Using OLS to Estimate and Test for Structural Changes in Models with Endogenous Regressors

We consider the problem of estimating and testing for multiple breaks in a single‐equation framework with regressors that are endogenous, i.e. correlated with the errors. We show that even in the presence of endogenous regressors it is still preferable, in most cases, to simply estimate the break dates and test for structural change using the usual ordinary least squares (OLS) framework. Except for some knife‐edge cases, it delivers estimates of the break dates with higher precision and tests with higher power compared to those obtained using an instrumental variable (IV) method. Also, the OLS method avoids potential weak identification problems caused by weak instruments. To illustrate the relevance of our theoretical results, we consider the stability of the New Keynesian hybrid Phillips curve. IV‐based methods only provide weak evidence of instability. On the other hand, OLS‐based ones strongly indicate a change in 1991:Q1 and that after this date the model loses all explanatory power. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing overidentifying restrictions with many instruments and heteroskedasticity

This paper gives a test of overidentifying restrictions that is robust to many instruments and heteroskedasticity. It is based on a jackknife version of the overidentifying test statistic. Correct asymptotic critical values are derived for this statistic when the number of instruments grows large, at a rate up to the sample size. It is also shown that the test is valid when the number of instruments is fixed and there is homoskedasticity. This test improves on recently proposed tests by allowing for heteroskedasticity and by avoiding assumptions on the instrument projection matrix. This paper finds in Monte Carlo studies that the test is more accurate and less sensitive to the number of instruments than the Hausman–Sargan or GMM tests of overidentifying restrictions.     

Factor‐Based Identification‐Robust Interference in IV Regressions

Robust methods for instrumental variable inference have received considerable attention recently. Their analysis has raised a variety of problematic issues such as size/power trade‐offs resulting from weak or many instruments. We show that information reduction methods provide a useful and practical solution to this and related problems. Formally, we propose factor‐based modifications to three popular weak‐instrument‐robust statistics, and illustrate their validity asymptotically and in finite samples. Results are derived using asymptotic settings that are commonly used in both the factor and weak‐instrument literature. For the Anderson–Rubin statistic, we also provide analytical finite‐sample results that do not require any underlying factor structure. An illustrative Monte Carlo study reveals the following. Factor‐based tests control size regardless of instruments and factor quality. All factor‐based tests are systematically more powerful than standard counterparts. With informative instruments and in contrast to standard tests: (i) power of factor‐based tests is not affected by k even when large; and (ii) weak factor structure does not cost power. An empirical study on a New Keynesian macroeconomic model suggests that our factor‐based methods can bridge a number of gaps between structural and statistical modeling. Copyright © 2015 John Wiley & Sons, Ltd.     

Hahn-Hausman test as a specification test

This paper develops a modified version of the Sargan [Sargan, J.D., 1958. The estimation of economic relationships using instrumental variables. Econometrica 26 (3), 393-415] restrictions, and shows that it is numerically equivalent to the test statistic of Hahn and Hausman [Hahn, J., Hausman, J., 2002. A new specification test for the validity of instrumental variables. Econometrica 70 (1), 163-189] up to a sign. The modified Sargan test is constructed such that its asymptotic distribution under the null hypothesis of correct specification is standard normal when the number of instruments increases with the sample size. The equivalence result is useful in understanding what the Hahn-Hausman test detects and its power properties.     

IV models of ordered choice

This paper studies single equation instrumental variable models of ordered choice in which explanatory variables may be endogenous. The models are weakly restrictive, leaving unspecified the mechanism that generates endogenous variables. These incomplete models are set, not point, identifying for parametrically (e.g. ordered probit) or nonparametrically specified structural functions. The paper gives results on the properties of the identified set for the case in which potentially endogenous explanatory variables are discrete. The results are used as the basis for calculations showing the rate of shrinkage of identified sets as the number of classes in which the outcome is categorised increases.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

Estimating Excess Sensitivity and Habit Persistence in Consumption Using Greenbook Forecasts

In this paper, we revisit the issue of excess sensitivity of consumption to income and address the weak instrument problem that is well documented in this literature. Using quarterly data for the US economy, we first highlight the weak instrument problem by showing that the use of conventional instruments tends to overestimate the share of rule-of-thumb consumers. To address this weak instrument problem, we propose a new instrument for endogenous disposable income growth in the consumption function, namely, the Greenbook forecast of real disposable income growth. We show that this instrument encompasses the information contained in the conventional set of instruments, and is a superior predictor of income growth. We find that using our proposed instrument ameliorates the weak instrument problem and provides a much smaller estimate for the rule-of-thumb consumers. We also extend our empirical framework to allow for habit persistence and provide an estimate for this important parameter of the consumption function. Finally, we use a time-varying specification of consumption function that allows for endogenous regressors, and document a decline in the share of rule-of-thumb consumers and a rise in the habit-persistence parameter in the US over our sample period.     

The exact distribution of exogenous variable coefficient estimators

This paper derives the exact probability density function of the instrumental variable (IV) estimator of the exogenous variable coefficient vector in a structural equation containing n + 1 endogenous variables and N degrees of overidentification. The derivations make use of an operator calculus which simplifies the algebra of invariant polynomials with multiple matrix arguments. A leading case of the general distribution that is more amenable to analysis and computation is also presented. Conventional classical assumptions of normally distributed errors and non-random exogenous variables are employed.     

