Inverting the indirect—The ellipse and the boomerang: Visualizing the confidence intervals of the structural coefficient from two-stage least squares

doi:10.1016/j.jeconom.2017.05.008

Journal of Econometrics

Volume 199, Issue 2, August 2017, Pages 173-183

https://doi.org/10.1016/j.jeconom.2017.05.008 Get rights and content

Abstract

In the just-identified model,the exact distribution of the two-stage least squares (2SLS) estimator of the coefficient of the endogenous regressor is a ratio of two normally distributed random variables. Basmann (1960), Basmann (1974) used Fieller’s 1932 result to derive the density function of the estimator. In this paper, we present a novel graphical exposition of Fieller’s 1954 technique to approximate the confidence interval for the 2SLS estimator. We use this approach to examine how the degree of endogeneity and instrument relevance influences the correspondence between the Fieller and traditional asymptotic confidence intervals for the estimator.

Introduction

Basmann (1974) found that when the structural equation is exactly identified, the 2SLS estimator of the coefficient of the endogenous regressor is a ratio of two normally distributed random variables and used Fieller’s result (1932) to derive the density function of the estimator. One of his motivations for deriving the exact distribution function was to be able to compute numerical approximations. Using results based on Fieller (1932), he showed that the exact distribution functions could be approximated using a normal distribution under certain conditions (Basmann, 1960). In 1954, Fieller developed a technique to approximate the confidence interval (CI) for the ratio that has become the most widely employed alternative to the usual asymptotic estimate. In this paper, we construct a new graphic method for comparing the Fieller and usual asymptotic CIs.

We demonstrate how the approximate CIs for the parameter estimate from a just-identified two-stage least squares (2SLS) estimate can be examined with a single diagram of a two dimensional shape that captures the primary characteristics of the estimating relationships. This graphic method allows for a direct comparison between the usual asymptotic (Delta) CI and the Fieller interval.¹ This graphic representation can be used to verify many of the results previously obtained from Monte Carlo studies that have investigated the properties of the 2SLS estimate in the just-identified case.

Although numerous diagrammatic approaches to the Fieller interval exist (see Fieller (1932), Fieller (1954), Creasy (1954), Guiard (1989), Von Luxburg and Franz (2004), Hirschberg and Lye (2010a), Hirschberg and Lye (2010b), Hirschberg and Lye (2010c)), our contribution is to construct an equivalent diagram for the Delta approximation of the bounds of the CI for the ratio of parameters which can be used to make comparisons between the two methods that incorporates the features of the just-identified 2SLS case.

The estimation of the just-identified model is widely used. In a survey of applications in which instrumental variable estimation is used for papers published in The American Economic Review, Journal of Political Economy and Quarterly Journal of Economics from 1999 to 2004, Chernozhukov and Hansen (2008) found that the bulk of estimated instrumental variables models employed exactly as many instruments as endogenous regressors. In addition, the advice from Angrist and Pischke (2009) for practitioners to avoid difficulties with weak instruments is to pick a single ‘best’ instrument and estimate the just-identified model.

This paper proceeds as follows. First, we review the equivalence of the indirect least squares and the 2SLS estimators to demonstrate that the usual 2SLS variance estimate is equivalent to the application of the Delta method for the estimate of the variance of a ratio of random variables. This is followed by the development of a unique graphical method for the comparison of the Delta to the Fieller CIs for the ratio. A number of simulated cases are then presented to demonstrate how the degree of endogeneity and instrument relevance influences the relationship between these CIs. We then present a simple empirical application along with two alternative graphical methods for comparing these intervals. We conclude with a summary of our findings and suggestions for further research in this area.

