Instrumental Variable Estimation of Nonparametric Models

In econometrics there are many occasions where knowledge of the structural relationship among dependent variables is required to answer questions of interest. This paper gives identification and estimation results for nonparametric conditional moment restrictions. We characterize identification of structural functions as completeness of certain conditional distributions, and give sufficient identification conditions for exponential families and discrete variables. We also give a consistent, nonparametric estimator of the structural function. The estimator is nonparametric two‐stage least squares based on series approximation, which overcomes an ill‐posed inverse problem by placing bounds on integrals of higher‐order derivatives.     

Identification of Nonseparable Models Using Instruments With Small Support

I consider nonparametric identification of nonseparable instrumental variables models with continuous endogenous variables. If both the outcome and first stage equations are strictly increasing in a scalar unobservable, then many kinds of continuous, discrete, and even binary instruments can be used to point‐identify the levels of the outcome equation. This contrasts sharply with related work by Imbens and Newey, 2009 that requires continuous instruments with large support. One implication is that assumptions about the dimension of heterogeneity can provide nonparametric point‐identification of the distribution of treatment response for a continuous treatment in a randomized controlled experiment with partial compliance.     

Nonparametric Identification of Finite Mixture Models of Dynamic Discrete Choices

In dynamic discrete choice analysis, controlling for unobserved heterogeneity is an important issue, and finite mixture models provide flexible ways to account for it. This paper studies nonparametric identifiability of type probabilities and type‐specific component distributions in finite mixture models of dynamic discrete choices. We derive sufficient conditions for nonparametric identification for various finite mixture models of dynamic discrete choices used in applied work under different assumptions on the Markov property, stationarity, and type‐invariance in the transition process. Three elements emerge as the important determinants of identification: the time‐dimension of panel data, the number of values the covariates can take, and the heterogeneity of the response of different types to changes in the covariates. For example, in a simple case where the transition function is type‐invariant, a time‐dimension of T = 3 is sufficient for identification, provided that the number of values the covariates can take is no smaller than the number of types and that the changes in the covariates induce sufficiently heterogeneous variations in the choice probabilities across types. Identification is achieved even when state dependence is present if a model is stationary first‐order Markovian and the panel has a moderate time‐dimension (T 6).     

Nonparametric Instrumental Regression

The focus of this paper is the nonparametric estimation of an instrumental regression function ϕ defined by conditional moment restrictions that stem from a structural econometric model E[Y−ϕ(Z)|W]=0, and involve endogenous variables Y and Z and instruments W. The function ϕ is the solution of an ill‐posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyzes identification and overidentification of this model, and presents asymptotic properties of the estimated nonparametric instrumental regression function.     

Instrumental Variable Models for Discrete Outcomes

Single equation instrumental variable models for discrete outcomes are shown to be set identifying, not point identifying, for the structural functions that deliver the values of the discrete outcome. Bounds on identified sets are derived for a general nonparametric model and sharp set identification is demonstrated in the binary outcome case. Point identification is typically not achieved by imposing parametric restrictions. The extent of an identified set varies with the strength and support of instruments, and typically shrinks as the support of a discrete outcome grows. The paper extends the analysis of structural quantile functions with endogenous arguments to cases in which there are discrete outcomes.     

The Realized Laplace Transform of Volatility

We introduce and derive the asymptotic behavior of a new measure constructed from high‐frequency data which we call the realized Laplace transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process. With a long span of data, that is, under joint long‐span and infill asymptotics, the statistic can be used to construct a nonparametric estimate of the volatility Laplace transform as well as of the integrated joint Laplace transform of volatility over different points of time. We derive feasible functional limit theorems for our statistic both under fixed‐span and infill asymptotics as well as under joint long‐span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes.     

Instrumental Variable Estimation of Nonlinear Errors‐in‐Variables Models

This paper establishes that instruments enable the identification of nonparametric regression models in the presence of measurement error by providing a closed form solution for the regression function in terms of Fourier transforms of conditional expectations of observable variables. For parametrically specified regression functions, we propose a root n consistent and asymptotically normal estimator that takes the familiar form of a generalized method of moments estimator with a plugged‐in nonparametric kernel density estimate. Both the identification and the estimation methodologies rely on Fourier analysis and on the theory of generalized functions. The finite‐sample properties of the estimator are investigated through Monte Carlo simulations.     

