Abstract. We provide new empirical evidence of the relationship between the availability of internal funds and firms' investment. By employing a semi-parametric fixed effect model, we estimate a U-shaped curve relating investment and internal funds. Our results highlight the importance of allowing for nonlinearities when modeling changes in internal funds and investment, and show that R&D expenses play a critical role on firms' investment under financial constraints.
Abstract. We propose a new class of estimators for a jump discontinuity on nonparametric regression. While there is a vast literature in econometrics that addresses this issue (e.g., Hahn et al., 2001; Porter, 2003; Imbens and Lemieux, 2008; Cattaneo and Escanciano, 2017), the main approach in these studies is to use local polynomial (linear) estimators on both sides of the discontinuity to produce an estimator for the jump that has desirable boundary properties. Our approach extends the regression from both sides of the discontinuity using a theorem of Hestenes (1941). The extended regressions are then estimated and used to construct an estimator for the jump discontinuity that solves the boundary problems normally associated with classical Nadaraya-Watson estimators. We provide asymptotic characterizations for the jump estimators, including bias and variance orders, and asymptotic distributions after suitable centering and normalization. Monte Carlo simulations show that our jump estimators can outperform those based on local polynomial (linear) regression.
Abstract. We consider nonparametric estimation of a distribution function \(F\) associated with a random variable \(X\) based on a random sample. First, for \(x\) a point of continuity of \(F\), we define a class of estimators for \(F(x)\) and obtain their rates of convergence. Contrary to the existing literature, we impose no restriction on the existence or smoothness of the derivatives of \(F\). The traditional kernel estimator for \(F(x)\) is a member of the class. Second, for \(x\) that is either the location of a jump discontinuity or an isolated point of the support, we define a class of estimators for the jump \(p_x=F(x)-F(x_-)\) and obtain their rates of convergence. Again, no additional restriction is imposed on \(F\) beyond right-continuity. Our results are of significant practical use as there are numerous examples in Economics, Finance and Biomedicine of distributions that have point masses. Our main insight is also applied to obtain new inversion theorems for characteristic functions and explicit estimates for convergence rates. A small simulation study provides some evidence on the finite sample properties of our proposed estimators and contrasts their performance with some existing alternatives. An empirical section illustrates the use of our estimators using data on global elevation and data associated with "P-hacking" in economics journals.
Abstract. We consider the estimation of the boundary of a set when it is known to be sufficiently smooth, to satisfy certain shape constraints and to have an additive structure. Our proposed method is based on spline estimation of a conditional quantile regression and is resistant to outliers and/or extreme values in the data. This work is a desirable extension of Martins-Filho and Yao (2007) and Wang and Xue (2018) and can also be viewed as an alternative to existing estimators that have been used in empirical analysis. The results of a Monte Carlo study show that the new method outperforms the existing methods when outliers or heterogeneity are present. Our theoretical analysis indicates that our proposed boundary estimator is uniformly consistent under a set of standard assumptions. We illustrate practical use of our method by estimating two production functions using real-world data sets.
Abstract. We propose kernel-based estimators for both the parametric and nonparametric components of a partially linear additive regression model where a subset of the covariates entering the nonparametric component are generated by the estimation of an auxiliary nonparametric regression. Both estimators are shown to be asymptotically normally distributed. The estimator for the finite dimensional parameter is shown to converge at the parametric \(\sqrt{n}\) rate and the estimator for the infinite dimensional parameter converges at a slower nonparametric rate that, as usual, depends on the rate of decay of the bandwidths and the dimensionality of the underlying regression. A small Monte Carlo study is conducted to shed light on the finite sample performance of our estimators and to contrast them with those of estimators available in the extant literature.
Abstract. Kernel density estimation in domains with boundaries is known to suffer from undesirable boundary effects. We show that in the case of smooth densities, a general and elegant approach is to estimate an extension of the density. The resulting estimators in domains with boundaries have biases and variances expressed in terms of density extensions and extension parameters. The result is that they have the same rates at boundary and interior points of the domain. Contrary to the extant literature, our estimators require no kernel modification near the boundary and kernels commonly used for estimation on the real line can be applied. Densities defined on the half-axis and in a unit interval are considered. The results are applied to estimation of densities that are discontinuous or have discontinuous derivatives, where they yield the same rates of convergence as for smooth densities on \(\Re\).
