Identification and Inference With Many Invalid Instruments

We study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome. To address the endogeneity, we exploit the presence of additional variables. Like conventional instrumental variables, these variables are correlated with the endogenous regressor. However, unlike conventional instrumental variables, they also have direct effects on the outcome, and thus are “invalid” instruments. Our novel identifying assumption is that the direct effects of these invalid instruments are uncorrelated with the effects of the instruments on the endogenous regressor. We show that in this case the limited-information-maximum-likelihood (liml) estimator is no longer consistent, but that a modification of the bias-corrected two-stage-least-square (tsls) estimator is consistent. We also show that conventional tests for over-identifying restrictions, adapted to the many instruments setting, can be used to test for the presence of these direct effects. We recommend that empirical researchers carry out such tests and compare estimates based on liml and the modified version of bias-corrected tsls. We illustrate in the context of two applications that such practice can be illuminating, and that our novel identifying assumption has substantive empirical content.     

Near exogeneity,weak identification and specification testing: Some asymptotic results

The paper studies the asymptotic size property of various specification tests in linear structural models where instrumental variables may locally violate the exclusion restrictions. Our results provide some new insights and extensions of earlier studies. In particular, we derive an explicit formula of the asymptotic size of the tests which shows clearly the factors that influence their size under instrument endogeneity. We show that all tests have correct asymptotic size when the usual orthogonality condition holds, but their asymptotic size can be arbitrary large even if only one instrument is slightly correlated with the error term. We present a Monte Carlo experiment that confirms our theoretical findings.     

The Effect of Pseudo-Exogenous Instrumental Variables on Hausman Test

Hausman test is popularly used to examine the endogeneity of explanatory variables in a regression model. To derive a well-defined asymptotic distribution of Hausman test, the correlation between the instrumental variables and the error term needs to converge to zero. However, it is possible that there remains considerable correlation in finite samples between the instruments and the error, even though their correlation eventually converges to zero. This article investigates the potential problem that such “pseudo-exogenous” instruments may create. We show that the performance of Hausman test is deteriorated when the instruments are asymptotically exogenous but endogenous in finite samples, through Monte Carlo simulations.     

Weak Instrumental Variables Models for Longitudinal Data

This article considers the estimation and testing of a within-group two-stage least squares (TSLS) estimator for instruments with varying degrees of weakness in a longitudinal (panel) data model. We show that adding the repeated cross-sectional information into a regression model can improve the estimation in weak instruments. Moreover, the consistency and limiting distribution of the TSLS estimator are established when both N and T tend to infinity. Some asymptotically pivotal tests are extended to a longitudinal data model and their asymptotic properties are examined. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed estimators.     

Statistical inference in dynamic panel data models

Anderson and his collaborators have made seminal contributions to inference with instrumental variables and to dynamic panel data models. We review these contributions and the extensive economic and statistical literature that these contributions spawned. We describe our recent work in these two areas, presenting new approaches to (a) making valid inferences in the presence of weak instruments and (b) instrument and model selection for dynamic panel data models. Both approaches use empirical likelihood and resampling. For inference in the presence of weak instruments, our approach uses model averaging to achieve asymptotic efficiency with strong instruments but maintain valid inferences with weak instruments. For instrument and model selection, our approach aims at choosing valid instruments that are strong enough to be useful.     

Sequential testing of gradual changes in the drift of a stochastic process

In this paper, some sequential monitoring procedures are constructed and analyzed for detecting a “gradual” change in the drift parameter of a general stochastic process satisfying a certain (weak) invariance principle. It is shown that the tests can be constructed such that the “false alarm rate” attains a prescribed level (say) α and that the tests have “asymptotic power 1”. A more precise analysis of the procedures under the alternative proves that the stopping times, suitably normalized, have a standard normal limiting distribution. A few results from a small simulation study are also presented in order to give an idea of the finite sample behaviour of the suggested procedures.     

