))N−k (ytbk(m))β,Rk(m;Ci) =N∑j=k+1(Bj(m)(Dj(x)−Dj−1(x))r − µ)(bk(m)Bj(m))β−(bk(m)fNi)β (fNi DN (x)r − µ − Ci).And the recursive equation (1.8) is the FOC to the above maximized valuefunction.For the second part of the theorem, we use the recursive equation (1.8) toprove the existence and uniqueness of b∗k(·), since the boundary problem (1.6− 1.7) is equivalent to the recursive equation (1.8). To prove this, we first usethe theorem from M. Eshaghi Gordji and Baghani (2011), which prove theexistence and uniqueness of the solution to the following nonhomogeneousnonlinear Volterra integral equation:b(c) = f(c) + ϕ(∫ cCLF (c, y, b(y))dy), (A.20)where c ∈ (CL, CU ), b ∈ X, f(·) : (CL, CU ) → R is a mapping, and F isa continuous function. X is a functional space. We take (X, d) to be acomplete metric space withd(f, g) = maxx∈(CL,CU ) | f(x)− g(x) |,for all f, g ∈ X and assume that ϕ is a linear bounded transformation onX.Theorem 14. Suppose the integral equation (A.20) satisfies the followingconditions77A.4. Proof of Theorem 11. ϕ : X → X is a bounded linear transformation,2. f : (CL, CU )→ R and F : (CL, CU )× (CL, c)×X are continuous,3. there exists a integrable function p : (CL, CU ) × (CL, CU ) → R suchthat,|F (c, y, u)− F (c, y, v)| ≤ p(c, y)φ(| u− v |),for each c, y ∈ (CL, CU ) and u, v ∈ X, where φ : [0,∞) → [0,∞)satisfies the following conditions:• φ is increasing,• for each x, φ(x) < x,• lim supx−→t+ φ(x)x < 1, ∀t ∈ [0,∞).4.supc∈(CL,CU )∫ CUCLp2(c, y)dy ≤ 1‖ϕ‖2 (CU − CL),then integral equation (A.20) has a unique fixed point in X.We will check each conditions of the above theorem for integral equation(1.8):bk(C)Dk(x)r − µ = C −Rk(C) +∫ CUC((b(C)b(y))β−(b(C)fNy)β)(1−G(y)1−G(C))N−kdy,whereRk(C) =N∑j=k+1(Bj(C)(Dj(x)−Dj−1(x))r − µ)(bk(C)Bj(C))β−(bk(C)fNC)β (fNC DN (x)r − µ − C),given BK+1(·), ..., BN (·) defined in (1.4 − 1.5).Conditions 1− 2 are straightforward. We will focus on conditions 3 and4. First, we define the following functions:(b1(C)b1(y))β−(b1(C)fNy)β≡ G(b1(c), b1(y), fNy ) ≡ G(p1, q1, fNy ),where G(p, q, fNy ) = pβ(1− (q/fNy )β)/qβ.78A.4. Proof of Theorem 1Similarly, we have(b2(C)b2(y))β−(b2(C)fNy)β≡ G(b2(c), b2(y), fNy ) ≡ G(p2, q2, fNy ).Since G(p, q, fy) is continuously differentiable in p and q, we haveG(p1, q1, fNy )−G(p2, q2, fNy )= G1(p?, q?, fNy )|p1 − p2|+G2(p?, q?, fNy )|q1 − q2|≤ (G1(p?, q?, fNy ) +G2(p?, q?, fNy )) maxy∈(CL,CU )|b1(y)− b2(y)|=G1(p?, q?, fNy ) +G2(p?, q?, fNy )µµ | b1(y∗)− b2(y∗) |,where G1 and G2 are the partial differential functions of G with respect tothe first and the second variable; p? and q? lie in [p1, p2] and [q1, q2] respec-tively; constant µ is chosen such that 0 < µ < 1; y∗ = argmaxy∈(CL,CU ) |b1(y)−b2(y)|. Now, we set φ(t) = µt andp(c, y) =G1(p?, q?, fNy ) +G2(p?, q?, fNy )µ(1−G(y)1−G(c))N−k.Hence| F (c, y, b1)− F (c, y, b2) |=| (G(p1, q1, fNy )−G(p2, q2, fNy ))(1−G(y)1−G(c))N−k|≤ p(c, y)µ | b1(y∗)− b2(y∗) |,then condition 3 is satisfied.To check condition 4, notesupc∈(CL,CU )∫ CUCLp2(c, y)dy ≤(G1(p?, q?, fNy¯ ) +G2(p?, q?, fNy¯ )µ)2(CU − CL) ,where y¯ lie in (CL, CU ). We will choose an interval (CL, CU ) large enoughand a value of µ to satisfy condition 4. Therefore, integral equation (1.8)has a unique fixed point.For part 3 of the theorem, we need to establish that setting m = Ci,i.e., truth-telling, always maximizes the expected payoff. Let V k(bk(m), Ci)be the expected payoff to player i, when his investment cost is Ci and his79A.4. Proof of Theorem 1threshold is bk(m). Assume that all of his rivals use thresholds bk(Cj), ∀j 6=i. By Theorem 1, we solve the threshold functions backward. Therefore wewill complete the proof by showing V k(bk(Ci), Ci) > Vk(bk(m), Ci), ∀k ∈{1, ..., N − 1}.Firstly, we recall thatV k(bk(m), Ci) =(1−W k(m))(bk(m)Dk(x)r − µ − Ci +Rk(m;Ci)),where1−W k(m) =(1−G(m)1−G(Cˆ))N−k (ytbk(m))β,Rk(m;Ci) =N∑j=k+1(Bj(m)(Dj(x)−Dj−1(x))r − µ)(bk(m)Bj(m))β−(bk(m)fNi)β (fNi DN (x)r − µ − Ci).Substituting the optimal threshold function bk(·) into V k(bk(m), Ci), wehaveV k(bk(m), Ci) =∫ CUmywk(y)dy −(1−W k(m))Ci,where wk(·) is W k(·)’s derivative. Hence,V k(bk(Ci), Ci)− V k(bk(m), Ci) = (m− Ci)W k(m) +∫ CimW (y)dy.Applying the mean value theorem, there exists a point x between Ci and m,such thatV k(bk(Ci), Ci)− V k(bk(m), Ci) = (m− Ci)(W k(m)−W k(x)) ≥ 0.since W k(·) is an increasing function and x is between Ci and m guaranteedby mean value theorem. Hence when m = Ci, bk(·) is a best responsefunction and bk(·) is a symmetric BNE.80Appendix BAppendix to Chapter 2B.1 Proof of Theorem 4Suppose that ξi(y) and ξj(y) are the break-even functions for each firm.We study the behaviour of both functions when the domain of the functionsare (0, fi). First, we rewrite these functions in the following formξi(y) = −aiyβ + ciy − 1,where the coefficients are given byai =1β − 1[β(1li− 1fi)(1fj)β−1+(1fi)β],ci =ββ − 11li.Similarly, we have the expression of ξj(y) for firm j by switching theindexes i and j in the expression of ξi(y).Clearly, ξi(y) and ξj(y) intersect at a unique non-zero pointyˆ =(cj − ciaj − ai) 1β−1,where ci > cj because of li < lj and sign(ai − aj) = sign(yˆ).We will show ξi(y) > ξj(y) holds when y ∈ (0, fi). Firstly, we will proveξi(fi) > 0 and ξj(fi) ≤ 0 as follows:ξi(fi) =1Ki[V Li (fi)− V Fi (fi)]=1Ki1fj[(D1 −D2r − µ)(fifj)−(D1 −D2r − µ)(fifj)β]> 0,(B.1)81B.2. Proof of Theorem 5ξj(fi) =1Kj[V Lj (fi)− V Fj (fi)]=1Kj[D2fir − µ −Kj −(D2fjr − µ −Kj)(fifj)β]≤ 0.(B.2)Note that (B.1) holds due to the following properties: D1 > D2, fi < fj , andβ > 1. And (B.2) is from the expression of the value function of firm j beinga follower: when evaluated at the time that y = fi < fj , the opportunityvalue before investment is larger than the value after the investment.As a result, ξi(fi) > ξj(fi) while is equivalent to the following expression(aj − ai)fβi + (ci − cj)fi > 0.Therefore, we must have fi <(cj−ciaj−ai) 1β−1= yˆ when aj − ai < 0. Thismeans ξi(y) and ξj(y) do not intersect on the interval (0, fi) and we musthave ξi(y) > ξj(y) when y ∈ (0, fi). However, when aj − ai > 0, since yˆ < 0and ξi(fi) > ξj(fi), we also have ξi(y) > ξj(y) when y ∈ (0, fi).Since bi and bj are the lowest value that ξi(y) = 0 and ξj(y) = 0 respec-tively when y ∈ (0, fi), bi is always lower than bj .B.2 Proof of Theorem 5Firstly, we show that bi < li < fi < fj . We evaluate firm i’s break-evenfunction at li,ξi(li)=1Ki(liD1(x)r − µ −Ki +fj(D2(x)−D1(x))r − µ(lifj)β−(fiD2(x)r − µ −Ki)(lifi)β)=1Ki(1β − 1Ki +fj(D2(x)−D1(x))r − µ(lifj)β− fiD2(x)β(r − µ)(lifi)β)=1β − 1(1−(D2(x)D1(x))β−1(β(fifj)β−1(1− D2(x)D1(x))+D2(x)D1(x)))>1β − 1(1−(D2(x)D1(x))β−1(β(1− D2(x)D1(x))+D2(x)D1(x))).82B.3. Proof of Theorem 6where the last inequality comes from the assumption that D2(x)/D1(x) < 1(negative externalities) and fi < fj (cost assumption of K˜i2 < K˜j2 whenKi < Kj), which implies (fi/fj)β−1 < 1 when β − 1 > 0.Denote x = D2(x)D1(x) and define a function g(x) as the following:g(x) = 1− (x)β−1(β(1− x) + x).We have g(1) = 0 and g(x)′ = β(β − 1)xβ−2(x − 1) < 0 when x < 1.Consequently, g(D2(x)/D1(x)) > 0 and ξ(li) > 0 implies bi < li.li < fi holds from the negative externalities assumption that D2(x) <D1(x). Also, bj < fi because firm j’s break-even function ξj(y) can only bepositive when y < fi. Therefore, inequalities bi < min(bj , li) < fi < fj holdin this game.By considering the thresholds inequalities and the properties that