Abstract. We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975) to give final estimators for CVaR and CES. We provide consistency and asymptotic normality of the proposed estimators under suitable normalization. We also present the results of a Monte Carlo study that sheds light on their finite sample performance. Empirical viability of the model and estimators is investigated through a backtesting exercise using returns on future contracts for five agricultural commodities.
Abstract. We define a new bandwidth-dependent kernel density estimator that improves existing convergence rates for the bias, and preserves that of the variation, when the error is measured in \(L_1\). No additional assumptions are imposed to the extant literature.
Abstract. Estimators for derivatives associated with a density function can be useful in identifying its modes and inflection points. In addition, these estimators play an important role in plug-in methods associated with bandwidth selection in nonparametric kernel density estimation. In this paper we extend the nonparametric class of density estimators proposed by Mynbaev and Martins-Filho (2010) to the estimation of m-order density derivatives. Contrary to some existing derivative estimators, the estimators in our proposed class have a full asymptotic characterization, including uniform consistency and asymptotic normality. An expression for the bandwidth that minimizes an asymptotic approximation for the estimators' integrated squared error is provided. A Monte Carlo study sheds light on the finite sample performance of our estimators and contrasts it with that of density derivative estimators based on the classical Rosenblatt-Parzen approach.
Abstract. In this paper we compare six nonparametric estimators for technical efficiency and use them to evaluate the efficiency of the banking sector in Brazil. The estimators considered are data envelopment analysis (DEA), free disposal hull (FDH), bias corrected FDH (FDHC), bias corrected DEA (DEAC), order-m and alpha-conditional quantile. Their theoretical properties are discussed and their implementation is illustrated using a sample of 184 Brazilian banks that extends from 1995 to 2004. The results indicate that these estimators can lead to significant discrepancy in estimated efficiency scores. Order-m and alpha-conditional quantile estimators have proven to be useful tools in identifying extreme values and are shown to be rather robust relative to DEA and FDH. Bias correction for both DEA and FDH was problematic, producing significant changes in firms rankings and estimated efficiencies.
Abstract. We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic's kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric \(R^2\) test for overall significance of a nonparametric regression model.
Abstract. This paper investigates the effects of the financial system on a firm’s investment decisions using data from 404 Brazilian firms over the 1998-2006 period. We also use country-level data and classify firms as financially constrained and unconstrained according to the KZ and WW indexes. The results show that financial development has a significant impact on a firm’s investment. Furthermore, the financial structure has an effect on the investment behavior of constrained firms even after controlling for the level of financial development. This finding points to a market-based financial system in order to reduce the constrained firms’ dependence on internal resources.
Abstract. Nonparametric prediction of a random variable Y conditional on the value of an explanatory variable X is a classical and important problem in Statistics. The problem is significantly complicated if there are heterogeneously distributed measurement errors on the observed values of X used in estimation and prediction. Carroll et al. (2009) have recently proposed a kernel deconvolution estimator and obtained its consistency. In this paper we use the kernels proposed in Mynbaev and Martins-Filho (2010) to define a class of deconvolution estimators for prediction that contains their estimator as one of its elements. First, we obtain consistency of the estimators under much less restrictive conditions. Specifically, contrary to what is routinely assumed in the extant literature, the Fourier transform of the underlying kernels is not required to have compact support, higher-order restrictions on the kernel can be avoided and fractional smoothness of the involved densities is allowed. Second, we obtain asymptotic normality of the estimators under the assumption that there are two types of measurement errors on the observed values of X . It is apparent from our study that even in this simplified setting there are multiple cases exhibiting different asymptotic behavior. Our proof focuses on the case where measurement errors are super-smooth and we use it to discuss other possibilities. The results of a Monte Carlo simulation are provided to compare the performance of the estimator using traditional kernels and those proposed in Mynbaev and Martins-Filho (2010).