Nonparametric Identification of Risk Aversion in First‐Price Auctions Under Exclusion Restrictions

This paper studies the nonparametric identification of the first‐price auction model with risk averse bidders within the private value paradigm. First, we show that the benchmark model is nonindentified from observed bids. We also derive the restrictions imposed by the model on observables and show that these restrictions are weak. Second, we establish the nonparametric identification of the bidders' utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders' participation, leading to a latent distribution of private values that is independent of the number of bidders. The key idea is to exploit the property that the bid distribution varies with the number of bidders while the private value distribution does not. We then extend these results to endogenous bidders' participation when the exclusion restriction takes the form of instruments that do not affect the bidders' private value distribution. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values, and asymmetric bidders. Last, possible estimation methods are proposed.     

Structural Nonparametric Cointegrating Regression

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill‐posed inverse problems. In functional cointegrating regressions where the regressor is an integrated or near‐integrated time series, it is shown here that inverse and ill‐posed inverse problems do not arise. Instead, simple nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving straightforward asymptotics that are useable in practical work. It is further shown that use of augmented regression, as is common in linear cointegration modeling to address endogeneity, does not lead to bias reduction in nonparametric regression, but there is an asymptotic gain in variance reduction. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series when there is a single integrated or near‐integrated regressor. The methods may be applied to a range of empirical models where functional estimation of cointegrating relations is required.     

Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain

We develop results for the use of Lasso and post‐Lasso methods to form first‐stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p. Our results apply even when p is much larger than the sample size, n. We show that the IV estimator based on using Lasso or post‐Lasso in the first stage is root‐n consistent and asymptotically normal when the first stage is approximately sparse, that is, when the conditional expectation of the endogenous variables given the instruments can be well‐approximated by a relatively small set of variables whose identities may be unknown. We also show that the estimator is semiparametrically efficient when the structural error is homoscedastic. Notably, our results allow for imperfect model selection, and do not rely upon the unrealistic “beta‐min” conditions that are widely used to establish validity of inference following model selection (see also Belloni, Chernozhukov, and Hansen (2011b)). In simulation experiments, the Lasso‐based IV estimator with a data‐driven penalty performs well compared to recently advocated many‐instrument robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the Lasso‐based IV estimator outperforms an intuitive benchmark. Optimal instruments are conditional expectations. In developing the IV results, we establish a series of new results for Lasso and post‐Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non‐Gaussian, heteroscedastic disturbances that uses a data‐weighted ℓ₁‐penalty function. By innovatively using moderate deviation theory for self‐normalized sums, we provide convergence rates for the resulting Lasso and post‐Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that logp = o(n^1/3). We also provide a data‐driven method for choosing the penalty level that must be specified in obtaining Lasso and post‐Lasso estimates and establish its asymptotic validity under non‐Gaussian, heteroscedastic disturbances.     

股指期权定价的非参数数值方法研究

扩散过程估计的参数化方法存在先入为主的不足,并且扩散项函数形式的设定十分困难,而非参数方法不需要数据产生过程的先验信息,直接从数据出发估计扩散函数,克服了以上不足。本文提出了一种基于连续时间过程的非参数股指期权定价模型。对于刻画基础资产动态行为特性的扩散函数不加任何函数形式限制,利用离散数据匹配密度函数构造它的非参数估计,进而计算股指期权的均衡价格。论文从理论上论证了扩散项估计的一致性和渐进正态性。实证研究表明,该方法对于实际市场价格具有较高的拟合效果,特别是在市场波动剧烈时期,非参数方法更优于经典B-S方法。     

Asymptotic Variance of Semiparametric Estimators With Generated Regressors

We study the asymptotic distribution of three‐step estimators of a finite‐dimensional parameter vector where the second step consists of one or more nonparametric regressions on a regressor that is estimated in the first step. The first‐step estimator is either parametric or nonparametric. Using Newey's (1994) path‐derivative method, we derive the contribution of the first‐step estimator to the influence function. In this derivation, it is important to account for the dual role that the first‐step estimator plays in the second‐step nonparametric regression, that is, that of conditioning variable and that of argument.     

Average and Quantile Effects in Nonseparable Panel Models

Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time‐homogeneity conditions that are like “time is randomly assigned” or “time is an instrument.” Partial‐identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed‐effects estimator is not a consistent estimator of the identified average effect, and a consistent estimator is given. A simple estimator of identified quantile treatment effects is given, providing a solution to the important problem of estimating quantile treatment effects from panel data. Bounds for overall effects in static and dynamic models are given. The dynamic bounds provide a partial‐identification solution to the important problem of estimating the effect of state dependence in the presence of unobserved heterogeneity. The impact of T, the number of time periods, is shown by deriving shrinkage rates for the identified set as T grows. We also consider semiparametric, discrete‐choice models and find that semiparametric panel bounds can be much tighter than nonparametric bounds. Computationally convenient methods for semiparametric models are presented. We propose a novel inference method that applies in panel data and other settings and show that it produces uniformly valid confidence regions in large samples. We give empirical illustrations.     