boththe leader’s and the follower’s value functions are strictly increasing (Pawlinaand Kort (2006), Janssens and Kort (2012)), it is better off for the low costfirm i to be the leader than be the follower. Therefore firm i’s optimalstrategy is to invest at min(bj , li).B.3 Proof of Theorem 6Firstly, we study the conditions when the sequential equilibrium (li, fj)is played, which is implied by li < bj . Since bj is the smallest value to makeξj(y) = 0, therefore we haveli < bj ⇔{ξj(li) < 0 , andξ′j(li) > 0.Substituting the expressions of ξj(li) and ξ′j(li) into the above two inequal-ities, we obtain(ββ − 1)lilj− 1 < 1β − 1[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi ,(ββ − 1)lilj>ββ − 1[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi .83B.4. A Three-player Real Options Game with Complete InformationCombining the above two inequalities, we have(ββ − 1)lilj− 1 < 1β − 1[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi<(1β − 1)lilj.(B.3)Next, we study the conditions that the preemptive equilibrium (bj , fj)is played, which means bj < li. Similarly, condition ξj(li) > 0 holds(ββ − 1)lilj− 1 > 1β − 1[β(1lj− 1fj)(1fi)β−1+(1fj)β]lβi . (B.4)Also note because of li < lj the following inequality always holds1β − 1lilj>(ββ − 1lilj− 1).Then, we must also have ξ′j(li) > 0 when ξj(li) > 0. As a result, bj < liwhen condition (B.4) holds.B.4 A Three-player Real Options Game withComplete InformationThis section develops a three-player real options game which is an ex-tension of the asymmetric duopoly game studied intensively by Pawlina andKort (2006) and Kong and Kwok (2007), etc. In the two-player game, theentry order is according to the efficiency order, such that the more efficientfirm with a lower entry cost always enters the market first. In the three-player game, however, the entry order is not necessarily the same as theefficiency order. Argenziano and Schmidt-Dengler (2012) show that the in-efficient firm can preempt the efficient firm in the three-player game evenwhen there are two types of players.B.4.1 SetupAssume there are three different risk-neutral developers who compete forthe optimal timing of real estate development projects. Each developer isassumed to have a perpetual investment opportunity on a development site.The investment decision is irreversible and the sunk cost of investment is84B.4. A Three-player Real Options Game with Complete Informationdifferent for each firm. Before the investment, the developers have an optionto wait for their optimal timing of investment and receive no profits. At anypoint in time, a firm i can choose to invest an amount of Ki, i = {A,B,C},and obtain an instantaneous profit flow ofpi(yt, Dj(x)) = ytDj(x), j = {1, 2, 3},where j stands for the total number of firms that have entered. For example,when j = 1, pi(yt, D1(x)) stands for the monopoly profit for firm i.The state variable yt represents demand shock of real estate market andfollows a Geometric Brownian Motion:dyt = µytdt+ σytdWt,where µ and σ are instantaneous drift parameter and instantaneous stan-dard deviation parameter. dt is the time increment and dWt is the Wienerincrement. Parameter µ can be interpreted as the industry growth rate andparameter σ as the industry volatility. We assume the stochastic processstarts from y0 low enough such that immediate investment is not optimalfor either firm.The deterministic multiplier Dj(xi), i = {A,B,C}, stands for the mar-ket externality. Note the firms are asymmetry in the sunk costs. xi arecovariates used for identification. We define the profit adjusted cost for firmi, i = {A,B,C}, as K˜ij = Ki/Dj(x), j = {1, 2, 3}.The following assumptions are made:Assumption 