Smooth minimum distance estimation and testing with conditional estimating equations: Uniform in bandwidth theory

To study the influence of a bandwidth parameter in inference with conditional moments, we propose a new class of estimators and establish an asymptotic representation of our estimator as a process indexed by a bandwidth, which can vary within a wide range including bandwidths independent of the sample size. We study its behavior under misspecification. We also propose an efficient version of our estimator. We develop a procedure based on a distance metric statistic for testing restrictions on parameters as well as a bootstrap technique to account for the bandwidth’s influence. Our new methods are simple to implement, apply to non-smooth problems, and perform well in our simulations.     

Instrumental variable estimation of nonseparable models

There are many environments where knowledge of a structural relationship is required to answer questions of interest. Also, nonseparability of a structural disturbance is a key feature of many models. Here, we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance, which disturbance is independent of instruments. This model leads to conditional quantile restrictions. We give local identification conditions for the structural equations from those quantile restrictions. We find that a modified completeness condition is sufficient for local identification. We also consider estimation via a nonparametric minimum distance estimator. The estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments. We show consistency of this estimator.     

Endogeneity in quantile regression models: A control function approach

This paper considers a linear triangular simultaneous equations model with conditional quantile restrictions. The paper adjusts for endogeneity by adopting a control function approach and presents a simple two-step estimator that exploits the partially linear structure of the model. The first step consists of estimation of the residuals of the reduced-form equation for the endogenous explanatory variable. The second step is series estimation of the primary equation with the reduced-form residual included nonparametrically as an additional explanatory variable. This paper imposes no functional form restrictions on the stochastic relationship between the reduced-form residual and the disturbance term in the primary equation conditional on observable explanatory variables. The paper presents regularity conditions for consistency and asymptotic normality of the two-step estimator. In addition, the paper provides some discussions on related estimation methods in the literature.     

A two-stage procedure for partially identified models

This paper studies a two-stage procedure for estimating partially identified models, based on Chernozhukov, Hong, and Tamer’s (2007) theory of set estimation and inference. We consider the case where a sub-vector of parameters or their identified set can be estimated separately from the rest, possibly subject to a priori restrictions. Our procedure constructs the second-stage set estimator and confidence set by taking appropriate level sets of a criterion function, using a first-stage estimator to impose restrictions on the parameter of interest. We give conditions under which the two-stage set estimator is a set-valued random element that is measurable in an appropriate sense. We also establish the consistency of the two-stage set estimator.     

Tikhonov regularization for nonparametric instrumental variable estimators

We study a Tikhonov Regularized (TiR) estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors. The nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives. After showing its consistency in the Sobolev norm and uniform consistency under an embedding condition, we derive the expression of the asymptotic Mean Integrated Square Error and the rate of convergence. The optimal value of the regularization parameter is characterized in two examples. We illustrate our theoretical findings and the small sample properties with simulation results. Finally, we provide an empirical application to estimation of an Engel curve, and discuss a data driven selection procedure for the regularization parameter.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.     

A test of cross section dependence for a linear dynamic panel model with regressors

This paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments (GMM). The test is valid when the cross-sectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in terms of deviations from time-specific averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results suggest that our tests perform well, particularly the version based on the [Blundell, R., Bond, S., 1998. Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87, 115–143] system GMM estimator. In addition, it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK firm data and the results show little evidence of heterogeneous error cross section dependence.     

Identification and information in monotone binary models

This paper considers binary response models where errors are uncorrelated with a set of instrumental variables and are independent of a continuous regressor v$v$, conditional on all other variables. It is shown that these exclusion restrictions are not sufficient for identification and that additional identifying assumptions are needed. Such an assumption, introduced by Lewbel [Semiparametric qualitative response model estimation with unknown heteroskedasticity or instrumental variables. Journal of Econometrics 97, 145–177], is that the support of the continuous regressor is large, but we show that it significantly restricts the class of binary phenomena which can be analysed. We propose an alternative additional assumption under which β$\beta$ remains just identified and the estimation unchanged. This alternative assumption does not impose specific restrictions on the data, which broadens the scope of the estimation method in empirical work. The semiparametric efficiency bound of the model is also established and an existing estimator is shown to achieve that bound. The efficient estimator uses a plug-in density estimate. It is shown that plugging in the true density rather than an estimate is inefficient. Extensions to ordered choice models are provided.     

