2024 Causal Inference in Microeconometrics Lecture 1 PDF

Causal Inference in Microeconometrics Causal Inference in Microeconometrics with Applications to Program Evaluation Lecture 1: Introduction Bart Cockx Ghent...

Causal Inference in Microeconometrics Causal Inference in Microeconometrics with Applications to Program Evaluation Lecture 1: Introduction Bart Cockx Ghent University 2024 Causal Inference in Microeconometrics Teaching and Evaluation Method Teaching Method Organization ▶ 11x3h20 on Fridays between 4/10 and 20/12/2024 (no lecture on November 1) ▶ 2x2 sessions of 50 min. (with 2 breaks of 15 min and a lunch break of 1h05): (1) 10:00-10:50; (2) 11:05-11:55 and (3) 13:00-13:50; (4) 14:05-14:55. ▶ Low quality recordings via MS Teams (as backup) ▶ On 29/11, 6/12 and 13/12 William Parienté (UCLouvain) on randomized controlled trials (RCT): Causal Inference in Microeconometrics Teaching and Evaluation Method Evaluation Method Evaluation Method 1. Presence in lectures is required! ▶ Penalty -1 points for each absence not justified in advance; ▶ Penalty -2 points for any absence from the 3rd absence onwards. 2. Computer and other assignments (10 points) ▶ Groups of max. 3 students (min. 2): Register on Ufora; ▶ Model solution will be posted on platform; ▶ Support: Natalia Bermudez and Giulia Tarullo. Causal Inference in Microeconometrics Teaching and Evaluation Method Evaluation Method Evaluation Method (2) 3. Presentation of paper or own research related to lectures with group discussion (10 points) ▶ Same groups of max. 3 students ▶ Paper related to own research (after approval) or chosen from list ▶ Aim: show understand lectures (re-explaining material of lectures in own words). ▶ Timing depends on # of students. Aim = 15 min. presentation + 5 minutes discussion. ▶ End of January (to be determined) Causal Inference in Microeconometrics Teaching and Evaluation Method Teaching Material Teaching Material ▶ Slides, documents, and recordings available on the Ufora. ▶ Non-UGent students need official registration. ▶ Registration on OASIS is required for those who want to take the exam and receive credit for this. ▶ Required reading: Blundell and Costa Dias (2009) ▶ General complementary readings (for specific ones, see lectures): 1. Imbens and Wooldridge (2009) or Angrist and Pischke (2009) 2. Book and other resources of Cunningham (2021) (for free). 3. Book of Imbens and Rubin (2015) thorough introduction 4. Abadie and Cattaneo (2018), Huber (2019), and Arkhangelsky and Imbens (2024) for recent overviews 5. Imbens (2022) and Imbens and Xu (2024) for history of and lessons from ’Credibility Revolution’ Causal Inference in Microeconometrics Introduction The Credibility Revolution The Credibility Revolution ▶ The Credibility Crisis: LaLonde (1986) ▶ Aimed at validating various non-experimental identification strategies by comparing their estimates using a non-experimental control group to those obtained using an experimental control group. ▶ Conclusion: Non-experimental evaluations may contain large and unknown biases resulting from specification errors. ▶ The Credibility Revolution: experiments + sensitivity analyses (Leamer, 1983) ▶ Use RCT’s ⇒ Banerjee/Duflo/Kremer (Nobel ‘19); ▶ Use ‘natural’ experiments ⇒ Ashenfelter/Card (Nobel ‘21) + Krueger (†); ▶ Methodological advancements ⇒ Angrist/Imbens (Nobel ‘21) ↔ Heckman (Nobel ‘00) Imbens and Xu (2024) replicate LaLonde with modern methods: still nuanced conclusion Causal Inference in Microeconometrics Introduction What is Causality? What is Causality? Consider the following questions ▶ Does smoking cause lung cancer? ▶ Does aspirin reduce the risk of heart attacks? ▶ Does training of unemployed workers increase employment and wages? ▶ Does an additional year of schooling increase future earnings? ▶ What’s the effect of FDI on profits/employment of domestic firms? ▶... The answers to these questions and to many others involve the identification of a causal relationship. What does this mean? Causal Inference in Microeconometrics Introduction An Example: Training of Unemployed Workers An Example: Training of Workers ▶ Causality The causal effect on earnings of participation in training of a worker is