17. D1(x) > D2(x) > D3(x).Assumption 18. KA < KB < KC .Assumption 19. K˜Aj < K˜Bj < K˜Cj, j = {1, 2, 3}.B.4.2 The Two-Firm SubgameIn this section, we use backward induction to analyze this complete in-formation game. Suppose that the first firm has already entered the market,and we consider the outcome of the ensuing two-firm subgame. This sub-game is analogous to the asymmetric duopoly game analyzed by Pawlinaand Kort (2006). The outcome is that the lower cost firm always entersthe market first. The optimal development time for the last player is de-rived in Appendix A.1. The optimal investment thresholds for the follower85B.4. A Three-player Real Options Game with Complete Informationi, i = {A,B,C}, is given asfi =ββ − 1(r − µ)KiD2(x).In this duopoly game, the leader i’s the investment trigger without pre-emption threat is given byli =ββ − 1(r − µ)KiD1(x). (B.5)This is analogous to the stand-alone investment trigger in Argenziano andSchmidt-Dengler (2012). Without preemption threat, the leader will act asa single decision maker and choose the optimal investment time to receivethe perpetual payment of ytD1(x) minus the investment cost Ki, and theperpetual dividend of yt(D2(x) − D1(x)), starting at Tj . And the followerwill invest at fjWith preemption threat, we have to consider each firm’s incentive to bethe leader in the two-firm subgame. As in Kong and Kwok (2007), we definethe functionξi(y) ≡ VLi (y)− V Fi (y)Ki, i = {A,B,C},as a measurement of the preemptive incentive of firm i. If ξi(y) is positive,then firm i prefers to be the leader rather than the follower. Define bi asthe lowest value of y such that ξi(y) = 0, and bi is the preemptive triggerof firm i, i = {A,B,C}. Therefore, the preemptive trigger is the first timea firm’s value function in a leader position exceeds in a follower position.Consequently, one firm’s preemptive trigger must occur before both its rivalfollowers’ triggers (when the firm is in a leader position) and the follower’strigger for itself (when the firm is in a follower position).B.4.3 Multiple Equilibria in the Three-Player GameBefore we analyze the equilibrium strategy of each firm in the three-firmgame, we need to identify the entry order first. However, once the the firstidentity is known, the following two firm enter the market according to theirefficiency order (lower cost firm enters first). Therefore, if firm A is the firstto invest, then the order of entry must be A− B − C; if firm B is the firstinvestor, then the entry order becomes B − A − C; and if firm C is theleader, then the order is C−A−B. Like in the two-player game, we need toanalyze each firm’s incentive to be the leader and determine their optimal86B.4. A Three-player Real Options Game with Complete Informationentry time accordingly.Firstly, we analyze the equilibrium conditions for the equilibrium resultA−B−C. We start with analyzing firm A’s incentive to be the first investor,given the strategies of B and C, which is the relative positions of B and C inthe entry order. We need to compare firm A’s leader value function and itsfollower valuer function, which is equivalent to compare the value functionof A in the entry order A−B −C and in the entry order B −A−C. Next,we need to make sure firm B has no incentive to be the first investor, sowe need to compare firm B’s leader value function and its follower valuerfunction, which is equivalent to compare the value function of B in the entryorder B − A − C and in the entry order A − B − C. Finally, we need toverify firm C has no incentive to deviate its follower’s position in the entryorder. So we need to compare firm C’s leader value function and its followervaluer function, which