Abstract. We consider the estimation of a high order quantile associated with the conditional distribution of a regressand in a nonparametric regression model. Our estimator is inspired by Pickands (1975) where it is shown that arbitrary distributions which lie in the domain of attraction of an extreme value type have tails that, in the limit, behave as generalized Pareto distributions (GPD). Smith (1987) has studied the asymptotic properties of maximum likelihood (ML) estimators for the parameters of the GPD in this context, but in our paper the relevant random variables used in estimation are standardized residuals from a first stage kernel based nonparametric estimation. We obtain convergence in probability and distribution of the residual based ML estimator for the parameters of the GPD as well as the asymptotic distribution for a suitably defined quantile estimator. A Monte Carlo study provides evidence that our estimator behaves well in finite samples and is easily implementable. Our results have direct application in finance, particularly in the estimation of conditional Value-at-Risk, but other researchers in applied fields such as insurance will also find the results useful.
Abstract. We consider the estimation of a nonparametric stochastic frontier model with composite error density which is known up to a finite parameter vector. Our primary interest is on the estimation of the parameter vector, as it provides the basis for estimation of firm specific (in)efficiency. Our frontier model is similar to that of Fan et al. (1996), but here we extend their work in that: a) we establish the asymptotic properties of their estimation procedure, and b) propose and establish the asymptotic properties of an alternative estimator based on the maximization of a conditional profile likelihood function. The estimator proposed in Fan et al. (1996) is asymptotically normally distributed but has bias which does not vanish as the sample size \(n \to \infty\). In contrast, our proposed estimator is asymptotically normally distributed and correctly centered at the true value of the parameter vector. In addition, our estimator is shown to be efficient in a broad class of semiparametric estimators. Our estimation procedure provides a fast converging alternative to the recently proposed estimator in Kumbhakar et al. (2007). A Monte Carlo study is performed to shed light on the finite sample properties of these competing estimators.
Abstract. In this paper we propose a local exponential estimator for a multiplicative nonparametric frontier model first introduced by Martins-Filho and Yao (2007). We improve their estimation procedure by adopting a variant of the local exponential smoothing introduced in Ziegelmann (2002). Our estimator is shown to be consistent and asymptotically normal under mild regularity conditions. In addition, due to local exponential smoothing, potential negativity of conditional variance functions that may hinder the use of Martins-Filho and Yao’s estimator is avoided. A Monte Carlo study is performed to shed light on the finite sample properties of the estimator and to contrast its performance with that of the estimator proposed in Martins-Filho and Yao (2007). We also conduct an empirical exercise in which a production function and associated efficiencies for branches of financial institutions in the United States are estimated.
Abstract. We review nonparametric estimation of efficiency and productivity, by which we mainly mean activity analysis, or data envelopment analysis (DEA). The review covers topics that we hope will be of special interest to those doing research in the realm of agriculture. We also include a brief appendix addressing nonparametric estimation from an econometric perspective.
Abstract. Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are non stochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result, nonparametric estimators are usually constructed using stochastic bandwidths. In this article, we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstochastic bandwidths. Our result extends previous work by Boente and Fraiman (1995) and Ziegler (2004).
Abstract. We propose a kernel-based estimator for a partially linear model in triangular systems where endogenous variables appear both in the nonparametric and linear component functions. Our estimator is easy to implement, has an explicit algebraic structure, and exhibits good finite sample performance in a Monte Carlo study.
Abstract. The introduction of directional distance func- tions has given researchers an alternative to Shephard distance functions. In this paper we conduct a Monte Carlo study to investigate the performance of distance functions as an approximation for models of technology. Our results indicate that quadratic representations of technology have better approximation properties than translog parameterizations.
Abstract. In this paper we propose a new nonparametric kernel-based estimator for a density function f which achieves bias reduction relative to the classical Rosenblatt–Parzen estimator. Contrary to some existing estimators that provide for bias reduction, our estimator has a full asymptotic characterisation including uniform consistency and asymptotic normality. In addition, we show that bias reduction can be achieved without the disadvantage of potential negativity of the estimated density – a deficiency that results from using higher order kernels. Our results are based on imposing global Lipschitz conditions on f and defining a novel corresponding kernel. A Monte Carlo study is provided to illustrate the estimator’s finite sample performance.
Abstract. The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.