defined as the change in earnings induced by externally changing her training status. Externally means holding constant other determinants of earnings/training. Examples of determinants: ▶ ind. characteristics: level of education, health, motivation,... ▶ predetermined endogenous factors: working time reduction, unemployment,... Note: Avoid to control for factors determined after assignment into training, because these may be part of the effect. ▶ The Fundamental Problem of Causal Inference We observe individuals only in one state (training or no training). Not being trained is counterfactual for the trainee. There is a problem of missing data. Causal Inference in Microeconometrics Introduction An Example: Training of Unemployed Workers The statistical solution to causal inference Identify the average causal effect rather than individual, but: ▶ Selection bias Replace missing information (= no training) by the average earnings of non-trainees, but can be problematic because 1. workers with high earnings capacity (in absence of training) are excluded from or not interested in training) (- bias); 2. workers with low earnings capacity face more hurdles to enter training (+ bias) Often unobserved and, hence, difficult to keep constant. ▶ Heterogeneous TE implies ATT ̸= ATE Average effect of randomly selected individual (ATE) ̸= average effect of trainee (ATT) 1. If high returns sort into training ⇒ ATT > ATE. 2. If only low returns are assigned to training ⇒ ATT < ATE Causal Inference in Microeconometrics Introduction An Example: Training of Unemployed Workers The statistical solution to causal inference ▶ Random assignment as solution to the selection problem ▶ Random assignment ≡ Assigning individuals of (a random sample of) a population with a predetermined probability (e.g. 50%) to a treatment or control group. ▶ By random assignment the composition of the two groups are on average the same, i.e. balanced, at least if the groups are large enough (LLN) ⇒ holds constant other determinants and thus eliminates selection bias ⇒ Difference between average earnings of treated (=trainees) and control group estimates the average causal effect of training of the population of interest. ▶ Issues: 1. Do we measure ATT or ATE? 2. What in case of non-compliance? If some controls get treated or some assigned to the treatment do not get the treatment? Causal Inference in Microeconometrics Introduction Randomization as “gold standard” Randomization as “gold standard” If randomization is not possible, the researcher should try to detect and exploit hidden forms of randomization (natural experiments) to identify causal effects by different methods: ▶ Unconfoundedness/conditional independence assumption (CIA): regression, (propensity score) matching, re-weighting ▶ Difference-in-differences (DiD) ▶ Regression Discontinuity Design (RDD) ▶ Instrumental Variables (IV) Together with RCT’s these are what Angrist and Pischke (2015) call the “furious five methods of causal inference” = introduction in this course. NOTE: Causal machine learning: ▶ NOT other identification method ▶ rather an adaptive non-parametric estimation (“supervised learning”) method resolving the “curse of dimensionality”. Causal Inference in Microeconometrics Introduction Randomization as “gold standard” Limitations and Key Ingredients of Modern Causal Inference ▶ No universal methods exists. Different methods exploit ̸= forms of hidden randomization. The choice of method depends on assignment mechanism. ▶ Non-experimental methods rely on some untestable assumptions. Essential for a researcher to convince that these are credible. ▶ In modern causal inference ▶ identification of TE is fundamentally non-parametric; ▶ TE heterogeneity is key: requires (1) a precise characterization of population for which the TE is estimated; (2) different estimation methods than traditional ones assuming TE homogeneity. Causal Inference in Microeconometrics Introduction Overview of identification methods Overview of identification methods: the training example Next we briefly provide an overview of the identification strategies followed in the methods discussed in this course. Causal Inference in Microeconometrics Introduction