is equivalent to compare the value function of C inthe entry order C −A−B and in the entry order A−B − C.Similarly we can analyze the equilibrium conditions for the other typesof equilibrium, B − A − C and C − A − B. But we can not separate thesethree different equilibria like in the duopoly case and as a result, we can notachieve point identification for all the model primitives.Suppose that we are considering the entry order i − j − k, then valuefunction of the first entry in the three-player game isV L1i (yt | y2j , y3k)=ytD1(x)r − µ −Ki +y2j(D2(x)−D1(x))r − µ(yty2j)β+y3k(D3(x)−D2(x))r − µ(yty3k)β if yt ≤ y2j ,ytD2(x)r − µ −Ki +y3k(D3(x)−D2(x))r − µ(yty3k)βif y2j < yt ≤ y3k,ytD2(x)r − µ −Ki if yt > y3k,(B.6)where y2j is the threshold of the second entry, and y3k is the thresholdof the third entry. y2j and y3k are derived as in the two-players subgame.After the first entry, if firm i enters the market as the second entry, then87B.4. A Three-player Real Options Game with Complete Informationfirm i’s value function isV L2i (yt | y3k) = ytD1(x)r−µ −Ki + y3k(D2(x)−D1(x))r−µ(yty3k)βif yt ≤ y3k,ytD2(x)r−µ −Ki if yt > y3k.(B.7)Finally, if firm i is the third entry, then its value function isV 3Fi (yt | x) =(y3iD3(x)r−µ −Ki)(yty3i)βif yt ≤ y3i,ytD3(x)r−µ −Ki if yt > y3i,(B.8)wherey3i =ββ − 1(r − µ)KiD3(x).Next we evaluate the incentive for each firm, such that i − j − k is theequilibrium entry order. For firm i, define i’s incentive function asξi(yt) = VL1i (yt)− V L2i (yt), yt < y2j .Suppose that b1i is the smallest value such that ξi(yt) = 0, which is thebreak-even value of firm i.Similarly, for firm j, define j’s incentive function asξj(yt) = VL1j (yt)− V L2j (yt), yt < y2i.Suppose that b1j is the smallest value such that ξj(yt) = 0, which is thebreak-even value of firm j.For firm k, define k’s incentive function asξk(yt) = VL1k (yt)− V 3Fk (yt), yt < y3k.Suppose that b1k is the smallest value such that ξk(yt) = 0, which is thebreak-even value of firm k.When b1i < b1j < b1k < y2j , then firm i’s optimal threshold ismin{li1, b1j},li1 =ββ − 1(r − µ)KiD1(x).And firm j and firm k’s optimal threshold are y2j and y3k respectively.When b1i < b1k < b1j < y2j , then firm i’s optimal threshold ismin{li1, b1k},and firm j and firm k’s optimal threshold are y2j and y3k respectively.88B.4. A Three-player Real Options Game with Complete InformationHowever, both case b1i < b1j < b1k < y2j and case b1i < b1k < b1j < y2jare overlapped in the parameter space of investment cost, and can be shownto be nonempty numerically (see Argenziano and Schmidt-Dengler (2012)and Bouis, Huisman and Kort (2009)). In fact, because of the entry order,now the parameter space of the model to be identified is larger than theduopoly case. In the duopoly case, we observe the actions of the players(TN , TN−1) and need to identify the parameters of the payoff structure,which includes two groups of parameters, (DN (x), µ, σ,G(·)) and DN−1(x).Now in the three-player case, we have to identify three groups of parametersin the payoff structure and two of the possible entry orders as well, whilewe only observe three possible actions (TN , TN−1, TN−2). Therefore theparameters of the model can not be identified. As a result, the identificationof this model has to be achieved in other ways.89Appendix CAppendix to Chapter 3C.1 Convergence of the DenominatorProof of Lemma 9: Recall thatDn,0(z) =1nn∑i=1Kh(Z2j(i) − z)(X1iXT1i X1iX2j(i)X2j(i)XT1i X22j(i)).Let Dn,0(j1,j2)(z) denote the (j1, j2)th block matrix element of Dn,0(z). Thenwe will prove that Dn,0(22)(z)p−→ f(z)Ω(22)(z) by showing that Dn,0(22)(z)converges to f(z)Ω(22)(z) in mean-square. The convergence of the rest