Abstract. Traditional estimators for nonparametric frontier models (DEA, FDH) are very sensitive to extreme values/outliers. Recently, Aragon et al. [2005. Nonparametric frontier estimation: a conditional quantile-based approach. Econometric Theory 21, 358–389] proposed a nonparametric \(\alpha\)-frontier model and estimator based on a suitably defined conditional quantile which is more robust to extreme values/outliers. Their estimator is based on a nonsmooth empirical conditional distribution. In this paper, we propose a new smooth nonparametric conditional quantile estimator for the \(\alpha\)-frontier model. Our estimator is a kernel based conditional quantile estimator that builds on early work of Azzalini [1981. A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68, 326–328]. It is computationally simple, resistant to outliers and extreme values, and smooth. In addition, the estimator is shown to be consistent and \(\sqrt{n}\) asymptotically normal under mild regularity conditions. We also show that our estimator’s variance is smaller than that of the estimator proposed by Aragon et al. A simulation study confirms the asymptotic theory predictions and contrasts our estimator with that of Aragon et al.
Abstract. In this article we define a class of estimators for a nonparametric regression model with the aim of reducing bias. The estimators in the class are obtained via a simple two-stage procedure. In the first stage, a potentially misspecified parametric model is estimated and in the second stage the parametric estimate is used to guide the derivation of a final semiparametric estimator. Mathematically, the proposed estimators can be thought as the minimization of a suitably defined Cressie–Read discrepancy that can be shown to produce conventional nonparametric estimators, such as the local polynomial estimator, as well as existing two-stage multiplicative estimators, such as that proposed by Glad (1998). We show that under fairly mild conditions the estimators in the proposed class are \(\sqrt{nh_n}\) asymptotically normal and explore their finite sample (simulation) behavior.
Abstract. This study examines how used vehicle markets responded to the automobile hydrocarbon emissions by linking used vehicle price to the large scale emission test data that contain 74 vehicle models manufactured over 18 years. An additive semiparametric hedonic model is estimated to analyze the relationship between vehicle price and hydrocarbon emissions. The estimation procedure is novel and involves a local polynomial estimator nested in a backfitting algorithm with the bandwidths chosen by a data driven plug-in method. The results indicate that hydrocarbon emissions have a significant negative impact on vehicle price, but the negative association is evident only at low emission levels. The price discount appears to be unrelated to the increased costs from recent emission regulations which mainly target high polluting vehicles.
Abstract. In this paper, we investigate the finite sample performance of four kernel-based estimators that are currently available for additive non-parametric regression models – the classic backfitting estimator (CBE), the smooth backfitting estimator, the marginal integration estimator, and two versions of a two-stage estimator of which the first is proposed by Kim, Linton and Hengartner (1999) and the second is proposed in this paper. The bandwidths are selected for each estimator by minimizing their respective asymptotic approximation of the mean average squared errors. In our simulations, we are particularly concerned with the performance of these estimators under this unified data-driven bandwidth selection method, since in this case both the asymptotic and the finite sample properties of all estimators are currently unavailable. The comparison is based on the estimators’ average squared error. Our Monte Carlo results seem to suggest that the CBE is the best performing kernel-based procedure.
Abstract. In this paper we propose a nonparametric regression frontier model that assumes no specific parametric family of densities for the unobserved stochastic component that represents efficiency in the model. Nonparametric estimation of the regression frontier is obtained using a local linear estimator that is shown to be consistent and \(\sqrt{nh_n}\) asymptotically normal under standard assumptions. The estimator we propose envelops the data but is not inherently biased as free disposal hull—FDH or data envelopment analysis—DEA estimators. It is also more robust to extreme values than the aforementioned estimators. A Monte Carlo study is performed to provide preliminary evidence on the estimator’s finite sample properties and to compare its performance to a bias corrected FDH estimator.
Abstract. We establish the \(\sqrt{n}\) asymptotic equivalence of V and U statistics when the statistic’s kernel depends on \(n\). Combined with a lemma of B. Lee this result provides conditions under which U statistics projections and V statistics are \(\sqrt{n}\) asymptotically equivalent. The use of this equivalence in nonparametric regression models is illustrated with several examples; the estimation of conditional variances, skewness, kurtosis and the construction of a nonparametric R-squared measure.