Overview of identification methods Methods Assuming CIA or unconfoundness ▶ Voluntary training for unemployed ▶ Suppose # of candidates > # available training slots. ▶ Suppose training is outsourced based only on observed attributes in admin files ((un)employment experience, age, gender,...) and capacity problem is resolved by assigning candidates randomly to training, conditional on these attributes. ▶ For a given set of attributes x, average earnings of non-participants is a good estimate of earnings of participants in counterfactual of no participation. ▶ Difference of average earnings between 2 groups identifies the AT T (x), conditional on attributes. Averaging over the distribution of attributes among participants yields the AT T. ▶ Note: Identified AT T ̸= AT E! [Can we identify AT E?] Causal Inference in Microeconometrics Introduction Overview of identification methods Difference-in-differences (DID) ▶ Mandatory training (with full compliance) introduced for unemployed high school drop-outs in a particular year t. ▶ Suppose that 1. prior to t (< t) the average earnings of these unemployed drop-outs and high school graduates evolves parallel (testable); 2. composition of these groups is not ̸= affected after t (e.g. to benefit from or to avoid training) (not testable!) ▶ Constant selection bias (SB) pre- (< t) and post-reform (≥ t). ▶ The average earnings of HS graduates post treatment (> t) + SB = estimate of missing counterfactual for treated drop-outs. ▶ DID can only identify ATT, not ATE. Why? Causal Inference in Microeconometrics Introduction Overview of identification methods Regression Discontinuity Design (RDD) ▶ In the absence of a natural barrier (river, mountain,...), living close at either side of the border is expected to be unrelated to the earnings of individuals, because: (i) employability changes only gradually with location; (ii) it is as if individuals are randomized close around the border. ▶ Suppose mandatory training (with full compliance) available only at one side of the border and cost of changing residence > net benefit of training (⇒ no sorting around border): participation rate in training changes abruptly at border. ▶ Then, if re-employment changes abruptly at the border = evidence of causal effect of training on re-employment. ▶ Only local ATE for the population close to the border: may not be representative for all trainees. Causal Inference in Microeconometrics Introduction Overview of identification methods Instrumental Variables (IV) (1) ▶ Assume that volunteers for training (T ∈ {0, 1}) are randomly assigned in a training (Z = 1) and control group (Z = 0) ⇒ Z satisfies the 2 classic assumptions of IV: 1. Z is a strong predictor of T , but not necessarily perfect because of non-compliance: P (T = 1|Z = 1) < 1 and/or P (T = 1|Z = 0) > 0. 2. Z is not directly related to Y : only indirectly through T. ▶ 2SLS = the ratio of 2 causal effects, i.e. bRF /bF S : 1. Z on Y = ’intent-to-treat’ effect (ITT) = E(Y |Z = 1) − E(Y |Z = 0) = bRF in the ’reduced form’ (RF) regression: Y = aRF + bRF Z + uRF to 2. Z on T = E(T |Z = 1) − E(T |Z = 0) = P (T = 1|Z = 1) − P (T = 1|Z = 0) = bF S in the ’first stage’ (FS) regression: T = aF S + bF S Z + uF S. Causal Inference in Microeconometrics Introduction Overview of identification methods Instrumental Variables (IV) (2): Identification ▶ What does 2SLS on regression of Y on T using Z as IV identify? ▶ Numerator = ∆ of Y induced by Z, but Z changes Y only for sub-population for which Z changes T (= ’compliers’) ⇒ requires scaling by the relative size of the complying sub-population, i.e. the denominator ⇒ 2SLS identifies 1. ATE = ATT only if TE = homogeneous (implicit in classic IV) 2. If TE = heterogeneous, i.e. TE for compliers ̸= non-compliers: (i) ATT if P (T = 1|Z = 0) = 0: no control enters training. (ii) Average TE for compliers = ’Local’ ATE = LATE (under additional assumptions) if P (T = 1|Z = 0) > 0 Causal Inference in Microeconometrics Introduction Overview of identification methods Instrumental Variables (IV) (3): Complications ▶ If TE = heterogeneous ⇒ 1. Causal interpretation requires stronger assumptions (i) Monotonicity of T in Z: Z makes T more (or less) likely