Estimation and inference of dynamic structural factor models with over-identifying restrictions

This paper develops a new estimator for the impulse response functions in structural factor models with a fixed number of over-identifying restrictions. The proposed identification scheme nests the conventional just-identified recursive scheme as a special case. We establish the asymptotic distributions of the new estimator and develop test statistics for the over-identifying restrictions. Simulation results show that adding a few more over-identifying restrictions can lead to a substantial improvement in estimation accuracy for impulse response functions at both zero and nonzero horizons. We estimate the effects of a monetary policy shock using a U.S. data set. The results show that our over-identified scheme can help to detect incorrect specifications that lead to spurious impulse responses.     

Semi-parametric estimation of American option prices

We introduce a novel semi-parametric estimator of American option prices in discrete time. The specification is based on a parameterized stochastic discount factor and is nonparametric w.r.t. the historical dynamics of the Markovian state variables. The historical transition density estimator minimizes a distance built on the Kullback–Leibler divergence from a kernel transition density, subject to the no-arbitrage restrictions for a non-defaultable bond, the underlying asset and some American option prices. We use dynamic programming to make explicit the nonlinear restrictions on the Euclidean and functional parameters coming from option data. We study asymptotic and finite sample properties of the estimators.     

Model averaging by jackknife criterion in models with dependent data

The past decade witnessed a literature on model averaging by frequentist methods. For the most part, the asymptotic optimality of various existing frequentist model averaging estimators has been established under i.i.d. errors. Recently, Hansen and Racine [Hansen, B.E., Racine, J., 2012. Jackknife model averaging. Journal of Econometrics 167, 38–46] developed a jackknife model averaging (JMA) estimator, which has an important advantage over its competitors in that it achieves the lowest possible asymptotic squared error under heteroscedastic errors. In this paper, we broaden Hansen and Racine’s scope of analysis to encompass models with (i) a non-diagonal error covariance structure, and (ii) lagged dependent variables, thus allowing for dependent data. We show that under these set-ups, the JMA estimator is asymptotically optimal by a criterion equivalent to that used by Hansen and Racine. A Monte Carlo study demonstrates the finite sample performance of the JMA estimator in a variety of model settings.     

On efficient estimation of the ordered response model

In this paper, we derive efficiency bounds for the ordered response model when the distribution of the errors is unknown. Furthermore, we develop an estimator that is efficient under suitable conditions. Interestingly, neither the bounds nor the estimator are trivial extensions of what has been proposed in the literature for the binary response model. The estimator is composed of quadratic B-splines, and estimation is performed by the method of sieves. In addition, the estimator of the distribution function is restricted to be a proper distribution function. An empirical example on the effect of fees on attendance rates at universities and community colleges is also included; we get substantively different results by relaxing the assumption that the distribution of the errors is normal.     

Testing and imposing Slutsky symmetry in nonparametric demand systems

Maximization of utility implies that consumer demand systems have a Slutsky matrix which is everywhere symmetric. However, previous non- and semi-parametric approaches to the estimation of consumer demand systems do not give estimators that are restricted to satisfy this condition, nor do they offer powerful tests of this restriction. We use nonparametric modeling to test and impose Slutsky symmetry in a system of expenditure share equations over prices and expenditure. In this context, Slutsky symmetry is a set of nonlinear cross-equation restrictions on levels and derivatives of consumer demand equations. The key insight is that due to the differing convergence rates of levels and derivatives and due to the fact that the symmetry restrictions are linear in derivatives, both the test and the symmetry restricted estimator behave asymptotically as if these restrictions were (locally) linear. We establish large and finite sample properties of our methods, and show that our test has advantages over the only other comparable test. All methods we propose are implemented with Canadian micro-data. We find that our nonparametric analysis yields statistically significantly and qualitatively different results from traditional parametric estimators and tests.     

A semiparametric model for binary response and continuous outcomes under index heteroscedasticity

This paper formulates a likelihood‐based estimator for a double‐index, semiparametric binary response equation. A novel feature of this estimator is that it is based on density estimation under local smoothing. While the proofs differ from those based on alternative density estimators, the finite sample performance of the estimator is significantly improved. As binary responses often appear as endogenous regressors in continuous outcome equations, we also develop an optimal instrumental variables estimator in this context. For this purpose, we specialize the double‐index model for binary response to one with heteroscedasticity that depends on an index different from that underlying the ‘mean response’. We show that such (multiplicative) heteroscedasticity, whose form is not parametrically specified, effectively induces exclusion restrictions on the outcomes equation. The estimator developed exploits such identifying information. We provide simulation evidence on the favorable performance of the estimators and illustrate their use through an empirical application on the determinants, and affect, of attendance at a government‐financed school. Copyright © 2009 John Wiley & Sons, Ltd.     

