Local structural quantile effects in a model with a nonseparable control variable

I consider a semiparametric version of the nonseparable triangular model of Chesher [Chesher, A., 2003. Identification in nonseparable models. Econometrica 71, 1405–1441]. The proposed model is linear in coefficients, where the coefficients are unknown functions of unobserved latent variables. Using a control variable idea and quantile regression methods, I propose a simple two-step estimator for the coefficients evaluated at particular values of the latent variables. Under the condition that the instruments are locally relevant (i.e. they affect a particular conditional quantile of interest of the endogenous variable) I establish consistency and asymptotic normality. Simulation experiments confirm the theoretical results.     

A smoothed least squares estimator for threshold regression models

We propose a smoothed least squares estimator of the parameters of a threshold regression model. Our model generalizes that considered in Hansen [2000. Sample splitting and threshold estimation. Econometrica 68, 575–603] to allow the thresholding to depend on a linear index of observed regressors, thus allowing discrete variables to enter. We also do not assume that the threshold effect is vanishingly small. Our estimator is shown to be consistent and asymptotically normal thus facilitating standard inference techniques based on estimated standard errors or standard bootstrap for the slope and threshold parameters.     

Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting

We consider semiparametric frequency domain analysis of cointegration between long memory processes, i.e. fractional cointegration, allowing derivation of useful long-run relations even among stationary processes. The approach is due to Robinson (1994b. Annals of Statistics 22, 515–539) and uses a degenerating part of the periodogram near the origin to form a narrow-band frequency domain least squares (FDLS) estimator of the cointegrating relation, which is consistent for arbitrary short-run dynamics. We derive the asymptotic distribution theory for the FDLS estimator of the cointegration vector in the stationary long memory case, thus complementing Robinson's consistency result. An application to the relation between the volatility realized in the stock market and the associated implicit volatility derived from option prices is offered.     

Nonparametric IV estimation of local average treatment effects with covariates

In this paper nonparametric instrumental variable estimation of local average treatment effects (LATE) is extended to incorporate covariates. Estimation of LATE is appealing since identification relies on much weaker assumptions than the identification of average treatment effects in other nonparametric instrumental variable models. Including covariates in the estimation of LATE is necessary when the instrumental variable itself is confounded, such that the IV assumptions are valid only conditional on covariates. Previous approaches to handle covariates in the estimation of LATE relied on parametric or semiparametric methods. In this paper, a nonparametric estimator for the estimation of LATE with covariates is suggested that is root-n asymptotically normal and efficient.     

Estimation and inference in unstable nonlinear least squares models

There is compelling evidence that many macroeconomic and financial variables are not generated by linear models. This evidence is based on testing linearity against either smooth nonlinearity or piece-wise linearity, but there is no framework that encompasses both. This paper provides an econometric framework that allows for both breaks and smooth nonlinearity in between breaks. We estimate the unknown break-dates simultaneously with other parameters via nonlinear least-squares. Using new central limit results for nonlinear processes, we provide inference methods on break-dates and parameter estimates and several instability tests. We illustrate our methods via simulated and empirical smooth transition models with breaks.     

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.     

Estimating semiparametric panel data models by marginal integration

We propose an alternative method for estimating the nonlinear component in semiparametric panel data models. Our method is based on marginal integration that allows us to recover the nonlinear component from an additive regression structure that results from the first differencing transformation. We characterize the asymptotic behavior of our estimator. We also extend the methodology to treat panel data models with two-way effects. Monte Carlo simulations show that our estimator behaves well in finite samples in both random effects and fixed effects settings.     

Nonparametric state price density estimation using constrained least squares and the bootstrap

The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions (such as monotonicity and convexity) within a single least squares procedure. The bootstrap is used to produce confidence intervals for the call function and its first two derivatives and to calibrate a residual regression test of shape constraints. We apply the techniques to option pricing data on the DAX.     

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.     

On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood

A number of information-theoretic alternatives to GMM have recently been proposed in the literature. For practical use and general interpretation, the main drawback of these alternatives, particularly in the case of conditional moment restrictions, is that they give up the computational and interpretational simplicity of quadratic optimization. The main contribution of this paper is to analyze the informational content of estimating equations within the unified framework of Chi-square distance. Improved inference by control variables, closed form formulae for implied probabilities and information-theoretic interpretations of continuously updated GMM are discussed in the two cases of unconditional and conditional moment restrictions.     