for all (see later) (ii) Not only Z not (directly) related to Y , but neither to the TE (i.e. to the ’return’ to T ) Example: Consider using distance (D) to training center as IV ⇒ Requires not only D ⊥ ⊥ Y , but also D ⊥ ⊥ ∆Y : * Residence choice should be unrelated to earnings capacity or return to training (OK for training, but not for education?) * Training centers should, e.g., not be located in ’low opportunity’ neighborhoods. ▶ If IV is multi-valued and TE heterogeneous 2SLS no longer identifies a causal effect ⇒ requires other estimation method. Causal Inference in Microeconometrics Introduction Outline of the Course Outline of the Course 1. The Problem of Causality in Microeconometrics 2. Methods Based on Unconfoudndedness or the CIA 3. Difference-in-Differences (“Natural Experiments”) (DiD) 4. Instrumental Variable Estimator (IV) 5. Regression Discontinuity Design (RDD) 6. Randomized Controlled Trials (RCTs) 7. Introduction to Causal Machine Learning Lecture on RCT’s given by William Parienté Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics Outline for the continuation of this lecture 1. The Problem of Causality 1.1 A Formal Framework to Think about Causality 1.2 The Fundamental Problem of Causal Inference 1.3 The Statistical Solution 1.4 RCTs as Solution 2. Causality in a Regression Framework 2.1 Specification of the Selection into Treatment 2.2 Problems with OLS Estimation 3. Is Selection Bias Likely? 3.1 The Generalized Roy (1951) Model 3.2 Alternative Reasons for Selection Bias 4. Limitations/Critiques of/on the Statistical Solution 5. Main Neglected Methods Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics Sources for the Continuation of this Lecture ▶ The presentation of the “statistical” approach to causal inference (Sections 1-3), the so-called Neyman (1923) - Rubin (1978) - Holland (1986) (NRH) model, is based on Ichino (2007), Blundell and Costa Dias (2009) ▶ Criticism from the “structural econometric approach” (Section 4) is based on Heckman and Vytlacil (2007). Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics A Formal Framework about Causality 1.1 A Formal Framework about Causality Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics A Formal Framework about Causality Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics The Fundamental Problem of Causal Inference 1.2 The Fundamental Problem of Causal Inference Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics The Statistical Solution 1.3 The Statistical Solution Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics The Statistical Solution Is Comparison by Treatment Status Informative for ATT? Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics The Statistical Solution Is Comparison by Treatment Status Informative for ATE? No, because of selection bias and TE heterogeneity. To see this, (i) observe for C ∈ {0, 1} : E(Yi (C)) = P (Di = 1)E{Yi (C)|Di = 1}+P (Di = 0)E{Yi (C)|Di = 0}; (ii) add and subtract E(Yi (1)) − E(Yi (0)) to Equation (4) to obtain after some calculations =ATE z }| { E{Yi |Di = 1} − E{Yi |Di = 0} = E(Yi (1)) − E(Yi (0)) +B + P (Di = 0) [E{Yi (1) − Yi (0)|Di = 1} −E{Yi (1) − Yi (0)|Di = 0}] where B ≡ E{Yi (0)|Di = 1} − E{Yi (0)|Di = 0}, i.e. the selection bias, and the third term is the difference in the TE between treated and control units. Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics RCTs as Solution 1.4 RCTs as Solution Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics Implementation Implementation 1. By lottery on eligible population ▶ Example: random assignment of eligible unemployed for training. 2. Phase in design ▶ Example: control group gets training later on. 3. Encouragement design ▶ Example: randomly assigned group of unemployed is encouraged to participate in training (e.g. by sending information only to this group). Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics Some Issues with Implementation of RCT’s Some Issues with Implementation 1. Noncompliance ▶ Example: ▶ Not all unemployed in treatment group enter training; ▶ Some unemployed in control group enter similar training. ▶ As long as # treated in treatment group > # treated in control group, a causal effect is identified: intention-to-treat (ITT) effect. ▶ Instrumental Variable (IV) can be used to go from ITT to LATE ̸= ATT or ATE. ▶ LATE = average effect of training for those who by the assignment to the treatment are induced to participate in training. 