A Framework for Eliciting, Incorporating, and Disciplining Identification Beliefs in Linear Models

1 Referring to more than 60 articles published in the top three empirical journals between 2002 and 2005, Moon and Schorfheide (2009 Moon, H. R., and Schorfheide, F. (2009), “Estimation With Overidentifying Inequality Moment Conditions,” Journal of Econometrics, 153, 136–154. DOI: 10.1016/j.jeconom.2009.05.003.[Crossref], [Web of Science ®] , [Google Scholar]) noted that “in almost all of the articles the authors explicitly stated their beliefs about the sign of the correlation between the endogenous regressor and the error term; yet none of the authors exploited the resulting inequality moment condition in their estimation.”

2 If $T^{*}=1$, the only way it can be mismeasured is downward: T = 0. If $T^{*}=0$ the only way it can be mismeasured is upward: T = 1. Hence, $\tilde{w}$ must be negatively correlated with $T^{*}$.

3 Such a decomposition is called a transparent parameterization in the statistics literature (see, e.g., Gustafson 2015).

4 See https://github.com/fditraglia/ivdoctr.

5 See (9) and the discussion immediately following it for details.

6 Note that our convention treats ζ as both a structural and reduced form error.

7 See the proof of Theorem 2.1 for details.

8 For related results, see DiTraglia and García-Jimeno (2019 DiTraglia, F., and García-Jimeno, C. (2019), “Identifying the Effect of a Mis-Classified, Binary, Endogenous Regressor,” Journal of Econometrics, 209, 376–390. DOI: 10.1016/j.jeconom.2019.01.007.[Crossref], [Web of Science ®] , [Google Scholar]) who derived the sharp identified set for a misclassified, binary endogenous regressor given a valid instrument with discrete support, in an additively separable model with arbitrary dependence on exogenous covariates.

9 This is called a transparent parameterization in the statistics literature (see, e.g., Gustafson 2015).

10 Additional results, available upon request, consider alternative specifications that include covariates. The results are essentially unchanged.

11 By Corollary 2.2, the identified set for β is $(-\infty ,\infty )$ unless ${\rho }_{u{\xi }^{*}}$ is restricted. Here we impose the researchers’ stated belief that ${\rho }_{u{\xi }^{*}}>0$ along with an extremely conservative upper bound for ${\rho }_{u{\xi }^{*}}$ of 0.9.

12 Footnote #19 of Acemoglu, Johnson, and Robinson (2001 Acemoglu, D., Johnson, S., and Robinson, J. A. (2001), “The Colonial Origins of Comparative Development: An Empirical Investigation,” The American Economic Review, 91, 1369–1401. DOI: 10.1257/aer.91.5.1369.[Crossref], [Web of Science ®] , [Google Scholar]) stated “We can ascertain, to some degree, whether the difference between OLS and 2SLS estimates could be due to measurement error by making use of an alternative measure of institutions …This suggests that ‘measurement error’ in the institutions variables …is of the right order of magnitude to explain the difference between the OLS and 2SLS estimates.”

13 Suppose T₁ and T₂ are two measures of institutions that are subject to classical measurement error: $T_1=T^{*}+w_1$ and $T_2=T^{*}+w_2$. Both T₁ and T₂ suffer from precisely the same degree of endogeneity, because they inherit this problem from $T^{*}$ alone under the assumption of classical measurement error. Thus, the OLS estimator based on T₁ converges to $\kappa (\beta +{\sigma }_{T^{*}u}/{\sigma }_{T^{*}}^2)$ while the IV estimator that uses T₂ to instrument for T₁ converges to $\beta +{\sigma }_{T^{*}u}/{\sigma }_{T^{*}}^2$. The ratio identifies κ: $0.52/0.87\approx 0.6$.

14 Based on footnote 19 of the article, he expressed the belief that at least 40% of the measured variation in quality of institutions was likely to be noise.

15 Note that under our Jeffreys’ prior the posterior mean equals the maximum likelihood estimator.

16 See Section 4.2.

17 See Section 4 for a detailed discussion of the difference between inference for the identified set and inference for the partially identified parameter.

18 Because the prior is uniform, “small” refers to the relative area of a region on the identified set: in Figure 2(a), for example, the red region is small compared to the blue and white regions.

19 In this exercise we include the controls listed in Section III of Becker and Woessmann (2009 Becker, S. O., and Woessmann, L. (2009), “Was Weber Wrong? A Human Capital Theory of Protestant Economic History,” Quarterly Journal of Economics, 124, 531–596. DOI: 10.1162/qjec.2009.124.2.531.[Crossref], [Web of Science ®] , [Google Scholar]), specifically: the fraction of the population younger than age 10, of Jews, of females, of individuals born in the municipality, of individuals of Prussian origin, the average household size, log population, population growth in the preceding decade, the fraction of the population with unreported education information, and fraction of the population that was blind, deaf-mute, and insane.

20 These are: an indicator for whether the girl is a child of the household head, the girl’s age, the number of years the household has lived in the village, a Farsi dummy, a Tajik dummy, a farmers dummy, the age of the household head, years of education of the household head, the number of people in the household, Jeribs of land, number of sheep, distance to the nearest formal school, and a dummy for Chagcharan province.