Lecture 7: Two-step methods 2...

Lecture 7: Two-step methods 2Pakes, Ostrovsky, and Berry (2007)

May 13, 2015

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POB: Motivation

• It is very hard to get data on the costs of entry and the sell-offvalues associated with exit.

• POB develop a method for estimation of dynamic discrete games,e.g. entry and exit game.

• The method consists of two steps:

1. Obtain a non-parametric estimates of the entry values andcontinuation values.

2. Using the first stage estimates as true values, obtain parameterestimates of the entry and sell-off value distributions.

• One of the key advantages of the method is its computationalsimplicity.

• Recall that multiplicity of equilibria in dynamic games poses asignificant problem for estimation. This is particularly relevant forentry and exit models.

• POB provide a set of assumptions under which there is only one setof equilibrium policies consistent with the data-generating process.

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POB: General idea

• To approximate entry value at all possible states use average of thediscounted value of net cash flows actually earned by entrants whoentered at these states.

• To approximate incumbents’ continuation value use average ofactual discounted values of net cash flows earned by incumbentswho did continue at these states.

• Assume that entry fees and sell-off values are independent drawsfrom distributions known up to parameter values.

• Then the probability of entry is the probability that an entrant getsan entry fee draw, which is less than the estimated entry value.

• Similarly, the probability of exit is the probability that an incumbentgets a sell-off value draw, which is greater than the estimatedcontinuation value.

• Then the parameters of the entry fee and sell-off value distributionsare estimated by matching entry and exit rates predicted by themodel to the entry and exit rates observed in the data.

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POB: General idea

• POB discuss computational and statistical properties of estimatorsobtained by using alternative metrics of “closeness” of modelpredictions to the observed data.

• Proposed estimators are checked for robustness to the presence ofserially correlated unobservables.

• POB model is similar to the framework developed by Ericson andPakes (1995).

• Major distinction is that POB allow for random entry fees and scrapvalues and do not consider continuous controls (e.g. investment).

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POB: Setup

• Notation:

◦ nt is the number of agents active at the beginning of each period.◦ zt is a vector of exogenous profit shifters.◦ π(nt , zt ; θ) represents a per-period profit function.◦ et and xt are the number of entrants and exitors respectively.

◦ kiid∼ F k(·|θ), with a lower bound of k > 0, is a random draw from a

distribution of entry fees.

◦ φ iid∼ Fφ(·|θ), with a non-negative support, is a random draw from adistribution of sell-off values.

◦ δ ∈ (0, 1) is the discount rate.◦ χ and χe represent exit and entry decisions by incumbent firms and

potential entrants where χ = 1 stands for an incumbent’s decision tocontinue and χe = 1 when a potential entrant decides to enter.

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POB: Setup

• Neither incumbent nor potential entrants know et and xt at the timethey make their decisions.

• Entry and exit decisions are made simultaneously in the beginning ofeach period.

• Incumbent firms are identical up to the realization of their scrapvalue.

• Potential entrants are identical up to the realization of their entrycost.

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POB: Setup

• Assumptions:

1. Fixed number of potential entrants in each period, E . Incumbentsand entrants know distributions F k(·|θ) and Fφ(·|θ) as well as ownrealizations of sell-off value and entry fee but do not know therealizations of their competitors.

2. Agents’ perceptions of the probabilities of entry and exit by theircompetitors in period t depend only on the publicly availableinformation summarized by (nt , zt).

3. The evolution of profit shifters, z , is governed by the Markov chainP ≡ {p(·|z), ∀z ∈ Z = (0, 1, . . . , z)}, and limn→∞ π(n, z , θ0) ≤ 0for every z ∈ Z , and π(·) is bounded.

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POB: Incumbent’s problem• In the beginning of each period an incumbent makes a decision of

whether to continue or exit the industry:

◦ In case of exit, the incumbent collects current profits and thediscounted scrap value.

◦ In case of staying, the incumbent obtain current profits and thediscounted continuation value.

• Let pc(e, x |n, z , χ = 1) denote an incumbent’s perception of theprobability distribution of the number of entrants and exitorsconditional on incumbent itself continuing.

