Partial identification in nonseparable binary response models with endogenous regressors

Question

1. What is the binary outcome variable in the model?

2. What are the limitations of the approaches studied in binary response models with endogenous regressors?

3. What is the structure of the paper?

4. What is the notation for the product s-algebra on X x X?

Accepted Answer

The binary outcome variable Y in the model is denoted as Y {0, 1}, representing a binary outcome variable. It takes the values 0 or 1, indicating the presence or absence of a certain event or condition. In the context of the model, Y is determined by the functions ph1 and ph2, which depend on the covariates X, latent variables U, and structural parameters th. The binary outcome variable plays a crucial role in analyzing the relationship between the covariates and the outcome, providing insights into the counterfactual parameters and their impact on the outcome.

Accepted Answer

All of these approaches have well-documented limitations. Linear probability models, maximum likelihood estimation, control function approaches, and approaches based on special regressors have been extensively studied. However, they each have their own limitations. For example, linear probability models can produce predicted probabilities outside the range of 0 to 1, and maximum likelihood estimation may not be robust to violations of model assumptions. Control function approaches can be sensitive to the choice of control variables, and approaches based on special regressors may not be applicable in all situations. These limitations highlight the need for alternative approaches to address the challenges posed by endogenous regressors in binary response models.

Accepted Answer

The paper is structured into seven sections. Section 2 introduces the theoretical framework and assumptions. Section 3 focuses on practical implementation and optimization-based bounding procedure. Section 4 incorporates functional form, independence, and monotonicity assumptions. Section 5 discusses estimation and inference. Section 6 applies the methodology to study health insurance impact. Section 7 concludes the paper. Appendices A, B, C, and D provide proofs, additional discussion, comparison, and supplementary material respectively.

Accepted Answer

The notation for the product s-algebra on X x X is denoted by B(X) B(X). This notation is used when dealing with two measurable spaces (X, B(X)) and (X, B(X)). The product s-algebra represents the combined s-algebra of both spaces, allowing for operations and calculations involving both X and X x X.

Accepted Answer

The purpose of defining the vector-valued function r(u, th) in the General Framework: Practical Considerations section is to make progress in the research. This function is defined as r(u, th) := 1{ph(x 1 , u, th) >= 0} 1{ph(x 2 , u, th) >= 0} . . . 1{ph(x m , u, th) >= 0}, where ph represents a structural function. The function r(u, th) represents response types, which are binary vectors that indicate the choices an individual with type indexed by (u, th) would have made under different counterfactual assignments. By partitioning the space of latent variables using response types, various counterfactual objects of interest can be written as a disjoint union of sets U(s, th) from (3.1). This decomposition helps in understanding the connection between response types and counterfactual choices, and it is essential for rationalizing a given collection of counterfactual conditional distributions.

Accepted Answer

Exogenous covariates are observed variables that are independent of latent variables. They induce variation in observed conditional probabilities without affecting the distribution of latent variables. These variables can be used as arguments in the index function, providing a way to produce a smaller identified set for counterfactual parameters. They are essential in research as they help in understanding the relationship between observed and latent variables, and can aid in improving the accuracy of statistical models.

Accepted Answer

Monotonicity Assumption states that for each tuple (j, k) in the set M, ph(x j , u, th) <= ph(x k , u, th). This assumption allows researchers to order the value of the structural function. It helps in ruling out response types with certain probabilities. For example, if ph(x j , u, th) <= ph(x k , u, th), then response types with 1{ph(x j , u, th) >= 0} > 1{ph(x k , u, th) >= 0} are automatically ruled out. Monotonicity assumptions are extensively explored in triangular systems by Heckman and Pinto (2018).

Accepted Answer

A weakly-measurable multifunction V: ohm - 2 V is defined on a probability space. It is considered weakly-measurable if for every A in O V, the set V - (A) is well-defined. This means that for any open set A in O V, the set of all ohm in V that belong to A is well-defined. In simpler terms, a weakly-measurable multifunction is a function that assigns a well-defined value to each open set in the collection O V.

Accepted Answer

The identified set under Assumption 4.1 is the set of all conditional distributions P * Yg |Y,W,Z,U satisfying P Yg |Y,W,Z,U (Y g = 1{ph(g(W, Z), U, th) >= 0} | Y = y, W = w, Z = z, U = u) = 1, and P Y,W,Z,U -a.s. for some pair (P U |Y,W,Z , th) I * Y,W,Z. This set is analogous to Definitions 2.1 and 2.2. The definition can be modified to accommodate any combination of Assumptions 4.1 and 4.2.

Accepted Answer

The key idea is to determine the values of th such that the function ph(X, th) : X - R induces a unique ordering of the points in the support X. With a scalar X variable, and ph(X, th) = Xth, Example 4 shows that only two orderings are possible, corresponding to the case when th >= 0 and th < 0. After the order is determined, we can immediately determine the set of response types that must be assigned zero probability by the distribution of U, and then impose these restrictions as an additional constraint in the bounding problems (3.12) and (3.13).

Partial identification in nonseparable binary response models with endogenous regressors

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Most frequently asked questions

1. What is the binary outcome variable in the model?

2. What are the limitations of the approaches studied in binary response models with endogenous regressors?

3. What is the structure of the paper?

4. What is the notation for the product s-algebra on X x X?

5. What is the purpose of defining the vector-valued function r(u, th) in the General Framework: Practical Considerations section?

6. What are exogenous covariates in research?

7. What is Monotonicity Assumption?

8. What is a weakly-measurable multifunction?

9. What is the identified set under Assumption 4.1?

10. What is the key idea behind implementing the approach when the index function is additively separable in U, as in Assumption B.2?

References

An IV Model of Quantile Treatment Effects

Reverse search for enumeration

Estimation and Confidence Regions for Parameter Sets in Econometric Models

Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity

Bounds on Treatment Effects from Studies with Imperfect Compliance

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