Abstract. We propose an estimation procedure for value-at-risk (VaR) and expected shortfall (TailVaR) for conditional distributions of a time series of returns on a financial asset. Our approach combines a local polynomial estimator of conditional mean and volatility functions in a conditional heterocedastic autoregressive nonlinear (CHARN) model with Extreme Value Theory for estimating quantiles of the conditional distribution. We investigate the finite sample properties of our method and contrast them with alternatives, including the method recently proposed by McNeil and Frey (2000), in an extensive Monte Carlo study. The method we propose outperforms the estimators currently available in the literature. An evaluation based on backtesting was also performed.
Abstract. We model a hedonic price function for housing as an additive nonparametric regression. Estimation is done via a backfitting procedure in combination with a local polynomial estimator. It avoids the pitfalls of an unrestricted nonparametric estimator, such as slow convergence rates and the curse of dimensionality. Bandwidths are chosen using a novel plug in method that minimizes the asymptotic mean average squared error (AMASE) of the regression. We compare our results to alternative parametric models and find evidence of the superiority of our nonparametric model. From an empirical perspective our study is interesting in that the effects on housing prices of a series of environmental characteristics are modeled in the regression. We find these characteristics to be important in the determination of housing prices.
Abstract. This paper considers the general problem of Feasible Generalized Least Squares Instrumental Variables (FGLS IV) estimation using optimal instruments. First we summarize the sufficient conditions for the FGLS IV estimator to be asymptotically equivalent to an optimal GLS IV estimator. Then we specialize to stationary dynamic systems with stationary VAR errors, and use the sufficient conditions to derive new moment conditions for these models. These moment conditions produce useful IVs from the lagged endogenous variables, despite the correlation between errors and endogenous variables. This use of the information contained in the lagged endogenous variables expands the class of IV estimators under consideration and thereby potentially improves both asymptotic and small-sample efficiency of the optimal IV estimator in the class. Some Monte Carlo experiments compare the new methods with those of Hatanaka (1976). For the DGP used in the Monte Carlo experiments, asymptotic efficiency is strictly improved by the new IVs, and experimental small-sample efficiency is improved as well.
Abstract. White’s (1984, Asymptotic Theory for Econometricians. Academic Press (Harcourt Brace Jovanovich), Orlando.) concept of asymptotic variance is shown to allow some ambiguities when used to study asymptotic efficiency. These ambiguities are resolved with some mild conditions on the estimators being studied, because then White’s asymptotic variance is an equivalence class in which efficiency conclusions are invariant across members of the class. Among the extant efficiency definitions, the liminf-based definition (White, 1994. Estimation, Inference and Specification Analysis. Econometric Society Monograph, vol. 22, Cambridge University Press, Cambridge. p. 136) is most informative even though identical conclusions can be obtained under our conditions with earlier definitions, but there are still some notions of efficiency allowed by White’s asymptotic variance that can only be detected by weaker efficiency definitions.
Abstract. Asymptotic equivalence of Aitken and feasible Aitken estimators in linear models with non scalar identity error covariance matrices is usually established in a tedious case-by-case manner. Some general sufficient conditions for this equivalence exist, but there are problems with the extant conditions. These problems are discussed, and new widely applicable sufficient conditions are presented and applied to a variety of error structures.
Abstract. We derive consistent, asymptotically efficient, and asymptotically normal estimators for SUR systems that have additive heteroscedastic contemporaneous correlation. Both our estimator for the location vector and the parameters of the covariance matrix possess these properties. The procedure is superior to other methods because we use GLS to estimate the parameters of the covariance matrix. Our method also permits the use of cross-equation parameter restrictions. We discuss how this type of heteroscedasticity arises naturally in share equation systems and random coefficient models, and how these models can be uniquely estimated with our two-step estimation technique.
Abstract. Although telephone pricing has received increasing attention in recent years, the geographic patterns of telephone pricing and the corresponding economic consequences of those patterns have remained perplexing to consumers and policymakers and largely unaddressed by economists. In this article we first specify a model of the demand for short (intraLATA) long distance calling. We then draw upon data made available by the recent adoption of extended area service (EAS) in four metropolitan areas to empirically measure the structure of inter-exchange telephone demand. Given these estimates, and a conceptual framework for analyzing the economic welfare effects, we are able to quantify the consumer-surplus effects of alternative pricing policies. The empirical results indicate that consumer surplus is noticeably enhanced by adopting EAS. But the net economic welfare effects are shown to be sensitive to, among other things, the level of price-cost margins prevailing prior to the implementation of EAS.