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This paper considers the finite-sample distribution of the 2SLS estimator and derives bounds on its exact bias in the presence of weak and/or many instruments. We then contrast the behavior of the exact bias expressions and the asymptotic expansions currently popular in the literature, including a consideration of the no-moment problem exhibited by many Nagar-type estimators. After deriving a finite-sample unbiased k-class estimator, we introduce a double-k-class estimator based on Nagar (1962) that dominates k-class estimators (including 2SLS), especially in the cases of weak and/or many instruments. We demonstrate these properties in Monte Carlo simulations showing that our preferred estimators outperform Fuller (1977) estimators in terms of mean bias and MSE.

This chapter studies the asymptotic properties of within-groups k-class estimators in a panel data model with weak instruments. Weak instruments are characterized by the coefficients of the instruments in the reduced form equation shrinking to zero at a rate proportional to nTδ, where n is the dimension of the cross-section and T is the dimension of the time series. Joint limits as (n,T)→∞ show that this within-group k-class estimator is consistent if 0≤δ<12 and inconsistent if 12≤δ<∞.

IV estimation is examined when some instruments may be invalid. This is relevant because the initial just-identifying orthogonality conditions are untestable, whereas their validity is required when testing the orthogonality of additional instruments by so-called overidentification restriction tests. Moreover, these tests have limited power when samples are small, especially when instruments are weak. Distinguishing between conditional and unconditional settings, we analyze the limiting distribution of inconsistent IV and examine normal first-order asymptotic approximations to its density in finite samples. For simple classes of models we compare these approximations with their simulated empirical counterparts over almost the full parameter space. The latter is expressed in measures for: model fit, simultaneity, instrument invalidity, and instrument weakness. Our major findings are that for the accuracy of large sample asymptotic approximations instrument weakness is much more detrimental than instrument invalidity. Also, IV estimators obtained from strong but possibly invalid instruments are usually much closer to the true parameter values than those obtained from valid but weak instruments.

Principal component (PC) techniques are commonly used to improve the small sample properties of the linear instrumental variables (IV) estimator. Carrasco (2012) argue that PC type methods provide a natural ranking of instruments with which to reduce the size of the instrument set. This chapter shows how reducing the size of the instrument based on PC methods can lead to poor small sample properties of IV estimators. A new approach to ordering instruments termed ‘normalized principal components’ (NPCs) is introduced to overcome this problem. A simulation study shows the favourable small samples properties of IV estimators using NPC, methods to reduce the size of the instrument relative to PC. Using NPC we provide evidence that the IV setup in Angrist and Krueger (1992) may not suffer the weak instrument problem.

This paper aims to investigate the effects of using poor-quality instruments to remedy endogeneity in logit-based demand models. Endogeneity problems in demand models occur when certain factors, unobserved by the researcher, affect both demand and the values of a marketing mix variable set by managers. For example, unobserved factors such as style, prestige or reputation might result in higher prices for a product and higher demand for that product. If not addressed properly, endogeneity can bias the elasticities of the endogenous variable and subsequent optimization of the marketing mix. In practice, instrumental variables (IV) estimation techniques are often used to remedy an endogeneity problem. It is well-known that, for linear regression models, the use of IV techniques with poor-quality instruments can produce very poor parameter estimates, in some circumstances even worse than those that result from ignoring the endogeneity problem altogether. The literature has not addressed the consequences of using poor-quality instruments to remedy endogeneity problems in non-linear models, such as logit-based demand models.

Using simulation methods, the authors investigate the effects of using poor-quality instruments to remedy endogeneity in logit-based demand models applied to finite-sample data sets. The results show that, even when the conditions for lack of parameter identification due to poor-quality instruments do not hold exactly, estimates of price elasticities can still be quite poor. That being the case, the authors investigate the relative performance of several non-linear IV estimation procedures utilizing readily available instruments in finite samples.

