Karim Chalak

Viewpoint: An extended class of instrumental variables for the estimation of causal effects

This paper examines the ways in which structural systems can yield observed variables, other than the cause or treatment of interest, that can play an instrumental role in identifying and estimating causal effects. We focus speciOcally on the ways in which structures determine exclusion restrictions and conditional exogeneity relations that act to ensure identification. We show that by carefully specifying the structural equations and by extending the standard notion of instrumental variables, one can identify and estimate causal effects in the endogenous regressor case for a broad range of economically relevant structures. Some of these have not previously been recognized. Our results there create new opportunities for identifying and estimating causal effects in non-experimental situations. Our results for more familiar structures provide new insights. For example, we extend results of Angrist, Imbens, and Rubin (1996) by taking into account an important distinction between cases where Z is an observed exogenous instrument and those where it is a proxy for an unobserved exogenous instrument. A main message emerging from our analysis is the central importance of sufficiently specifying the causal relations governing the unobservables, as these play a crucial role in creating obstacles or opportunities for identification. Because our results exhaust the possibilities for identification, we ensure that there are no other opportunities for identification based on exclusion restrictions and conditional independence relations still to be discovered. To accomplish this characterization, we introduce notions of conditioning and conditional extended instrumental variables (EIVs). These are not proper instruments, as they are endogenous. They nevertheless permit identification and estimation of causal effects. We analyze methods using these EIVs either singly or jointly.

Identification and Identification Failure for Treatment Effects Using Structural Systems

We provide necessary and sufficient conditions for effect identification, thereby characterizing the limits to identification. Our results link the nonstructural potential outcome framework for identifying and estimating treatment effects to structural approaches in economics. This permits economic theory to be built into treatment effect methods. We elucidate the sources and consequences of identification failure by examining the biases arising when the necessary conditions fail, and we clarify the relations between unconfoundedness, conditional exogeneity, and the necessary and sufficient identification conditions. A new quantity, the exogeneity score, plays a central role in this analysis, permitting an omitted variable representation for effect biases. This analysis also provides practical guidance for selecting covariates and insight into the price paid for making various identifying assumptions and the benefits gained.

Viewpoint: An Extended Class of Instrumental Variables for the Estimation of Causal Effects (Une Classe Tendue De Variables Instrumentales Pour L'Estimation Des Effets De Causalit)

This paper examines the ways in which structural systems can yield observed variables, other than the cause or treatment of interest, that can play an instrumental role in identifying and estimating causal effects. We focus speciOcally on the ways in which structures determine exclusion restrictions and conditional exogeneity relations that act to ensure identification. We show that by carefully specifying the structural equations and by extending the standard notion of instrumental variables, one can identify and estimate causal effects in the endogenous regressor case for a broad range of economically relevant structures. Some of these have not previously been recognized. Our results there create new opportunities for identifying and estimating causal effects in non-experimental situations. Our results for more familiar structures provide new insights. For example, we extend results of Angrist, Imbens, and Rubin (1996) by taking into account an important distinction between cases where Z is an observed exogenous instrument and those where it is a proxy for an unobserved exogenous instrument. A main message emerging from our analysis is the central importance of sufficiently specifying the causal relations governing the unobservables, as these play a crucial role in creating obstacles or opportunities for identification. Because our results exhaust the possibilities for identification, we ensure that there are no other opportunities for identification based on exclusion restrictions and conditional independence relations still to be discovered. To accomplish this characterization, we introduce notions of conditioning and conditional extended instrumental variables (EIVs). These are not proper instruments, as they are endogenous. They nevertheless permit identification and estimation of causal effects. We analyze methods using these EIVs either singly or jointly.

Causality, conditional independence, and graphical separation in settable systems

We study the connections between causal relations and conditional independence within the settable systems extension of the Pearl causal model (PCM). Our analysis clearly distinguishes between causal notions and probabilistic notions, and it does not formally rely on graphical representations. As a foundation, we provide definitions in terms of suitable functional dependence for direct causality and for indirect and total causality via and exclusive of a set of variables. Based on these foundations, we provide causal and stochastic conditions formally characterizing conditional dependence among random vectors of interest in structural systems by stating and proving the conditional Reichenbach principle of common cause, obtaining the classical Reichenbach principle as a corollary. We apply the conditional Reichenbach principle to show that the useful tools of d-separation and D-separation can be employed to establish conditional independence within suitably restricted settable systems analogous to Markovian PCMs.

Instrumental Variables Methods With Heterogeneity And Mismeasured Instruments

We study the consequences of substituting an error-laden proxy W for an instrument Z on the interpretation of Wald, local instrumental variable (LIV), and instrumental variable (IV) estimands in an ordered discrete choice structural system with heterogeneity. A proxy W need only satisfy an exclusion restriction and that the treatment and outcome are mean independent from W given Z. Unlike Z, W need not satisfy monotonicity and may, under particular specifications, fail exogeneity. For example, W could code Z with error, with missing observations, or coarsely. We show that Wald, LIV, and IV estimands using W identify weighted averages of local or marginal treatment effects (LATEs or MTEs). We study a necessary and sufficient condition for nonnegative weights. Further, we study a condition under which the Wald or LIV estimand using W identifies the same LATE or MTE that would have been recovered had Z been observed. For example, this holds for binary Z and therefore the Wald estimand using W identifies the same “average causal response,†or LATE for binary treatment, that would have been recovered using Z. Also, under this condition, LIV using W can be used to identify MTE and average treatment effects for e.g., the population, treated, and untreated.

Measurement Error Without Exclusion: The Returns to College Selectivity and Characteristics

This paper studies the identification of the coefficients in a linear equation when data on the outcome, covariates, and an error-laden proxy for a latent variable are available. We maintain that the measurement error in the proxy is classical and relax the assumption that the proxy is excluded from the outcome equation. This enables the proxy to directly affect the outcome and allows for differential measurement error. Without the proxy exclusion restriction, we first show that the effects of the latent variable, the proxy, and the covariates are not identified. We then derive the sharp identification regions for these effects under any configuration of three auxiliary assumptions. The first weakens the assumption of no measurement error by imposing an upper bound on the noise to signal ratio. The second imposes an upper bound on the outcome equation coefficient of determination that would obtain had there been no measurement error. The third weakens the proxy exclusion restriction by specifying whether the latent variable and its proxy affect the outcome in the same or the opposite direction, if at all. Using the College Scorecard aggregate data, we illustrate our framework by studying the financial returns to college selectivity and characteristics and student characteristics when the average SAT score at an institution may directly affect earnings and serves as a proxy for the average ability of the student cohort.

Identification Without Exogeneity Under Equiconfounding in Linear Recursive Structural Systems

This chapter obtains identification of structural coefficients in linear recursive systems of structural equations without requiring that observable variables are exogenous or conditionally exogenous. In particular, standard instrumental variables and control variables need not be available in these systems. Instead, we demonstrate that the availability of one or two variables that are equally affected by the unobserved confounder as is the response of interest, along with exclusion restrictions, permits the identification of all the system’s structural coefficients. We provide conditions under which equiconfounding supports either full identification of structural coefficients or partial identification in a set consisting of two points.