Identification of Discrete Choice Models Using Moment Inequalities: Theory and Application

Transcription

1 Identification of Discrete Choice Models Using Moment Inequalities: Theory and Application Eduardo Morales Columbia University & Princeton University April 24, 2012

2 Topics Mapping between statistical and behavioral discrete choice models. Pakes (2010); Pakes et al. (2011); Dickstein and Morales (2012). Applications. Ho (2009), Holmes (2011), Morales et al. (2011), Ho and Pakes (2011), Crawford and Yurukoglu (2011), Ishii (2012).

3 MAPPING BETWEEN STATISTICAL AND BEHAVIORAL MODELS

4 Discrete Choice Problem Utility of agent i for alternative j is: U ij = βx ij + ν ij, j = 1,..., J, x ij X R 2. Choice of individual i is captured in d i and d ij = 1[U ij U ij, for j = 1..., J], d i = d(x i, ν i ) We observe a random sample of vectors (d i, x i, z i ) P, i = 1,..., N, with x i1 = x i1 + ɛx i1, z i1 = x i1 + ɛz i1, x i2 = x i2. Assumptions on (ν i, ɛ x i1, ɛz i1 ): structural error: νi F ν(ν i xi, ɛ z i, ɛ x i ) = F ν(ν i xi ) measurement errors: E[ɛ x i xi, ν i, ɛ z i ] = E[ɛ x i xi ] = 0 Note: we omit subindex i from here onwards.

5 Example I: Estimation of Structural Parameters Consumer Choice Problem Each individual maximizes a utility function: U j = β 1 E[x j J ] + β 2 p j + ν j, j = 1,..., J. Instead of agent s i subjective expectations, the econometrician observes both the realized values, x j = E[x j J ] + ɛ x j, E[ɛ x j J ] = 0, and an additional shifter of agent s i expectations: z j = E[x j J ] + ɛ z j. The expectational error corresponds to the classical measurement error in an explanatory variable (errors-in-variables).

6 Example II: Estimation of Structural Parameters Two-period Entry Problem Each firm decides where to locate a plant: U j = E[R j J ] + β 2 F j + ν j, j = 1,..., J. Instead of agent s i subjective expectations, the econometrician observes realized values: R j = E[R j J ] + ɛ R j. The econometrician also observes Z such that Z j = E[R j J ] + ɛ Z j, Similar to previous example but with structural restriction: β 1 = 1. Standard application of the moment inequalities estimator: Ho (2009), Holmes (2011), Morales et al. (2011), Ishii (2012).

7 Example III: Estimation of Reduced Form Parameters General Entry Problem Each firm decides where to locate a plant. Period t profits of locating plant in location j are: J π jt = R jt F jj d ij t 1 j =1 The expected present value of locating plant in location j is: U jt = E[(π jt + δ s t d js π js ) J t, d jt = 1], s=t+1 J = V jt F jj d j t 1, j =1 with V jt = R jt + E[( δ s t d js π js ) J t, d jt = 1]. s=t+1

8 Example III: Estimation of Reduced Form Parameters General Entry Problem (cont.) Assume a projection of V jt on a set of observable covariates: Assume a structural form for F jj : V jt = βx jt + ν jt, (x t, ν t ) J it. F jj = θ L j L j, with some measure of distance, and L j an indicator of location j. Therefore: J U jt = βx jt + θ L j L j d j t 1 + ν jt. j =1 Additionally, we can allow for measurement error in x jt.

9 Maximum Likelihood Estimation Model 1: Assume (up to a finite parameter vector) the distributions: {F ν(ν x ), F ɛ(ɛ x ), P x (x )} The individual i likelihood function is: with L(d x, z) = P(d j = 1 x, z) = E(1{d j = 1} x, z) = E(E(1{d j = 1} x, z, x ) x, z) = E(E(1{d j = 1} x ) x, z) Z = h Z i 1{U j max{u j }}df ν(ν x ) df x (x x, z), j J x ν U j = βx j + ν j.

10 Maximum Likelihood Estimation Model 2: Assume (up to a finite parameter vector) the distributions: {F ν(ν x ), F ɛ(ɛ x x ), P x (x )} such that: F ν(ν x ) = F ν(ν), and F ɛ(ɛ x) = F ɛ(ɛ) The individual i likelihood function is: with L(d x) = P(d j = 1 x) = E(1{d j = 1} x) Z = 1{U j max {U j j }}dfν+ɛ(ν + J ɛx ), ν+ɛ U j = βx j + ν j βɛ x j.

11 Maximum Likelihood Estimation Model 3: Assume (up to a finite parameter vector) the distribution F ν(ν x ), and assume x = x. The individual i likelihood function is: with L(d x) = P(d j = 1 x) = E(1{d j = 1} x) = E(1{d j = 1} x ) Z = 1{U j max{u j }}df ν(ν x ), j J ν U j = βx j + ν j.

12 Summary of MLE (similar in GMM) Dealing with structural and measurement error in MLE or GMM requires: Assuming both the marginal distribution of the unobserved true covariates, and the distribution of both measurement and structural error conditional on these covariates (Model 1); or, Assuming the structural error is independent of the true covariates, and the measurement error is independent of the observed covariates (Model 2): Only in this case the difference between structural and measurement error is irrelevant!. We can think of a single error, η, such that: η = ν + ɛ and assume a single distribution F η(η); or, Assuming the distribution of the structural error conditional on the true covariates, and assuming that these are measured without error (Model 3).

13 Moment Inequalities Introduction Given our discrete choice problem, we can derive conditional moment inequalities. A conditional moment inequality is: E[m(d, x, z, j, j ; β 0 ) x, z] 0, where d j = 1, and j j. For simplicity, in these slides we base identification on a set of unconditional moment inequalities: E[m s (d, x, z, j, j ; β 0 )] 0, s = 1,..., S. Moment inequalities will generically lead to set identification. The identified set is: {β B : (min{m s (d, x, z, j, j ; β), 0} 2 )dp(d, x, z) = 0} s S j J j j We denote the identified set as B M (P).

14 Moment Inequalities Deriving Moment Inequalities from our Discrete Choice Problem: Model 1 Assumptions: (ɛ x j, ν j ) = (0, 0), for every j J. Moment inequalities. m s (d, x, z, j, j ; β 0 ) = 1{d j = 1}β 0 x jj 0, Normalization by scale: (1) β 01 = 1; (2) β 0 = 1. Completely deterministic model; very likely rejected by the data. Contradictory inequalities derived from the model.

15 Moment Inequalities Deriving Moment Inequalities from our Discrete Choice Problem: Model 2 Assumptions: No structural error: νj = 0, for every j J; Measurement error indep. of true covariate: Fɛ[ɛ x 1 x ] = F ɛ[ɛ x 1]; No parametric assumption on F ɛ[ɛ x 1 ] needed. Structural restriction: β01 = 1. Moment inequalities. with E[1{d j = 1}χ s ( x 2jj )β 0 xjj ] 0, E[1{d j = 1}χ s ( x 2jj )(β 0 x jj β 0 ɛ x 1jj )] 0, No need to normalize by scale. E[1{d j = 1}χ s ( x 2jj )β 0 x jj ] 0, χ 1 ( x 2jj ) = 1{ x 2jj 0}, χ 2 ( x 2jj ) = 1{ x 2jj < 0}.

16 Moment Inequalities Deriving Moment Inequalities from our Discrete Choice Problem: Model 3 Assumptions: No structural error: νj = 0, for every j J; Measurement error indep. of true covariate: Fɛ[ɛ x 1 x ] = F ɛ[ɛ x 1] No parametric assumption on F ɛ[ɛ x 1 ] needed. Additional indicator of the covariate measured with error: z1. Moment inequalities. with E[1{d j = 1}χ s ( z 1jj, x 2jj )β 0 xjj ] 0, E[1{d j = 1}χ s ( z 1jj, x 2jj )(β 0 x jj β 0 ɛ x 1jj )] 0, E[1{d j = 1}χ s ( z 1jj, x 2jj )β 0 x jj ] 0, χ 1 ( z 1jj, x 2jj ) = 1{ z 1jj 0}1{ x 2jj 0}, χ 2 ( z 1jj, x 2jj ) = 1{ z 1jj 0}1{ x 2jj < 0}, χ 3 ( z 1jj, x 2jj ) = 1{ z 1jj < 0}1{ x 2jj 0}, χ 4 ( z 1jj, x 2jj ) = 1{ z 1jj < 0}1{ x 2jj < 0}.

17 Moment Inequalities Deriving Moment Inequalities from our Discrete Choice Problem: Model 4 Assumptions: Structural error organized in nests: ( 0 if G(j) = G(j ), ν j ν j = R if G(j) G(j ), where G(j) denotes a particular subset of J. No parametric assumption on F ν[ν x] needed. No measurement error. Moment inequalities. E[1{d j = 1}1{G(j) = G(j )}χ s ( x jj )(β 0 x jj + ν jj )] 0, E[1{d j = 1}1{G(j) = G(j )}χ s ( x jj )β 0 x jj ] 0. Trivial to allow for measurement error indep. of the true covariate. impose β01 = 1 as a structural assumption (as in Model 2); or, use an additional indicator z1 (as in Model 3).

18 Moment Inequalities Deriving Moment Inequalities from our Discrete Choice Problem: Additional Models See Dickstein and Morales (2012) for guidance on how to build moment inequalities for the following models: Model 5: Assume a parametric distribution for νi : F ν[ν x; σ] Allow for distribution free measurement error in two cases: multiple indicator assumption; fixed parameter on the variable measured with error. Model 6: Assume νi is distributed independently of x. No parametric assumption on F ν[ν x] needed. Allow for distribution free measurement error in two cases: multiple indicator assumption; fixed parameter on the variable measured with error.

19 Summary of Moment Inequalities In words, dealing with both structural and measurement error using moment inequalities requires: In order to deal with measurement error, we may apply any of the usual IV solutions to measurement error problems in linear regression models. In particular, no parametric assumption on its distribution function is needed. In order to deal with structural error, one needs to: Assume it away; or, Assume that it is an individual effect common to a subset of choices; or, Assume a parametric distribution on it; or, Assume that it is distributed independently of x (no parametric assumption needed).

20 APPLICATION OF MOMENT INEQUALITIES: RATIONAL EXPECTATIONS SINGLE AGENT DYNAMIC DISCRETE CHOICE MODELS. Gravity and Extended Gravity: Estimating a Structural Model of Export Entry. Eduardo Morales Gloria Sheu Andrés Zahler

21 Motivation Firm 1 (Compañía Manufacturera de Aconcagua): standard gravity factors Trade models based on gravity imply that firms should enter first bordering countries... Figure: Year 1995

22 Motivation Firm 1 (Compañía Manufacturera de Aconcagua): standard gravity factors... or markets that are geographically close and share the same official language as the domestic country... Figure: Year 2000

23 Motivation Firm 1 (Compañía Manufacturera de Aconcagua): standard gravity factors... and only later they overcome larger entry costs and access countries that are further away and in which different languages are spoken... Figure: Year 2005

24 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity factors... but, some firms, even though they follow this rule... Figure: Year 1995

25 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity factors... at the early stages of their export history... Figure: Year 2000

26 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity equation... suddenly access countries that are very different from the country of origin but similar to previous export destinations... Figure: Year 2002

27 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity equation... and they don t stay for a long time in these countries... Figure: Year 2003

28 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity equation... and they never export high volumes to them. Figure: Year 2004

29 Motivation Firm 2 (Industrias Cerecita SA): Extended gravity equation Are firm 2 s entry costs in these far away countries as high as standard gravity factors would predict? Figure: Year 2005

30 Aim of the paper Empirical analysis of country-specific firm export dynamics. Analyse the determinants of firm entry and exit into spatially related markets. Entry and exit into each potential destination country may depend on: Similarity between the importing country and firm s home country. Gravity: closeness between home and destination markets. Similarity between the current importing country and prior destinations of the firm s exports. Extended gravity: similarities between two receiving countries. Quantify how strong gravity and extended gravity effects are in determining firms country-specific entry and exit decisions. Structural estimation of destination-specific costs of exporting that depend on gravity and extended gravity factors. Measurement paper.

31 Quick summary of model and estimation Multi-period, multi-country firm-level trade model. Single-agent dynamic model with multiple spatially related markets. Four different type of costs: Gravity Extended Gravity Marginal Costs Fixed Costs Sunk Costs Estimation based on a moment inequalities. Application of an analogue of Euler s perturbation method.

32 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

33 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

34 Data sources Data sources: Chilean customs database: revenue at the country-firm-year level. Chilean industrial survey: sector (4 digit ISIC), domestic sales, value added, proportion of skilled workers, average wage at the firm-year level. CEPII: geographical location and official language at the country level. WDI: GDP and GDP per capita at the country-year level. Unbalanced panel of firms for the sample period: Sector: Manufacture of chemicals and chemical products (18% of manufacturing exports).

35 Transition matrices Sector: Manufacture of chemicals and chemical products Strong effect of export history: Conditional on t 1 Conditional on t 2 All Countries but South American Entry No Entry Entry No Entry Options Relative Shares Continent to Prior Bundle Does Not All Countries but Spanish Speaking Entry No Entry Entry No Entry Options Relative Shares Language to Prior Bundle Does Not All Countries but U Middle Income Entry No Entry Entry No Entry Options Relative Shares Income Group to Prior Bundle Does Not All Countries but Bordering Chile Entry No Entry Entry No Entry Options Relative Shares Border to Prior Bundle Does Not

36 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

37 Model Demand Utility function for the representative consumer of every country j in every sector: Budget constraint is: [ Q jt = i A i A η 1 η qijt ] η η 1 di p ijt q ijt di = C jt

38 Model Supply Every firm i is the single supplier of a variety in any country. Every firm i faces a constant marginal cost of supplying any market j. mc ijt = f it g M jt ɛ M ijt Fixed costs of exporting: Sunk costs of exporting: fc ijt = g F j + ɛ F ijt Basic costs of exporting: Terms: sc ijbt 1 t = g S j e S jb t 1 + ɛ S ijt bc it = µ B 0 + ɛ B it g M jt, gj F, gj S : gravity equation terms e S jbt 1 : EXTENDED gravity equation terms (ɛ M ijt, ɛ F ijt, ɛ S ijt, ɛ B it ): unobservable variables.

39 Interactions between destination countries How do we measure those interactions between destination countries? Dummy variable for exporting at t 1 to at least one country that: shares a border with j (and not with Chile). has the same official language as j (except Spanish). belongs to the same continent as j (except South America). belongs to the same GDPpc bracket as j (except upper middle income countries). Therefore, a firm will not benefit from those interactions when entering j if: It did not export in the previous period. It exported in the previous period to countries that are very different from j. Country j is very similar to Chile.

40 Static problem of the firm Given the demand and supply assumptions indicated above, the potential revenue from exporting to country j for firm i at period t (expressed in logs) is: log(r ijt ) = c + (1 η) ln(mc ijt ) + ln(c jt ) (1 η) ln(p jt ) where P j is the price index of the sector in the destination country j. The expression for the (log of) gross profits from exporting is: log(v ijt ) = log(η) + log(r ijt )

41 Dynamic problem of the firm Possibly nonzero sunk costs of exporting make the entry and exit in each country j the outcome of a dynamic optimization problem. Possibly nonzero extended gravity effects in sunk costs make the entry and exit in each country j dependent on the entry and exit decision in every other country j. In every period t, firm j is going to pick a bundle of export destinations taking into account the effect on future export profits. The size of the choice set ( ) makes it impossible to use any standard discrete choice model in the estimation. We base our estimation strategy on moment inequalities derived from the previous model. We apply Euler s perturbation method (one period deviations) to build these moment inequalities.

42 Dynamic problem of the firm The net profits from exporting are: π ibt b t 1 t = X j b t π ijbt 1 t 1{b t 1 = }bc it π ijbt 1 t = v ijt fc ijt 1{j / b t 1}sc ijbt 1 t Assumption 1: Firms Maximize Profits Let s denote by o T 1 where = {o 1, o 2,..., o T } an observed sequence of bundles, then: o t = argmax E iˆπibt o t 1 t J it b t B iot 1 t Π ibt o t 1 t = π ibt o t 1 t + δπ ibt+1 b t t+1 + ω ibt+1 t+2, ω ibt+1 t+2 = ω it+2 (δ, π ibt+2 b t+1 t+2, π ibt+3 b t+2 t+3,... ), b t+s = t = 1, 2,..., T argmax E iˆπibt+s b t+s 1 t+s J it+s, s 1. b t+s B ibt+s 1 t+s

43 Dynamic problem of the firm Assumption 1 is compatible with firms being forward-looking in many different degrees: Firms may take into account only one period ahead: ω ibt+1 t+2 = 0. Firms may consider any finite number p of periods ahead: ω ibt+1 t+2 = δ 2 π ibt+2 b t+1 t+2 + δ 3 π ibt+3 b t+2 t δ p π ibt+pb t+p 1 t+p; Firms may consider an infinite number of period ahead (perfectly forward looking firms): ω ibt+1 t+2 = E iˆπibt+2 b t+1 t J it+2.

44 Instead of deriving a likelihood function from the model... Deriving choice probabilities from Assumption 1 requires: Specify firms choice sets: {Biot 1 t} I,T i=1,t=1. Specify firms conditional expectation functions: {Ei [ J it ]} I i=1. Specify firms information sets: {Jit } I,T Specify function ωibt+1 t+2. i=1,t=1. Specify distribution function for the unobservable vector: (ɛ M ijt, ɛ F ijt, ɛ S ijt, ɛ B ijt). Specify stochastic process for the state variables. Solve the dynamic optimization problem: Problems with this approach: o t = argmax E iˆπibt o t 1 t J it. b t B iot 1 t We impose assumptions on objects on which we have no information. Solving the dynamic optimization problem is computationally very complicated.

45 Deriving moment inequalities: one-period deviations t-1 t t+1 t+2 0 E i [(π iata t 1 π ibta t 1 ) δ(π ibt+1a t π ibt+1b t ) J it ]

46 Deriving moment inequalities: one-period deviations Proof Given Assumption 1 and assuming o t B iot 1t: E i [ πioto t 1t + δπ iot+1o tt J it ] E i [ πio t o t 1t + δπ iot+1o t t J it ] o t+1 = Simplifying notation: [ ] argmax E i Πibt+1ott+1 J it+1 b t+1 B io t t+1 E i [ πidot+1t J it ] 0 π idot+1t = (π ioto t 1t π io t o t 1t) + δ(π iot+1o tt+1 π iot+1o t t+1 ) Given that this inequality holds for every deviation, firm, and time period, the model predicts that: 1 D k I D T k it E i [π idot+1t J it ] 0. i=1 t=1 d=1

47 Deriving moment inequalities: one-period deviations Finally, we impose some restrictions on the expectations of the agents: Assumption 2: Firms are Right on Average There is a positive valued function g kl ( ) and an x it J it such that: I D T k it [ I E i [π idot+1t J it ] 0 E i=1 t=1 d=1 D T k it i=1 t=1 d=1 ] π idot+1tg kl (x idt ) 0 where E[ ] denotes the statistical expectation or expectation with respect to the data generation process.

48 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

49 List of parameters The list of parameters to estimate through moment inequalities is: Fixed cost parameters: constant z} { µ F 0, µ F cont, µ F lan, {z µ F gdppc } gravity effect Basic cost parameters: constant z} { µ B 0 Sunk cost parameters: constant z} { µ S 0, µ S cont, µ S lan, {z µ S gdppc, } gravity effect ζbord, S ζcont, S ζlan, S {z ζgdppc S } extended gravity effect

50 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

51 An Example: µ F 0 Imagine we observe the following export history for firm i in country j: Year Profits v ij1 v ij2 v ij3 v ij4 v ij5 Exports Using the following counterfactual strategy: Year Actual Counterfactual we get the following difference in profits: π ido5 4 = v ij4 + fc ij4 If j is in South America, speaks Spanish, and has an upper middle GDPpc: π ido5 4 = v ij4 + µ F 0 + ɛ F ij4

52 An Example: ζ S lan Assume the following stream of profits and export strategies in countries j and j : Actual Counterfactual Year Country j Country j Country j Country j We get the following difference in profits: π ido98 = v ij8 v ij 8 (fc ij8 fc ij 8) (sc ij8 sc ij 8) If i exports at year 7 to a country that shares official language with j (and nothing with j ) and neither j nor j are in South America, speak Spanish or have an upper middle GDPpc, then the moment inequality simplifies to: π ido98 = v ij8 v ij 8 + ζ S lan

53 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

54 Estimation strategy: linear moment inequalities estimation Mom.Ineq.Framework Our model provides S moment inequalities, each indexed by a pair (k, l): [ 1 I m s (θ) = E D k D T k it i=1 t=1 d=1 ] π idot+1tg kl (x idt ) 0 with θ = (β, µ F, µ S, µ B, ζ S ), and: π idot+1t = v idt (β) g F d (µ F ) (g S do t+1 (µ S ) e S do t+1 (ζ S )) bc dot+1 (µ B ) ɛ 2idt with g F ( ), g S ( ), e S ( ), and bc( ) linear functions and v( ) is loglinear. We follow a two-stage estimation strategy: First stage: linear panel data estimates of β. Second stage: linear moment inequality estimates of (µ F, µ S, µ B, ζ S ) are obtained conditional on the first stage estimates ˆβ.