Determinacy,indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model may have a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). The testing problem is addressed by a misspecification-type approach in which the overidentifying restrictions test obtained from the estimation of the system of Euler equations of the LRE model through the generalized method of moments is combined with a likelihood-based test for the cross-equation restrictions that the model places on its reduced form solution under determinacy. The resulting test has no power against a particular class of indeterminate equilibria, hence the non rejection of the null hypothesis can not be interpreted conclusively as evidence of determinacy. On the other hand, this test (i) circumvents the nonstandard inferential problem generated by the presence of the auxiliary parameters that appear under indeterminacy and that are not identifiable under determinacy, (ii) does not involve inequality parametric restrictions and hence the use of nonstandard inference, (iii) is consistent against the dynamic misspecification of the LRE model, and (iv) is computationally simple. Monte Carlo simulations show that the suggested testing strategy delivers reasonable size coverage and power against dynamic misspecification in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model of the US economy.     

Estimation of conditional moment restrictions without assuming parameter identifiability in the implied unconditional moments

A well-known difficulty in estimating conditional moment restrictions is that the parameters of interest need not be globally identified by the implied unconditional moments. In this paper, we propose an approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. These unconditional moments depend on the “instruments” generated from a “generically comprehensively revealing” function, and they are further projected along the exponential Fourier series. The objective function is based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that the full continuum of unconditional moments is incorporated into each Fourier coefficient. We show that, when the number of Fourier coefficients in the objective function grows at a proper rate, the proposed estimator is consistent and asymptotically normally distributed. An efficient estimator is also readily obtained via the conventional two-step GMM method. Our simulations confirm that the proposed estimator compares favorably with that of Domínguez and Lobato (2004, Econometrica) in terms of bias, standard error, and mean squared error.     

Efficient information theoretic inference for conditional moment restrictions

The generalised method of moments estimator may be substantially biased in finite samples, especially so when there are large numbers of unconditional moment conditions. This paper develops a class of first-order equivalent semi-parametric efficient estimators and tests for conditional moment restrictions models based on a local or kernel-weighted version of the Cressie–Read power divergence family of discrepancies. This approach is similar in spirit to the empirical likelihood methods of Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restrictions models. Econometrica 72, 1667–1714] and Tripathi and Kitamura [2003. Testing conditional moment restrictions. Annals of Statistics 31, 2059–2095]. These efficient local methods avoid the necessity of explicit estimation of the conditional Jacobian and variance matrices of the conditional moment restrictions and provide empirical conditional probabilities for the observations.     

Learning in a multilateral bargaining experiment

This paper analyzes data from an investigation of a majoritarian bargaining experiment. A learning model is proposed to account for the evolution of play in this experiment. It is also suggested that an adjustment must be made to account for the panel structure of the data. Such adjustments have been used in other fields and are known to be important as unadjusted standard errors may be severely biased downward. These results indicate that this adjustment also has an important effect in this application. Furthermore, an efficient estimator that takes into account heterogeneity across players is proposed. A unique learning model to account for the paths of play under two different amendment rules cannot be rejected with the standard estimator with adjusted standard errors, however it can be rejected using the efficient estimator. The data and the estimated learning model suggest that after proposing “fair” divisions, subjects adapt and their proposals change rapidly in the treatment where uneven proposals are almost always accepted. Their beliefs in the estimated learning model are influenced by more than just the most recent outcomes.     

GMM Estimation with Non‐causal Instruments under Rational Expectations*

Lanne and Saikkonen [Oxford Bulletin of Economics and Statistics (2011a) Vol. 73, pp. 581–592], show that the generalized method of moments (GMM) estimator is inconsistent, when the instruments are lags of variables that admit a non‐causal autoregressive representation. This article argues that this inconsistency depends on distributional assumptions, that do not always hold. In particular under rational expectations, the GMM estimator is found to be consistent. This result is derived in a linear context and illustrated by simulation of a nonlinear asset pricing model.