• Then, an incumbent Bellman equation is

V (n, z , φ; θ) = max{π(n, z ; θ) + δφ, π(n, z ; θ) + δVC (n, z ; θ)}

where VC (·) is defined as follows

VC(n, z) ≡∑e,x,z′

∫φ′

V (n+e−x , z ′, φ′; θ)p(dφ′|θ)pc(e, x |n, z , χ = 1)p(z ′|z)

• Incumbent exits whenever the first term under the max is greaterthan the second.

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POB: Entrant’s problem

• In the beginning of each period each entrant (from the pool E)decides whether to enter the industry.

• In the entry period an incumbent does not earn any profits, i.e. ittakes exactly one period to start up production.

• POB assume short-lived potential entrants, i.e. there is nopossibility of waiting to enter in some future period.

• Let pe(e, x |n, z , χe = 1) denote an entrant’s perception of theprobability distribution of the number of entrants and exitorsconditional on the entrant itself entering.

• POB define value of entry as

VE(n, z) ≡∑e,x,z′

∫φ′

V (n+e−x , z ′, φ′; θ)p(dφ′|θ)pe(e, x |n, z , χe = 1)p(z ′|z)

• A potential entrant enters if δVE (n, z ; θ) ≥ k , where k is its drawfrom the distribution of entry costs.

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POB: Equilibrium behavior

• POB consider Markov perfect equilibrium.

• Recall that in MPE each agent’s (incumbents and potentialentrants) optimal policies are chosen given its perceptions ofprobability distribution over possible future industry structures andthese perceptions must be consistent with the behavior of each ofthe agent’s competitors.

• In equilibrium, incumbents’ behavior is based on their perception ofthe probability distribution over entry and exit

pc(e, x |n, z , χ = 1) = bx(x , n − 1|n, z , θ)pc(e|n, z , χ = 1),

with

bx(x , n−1|n, z ; θ) =

(n − 1

x

)Fφ(VC(n, z ; θ)|θ)n−1−x [1−Fφ(VC(n, z ; θ)|θ)]x

and, in equilibrium, pc(e|n, z , χ = 1) is consistent with the behaviorof entrants

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• Note that, although existence of MPE is guaranteed, there are mightbe more than one equilibria.

• There is n such that we never observe n > n, i.e. each equilibriumgenerates a finite-state Markov chain in (n, z).

• The finiteness ensures that there exist a recurrent class R (a subsetof all possible (n, z)) such that any possible sequence of {(nt , zt)}will eventually arrive into R, and once (nt , zt) is in the recurrentclass, it stays in it forever.

• All states in R communicate with each other (there is a positiveprobability of transiting from one state to another in finite numberof periods) and each of them is visited infinitely often.

• POB show that there is only one profile of equilibrium policies thatis consistent with a given data-generating process in the recurrentclass.

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POB: Estimation

• POB estimation is based on a two-step approach

1. Compute averages of the realized continuation and entry values forall firms that did continue and enter at alternative values of (n, z).

2. Using the estimates from the first step as the actual continuationand entry values estimate model’s parameters by matching modelpredictions (conditional on alternative parameter values) for entryand exit to the data.

• Note that averages in the first step must be consistent with theagents’ expectations, therefore these averages will converge to thetrue expected continuation and entry values.

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POB: Estimating continuation values

• Estimation of VC (·)◦ Expected continuation value for an incumbent in state (n, z) is given

by

VC(n, z ; θ) = E cn′,z′

[π(n′, z ′) + δEφ′ [max{VC(n′, z ′; θ), φ′}|n′, z ′]

],

where E cn′,z′ [ · ] is the expectation of the future state conditional on

the incumbent itself continuing.

◦ The incumbent exits if φ′ > VC(n′, z ′; θ), thereforeEφ′ [max{VC(n′, z ′; θ), φ′}|n′, z ′] can be written

Eφ′ [max{VC(n′, z′; θ), φ′}|n′, z′] = Pr(φ′< VC(n′, z′; θ)) · VC(n′, z′; θ)+

Pr(φ′> VCp(n′, z′; θ)) · E [φ′|φ′

> VC(n′, z′; θ)]

◦ Let px(n′, z ′) ≡ Pr(φ′ > VC (n′, z ′; θ)) be the probability of exit.

