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**1 - 4**of**4**### 2009. From association to causation via a potential outcomes approach

- Informat. Systems Res

"... doi 10.1287/isre.1080.0184 ..."

### BMC Public Health BioMed Central

, 2008

"... Research article Implications of the HIV testing protocol for refusal bias in seroprevalence surveys ..."

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Research article Implications of the HIV testing protocol for refusal bias in seroprevalence surveys

### Endogeneity in Probit Response Models

"... Suppose a linear regression model describes responses to treatment and to covariates. If subjects self-select into treatment, the process being dependent on the error term in the model, endogeneity bias is likely. Similarly, we may have a linear model that is to be estimated on sample data; if subje ..."

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Suppose a linear regression model describes responses to treatment and to covariates. If subjects self-select into treatment, the process being dependent on the error term in the model, endogeneity bias is likely. Similarly, we may have a linear model that is to be estimated on sample data; if subjects self-select into the sample, endogeneity becomes an issue. Heckman (1978, 1979) suggested a simple and ingenious two-step method for taking care of endogeneity, which works under the conditions described in those papers. This method is widely used. Some researchers have applied the method to probit response models. However, the extension is unsatisfactory. The non-linearity in the probit model is an essential difficulty for the two-step correction, which will often make bias worse. Likelihood techniques are to be preferred, although the numerics are delicate. To define the models and estimation procedures more formally, consider n subjects, indexed by i = 1,...,n. Subjects are assumed to be independent and identically distributed. For each subject, there are two manifest variables Xi,Zi and two latent variables Ui,Vi. Assume that (Ui,Vi) are bivariate normal, with mean 0, variance 1, and correlation ρ. Assume further that (Xi,Zi) is independent of (Ui,Vi), i.e., the manifest variables are exogenous. For ease of exposition, we take

### doi:10.1093/pan/mpp037 Endogeneity in Probit Response Models

, 2010

"... We look at conventional methods for removing endogeneity bias in regression models, including the linear model and the probit model. It is known that the usual Heckman twostep procedure should not be used in the probit model: from a theoretical perspective, it is unsatisfactory, and likelihood metho ..."

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We look at conventional methods for removing endogeneity bias in regression models, including the linear model and the probit model. It is known that the usual Heckman twostep procedure should not be used in the probit model: from a theoretical perspective, it is unsatisfactory, and likelihood methods are superior. However, serious numerical problems occur when standard software packages try to maximize the biprobit likelihood function, even if the number of covariates is small. We draw conclusions for statistical practice. Finally, we prove the conditions under which parameters in the model are identifiable. The conditions for identification are delicate; we believe these results are new. 1