A Three‐step Method for Choosing the Number of Bootstrap Repetitions

This paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, confidence regions, hypothesis tests, p‐values, and bias correction. For each of these problems, the paper provides a three‐step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval length, test's critical value, test's p‐value, or bias‐corrected estimate based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B=∞. The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals. Monte Carlo simulations show that the proposed methods work very well.     

Firm‐level Evidence on Gender Wage Discrimination in the Belgian Private Economy

In this paper we explore a matched employer–employee data set to investigate the presence of gender wage discrimination in the Belgian private economy labour market. Contrary to many existing papers, we analyse gender wage discrimination using an independent productivity measure. Using firm‐level data, we are able to compare direct estimates of a gender productivity differential with those of a gender wage differential. We take advantage of the panel structure to identify gender‐related differences from within‐firm variation. Moreover, inspired by recent developments in the production function estimation literature, we address the problem of endogeneity of the gender mix using a structural production function estimator alongside instrumental variable‐general method of moments (IV‐GMM) methods where lagged value of labour inputs are used as instruments. Our results suggest that there is no gender wage discrimination inside private firms located in Belgium, on the contrary.     

Optimal policy for imperfect inspection in discrete time

Abstract

The problem of finding the optimal inspection interval for a fallible system that operates in discrete time when inspection is subject to errors was addressed. A Markov chain model of the system state was developed to formulate the objective function, which is the minimum expected total cost per lime unit. A numerical search procedure was used to find local minima for a representative range of parameter values. The objective function turned out to be quite insensitive to deviations in the inspection interval in the neighbourhood of the minimum, so an approximate method based on curve fitting was developed and tested, yielding good results in the representative range.     

Efficient Semiparametric Estimation of Censored and Truncated Regressions via a Smoothed Self‐Consistency Equation

An asymptotically efficient likelihood‐based semiparametric estimator is derived for the censored regression (tobit) model, based on a new approach for estimating the density function of the residuals in a partially observed regression. Smoothing the self‐consistency equation for the nonparametric maximum likelihood estimator of the distribution of the residuals yields an integral equation, which in some cases can be solved explicitly. The resulting estimated density is smooth enough to be used in a practical implementation of the profile likelihood estimator, but is sufficiently close to the nonparametric maximum likelihood estimator to allow estimation of the semiparametric efficient score. The parameter estimates obtained by solving the estimated score equations are then asymptotically efficient. A summary of analogous results for truncated regression is also given.     

Using Probability Boxes to Model Elicited Information: A Case Study

Governments are responsible for making policy decisions, often in the face of severe uncertainty about the factors involved. Expert elicitation can be used to fill information gaps where data are not available, cannot be obtained, or where there is no time for a full‐scale study and analysis. Various features of distributions for variables may be elicited, for example, the mean, standard deviation, and quantiles, but uncertainty about these values is not always recorded. Distributional and dependence assumptions often have to be made in models and although these are sometimes elicited from experts, modelers may also make assumptions for mathematical convenience (e.g., assuming independence between variables). Probability boxes (p‐boxes) provide a flexible methodology to analyze elicited quantities without having to make assumptions about the distribution shape. If information about distribution shape(s) is available, p‐boxes can provide bounds around the results given these possible input distributions. P‐boxes can also be used to combine variables without making dependence assumptions. This article aims to illustrate how p‐boxes may help to improve the representation of uncertainty for analyses based on elicited information. We focus on modeling elicited quantiles with nonparametric p‐boxes, modeling elicited quantiles with parametric p‐boxes where the elicited quantiles do not match the elicited distribution shape, and modeling elicited interval information.     

Applied Nonparametric Instrumental Variables Estimation

Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used to identify and estimate these parameters. However, linear and other finite‐dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators.     

Identification in Auctions With Selective Entry

This paper considers nonparametric identification of a two‐stage entry and bidding game we call the Affiliated‐Signal (AS) model. This model assumes that potential bidders have private values, observe signals of their values prior to entry, and then choose whether to undertake a costly entry process, but imposes only minimal structure on the relationship between signals and values. It thereby nests a wide range of entry processes, including in particular the Samuelson (1985) and Levin and Smith (1994) models as special cases. Working within the AS model, we map variation in factors affecting entry behavior (potential competition or entry costs) into identified bounds on model fundamentals. These bounds are constructive, collapse to point identification when available entry variation is continuous, and can readily be refined to produce the pointwise sharp identified set. We then extend our core results to accommodate nonseparable unobserved auction‐level heterogeneity and potential endogeneity of entry shifters, thereby establishing a formal identification framework for structural analysis of auctions with selective entry.