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◦ If Fφ(φ|θ) = 1− exp(− 1σφ), then

E [φ|φ > VC(n′, z ′; θ)] = VC(n′, z ′; θ) + σ

and

VC(n, z ; θ) = E cn′,z′

[π(n′, z ′) + δEφ′ [max{VC(n′, z ′; θ), φ′}|n′, z ′]

]= E c

n′,z′

[π(n′, z ′) + δ

((1− px(n′, z ′))VC(n′, z ′; θ)+

px(n′, z ′)(VC(n′, z ′; θ) + σ)

)]= E c

n′,z′[π(n′, z ′) + δVC(n′, z ′; θ) + δpx(n′, z ′)σ

]

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• Define

◦ A vector of continuation values at each point in the state VC(θ) ;◦ A vector of exit probabilities px ;◦ A vector of per-period profits π; and◦ A matrix of the incumbent’s perceived transition probabilities Mc .

• Then continuation value for an incumbent is given by

VC (θ) = Mc [π + δVC (θ) + δσpx ]

• Solving for VC (θ) gives

VC (θ) = (I − δMc)−1Mc(π + δσpx)

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• The solution above can also be obtained by the following steps

VC (θ) = Mc [π + δσpx ] + δMc [VC (θ)]

= Mc [π + δσpx ] + δM2c [π + δσpx ] + δ2M2

c [VC (θ)]

· · ·

= Mc [π + δσpx ] +∞∑τ=1

δτMτc Mc [π + δσpx ]

which shows that continuation value can be computed by finding theexpected discounted future returns that the firm would earn onalternative possible future sample paths.

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• Define T (n, z) = {t : (nt , zt) = (n, z)}, i.e. the set of periods withthe same state (n, z) and let #T (n, z) be the number of times state(n, z) is visited in the data.

• Then, sample analog of exit probability at state (n, z) is given by

px(n, z) =1

#T (n, z)

∑t∈T (n,z)

xtn

• Note thatpx(n, z) −→

#T (n,z)→∞px(n, z)

• Let Mc,(n,z),(n′,z′) be an incumbent’s perceived probability oftransiting to state (n′, z ′), conditional on continuing in state (n, z).

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• Then

Mc,(n,z),(n′,z′) =

∑t∈T (n,z)(n − xt)1((nt+1, zt+1) = (n′, z′))∑

t∈T (n,z)(n − xt)−→

#T (n,z)→∞Mc,(n,z),(n′,z′)

• Each occurrence is weighted by the corresponding number ofincumbents who actually continue.

• By replacing transition and exit probabilities with their estimates wecan write

VC (θ) = (I − δMc)−1Mc(π + δσpx)

= Aπ + aσ,

where A = (I − δMc)−1Mc and a = δ(I − δMc)−1Mc px .

• Note that both A and a are independent of the parameter vectorand can be computed once at the beginning of estimation, i.e. givenprofits, the first stage estimates of continuation values are linear inparameters.

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POB: Estimation entrants’ value

• Expected value of entry for a potential entrant in state (n, z) isgiven by

VE (n, z ; θ) = E en′,z′ [π(n′, z ′) + δVC (n′, z ′; θ) + δpx(n′, z ′)]

which, in matrix notation, is

VE (θ) = Me(π + δVC (θ) + δpxσ)

where Me , (n, z), (n′, z ′) is a potential entrant’s perceived probabilityof starting operations at state (n′, z ′) conditional on entering instate (n, z).

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POB: Estimation entrants’ value

• Similar to the incumbent’s perceived transition probability POBderives an estimate of Me , (n, z), (n′, z ′), i.e.

Me,(n,z),(n′,z′) =

∑t∈T (n,z) et1((nt+1, zt+1) = (n′, z ′))∑

t∈T (n,z) et

• Then a consistent estimate of VE can be written compactly as

VE (θ) = Bπ + bσ,

where B = Me + δMeA and b = δMe a + δMe px .

• Similar to the incumbent’s value, entrant’s valuation is linear inparameters.

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POB: Estimating value functions

• First-stage estimators discussed above are based on empiricaltransition matrices and matrix inversion to compute value functions.Note that the latter requires specific assumptions on the distributionof scrap values.