Section snippets

Tests of significance and CIs in the exactly identified case: A recap

Suppose we are interested in estimating the single parameter $β$ from a structural equation in a simultaneous equation model,² $Y = X β + u$ where $Y$ is a $T$ by 1 vector of the dependent variable of interest, $X$ is a $T$ by 1 endogenous regressor vector, and $u$ is the $T$ by 1 vector of

Employing a constrained optimization to compare the Fieller and Delta intervals

In Hirschberg and Lye (2010b) (henceforth referred to as HL), we present a method for the graphical comparison of the Fieller and the Delta CIs. In that paper, we demonstrate that the intervals from those two methods could be found by a graphic solution to a constrained optimization problem as proposed by Durand (1954), Scheffé (1959, appendix III) and in the econometrics literature by Leamer (1978, Theorem 5.4). These earlier contributions demonstrate that $100 (1 - α)$ % confidence bounds for a

The Fieller and Delta intervals in the case of the just-identified simultaneous equations model

In order to make comparisons between the Delta and Fieller CIs for the parameter $β$ from the structural equation $Y = X β + u$ , we consider the parameters from the reduced form Eqs. (2), (4). $β$ is estimated by ILS as the ratio of the estimates of $θ$ as $h$ and $γ$ as $g$ , where $g = {(Z^{T} Z)}^{- 1} Z^{T} X$ and $h = {(Z^{T} Z)}^{- 1} Z^{T} Y$ . Recall from (5) , we have that $(\begin{matrix} v_{i} \\ ε_{i} \end{matrix}) \sim i.i.d. (0, Ω)$ .

In a series of Monte Carlo studies, Zivot et al. (1998) consider this model in order to examine the implications of various scenarios with respect to the

An empirical example

A widely cited example of a case where the implications of a just-identified system of equations are of interest is the paper by Acemoglu et al. (2001) in which they employ a cross country data set to examine the effect of institutions on economic performance. In order to account for the endogeneity of the “average index of protection against expropriation risk from 1985 to 1995” or IPAE in explaining a nation’s per capita income, they use the mortality rates among European colonists as an

Conclusions

We have demonstrated that the Delta and Fieller CIs for structural equation parameter in the just-identified simultaneous equations can be compared by the examination of two appropriately defined constraint shapes. By controlling the levels of endogeneity and the relevance of the instrument, it is possible to demonstrate how these two characteristics influence the degree to which these estimated CIs agree or diverge without the need for a simulation.

Our findings indicate that when the degree of

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^☆: We have greatly benefited by the comments of the referees for this paper. The usual caveat holds.

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Inverting the indirect—The ellipse and the boomerang: Visualizing the confidence intervals of the structural coefficient from two-stage least squares☆

Abstract

Introduction

Section snippets

Tests of significance and CIs in the exactly identified case: A recap

Employing a constrained optimization to compare the Fieller and Delta intervals

The Fieller and Delta intervals in the case of the just-identified simultaneous equations model

An empirical example

Conclusions

The colonial origins of comparative development: An empirical investigation

Amer. Econ. Rev.

The asymptotic properties of estimates of the parameters of a single equation in a complete system of stochastic equations

Ann. Math. Stat.

Inference with weak instruments

Mostly Harmless Econometrics: An Empiricist’s Companion

An expository note on estimation of simultaneous structural equations

Biometrics

Exact finite sample distribution for some econometric estimators and test statistics: a survey and appraisal

An experimental study of structural estimators and test statistics associated with dynamical econometric models

Econometrica

The reduced form: A simple approach to inference with weak instruments

Econom. Lett.

Confidence curves in nonlinear regression

J. Amer. Statist. Assoc.

Limits for the ratio of means

J. R. Stat. Soc. Ser. B Stat. Methodol.

Some impossibility theorems in econometrics with applications to structural and dynamic models

Econometrica

Joint confidence regions for multiple regression coefficients

J. Amer. Statist. Assoc.

The distribution of the index in a normal bivariate population

Biometrika

Some problems in interval estimation

J. R. Stat. Soc. Ser. B Stat. Methodol.

Some remarks on the estimation of the ratio of the expected values of a two-dimensional normal random variable (correction of the Theorem of Milliken)

Biom. J.