Abstract not found

This paper provides sufficient conditions for the nonparametric identification of the regression function m (·) in a regression model with an endogenous regressor x and an instrumental variable z. It has been shown that the identification of the regression function from the conditional expectation of the dependent variable on the instrument relies on the completeness of the distribution of the endogenous regressor conditional on the instrument, i.e., f(x|z). We provide sufficient conditions for the completeness of f(x|z) without imposing a specific functional form, such as the exponential family. We show that if the conditional density f(x|z) coincides with an existing complete density at a limit point in the support of z, then f(x|z) itself is complete, and therefore, the regression function m (·) is nonparametrically identified. We use this general result provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable z.

In this selective review, we …rst provide some empirical examples that motivate the usefulness of semi-nonparametric techniques in modelling economic and …nancial time series. We describe popular classes of semi-nonparametric dynamic models and some temporal dependence properties. We then present penalized sieve extremum (PSE) estimation as a general method for semi-nonparametric models with cross-sectional, panel, time series, or spatial data. The method is especially powerful in estimating di ¢ cult ill-posed inverse problems such as semi-nonparametric mixtures or conditional moment restrictions. We review recent advances on inference and large sample properties of the PSE estimators, which include (1) consistency and convergence rates of the PSE estimator of the nonparametric part; (2) limiting distributions of plug-in PSE estimators of functionals that are either smooth (i.e., root-n estimable) or non-smooth (i.e., slower than root-n estimable); (3) simple criterion-based inference for plug-in PSE estimation of smooth or non-smooth functionals; and (4) root-n asymptotic normality of semiparametric two-step estimators and their consistent variance estimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH,

Abstract not found

This paper considers inference on functionals of semi/nonparametric conditional moment re-strictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) non-parametric instrumental variables (IV) as special cases. For these models it is often difficult to verify whether a functional is regular (i.e., root-n estimable) or irregular (i.e., slower than root-n estimable). We provide computationally simple, unified inference procedures that are asymptot-ically valid regardless of whether a functional is regular or not. We establish the following new useful results: (1) the asymptotic normality of a plug-in penalized sieve minimum distance (PSMD) estimator of a (possibly irregular) functional; (2) the consistency of simple sieve variance estimators of the plug-in PSMD estimator, and hence the asymptotic chi-square distribution of the sieve Wald statistic; (3) the asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non-optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) Wilks phenomenon of the sieve QLR test of hypothesis with increasing di-

We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a gen-eral upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly depen-dent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L2-norm rates for severely ill-posed problems, and are power of log(n) slower than the optimal L2-norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided.

Abstract: We develop an econometric model of market equilibrium with endogenous entry to analyze rail transportation markets for coal in the Powder River Basin of Wyoming and Montana. Estimation is performed using nonparametric techniques to obtain consistent estimates of the effect of entry on prices. We illustrate a new approach to measuring equilibrium competitive behavior by determining how prices have been affected when a rail carrier enters a monopoly market and find that such entry has caused prices to fall substantially and close to the competitive price. We identify features of coal transportation markets that facilitate this outcome.

This paper considers the problem of choosing the regularization parameter and the smoothing parameter in nonparametric instrumental variables estimation. We propose a simple Mallows’ Cp-type criterion to select these two parameters simultaneously. We show that the proposed selection criterion is optimal in the sense that the selected estimate asymptotically achieves the lowest possible mean squared error among all candidates. To account for model uncertainty, we introduce a new model averaging estimator for nonparametric instrumental variables regres-sions. We propose a Mallows criterion for the weight selection and demonstrate its asymptotic optimality. Monte Carlo simulations show that both selection and averaging methods generally achieve lower root mean squared error than other existing methods. The proposed methods are applied to two empirical examples, the class size question and Engel curve.

The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that re-stricting the problem to models with monotone regression functions and monotone instruments significantly weakens the ill-posedness of the problem. Under these two monotonicity assumptions, we establish that the constrained estimator that imposes monotonicity possesses the same asymptotic rate of convergence as the un-constrained estimator, but the finite-sample behavior of the constrained estimator (in terms of error bounds) is much better than expected from the asymptotic rate of convergence when the regression function is not too steep. In the absence of the point-identifying assumption of completeness, we also derive non-trivial iden-tification bounds on the regression function as implied by our two monotonicity assumptions. Finally, we provide a new adaptive test of the monotone instrument assumption and a simulation study that demonstrates significant finite-sample per-formance gains from imposing monotonicity.

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression