From Reduced-Form to Structural Evaluation: Expanding Financial Infrastructure and Impact Robert M. Townsend MIT March 2013 Robert M. Townsend (Institute) Structural 03/13 1 / 65
What is this lecture about? We illustrate the limitations of reduced-form and IV analysis. Highlight the bene…t of using reduced-form and structural analysis together Based on Urzua and Townsend (2009) paper. Turn to a detailed structural analysis in Keniston et al.(2012) - Using BBL methodology This methodology allows to estimate determinants of costs and demands of a player, i.e., a bank, without having to solve for all strategies of all players (even o¤ equilibrium). We then illustrate limitations of this approach when there is a need to know counterfactual strategies (o¤-equilibrium). Robert M. Townsend (Institute) Structural 03/13 2 / 65
Urzua and Townsend (2009) What is the impact of …nancial intermediation on productivity? A¤ects occupational choice as well as allocation of risk Consider both static and dynamic structural models, and IV and OLS. Goal is to bridge the structural approach and the reduced-form IV approach Highlight that under strong assumptions, IV can recover the true LATE, but even then it can be very di¤erent from the Average Treatment E¤ect (ATE), or the Treatment on the Treated (TT) e¤ect. This is driven by the presence of heterogeneity in the population Having more margins of decisions, as well as more periods in a dynamic contract increases di¢culty to interpret IV. Robert M. Townsend (Institute) Structural 03/13 3 / 65
Standard Model of Occupational Choice Individual preferences: u ( c ) = c Beginning of period wealth b i (observed by econometrician) Cost of entry into business θ E i (private information, with density f θ E ) W Talent as a wage earner θ i (private information, with density f W ), θ E θ independent of End of period wealth: W W i = w + θ i + b i if wage earner � � E W = π θ i , b i , w + b i if entrepreneur i where pro…ts come from � � E E π θ i , b i , w = max f ( k , l ) � wl � k � θ i f k , l g E : 0 � k � b i � θ i s . t Robert M. Townsend (Institute) Structural 03/13 4 / 65
Standard Econometric Approach for e¤ects of occupational choice Decision rule D = 1 if person becomes entrepreneur � � � � E W E W D θ i , θ i , b i , w = 1 if π θ i , b i , w > w + θ i = 0 else Can reduced form approach identify e¤ects of occupational choice? W Econometrician observes income (either π + b i or b i + w + θ i depending on occupation). End of period income is: � � � � � � E W Y i = D i π θ i , b i , w + b i + ( 1 � D i ) w + θ l + b i E If assume linear separable model, π = φ w w + φ θ θ i + φ b b i then: Y i = w + b i + ( φ b b i + ( φ w � 1 ) w ) D i + ε i � � W E W where ε i = θ i + φ θ θ i � θ i D i , is correlated with D i (so simple OLS produces biased estimators). Robert M. Townsend (Institute) Structural 03/13 5 / 65
Standard Econometric Approach for e¤ects of occupational choice II Instead, use IV. Instrument: reandomly assigned subsidy that increases profts by ψ (only conditional on setting up a …rm, cannot be used to …nance k ). � � � � E W E W New decision rule: D θ i , θ i , b i , w = 1 if π θ i , b i , w + ψ i > w + θ i , and D i = 0 else. Subsidy is valid instrument: A¤ects choice of occupation but not potential outcome Satis…es monotonicity assumption: for each individual, an increase in subsidy increases chance of becoming entrepreneur ¯ 0 then ¯ and ψ If subsidy can take two values, ψ � � � E ( Y i j ψ i = ψ 0 , b i = b ¯ i ¯ , E Y i j ψ i = ψ i b i = b ) ∆ IV = � � � E ( D i j ψ i = ψ 0 , b i = b ¯ i ¯ , D i j ψ i = ψ i b i = b ) E which is also equal to the local average treatment e¤ect (LATE): h � � i � ¯ 0 � = 1 , D i ( ψ ∆ LATE = E E W ¯ ) = 0, b i = b π θ i , b i , w � w � θ i j D i ψ Robert M. Townsend (Institute) Structural 03/13 6 / 65
Standard Econometric Approach for e¤ects of occupational choice III Treatment on the treated (TT): average bene…t of becoming an entrepreneur for those who actually become entrepreneurs � � � � � � ∆ TT ( b ) = E E W π θ i , b i , w � w + θ i j D i = 1 , b i = b Average treatment e¤ect (ATE): e¤ect of becoming entrepreneur versus wage earner for the entire population � � � � � � ∆ ATE ( b ) = E E W π θ i , b i , w � w + θ i j b i = b If no heterogeneity, or all heterogeneity observed, then = ∆ TE = ∆ TT . Else, di¢cult to estimate ∆ ATE and ∆ TT . ∆ LATE A Robert M. Townsend (Institute) Structural 03/13 7 / 65
Parametric estimation for e¤ects of occupational choice (ATE and TT) Can …nd ATE and TT if make additional parametric assumptions on functional forms for pro…ts ( f quadratic) and distribution functions of θ s (normally distributed) Probability of being entrepreneur: � � � � E E Pr π θ i , b i , w + ψ i > w + θ i � � E W = Pr φ w + φ θ θ i + φ b b i + ψ i > w + θ i w 0 1 Φ @ ( φ w � 1 ) w + φ b b i + ψ i A q = 2 2 2 σ + φ σ W θ E 2 2 W E σ σ are the variances of θ and θ , respectively. where and W E Robert M. Townsend (Institute) Structural 03/13 8 / 65
Parametric estimation for e¤ects of occupational choice (ATE and TT) II Expected pro…ts conditional on being an entrepreneur: � � � � E E π θ i , b i , w j D i = 1 , b i , ψ i (1) 0 1 2 2 φ σ @ φ w � 1 ) w + φ b b i + ψ i ( θ E A q q = φ w w + φ b b i � λ 2 2 2 2 2 2 σ + φ σ σ + φ σ W θ E W θ E where λ () is a function (the Mills’ ratio). Hence, correct regression is of pro…ts/earnings onto the wage, b i , and λ - 2 note that φ θ and σ cannot be separately identi…ed. E Robert M. Townsend (Institute) Structural 03/13 9 / 65
Parametric estimation for e¤ects of occupational choice (ATE and TT) III Average wages among entrepreneurs (unobserved of course): � � E E w + θ i j D i = 1 , b i , ψ i (2) 0 1 2 σ @ φ w � 1 ) w + φ b b i + ψ i ( W A + q q = w λ 2 2 2 2 2 2 σ + φ σ σ + φ σ W θ E W θ E which depends only on identi…ed parameters (from the probit), so can be constructed for all b i and ψ i values. Robert M. Townsend (Institute) Structural 03/13 10 / 65
Parametric estimation for e¤ects of occupational choice (ATE and TT) IV Hence, can compute: ∆ TT ( b , ψ ) � � � � E = π θ i , b i , w j D i = 1 , b i = b , ψ i = ψ E | {z } identi…ed from ( 1 ) � � w + θ E � E i j D i = 1 , b i = b , ψ i = ψ | {z } identi…ed from ( 2 ) � � � � � � ∆ ATE E E ( b ) = E π θ i , b i , w � w + θ j b i = b = ( φ w � 1 ) w + φ b b i To get unconditional version, just integrate over b and ψ over appropriate region. Robert M. Townsend (Institute) Structural 03/13 11 / 65
Further Note on Heterogeneous Treatment e¤ects I Method by Heckman and Vytlacil (2001) Compute the Local IV estimator, ∆ LIV : ∂ E ( Y i j p i , b i = b ) LIV ∆ ( p , b ) = j i = p p ∂ p i W E i = θ i � φ θ θ i . where p i is the propensiry score, here p This can identify the treatment parameter � � � � � � ∆ MTE ( p , b ) = E E W W E π θ i , b i , w � w + θ i j b i = b , θ i � φ θ θ i = p (treatment e¤ect for those individuals indi¤erent between occupations, given p and b ). Robert M. Townsend (Institute) Structural 03/13 12 / 65
Further Note on Heterogeneous Treatment e¤ects II Can then obtain ∆ ATE and ∆ TT as weighted averages of ∆ MTE : Z TE ( u , b ∆ TT M TT ( b ) = ∆ ) ω ( u , b ) du Z ∆ ATE b ∆ MTE ATE ( ) = ( u , b ) ω ( u ) du ATE ω ( u ) = 1, where ( u , b ) = Pr ( p ( w , b , ψ ) > u ) / R TT ω Pr ( p ( w , b , ψ ) > u ) du To compute ∆ LIV ( p , b ) , can approximate it by a polynomial on p i . Robert M. Townsend (Institute) Structural 03/13 13 / 65
Measuring Impact of Occupations on Income I Directly simulate data from model to compare di¤erent estimates. Parameterize model (see table 1 in paper for all details). How do we do this? We …x some parameters for the full model (’calibrate’ it), randomly assign a subsidy to some agents. Model then tells us what occupation each agent chooses and what his realized income is. We also know what his counterfactual would have been without the subsidy. Directly estimate the e¤ects of occupation by directly looking at income before and after the subsidy for the same individual. Then try to directly run the IV regression on the model-generated data (see the next slide). Robert M. Townsend (Institute) Structural 03/13 14 / 65
Measuring Impact of Occupations on Income II Subsidy = f 0 , 1 g . Suppose researcher tries to estimate e¤ect of occupational choice from κ 3 + κ 2 b i below: Y i = κ 0 + κ 1 b i + κ 2 b i D i + κ 3 D i + ε i Can use subsidy as IV for D i . OLS and IV are very di¤erent (see next slide): IV shows negative impact, OLS positive - because occupational choice is related to unobserved talent, hence endogenous. IV is ’correct’: individuals who switch occupation as result of subsidy are those with lower pro…ts and higher wages (than those who already are entrepreneurs). Robert M. Townsend (Institute) Structural 03/13 15 / 65
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