• POB considers two alternative ways of calculating transitionmatrices and value functions:

1. Structural transition matrices, and2. Nested fixed point algorithm for computing value functions

• Both methods increase computation time, while Monte-Carloexperiments do not reveal any considerable advantages over theoriginal estimators.

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POB: Estimating parameters of the model

• POB consider three different objective functions for the second stepestimation procedure:

1. A pseudo log-likelihood function

2. A pseudo minimum χ2 estimator: a method of moments estimatorthat minimizes a norm in the difference between the data on thestate-specific entry and exit rates and the entry and exit ratespredicted by the model

3. A method of moments estimator that minimizes a norm in theaverage over all states of the difference between the actual entry andexit rates and the entry and exit rates predicted by the model fordifferent values of θ.

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POB: Discussion

• POB discusses several alternative estimators of of transitionprobabilities and continuation and entry values. When usingsemi-parametric estimators with the first step estimators entering ina non-linear way, POB emphasize the importance of considering finalsample bias.

• Method of moments estimator based on highly aggregated momentconditions helps to deal with the problem of finite sample bias.

• POB estimator has advantage that its simplicity should facilitate ananalysis of the impacts of different profit functions or distributionalassumptions.

• Note that, differently from other approaches, POB do not “invertout” CCP’s to get continuation values, but rather estimate themdirectly from the average of the discounted values of realized netcash flows.

• Under specific assumptions on the distribution of sell-off values,POB first stage estimates of the value functions are linear inparameters. This considerably reduces computational burden.

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POB: Discussion

• POB framework could be generalized to multiple entry locations andrandom number of entrants.

• Although one could obtain estimates of other primitives of themodel (e.g. revenue and cost function parameters) using justentry/exit data, their identification may be questionable.

⇒ More robust estimation strategies would require using additionaldata. In many cases, profit function parameters can be recoveredwithout solving/estimating continuation values.

• Simple form of the first stage estimates (VC (θ), VE (θ)) does notdepend on the distributional assumption on the entry cost. Thus,one can make the model more general by allowing for multiple entrylocations with realistic joint distributions of entry costs (e.g. theymay correlate across locations).

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POB: Discussion

• Exponential assumption on the distribution of scrap values impliesthat E [φ|φ > VC (θ)] is linear in VC (θ). This, in turn, allows POB

to use matrix inversion when solving for VC (θ).

• More flexible distributional assumption for Fφ(·) would result in the

following expression for VC (θ)

VC (θ) = Mc [π + δ(1− px). ∗ VC (θ) + δpx . ∗ E [φ|φ > VC (θ)]

where “.∗” stands for element-by-element multiplication.

POB shows that if the distribution Fφ is log-concave, the equationsystem is contraction mapping and therefore is easy to solve.

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POB: Discussion

• Under POB assumptions the model “picks out” the equilibrium thatwas played in the past. This, however, does not help in equilibriumselection resulting from a policy change, when simulatingcounterfactual scenarios.

⇒ When conducting policy analysis it is important to account forthe equilibrium selection when using the parameter estimates of theentry fee and sell-off value distributions.

• In case of panel data, one should ensure (assume) that the sameequilibrium was played in every market.

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POB: Discussion

• POB focus on identifying parameters of the entry and sell-offdistributions leaving aside identification of other primitives of themodel, which typically could be identified using alternative datasources.

• Reduction in computation time: Under convenient specifications forthe distribution of exit values (regardless of the distributionalassumption on entry values), the estimates of VC (·) and VE (·) arelinear in parameters and there is no need in iterative datatransformations.

• POB discuss robustness of their estimators to the presence of seriallycorrelated unobserved state variables, i.e. availability of alternativemeasures of profits conditional on the state of the system helps a lotin this respect.

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Richard Ericson and Ariel Pakes. Markov-perfect industry dynamics: a framework for empirical work. The Review of Economic Studies, 62(1):53–82, January 1995.

Ariel Pakes, Michael Ostrovsky, and Steven Berry. Simple estimators for the parameters of discrete dynamic games (with entry/exitexamples). RAND Journal of Economics, 38(2):373–399, 2007.

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