The study highlights the attractiveness of the control function approach (Petrin and Train, 2010) and readily available instruments, which together reduce the mean squared elasticity errors substantially for experimental conditions in which the theory-backed instruments are poor in quality. The authors find important effects for sample size, in particular for the number of brands, for which it is shown that endogeneity problems are exacerbated with increases in the number of brands, especially when poor-quality instruments are used. In addition, the number of stores is found to be important for likelihood ratio testing. The results of the simulation are shown to generalize to situations under Nash pricing in oligopolistic markets, to conditions in which cross-sectional preference heterogeneity exists and to nested logit and probit-based demand specifications as well. Based on the results of the simulation, the authors suggest a procedure for managing a potential endogeneity problem in logit-based demand models.

The literature on demand modeling has focused on deriving analytical results on the consequences of using poor-quality instruments to remedy endogeneity problems in linear models. Despite the widespread use of non-linear demand models such as logit, this study is the first to address the consequences of using poor-quality instruments in these models and to make practical recommendations on how to avoid poor outcomes.

The authors propose inference methods for endogeneity parameters in linear simultaneous equation models allowing for weak identification and missing instruments. Endogeneity parameters measure the impact of unobserved variables which may be correlated with observed explanatory variables, and play a central role in determining the “bias” associated with endogeneity and measurement errors in structural equations. These results expand, in several ways, the finite-sample theory in Doko Tchatoka and Dufour (2014) for this problem. The latter theory relies on relatively restrictive assumptions, in particular the hypothesis that the reduced form is complete (e.g., contains all the relevant instruments), which is questionable in many practical situations. While the new proposed inference methods retain identification robustness, they also allow the reduced form to be incomplete, for example, due to missing instruments. The authors propose easily applicable inference methods for endogeneity parameters – in particular, two-stage procedures (similar to those in Dufour, 1990). An application to a model of returns to schooling is presented.

This chapter shows how a weighted average of a forward and reverse Jackknife IV estimator (JIVE) yields estimators that are robust against heteroscedasticity and many instruments. These estimators, called HFUL (Heteroscedasticity robust Fuller) and HLIM (Heteroskedasticity robust limited information maximum likelihood (LIML)) were introduced by Hausman, Newey, Woutersen, Chao, and Swanson (2012), but without derivation. Combining consistent estimators is a theme that is associated with Jerry Hausman and, therefore, we present this derivation in this volume. Additionally, and in order to further understand and interpret HFUL and HLIM in the context of jackknife type variance ratio estimators, we show that a new variant of HLIM, under specific grouped data settings with dummy instruments, simplifies to the Bekker and van der Ploeg (2005) MM (method of moments) estimator.

This chapter proposes a simple, fairly general, test for global identification of unconditional moment restrictions implied from point-identified conditional moment restrictions. The test is a Hausman-type test based on the Hausdorff distance between an estimator that is consistent even under global identification failure of the unconditional moment restrictions, and an estimator of the identified set of the unconditional moment restrictions. The proposed test has a χ2 limiting distribution and is also able to detect weak identification. Some Monte Carlo experiments show that the proposed test has competitive finite sample properties already for moderate sample sizes.

We show that, in the US telecommunications industry, market participants have a sophisticated understanding of the political process, and behave strategically in their allocation of contributions to state legislators as if seeking to purchase influence over regulatory policy. We find that interests respond defensively to contributions from rivals, take into account the configuration of support available to them in both the legislature and the regulatory commission, and vary their contributions according to variations in relative costs for influence by different legislatures. This strategic behavior supports a theory that commercially motivated interests contribute campaign resources in order to mobilize legislators to influence the decisions of regulatory agencies. We also report evidence that restrictions on campaign finance do not affect all interests equally. The paper therefore provides positive evidence on the nature and effects of campaign contributions in regulated industries where interest group competition may be sharp.

Although the theoretical trade-off between the quantity and quality of children is well established, empirical evidence supporting such a causal relationship is limited. This chapter applies a recently developed nonparametric estimator of the conditional local average treatment effect to assess the sensitivity of the quantity–quality trade-off to functional form and parametric assumptions. Using data from the Indonesia Family Life Survey and controlling for the potential endogeneity of fertility, we find mixed evidence supporting the trade-off.