2. Other issues: Hawthorne effects, external validity, spillover effects,... Causal Inference in Microeconometrics The Problem of Causality in Microeconometrics Some Issues with Implementation of RCT’s Causal Inference in Microeconometrics Causality in a regression framework Causal Inference in Microeconometrics Causality in a regression framework Specification of the selection into treatment 2.1 Specification of the selection into treatment Causal Inference in Microeconometrics Causality in a regression framework Specification of the selection into treatment Causal Inference in Microeconometrics Causality in a regression framework Specification of the selection into treatment Causal Inference in Microeconometrics Causality in a regression framework Problems with OLS Estimation 2.2.0 What the Coefficient in OLS Measures Consider the following regression equation: Yi = β + αDi + ϵi Then for the sample size tending to infinity Cov(Yi , Di ) E(Yi Di ) − E(Yi )E(Di ) α̂OLS = = 2 V ar(Di ) E(Di2 ) − [E(Di )] Now using the LIE and that E(Di ) = P (Di )1 + [1 − P (Di )] 0 = P (Di ) ≡ p we obtain: E(Yi Di ) = E(Yi |Di = 1)p, E(Yi )E(Di ) = [E(Yi |Di = 1)p + E(Yi |Di = 0)(1 − p)] p, 2 E(Di2 ) − [E(Di )] = p − p2 = p(1 − p) so that we obtain [E(Yi |Di = 1) − E(Yi |Di = 0)] p(1 − p) α̂OLS = p(1 − p) = E(Yi |Di = 1) − E(Yi |Di = 0) Causal Inference in Microeconometrics Causality in a regression framework Problems with OLS Estimation The Law of Iterated Expectations (LIE) Consider a random vector w such that x = f (w) (e.g. x is a subset of w). Then for a random variable y: 1. E(y|x) = E[E(y|w)|x] 2. E(y|x) = E[E(y|x)|w] ”The smaller information set always dominates” Special case: the empty set is a subset of x ⇒ E(y) = E(E(y|x)) For more discussion see e.g. Section 2.2.3-2.2.4 in Wooldridge (2010) (see Ufora). Causal Inference in Microeconometrics Causality in a regression framework Problems with OLS Estimation Causal Inference in Microeconometrics Causality in a regression framework Problems with OLS Estimation Causal Inference in Microeconometrics Causality in a regression framework Problems with OLS Estimation Causal Inference in Microeconometrics Is selection bias likely? The Generalized Roy (1951) Model The Generalized Roy Model Consider the training example, but, to allow for linear model, outcome, Y = earnings. Earnings for untrained and trained individuals is modeled as Y1 = Xβ1 + U1 (1) Y0 = Xβ0 + U0 (2) The cost of training depends on W (e.g. distance to training center, subsidy,...) and specified C = W βC + UC (3) The net benefit to training is then R = X(β1 − β0 ) − W βC + U1 − U0 − UC = Zγ + ν (4) where ν = (U1 − U0 − UC ), Z = (X, W ) and γ = (β1′ − β0′ , −βC ′ ) Causal Inference in Microeconometrics Is selection bias likely? The Generalized Roy (1951) Model ▶ The generalized Roy model assumes 1. Z⊥(U0 , U1 , UC ) ⇔ linear specification is correct 2. (U0 , U1 , UC ) ∼ N ormal(0, ΣGR ) ▶ The probability of self-selecting into training is     ν −zγ zγ P r(R ≥ 0|Z = z) = P r(ν ≥ −zγ) = P r ≥ =Φ σν σν σν where Φ is the cumulative standard normal distribution. ▶ Since ν = (U1 − U0 − UC ), participation in training is correlated with the idiosyncratic return to training, U1 − U0. Therefore, E (U1 − U0 |D = 1) = E (U1 − U0 |ν ≥ −zγ) = Cov (U1 − U0 , ν) λ(zγ) ϕ(.) where λ(.) = Φ(.) the inverse Mill’s ratio Causal Inference in Microeconometrics Is selection bias likely? The Generalized Roy (1951) Model ▶ Therefore in the case of the simple Roy model (C = 0) or the extended Roy model (UC = 0) we obtain E (U1 − U0 |D = 1) = V ar (U1 − U0 ) λ(zγ) > 0 i.e. individuals with high returns self-select into training. ▶ Since usually unobserved determinants of income in the absence of training (U0 ) are correlated with ability, and ability and investment in human capital are complements, we also expect a positive selection on the untreated outcome: E [U0 |D = 1] > E [U0 |D = 0] (5) ▶ ⇒ OLS estimator overestimates both ATE and ATT, unless Cov (U1 − U0 , UC ) >> 0, i.e. if for individuals with high idiosyncratic returns (U1 − U0 ) the cost (UC ) is even higher, e.g. because of high time opportunity costs. Causal Inference in Microeconometrics Is selection bias likely? Alternative reasons for selection bias Alternative reasons for selection bias ▶ The Roy model suggests that if participation to treatment is voluntary, selection on (expected) gains is likely Note: ▶ If no selection on untreated outcome, regression identifies ATT ⇒ RCT among voluntary candidates identifies ATT (if full compliance in control group); ▶ RCT on full population cannot identify ATE, unless participation is enforced (because otherwise only high return individuals participate). ▶ Selection induced by program administrators ▶ Employment Agencies “cream skim” if they are financed according to placement rate ▶ “Positive discrimination” may lead to negative selection Note: Unless participation is mandatory, self-selection still interferes with selection process Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques of/on the NRH Causal Model 1. Potential outcomes Yi (Zi , Di ) are invariant to the assignment mechanism Zi :∀Zi : Yi (Zi , Di ) = Yi (Di ) (Cf. “excl restrict”) Examples: 0.1 If assignment to training is randomized in social experiment, behavior may change since one is “observed”, which may affect the likelihood of being employed (irrespective of being trained) (= “Hawthorne” effect) 0.2 If centers are located closer to places where more jobs are available, the distance between the residence and training center is an invalid assignment mechanism because distance also has a direct effect on employment/earnings. Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques (2) 2. No social interactions effects or general equilibrium effects for the outcomes (Cf. Stable Unit Treatment Value Assumption: “SUTVA”) Example: Training may not generate employment at the expense of an untrained worker (employed or unemployed). One identifies the effect of “small” programs only. Exception: Crépon et al. (2013) follow a two step procedure to evaluate displacement effects of job placement assistance for young unemployed job seekers: 1. randomly assign treatment intensities across labor markets; 2. within each labor market treatment was assigned according to predetermined proportions. Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques (3) 3. Focus on objective outcomes realized ex post ⇒ 1. causal effects on subjective evaluation of an outcome (individual and social welfare) is not considered 2. ex ante effects, neither regret on welfare and choices of individuals is considered. 4. There is no simultaneity in causal effects, i.e., outcomes cannot cause each other reciprocally Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques (4) 5. The causal effect is a “black box”: no theory of the mechanism required, only randomization ⇒ 1. Holland states that “there can not be a causal effect of gender on earnings because analysts cannot randomly assign gender” ⇒ confounds definition and identification of a causal effect (⋆). 2. Causal effects are defined in terms of what can be identified and not what is of theoretical interest (see discussion on IV and LATE) 3. together with previous critique this implies that one can only evaluate historical interventions in a given environment (internal validity ) and not forecast impacts in another environment (external validity ) or to never historically experienced environments. 4. theory cannot be used to (help) identifying causal effects (see next point) (⋆) Note: Pearl (2000) uses causal graphs to clarify relation between causal identification and required independence Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques (5) 6. Only effects involving marginal distributions of potential outcomes can be identified ⇒ 1. Distribution of gains (F (Y1 − Y0 )) or proportion that gains (P r(Y 1 > Y0 )) cannot be identified, because these depend on the joint distribution (F (Y0 , Y1 )). 2. Additional information is required for identification: 2.1 Treatment effect homogeneous in terms of unobservables: Y1 − Y0 = α(X) ⇒ degenerate distribution of gains. 