Intro DAG Fix Do Exercise Roy Conclusion The Roy Model and Pearl’s Do Calculus: What “Do” Cannot Do James Heckman University of Chicago Rodrigo Pinto University of Chicago Econ 312, Spring 2019 James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion 1. Causal Effects and the Do-calculus James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Identification of Treatment Effects of a DAG Pearl’s Do-Calculus: Purpose: Identify casual effects from non-experimental data. 1 James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Identification of Treatment Effects of a DAG Pearl’s Do-Calculus: Purpose: Identify casual effects from non-experimental data. 1 Application: Bayesian network structure, i.e., Directed Acyclic 2 Graph (DAG) that represents causal relationships. James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Identification of Treatment Effects of a DAG Pearl’s Do-Calculus: Purpose: Identify casual effects from non-experimental data. 1 Application: Bayesian network structure, i.e., Directed Acyclic 2 Graph (DAG) that represents causal relationships. Tools: Three inference rules that translate graphical relations 3 of a DAG into causal independence conditional relations (Pearl 1995, and Pearl 2000). James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Identification of Treatment Effects of a DAG Pearl’s Do-Calculus: Purpose: Identify casual effects from non-experimental data. 1 Application: Bayesian network structure, i.e., Directed Acyclic 2 Graph (DAG) that represents causal relationships. Tools: Three inference rules that translate graphical relations 3 of a DAG into causal independence conditional relations (Pearl 1995, and Pearl 2000). Identification method: Iteration of do-calculus rules to 4 generate a function that describes treatment effects statistics as a function of the observed variables only (Tian and Pearl 2002, Tian and Pearl 2003). James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Characteristics of Pearl’s Do-Calculus Completeness If some causal effect of a DAG is identifiable , then there exists a sequence of application of the Do-Calculus rules that can generate a formula that translates causal effects into an equation that only relies on observational quantities (Huang and Valtorta 2006, Shpitser and Pearl 2006). Limitation Only works for DAGs. Does not allow for additional information outside the DAG framework that could generate identification of causal distributions. Only applies to the information content of a DAG. James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion 2. Statistical Tools for DAGs James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Markovian Model A Markovian Model (Tian and Pearl, 2003) is defined by four elements: � � M = N , U , G , P ( V i | pa ( V i )) where: N = { N 1 , . . . , M m } is a set of observed variables; 1 U = { U 1 , . . . , U n } is a set of unobserved variables; 2 G is a direct acyclic graph with nodes corresponding to the 3 variables V i in V = N ∪ U ; V comprises both observed and unobserved variables. 4 P ( V i | pa ( V i )) is the conditional probability of a variable V i ∈ V 5 given its parents pa ( V i ) ⊂ V . James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Factorization of a Markovian Model Joint Probability Pr( V 1 , . . . , V n + m ) can be factorized as: � Pr( V 1 , . . . V n + m ) = Pr( V i | pa ( V i )) V i ∈ V Causal Interpretation – Structural Equations V i = f ( pa ( V i )) e.g. Y = f ( X , U ) Direction: Variables pa ( V i ) cause V i (e.g. X , U cause Y ) and not the contrary. Autonomy: Structural equation f is a deterministic function ( y = f ( x , u )) that is invariant to changes in x or u (Frisch, 1938). James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation Parents: Variables pa ( Y ) ⊂ V are termed parents of Y ∈ V . 