ofDn,0(j1,j2)(z) can be shown in a similar way.Before we proceed, we first outline some properties of Z2j(i), i ∈ {1, ..., n}.Let fj(i)(·), i ∈ {1, ..., n} denote the density of Z2j(i), i ∈ {1, ..., n}, which isalso the distribution of Z2j conditional on it being a nearest match to Z1i,j ∈ {1, ...m}. Because the density of Z2j is f(z), thenfj(i)(z) =m∑j=1Pr(j(i) = j | Z2j = z)f(z)= f(z) ·m∑j=1(1m+ o(1m))= f(z)(1 + o(1)),where the second equality comes from the conditional probability resultderived in Abadie and Imbens (2006)’s Additional Proofs on page 5. Theabove implies that the density of any Z2j(i) differs from another density ofZ2j(k) and the population density by o(1), i, k ∈ {1, ..., n}, k 6= i. Also, C(j)is defined as the number of times that unit j in the sample {X2j , Z2j}mj=1 is90C.1. Convergence of the Denominatorused as a match to unit i in the sample {Yi, X1i, Z1i}ni=1,C(j) =n∑i=11(j = j(i)), j ∈ {1, ...m}where 1(·) is the indicator function, equal to one if j = j(i) is true and zerootherwise. Then E(C(j) | Z2j) = nm(1+o(1)) is given in Abadie and Imbens(2006)’s Additional Proofs on page 11.Let Z1,2 denote all the Z from the two samples. Note that, the condi-tional expectation of Dn,0(22)(z) isE(Dn,0(22)(z) | Z1,2)=1nn∑i=1Kh(Z2j(i) − z)Ω(22)(Z2j(i))=1nm∑j=1C(j)Kh(Z2j − z)Ω(22)(Z2j).And the unconditional expectation isE(Dn,0(22)(z))= E(1nn∑i=1Kh(Z2j(i) − z)Ω(22)(Z2j(i)))= E 1nm∑j=1C(j)Kh(Z2j − z)Ω(22)(Z2j)=1nm∑j=1E(E(C(j) | Z2j)Kh(Z2j − z)Ω(22)(Z2j))=mnE( nm(1 + o(1))Kh(Z2j − z)Ω(22)(Z2j))= f(z)Ω(22)(z)(1 + o(1) +O(h2)).Next we are going to show the convergence of the variance of Dn,0(22)(z).Apply the law of total variance,V ar(Dn,0(22)(z))= E(V ar(Dn,0(22)(z) | Z1,2))+ V ar(E(Dn,0(22)(z) | Z1,2)).First we consider V ar(E(Dn,0(22)(z) | Z1,2)). Since Ω(z) = E(XX ′ | z) is91C.1. Convergence of the Denominatora matrix, we transform it into a vector for its convergence. Let Ω(22)(z) bethe (2, 2)th block matrix element of Ω(z). DenoteAj = vec(Kh(Z2j − z)Ω(22)(Z2j)− f(z)Ω(22)(z)),Aj(i) = vec(Kh(Z2j(i) − z)Ω(22)(Z2j(i))− f(z)Ω(22)(z)).Then V ar(E(Dn,0(22)(z) | Z1,2))can be rewritten asV ar(E(Dn,0(22)(z) | Z1,2))= E((1nn∑i=1Aj(i))(1nn∑i=1Aj(i))T)=1n2E(n∑i=1n∑t=1Aj(i)ATj(t))=1n2E n∑i=1∑j(t)=j(i)t∈{1,...,n}Aj(i)ATj(t)+ 1n2E n∑i=1∑j(t)6=j(i)t∈{1,...,n}Aj(i)ATj(t)=1n2E(n∑i=1Aj(i)C(j(i))ATj(i))+1n2n∑i=1∑j(t)6=j(i)t∈{1,...,n}E(Aj(i))E(ATj(t))≤ 1n2E(n∑i=1( maxj=1,...,mC(j))Aj(i)ATj(i))+n2 − nn2(o(1)2 +O(h2)2)=1n2E m∑j=1( maxj=1,...,mC(j))C(j)AjATj+O(h4) + o(1)2=1√n(mn) 32 E((1√mmaxj=1,...,mC(j))C(j)AjATj)+O(h4) + o(1)2=1√n(mn) 32 E(1√m( maxj=1,...,mC(j))2AjATj)+O(h4) + o(1)2≤ 1√n(mn) 32 E(1√m( maxj=1,...,mC(j))2)µ¯2 +O(h4) + o(1)2,where µ¯2 = supZj ‖AjATj ‖. µ¯2 is finite due to Assumption 14 and As-sumption 15, which implies that Ω and Kh are continuous on the finitesupport of Z, and in turn implies that Kh and Ω satisfy the Lipschitz condi-92C.1. Convergence of the Denominatortion. Next we are going to prove that E(1√m(maxj=1,...,mC(j))2)is boundeduniformly in m, which implies that V ar(E(Dn,0(22)(z) | Z1,2))convergesto zero matrix as n,m go to infinity of the same order.Following the proofs on Page 23 in Abadie and Imbens (2006)’s Addi-tional Proofs, by Bonferroni’s inequality:E((1√m( maxj=1,...,mC(j))2)2)=1mE(maxj=1,...,mC(j)4)=1m∞∑N=0Pr( maxj=1,...,mC(j)4 > N)≤ 1m∞∑N=0mPr(C(j)4 > N)≤ E (C(j)4)(C.1)The first equation holds because C(j) is a positive integer for all j =1, . . . ,m. According to Lemma 3 in Abadie and Imbens (2006) , givenAssumption 13 and part 1 of Assumption 14, E (C(j)q) is bounded uni-formly in m for all q > 0. SinceE(1√m( maxj=1,...,mC(j))2)2≤ E((1√m( maxj=1,...,mC(j))2)2), (C.2)which implies the