2.2 Impose a particular (e.g. perfect) rank correlation between the 2 counterfactual outcome distributions F (Y0 ) and F (Y1 ); 2.3 Assume that the gain (Y1 − Y0 ) is independent of Y0 : gain cannot be forecast when participation decision is taken; 2.4 Restrictions imposed by theory: e.g. the (extended) Roy model (see below). 2.5 See Heckman et al. (1997) for more discussion. Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Limitations/Critiques (5) Quantile treatment effects ̸= quantile of gains to treatment! ▶ Quantile treatment effect = Q(Y1 ) − Q(Y0 ) ̸= Q(Y1 − Y0 ) = quantile of the gains to treatment; ▶ Equality only in case of perfect rank correlation ▶ Quantile treatment effects make sense when one aims at comparing 2 counterfactual distributions with an anonymous (symmetric) social welfare function, but not to evaluate individual welfare effects of policy. Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Identification of F (Y1 − Y0 ) in the (extended) Roy model ▶ From Equations (1) and (2), the properties of the truncated Normal and by normalizing σν = 1, we obtain: E (Y1 |D = 1, X = x, Z = z) = xβ1 + Cov(U1 , ν)λ(zγ) (6) E (Y0 |D = 0, X = x, Z = z) = xβ0 + Cov(U0 , ν)λ̃(zγ) (7) ϕ(zγ) where λ̃(zγ) = − 1−Φ(zγ). ▶ Identification can be assured since: 1. A Probit on participation identifies λ(zγ) and λ̃(zγ); 2. A linear regression of Y on xd, x(1 − d), λ(zγ) and λ̃(zγ) identifies β1 , β0 , Cov(U1 , ν) and Cov(U0 , ν); 3. The residuals of the regression identify V ar(U0 ) and V ar(U1 ); 4. In the (extended) Roy model UC = 0, so that Cov(Ud , ν) = Cov(Ud , U1 − U0 ) (d = 0, 1) identify Cov(U0 , U1 ), and, hence, F (Y0 , Y1 ) and F (Y1 − Y0 ), since V ar(U0 ) and V ar(U1 ) are known. Causal Inference in Microeconometrics Limitations/Critiques of/on the Statistical Solution to Causal Inference Identification of F (Y1 − Y0 ) in the (extended) Roy model Implication: In the Roy model, where C = 0, Φ(zγ) = P r(Y1 > Y0 |Z = z) identifies the proportion of individuals who benefit from treatment. For further discussion on the debate between the “statistical” and “econometric” approach see: 1. Heckman and Vytlacil (2007), Deaton (2009), and more recently, Deaton and Cartwright (2016) for further critique; 2. Heckman and Pinto (2024) criticize statistical approach of NR and the Directed Acyclic Graphs (DAGs) of Pearl (2000) in computer science and argue for the econometric framework of Frisch and Haavelmo; 3. Imbens (2009) for a reply. Causal Inference in Microeconometrics Some Important Neglected Methods Some Important Neglected Methods 1. Statistical Bounds: Manski (1990), Horowitz and Manski (2000), Lee (2009). 2. Bunching: Kleven (2016), Berthanha et al. (2023). 3. Timing of Events: Abbring (2003), Abbring and van den Berg (2004), Lomabardi and Gerard J. van den Berg (2024). 4. Designed-Based Identification (“Shift-Share” instruments): Goldsmith-Pinkham et al. (2020), Borusyak et al. (2022, 2023). 5. Local Projections (related to vector autoregressions (VAR)) Jorda (2024); Dube et al. (2023) relates to event studies. 6. Matrix Completion (generalizes TWFE): E.g. Bai (2009). Causal Inference in Microeconometrics References Abadie, A. and M. D. Cattaneo (2018). Econometric Methods for Program Evaluation. Annual Review of Economics 10, 465–503. Abbring, J. H., G. J. v. d. B. (2003). The nonparametric identification of treatment effects in duration models. Econometrica 71 (5), 1491–1517. Abbring, J. H. and G. van den Berg (2004). Analyzing the effect of dynamically assigned treatments using duration models, binary treatment models, and panel data models. Empirical Economics 29, 5–20. Angrist, J. D. and J.-S. Pischke (2009). Mostly Harmless Econometrics. An Empiricist’s Companion. New Jersey: Princeton University Press. Angrist, J. D. and J.-S. Pischke (2015). Matering ’Metrics. New Jersey: Princeton University Press. Arkhangelsky, D. and G. W. Imbens (2024). Causal models for longitudinal and panel data: a survey. The Econometrics Journal utae014. Causal Inference in Microeconometrics References Bai, J. (2009). Panel data models with interactive fixed effects. Econometrica 77 (4), 1229–1279. Berthanha, M., C. Caetano, H. Jales, and N. 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