1 D ( Y ) = ∪ | V | j =1 pa − j ( T ) , where pa − ( k +1) ( G ) = pa − 1 ( par − k ( G )) , pa − 1 ( G ) = ∪ T ∈G pa − 1 ( T ) such that G ⊂ V and pa − 1 ( T ) = { Y ∈ V ; T ∈ pa ( Y ) } . James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation Parents: Variables pa ( Y ) ⊂ V are termed parents of Y ∈ V . 1 If pa ( Y ) = ∅ , the Y is not caused by any variable in the model. D ( Y ) = ∪ | V | j =1 pa − j ( T ) , where pa − ( k +1) ( G ) = pa − 1 ( par − k ( G )) , pa − 1 ( G ) = ∪ T ∈G pa − 1 ( T ) such that G ⊂ V and pa − 1 ( T ) = { Y ∈ V ; T ∈ pa ( Y ) } . James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation Parents: Variables pa ( Y ) ⊂ V are termed parents of Y ∈ V . 1 If pa ( Y ) = ∅ , the Y is not caused by any variable in the model. In such cases, Y is termed external variable . D ( Y ) = ∪ | V | j =1 pa − j ( T ) , where pa − ( k +1) ( G ) = pa − 1 ( par − k ( G )) , pa − 1 ( G ) = ∪ T ∈G pa − 1 ( T ) such that G ⊂ V and pa − 1 ( T ) = { Y ∈ V ; T ∈ pa ( Y ) } . James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation Parents: Variables pa ( Y ) ⊂ V are termed parents of Y ∈ V . 1 If pa ( Y ) = ∅ , the Y is not caused by any variable in the model. In such cases, Y is termed external variable . Descendants: Variables D ( Y ) ⊂ V that are caused by Y 2 (directly or indirectly) are termed descendants of Y ∈ V . D ( Y ) = ∪ | V | j =1 pa − j ( T ) , where pa − ( k +1) ( G ) = pa − 1 ( par − k ( G )) , pa − 1 ( G ) = ∪ T ∈G pa − 1 ( T ) such that G ⊂ V and pa − 1 ( T ) = { Y ∈ V ; T ∈ pa ( Y ) } . James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion 3. Fixing versus Conditioning James Heckman Roy Does Not
Intro DAG Fix Do Exercise Roy Conclusion Pearl’s Definition of Causal Effects, the Do-operator The Do-operator is based on the Truncated Factorization of the probability factor of the fixed variable is deleted: Let X ⊂ V : Then Pr( V ( x ) = v ) = Pr( V 1 = v 1 , . . . , V m + n = v m + n , | do ( X ) = x ) and: � � V i ∈ V \ X P ( V i = v i | pa ( V i )) if v is consistent with x ; Pr( V ( x ) = v ) = 0 if v is inconsistent with x . James Heckman Roy Does Not
Example of the Do-operator Z X Y Variables: Y , X , Z Factorization: Pr( Y , X , Z ) = Pr( Y | Z , X ) Pr( X | Z ) Pr( Z ) = Pr( Y | X ) Pr( X | Z ) Pr( Z ) Do-operator: Pr( Z , Y | do ( X ) = x ) = Pr( Y | X = x ) Pr( Z ) Conditional operator: Pr( Y , Z | X = x ) = Pr( Y | Z , X = x ) Pr( X | Z , X = x ) Pr( Z | X = x ) = Pr( Y | X = x ) Pr( Z | X = x ) Do-operator targets variables, not causal links.
Example of the Do-operator V U X Y Variables: Y , X , U , V Factorization: Pr( V , U , X , Y ) = Pr( Y | U , X ) Pr( X | V ) Pr( U | V ) Pr( V ) Do-operator: Pr( V , U , Y | do ( X ) = x ) = Pr( Y | U , X = x ) Pr( U | V ) Pr( V ) Conditional operator: Pr( V , U , Y | X = x ) = Pr( Y | U , V , X = x ) Pr( U | V , X = x ) Pr( V | X = x ) = Pr( Y | U , X = x ) Pr( U | V ) Pr( V | X = x )
Intro DAG Fix Do Exercise Roy Conclusion Empirical and Hypothetical Models Heckman and Pinto (2013) Empirical Model Hypothetical Model V U V U ~ X Y X Y X James Heckman Roy Does Not
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation;
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters; Causal parameters are defined under the hypothetical model; 1
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters; Causal parameters are defined under the hypothetical model; 1 Observed data is generated through empirical model; 2
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters; Causal parameters are defined under the hypothetical model; 1 Observed data is generated through empirical model; 2 Distinguish definition from identification;
Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters; Causal parameters are defined under the hypothetical model; 1 Observed data is generated through empirical model; 2 Distinguish definition from identification; Identification requires to connect the hypothetical and 1 empirical models such that
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