Tighter bounds in triangular systems

We study a nonparametric triangular system with (potentially discrete) endogenous regressors and nonseparable errors. Like in other work in this area, the parameter of interest is the structural function evaluated at particular values. We impose a global exclusion and exogeneity condition, in contrast to Chesher (2005), but develop a rank condition which is weaker than Chesher’s. The alternative rank condition can be satisfied for binary endogenous regressors, and it often leads to an identified interval tighter than Chesher (2005)’s minimum length interval. We illustrate the potential of the new rank condition using the Angrist and Krueger (1991) data.     

Functional coefficient instrumental variables models

We consider estimation of nonparametric structural models under a functional coefficient representation for the regression function. Under this representation, models are linear in the endogenous components with coefficients given by unknown functions of the predetermined variables, a nonparametric generalization of random coefficient models. The functional coefficient restriction is an intermediate approach between fully nonparametric structural models that are ill posed when endogenous variables are continuously distributed, and partially linear models over which they have appreciable flexibility. We propose two-step estimators that use local linear approximations in both steps. The first step is to estimate a vector of reduced forms of regression models and the second step is local linear regression using the estimated reduced forms as regressors. Our large sample results include consistency and asymptotic normality of the proposed estimators. The high practical power of estimators is illustrated via both a Monte Carlo simulation study and an application to returns to education.     

Local multiplicative bias correction for asymmetric kernel density estimators

We consider semiparametric asymmetric kernel density estimators when the unknown density has support on [0,∞)$[0,\infty )$. We provide a unifying framework which relies on a local multiplicative bias correction, and contains asymmetric kernel versions of several semiparametric density estimators considered previously in the literature. This framework allows us to use popular parametric models in a nonparametric fashion and yields estimators which are robust to misspecification. We further develop a specification test to determine if a density belongs to a particular parametric family. The proposed estimators outperform rival non- and semiparametric estimators in finite samples and are easy to implement. We provide applications to loss data from a large Swiss health insurer and Brazilian income data.     

Nonparametric estimation of distributional policy effects

This paper proposes a fully nonparametric procedure to evaluate the effect of a counterfactual change in the distribution of some covariates on the unconditional distribution of an outcome variable of interest. In contrast to other methods, we do not restrict attention to the effect on the mean. In particular, our method can be used to conduct inference on the change of the distribution function as a whole, its moments and quantiles, inequality measures such as the Lorenz curve or Gini coefficient, and to test for stochastic dominance. The practical applicability of our procedure is illustrated via a simulation study and an empirical example.     

Approximating the distribution of the two-stage least squares estimator when the concentration parameter is small

This paper presents a new approximation to the exact sampling distribution of the instrumental variables estimator in simultaneous equations models. It differs from many of the approximations currently available, Edgeworth expansions for example, in that it is specifically designed to work well when the concentration parameter is small. The approximation is remarkable in that simultaneously: (i) it has an extremely simple final form; (ii) in situations for which it is designed it is typically much more accurate than is the large sample normal approximation; and (iii) it is able to capture most of those stylized facts that characterize lack of identification and weak instrument scenarios. The development leading to the approximation is also novel in that it introduces techniques of some independent interest not seen in this literature hitherto.     

Local polynomial estimation of nonparametric simultaneous equations models

We define a new procedure for consistent estimation of nonparametric simultaneous equations models under the conditional mean independence restriction of Newey et al. [1999. Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565–603]. It is based upon local polynomial regression and marginal integration techniques. We establish the asymptotic distribution of our estimator under weak data dependence conditions. Simulation evidence suggests that our estimator may significantly outperform the estimators of Pinkse [2000. Nonparametric two-step regression estimation when regressors and errors are dependent. Canadian Journal of Statistics 28, 289–300] and Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578].     

Semiparametric bounds on treatment effects

We present a variety of semiparametric models that produce bounds on the average causal effect of a binary treatment on a binary outcome. The semiparametric assumptions exploit variation in observable covariates to narrow the bounds. In our main model, the outcome is determined by a generalized linear model, but the treatment may be arbitrarily endogenous. Our bounding strategy does not require the existence of an instrument, but incorporating an instrument narrows the bounds. The bounds are further improved by combining the semiparametric model with the joint threshold-crossing assumption of Shaikh and Vytlacil (2005).