Testing parameter significance in instrumental variables probit estimators: some simulation

Binary choice models that contain endogenous regressors can now be estimated routinely using modern software. Each of the two packages, Stata 11 [Stata Statistical Software: Release 11, StataCorp LP, College Station, TX, 2009] and Limdep 9 [Econometric Software Inc., Plainview, NY, 2008], contains two estimators that can be used to estimate such a model. This paper compares the performance of maximum likelihood, Newey's Amemiya's generalized least-squares (AGLS) estimator, an instrumental variables plug-in estimator and others in samples of sizes 200 and 1000 using simulation. Specifically, this paper focuses on tests of parameter significance under various degrees of instrument strength and severity of endogeneity. Although the maximum-likelihood estimator performs well in large samples, there is some evidence that the more computationally robust AGLS estimator may perform better in smaller samples when instruments are weak. It also appears that instruments in endogenous probit estimation need to be even stronger than when used in linear instrumental variables (IV) estimation.     

Asymptotic optimality of estimators of a linear functional eeiation if the ratio of the error variances is known 2

For the problem of estimating a linear functional relation when the ratio of the error variances is known a general class of estimators is introduced. They include as special cases the instrumental variable and replication cases and some others. Conditions are given for consistency, asymptotic normality and asymptotic optimality within this class based on the variance of the limit distribution. Fisheb's lower bound for asymptotic variances is established, and under normality the asymptotically optimal estimators are shown to be best asymptotically normal. For an inhomogeneous linear relation only estimators which are invariant with respect to a translation of the origin are considered, and asymptotically optimal invariant and, under normality, best asymptotically normal invariant estimators are obtained. Several special cases are discussed.     

Generalized minimum distance estimators of a linear model with correlated errors

R^p of a linear regression model of the type Y = Xθ + ɛ, where X is the design matrix, Y the vector of the response variable and ɛ the random error vector that follows an AR(1) correlation structure. These estimators are asymptotically analyzed, by proving their strong consistency, asymptotic normality and asymptotic efficiency. In a simulation study, a better behaviour of the Mean Squared Error of the proposed estimator with respect to that of the generalized least squares estimators is observed. Received: November 16, 1998; revised version: May 10, 2000     

Limit Theory for Random Coefficient First-Order Autoregressive Process

Random coefficient AR(1) processes are investigated where the random coefficient satisfies some suitable conditions. Conditional least squares estimator is shown to be consistent and to be asymptotically normality distributed. This extends the limit theory for stationary and near-stationary cases. Only second moments are assumed, as in the case of stationary autoregression models with fixed coefficient |ρ| < 1.     

Unconditional Quantile Treatment Effects Under Endogeneity

This article develops estimators for unconditional quantile treatment effects when the treatment selection is endogenous. We use an instrumental variable (IV) to solve for the endogeneity of the binary treatment variable. Identification is based on a monotonicity assumption in the treatment choice equation and is achieved without any functional form restriction. We propose a weighting estimator that is extremely simple to implement. This estimator is root n consistent, asymptotically normally distributed, and its variance attains the semiparametric efficiency bound. We also show that including covariates in the estimation is not only necessary for consistency when the IV is itself confounded but also for efficiency when the instrument is valid unconditionally. An application of the suggested methods to the effects of fertility on the family income distribution illustrates their usefulness. Supplementary materials for this article are available online.     

Optimal Bayesian strategies for the infinite-armed Bernoulli bandit

We consider the bandit problem with an infinite number of Bernoulli arms, of which the unknown parameters are assumed to be i.i.d. random variables with a common distribution F. Our goal is to construct optimal strategies of choosing “arms” so that the expected long-run failure rate is minimized. We first review a class of strategies and establish their asymptotic properties when F is known. Based on the results, we propose a new strategy and prove that it is asymptotically optimal when F is unknown. Finally, we show that the proposed strategy performs well for a number of simulation scenarios.     

A note on properties of spatial yule-walker estimators

For two-dimensional spatial autoregressive (AR) models, asymptotic properties of the spatial Yule-Walker (YW) estimators (Tjøstheim, 1978) are studied. These estimators although consistent, are shown to be asymptotically biased. Estimators from the first-order spatial bilateral AR model are looked at in more detail and the spatial YW estimators for this model are compared with the exact maximum likelihood estimators. Small sample properties of both estimators are also discussed briefly and some simulation results are presented.     

Locally asymptotically optimal tests for AR(p) against diagonal bilinear dependence

The problem of testing linear AR(p1) against diagonal bilinear BL(p1, 0; p2, p2) dependence is considered. Emphasis is put on local asymptotic optimality and the nonspecification of innovation densities. The tests we are deriving are asymptotically valid under a large class of densities, and locally asymptotically most stringent at some selected density f. They rely on generalized versions of residual autocorrelations (the spectrum), and generalized versions of the so-called cubic autocorrelations (the bispectrum). Local powers are explicitly provided. The local power of the Gaussian Lagrange multipliers method follows as a particular case.     

A MONTE CARLO COMPARISON OF VARIOUS ASYMPTOTIC APPROXIMATIONS TO THE DISTRIBUTION OF INSTRUMENTAL VARIABLES ESTIMATORS

We examine empirical relevance of three alternative asymptotic approximations to the distribution of instrumental variables estimators by Monte Carlo experiments. We find that conventional asymptotics provides a reasonable approximation to the actual distribution of instrumental variables estimators when the sample size is reasonably large. For most sample sizes, we find Bekker[11] asymptotics provides reasonably good approximation even when the first stage R² is very small. We conclude that reporting Bekker[11] confidence interval would suffice for most microeconometric (cross-sectional) applications, and the comparative advantage of Staiger and Stock[5] asymptotic approximation is in applications with sample sizes typical in macroeconometric (time series) applications.     

Some properties of tests for parameters that can be arbitrarily close to being unidentified

Confidence intervals for parameters that can be arbitrarily close to being unidentified are unbounded with positive probability [e.g. Dufour, J.-M., 1997. Some impossibility theorems in econometrics with applications to instrumental variables and dynamic models. Econometrica 65, 1365–1388; Pfanzagl, J. 1998. The nonexistence of confidence sets for discontinuous functionals. Journal of Statistical Planning and Inference 75, 9–20], and the asymptotic risks of their estimators are unbounded [Pötscher, B.M., 2002. Lower risk bounds and properties of confidence sets for ill-posed estimation problems with applications to spectral density and persistence estimation, unit roots, and estimation of long memory parameters. Econometrica 70, 1035–1065]. We extend these “impossibility results” and show that all tests of size α concerning parameters that can be arbitrarily close to being unidentified have power that can be as small as α for any sample size even if the null and the alternative hypotheses are not adjacent. The results are proved for a very general framework that contains commonly used models.     

Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

This article deals with quasi- and pseudo-likelihood estimation for a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.     

Rank tests for the k-sample problem with restricted alternatives

An asymptotically maximin most powerful rank test among somewhere asymptotically most powerful linear rank tests with scores generating function cf> is derived for each of the simple order alternative, the simple loop alternative and the simple tree alternative in the k-sample problem. The comparisons of the tests obtained with the rank analogues of the Bartholomew's xv tests are made in terms of local asymptotic relative efficiency. It is found that our tests are better than the rank analogues of the xk tests. Furthermore, the asymptotic equivalence of the ranking by the pooled sample to the ranking in pairs are discuss¬ed and the tests which are asymptotically equivalent to ours are given.     

Double filter instrumental variable estimation of panel data models with weakly exogenous variables

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.     

Empirical Bayes estimation in finite population sampling under functional measurement error models

The paper considers simultaneous estimation of finite population means for several strata. A model-based approach is taken, where the covariates in the super-population model are subject to measurement errors. Empirical Bayes (EB) estimators of the strata means are developed and an asymptotic expression for the MSE of the EB estimators is provided. It is shown that the proposed EB estimators are “first order optimal” in the sense of Robbins [1956. An empirical Bayes approach to statistics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, pp. 157–164], while the regular EB estimators which ignore the measurement error are not.