Bootstrap validity for the score test when instruments may be weak

It is well-known that size adjustments based on bootstrapping the t$t$-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. In this paper, we provide a theoretical proof that guarantees the validity of the bootstrap for the score statistic. This theory does not follow from standard results, since the score statistic is not a smooth function of sample means and some parameters are not consistently estimable when the instruments are uncorrelated with the explanatory variable.     

Estimating an economic model of crime using panel data from North Carolina

This paper replicates the Cornwell and Trumbull ( 1994 ) estimation of a crime model using panel data on 90 counties in North Carolina over the period 1981–1987. While the Between and Within estimates are replicated, the fixed effects 2SLS as well as the 2SLS estimates are not. In fact, the fixed effects 2SLS estimates turn out to be insignificant for all important deterrent variables as well as legal opportunity variables. We argue that the usual Hausman test, based on the difference between fixed effects and random effects, may lead to misleading inference when endogenous variables of the conventional simultaneous equation type are among the regressors. We estimate the model using random effects 2SLS and perform a Hausman test based on the difference between fixed effects 2SLS and random effects 2SLS. We cannot reject the consistency of the random effects 2SLS estimator and this estimator yields plausible and significant estimates of the crime model. This result should be tempered by the legitimacy of the chosen instruments. Copyright © 2006 John Wiley & Sons, Ltd.     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Tests with correct size when instruments can be arbitrarily weak

This paper applies classical exponential-family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced-form variance matrix. The results can be divided into the limited-information and full-information categories. In the limited-information model, it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable. In the full-information framework, this paper proposes a family of similar tests for subsets of endogenous variables’ coefficients. For both limited- and full-information models, there exist power upper bounds for unbiased tests. When the model is just-identified, the Anderson–Rubin, score, and (pseudo) conditional likelihood ratio tests are optimal. When the model is over-identified, the (pseudo) conditional likelihood ratio test has power close to the power envelope when identification is strong.     

An econometric approach to estimating support prices and measures of productivity change in public hospitals

In industry sectors where market prices for goods and services are unavailable, it is common to use estimated output and input distance functions to estimate rates of productivity change. It is also possible, but less common, to use estimated distance functions to estimate the normalised support (or efficient) prices of individual inputs and outputs. A problem that arises in the econometric estimation of these functions is that more than one variable in the estimating equation may be endogenous. In such cases, maximum likelihood estimation can lead to biased and inconsistent parameter estimates. To solve the problem, we use linear programming to construct a quantity index. The distance function is then written in the form of a conventional stochastic frontier model where the explanatory variables are unambiguously exogenous. We use this approach to estimate productivity indexes, measures of environmental change, levels of efficiency, and support prices for a sample of Australian public hospitals.     

Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables

The aim of this paper is to convey to a wider audience of applied statisticians that nonparametric (matching) estimation methods can be a very convenient tool to overcome problems with endogenous control variables. In empirical research one is often interested in the causal effect of a variable X on some outcome variable Y . With observational data, i.e. in the absence of random assignment, the correlation between X and Y generally does not reflect the treatment effect but is confounded by differences in observed and unobserved characteristics. Econometricians often use two different approaches to overcome this problem of confounding by other characteristics. First, controlling for observed characteristics, often referred to as selection on observables, or instrumental variables regression, usually with additional control variables. Instrumental variables estimation is probably the most important estimator in applied work. In many applications, these control variables are themselves correlated with the error term, making ordinary least squares and two-stage least squares inconsistent. The usual solution is to search for additional instrumental variables for these endogenous control variables, which is often difficult. We argue that nonparametric methods help to reduce the number of instruments needed. In fact, we need only one instrument whereas with conventional approaches one may need two, three or even more instruments for consistency. Nonparametric matching estimators permit     consistent estimation without the need for (additional) instrumental variables and permit arbitrary functional forms and treatment effect heterogeneity.     

信贷合约内部机理及关键影响因素研究

以信号理论、信息不对称理论、关系型信贷理论以及债务代理成本理论等信贷合约相关理论为基础,围绕信贷合约的内生变量利率、担保和期限三者之间的相互关系以及影响信贷合约的关键外生变量,提出了若干核心理论假设;在考虑信贷合约要素利率、担保和期限内生性的情况下建立了结构方程模型,同时对结构方程模型解的识别进行了验证,并对结构方程模型进行了理论求解。     

On the Practice of Lagging Variables to Avoid Simultaneity

A common practice in applied economics research consists of replacing a suspected simultaneously determined explanatory variable with its lagged value. This note demonstrates that this practice does not enable one to avoid simultaneity bias. The associated estimates are still inconsistent, and hypothesis testing is invalid. An alternative is to use lagged values of the endogenous variable in instrumental variable estimation. However, this is only an effective estimation strategy if the lagged values do not themselves belong in the respective estimating equation, and if they are sufficiently correlated with the simultaneously determined explanatory variable.     

Increasing the credibility of the twin birth instrument

Twin births are an important instrument for the endogenous fertility decision. However, twin births are not exogenous either as dizygotic twinning is correlated with maternal characteristics. Following the medical literature, we assume that monozygotic twins are exogenous, and construct a new instrument, which corrects for the selection although monozygotic twinning is usually unobserved in survey and administrative datasets. Using administrative data from Sweden, we show that the usual twin instrument is related to observed and unobserved determinants of economic outcomes, while our new instrument is not. In our applications we find that the classical twin instrument underestimates the negative effect of fertility on labor income. This finding is in line with the observation that high earners are more likely to delay childbearing and hence have a higher risk to get dizygotic twins.