55 First stage estimation In order to estimate β we use the revenue equation that arises from solving the static problem of the firm. log(r ijt ) = βz ijt + (1 η)ɛ M ijt Results and we define an approximation to gross profits from exporting as: ˆv ijt = 1ˆη ˆα exp(ˆθ 1 Z ijt ) where we borrow the estimate ˆη from Broda, Greenfield, and Weinstein (2006), and ˆα accounts for the effect of higher order moments of ɛ M ij on the expected value of v ij. We define the first stage error as: ɛ 1idt = v idt ˆv idt

56 Second stage estimation In order to obtain estimates for (µ F, µ S, µ B, ζ S ) we build sample analogues of the moment inequalities derived from the model. m s (µ F, µ S, µ B, ζ S ) = 1 D k D I T k it π idot+1tg kl (x idt ) 0 i=1 t=1 d=1 and π idot+1t = ˆv idt g F d (µ F ) (g S do t+1 (µ S ) e S do t+1 (ζ S )) bc dot+1 (µ B ) = π idot+1t(θ) ɛ idt with ɛ idt = ɛ 1idt + ɛ 2idt

57 Estimation strategy Second stage estimation Our linear moment inequality estimate is consistent as long as the following assumption holds: Assumption 3: Orthogonality 1 D k I D T k it ɛ idt g kl (x idt ) Dk 0. i=1 t=1 d=1 Caveat: Heterogeneity vs. state-dependence? Correcting for selection in ɛidt (see Dickstein and Morales (2012))

58 Overview First look at the data Model List of parameters Building our moment inequalities: examples Estimation method Results

59 Combining parameters Manufacture of chemicals and chemical products: narrow version If the firm at t If the firm at t-1 Estimates Mean profits/cost exports to... exported to... Lower Upper Midpoint Exporters All US Brazil Colombia US 223, , , Canada 319, , , Mexico 379, , , UK 319, , , Colombia 387, , , Chile 387, , , Brazil 219, , , Argentina 277, , , Portugal 223, , , Ecuador 283, , , Chile 283, , , Colombia 215, , , Venezuela 264, , , Argentina 277, , , Chile 277, , ,

60 Summary Moment inequalities allow for identification of structural parameters in single agent dynamic models that are computationally impossible to estimate using standard estimation methods. Possibility of dealing very large discrete choice sets. This is possible because we do not need to solve the model for identification and estimation. Besides, we incorporate distribution-free measurement error. Drawbacks: Strong assumptions on structural errors. Impossibility of performing counterfactuals without previously solving the model.

61 Proof Back From Assumption 1 we know that: E i [π iot o t 1 t+δπ iot+1 o t t+1+ω iot+1 t+2 J it ] E i [π io t o t 1 t+δπ io t+1 o t t+1 +ω io t+1 t+2 J it ], with and o t+1 = o t+1 = By transitivity of preferences, argmax E iˆπibt+1 o t t+1 J it+1, b t+1 B io t t+1 argmax E iˆπibt+1 o t b t+1 B t+1 J it+1. io t t+1 E i [π iot o t 1 t+δπ iot+1 o t t+1+ω iot+1 t+2 J it ] E i [π io t o t 1 t+δπ iot+1 o t t+1 +ω iot+1 t+2 J it ], Canceling terms on both sides: E i [π iot o t 1 t + δπ iot+1 o t t+1 J it ] E i [π io t o t 1 t + δπ iot+1 o t t+1 J it ]. Q.E.D.

62 Estimation strategy First stage results: ˆθ 1 Independent Variable Log Revenue Regression domsalesit 0.431*** (0.040) avgskillit *** (0.150) lavgwageit 0.322*** (0.077) lavgvalueaddit *** (0.057) borderjt 0.409*** (0.084) contjt (0.084) langjt * (0.103) gdppcjt 0.179*** (0.068) lgdpjt 0.250*** (0.022) avglrerj *** (0.012) devlrerjt 0.610*** (0.259) legalj *** (0.020) N Observations 6,253 Notes: * denotes 10% significance, ** denotes 5% significance, *** denotes 1% significance. Robust standard errors are in parentheses. The dependent variable is log revenue. Year fixed effects are included