boundedness of E(1√m(maxj=1,...,mC(j))2), and in turnimplies that V ar(E(Dn,0(22)(z) | Z1,2))converges to zero matrix as n,mgo to infinity of the same order. By the same method, we can easilyshow that under certain finite moment assumptions, the expectation ofV ar(Dn,0(22)(z) | Z1,2)goes to zero as sample sizes go to infinity.To conclude, we have showed that V ar(Dn,0(22)(z))goes to zero assample sizes go to infinity and therefore Dn,0(22)(z) converges to f(z)Ω(22)(z)in mean-square. Similarly, we can show that all the block matrix elementsof Dn,0(z) converges to the respective block matrix elements of f(z)Ω(z) inmean-square. The convergence of Dn(z) to f(z)Ω(z)⊗( 1 00 µ2)is provedin the same line of arguments.93C.2. Convergence of the NumeratorC.2 Convergence of the NumeratorProof of Lemma 10: We approximate the expression of Y = XTi β(Z1i)+ui ' XTi β(Z2j(i)) + ui by a Taylor expansion in the neighbourhood of|Z1i − z| < h and |Z2j(i) − z| < h.XTi β(Z1i) = XTi β(z) + (Z2j(i)− z)XTi β′(z) +h22(Z2j(i) − zh)2β(z)′′+ o(h2) a.s.,where β′(z) and β(z)′′ are the vectors consisting of the first and the secondderivatives of the function β(z).Then we substitute the approximate expression of Y into the expressionof Nn(z).Nn(z) =1n(DXm)TWY=1n(DXm)TW(DX(β(z)hβ′(z))+h22AzXTβ′′(z) + u+ o(h2)),whereDXm =XTj(1) XTj(1)Z2j(1)−zh... ...XTj(n) XTj(n)Z2j(n)−zh ,W = diag(Kh(Z2j(1) − z), ...,Kh(Z2j(n) − z)),DX =XT1 XT1Z2j(1)−zh... ...XTn XTnZ2j(n)−zh ,andAz = diag((Z2j(1) − zh)2, ..., (Z2j(n) − zh)2).Recall that Xj(i) = (XT1i, X2j(i))T is the matching pair for Xi = (XT1i, X2i)T ,i ∈ {1, ...n}.Next we decompose Nn(z)− 1n(DXm)TWDXm(β(z)hβ′(z))= I1 + I2 + I3 +I4, whereI1 =1n(DXm)TW(DX(β(z)hβ′(z))− E(DX | Z1,2)(β(z)hβ′(z))),94C.2. Convergence of the NumeratorI2 =1n(DXm)TW(E(DX | Z1,2)(β(z)hβ′(z))−DXm(β(z)hβ′(z))),I3 =1n(DXm)TWh22AzXTβ′′(z),I4 =1n(DXm)TWu.Firstly, we consider I1. SinceDX − E(DX | Z1,2) = (X1 − g(Z11))T (X1 − g(Z11))T(Z2j(1)−z)h... ...(Xn − g(Z1n))T (Xn − g(Z1n))T (Z2j(n)−z)h ,thenI1 =(1n∑ni=1Xj(i) (Xi − g (Z1i))T Kh(Z2j(i) − z)1nh∑ni=1Xj(i) (Xi − g (Z1i))T(Z2j(i) − z)Kh(Z2j(i) − z) )β(z)+(1nh∑ni=1Xj(i) (Xi − g (Z1i))T(Z2j(i) − z)Kh(Z2j(i) − z)1nh2∑ni=1Xj(i) (Xi − g(Z1i))T(Z2j(i) − z)2Kh(Z2j(i) − z) )hβ′(z).We will show thatI1p−→ f(z)(Σ11 Σ120 0)β(z)⊗(10)+ f(z)(Σ11 Σ120 0)hβ′(z)⊗( 0µ2).DenoteI10 =1nn∑i=1Xj(i)(Xi − g(Z1i))TKh(Z2j(i) − z),I11 =1nhn∑i=1Xj(i)(Xi − g(Z1i))T (Z2j(i) − z)Kh(Z2j(i) − z),andI12 =1nhn∑i=1Xj(i)(Xi − g(Z1i))T (Z2j(i) − z)2Kh(Z2j(i) − z).Next, we will show that I10p−→ f(z)(Σ11(z) Σ12(z)0 0). Suppose thatj(i) = j, then X2j is the match to X2i. Denote X(i,j) = (XT1i, XT2j)T as thematch to Xi = (XT1i, X2i), when j(i) = j.95C.2. Convergence of the NumeratorE(I10 | Z1,2) = 1nm∑j=1E ∑i∈A(j)X(i,j)(Xi − g(Z1i))TKh(Z2j − z) | Z1,2=1nm∑j=1Kh(Z2j − z)E ∑i∈A(j)X(i,j)(Xi − g(Z1i))T | Z1,2=1nm∑j=1Kh(Z2j − z)∑i∈A(j)E((X1iX2j)(v1i, v2i)T | Z1,2)=1nm∑j=1Kh(Z2j − z)∑i∈A(j)(Σ11(Z1i) Σ12(Z1i)0 0)=1nm∑j=1Kh(Z2j − z)C(j)(Σ11(Z2j) Σ12(Z2j)0 0)(1 + o(1)) .The next to the last equality is from the fact that {X1i}ni=1 and {X2j}mj=1 aretwo independent samples from the same population. Since A(j) is definedas the subset of the index i, i ∈ {1, ..., n}, such that j is used as a matchto each indexed observation, then the number of elements in the set A(j) isC(j), j ∈ {1, ...m}. Also C(j) is nonstochastic conditional on Z1,2. As wediscussed before, for a pair of match, Z2j(i) and Z1i, their density only differby o(1). So for all i ∈ A(j), the density of Z1i and Z2j differ by o(1) as well.Therefore, the last equality holds.To compute the unconditional expectation of I10, we apply the result ofE(C(j)|Z1,2) = nm(1 + o(1)) from Abadie and Imbens (2006). ThenE(I10) = f(z)(Σ11(z) Σ12(z)0 0).The convergence of the variance V ar(I10) is in the Appendix C.3. Asa result, we have the convergence result of I10. Similarly, we can show theconvergence of I11 and I12. And then the convergence of I1 is straightfor-ward.Next we consider the convergence of I2. SinceE(DX | Z1,2)−DXm = (g(Z11)−Xj(1))T (g(Z11)−Xj(1))T(Z1j(1)−z)h... ...(g(Z1n)−Xj(n))T (g(Z1n)−Xj(n))T (Znj(n)−z)h ,96C.3. Convergence of V ar(I10)thenI2 =(1n∑ni=1Xj(i)(g(Z1i)−Xj(i))TKh(Z2j(i) − z)1nh∑ni=1Xj(i)(g(Z1i)−Xj(i))T (Z1i − z)Kh(Z2j(i) − z))β(z)+(1nh∑ni=1Xj(i)(g(Z1i)−Xj(i))T (Z1i − z)Kh(Z2j(i) − z)1nh2∑ni=1Xj(i)(g(Z1i)−Xj(i))T (Z1i − z)2Kh(Z2j(i) − z))hβ′(z).Following a similar argument, we can show thatI2p−→ f(z)( −Σ11 00 −Σ22)β(z)⊗(10)+ f(z)( −Σ11 00 −Σ22)hβ′(z)⊗( 0µ2)Finally, we can compute in a similar way to prove that I3 = o(h2) andI4 = op(1).C.3 Convergence of V ar(I10)Let vec(·) denote the vectorization transformation of matrix and V I10 =vec(I10).Consider the variance decomposition,V ar (V I10) = E(V ar(V I10 | Z1,2))+ V ar(E(V I10 | Z1,2)).In the following, we will prove that the convergence of both E(V ar(V I10 | Z1,2))and V ar(E(V I10 | Z1,2))as sample sizes go to infinity.First consider E(V ar(V I10 | Z1,2)). LetB(i,j) = Kh(Zj(i) − z) · vec(Xj(i)(Xi − g(Zi))T −(Σ11(Zi) Σ12(Zi)0 0)),B(i,i) = Kh(Zi − z) · vec(Xi(Xi − g(Zi))T − Σ).97C.3. Convergence of V ar(I10)ThenE(V ar(V I10 | Z1,2))= E((V I10 − E(V I10 | Z1,2)) (V I10 − E(V I10 | Z1,2))T)= E((1nn∑i=1B(i,j))(1nn∑i=1B(i,j))T)≤ 1√n(mn) 32 E(1√m( maxj=1,...,mC(j))2B(i,i)BT(i,i))+O(h4) + o(1)2Following the same assumptions and argument in proof of Lemma 9, wehave that E(1√m(maxj=1,...,mC(j))2B(i,i)BT(i,i))is bounded uniformly in allm and therefore E(V ar(V I10 | Z1,2))converges to zero matrix as samplesize n,m go to infinity of the same order.Before we proceed, we introduce a useful lemma from Abadie and Imbens(2006) about the distribution of the matching discrepancy. Suppose that wehave a random sample Z1, . . . , ZN , with density f over bounded supportZ. Now consider the closest match to a z ∈ Z in the sample. Let j1 =argminj=1,...,N‖Zj − z‖ and let Uj1 = Zj1 − z be the matching discrepancy.Lemma 15. Matching Discrepancy - Asymptotic Properties: Sup-pose that f is differentiable in a neighbourhood of z. Then Uj1 = Op(N−1).Moreover, the first two moments of Uj1 are O(N− 12 ).Now consider V ar(E(V I10 | Z1,2)). DefineAi,j(i) = Kh(Zj(i)−z)·vec(Σ11(Zi) Σ12(Zi)0 0)−fz(z)·vec(Σ11(z) Σ12(z)0 0),Ai,i = Kh(Zi−z)·vec(Σ11(Zi) Σ12(Zi)0 0)−fz(z)·vec(Σ11(z) Σ12(z)0 0).By Lemma 15 and Lipschitz assumption on Kh, ‖Ai,j(i)−Ai,i‖ = Op(N−1).Also note that by simple calculation, we haveE(Kh(Zi − z)(Σ11(Zi) Σ12(Zi)0 0))= fz(z)(Σ11(z) Σ12(z)0 0)+O(h2),98C.4. Boundedness of V (z)which implies that E(Ai,i) = O(h2). Therefore,V ar(E(V I10 | Z1,2))= E((1nn∑i=1Ai,j(i))(1nn∑i=1Ai,j(i))T)=1√n(mn) 32 E(1√m( maxj=1,...,mC(j))2A(i,i)AT(i,i))+O(h4) + o(1)2Note that under the Lipschitz assumption onKh and Σ, V ar(E(V I10 | Z1,2))goes to zero matrix as sample sizes go to infinity. Therefore we have demon-strated that I10 converges in probability to f(z)(Σ11(z) Σ12(z)0 0).C.4 Boundedness of V (z)The following shows that the expectation of V (z) is finite. The resultfollows Lemma 3 in Abadie and Imbens (2006).V (z) =m∑j=1σ2(C(j)Kh(Zj − z))2Ω(Zj)⊗( 1 (Zj − z)/h(Zj − z)/h ((Zj − z)/h)2).From Lemma 3 in Abadie and Imbens (2006), C(j) is O(1) , j ∈ {1, ...,m}.By Assumptions 15−16, the kernel functions we consider are Lipschitzcontinuous on a compact set, and νj =∫ujK2(u)du, j = 1, 2, 3 are alsobounded, therefore V (z) is also bounded.99

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