A flexible parametric approach for estimating switching regime models and treatment effect parameters

In this paper, we propose a flexible, parametric class of switching regime models allowing for both skewed and fat-tailed outcome and selection errors. Specifically, we model the joint distribution of each outcome error and the selection error via a newly constructed class of multivariate distributions which we call generalized normal mean–variance mixture distributions. We extend Heckman’s two-step estimation procedure for the Gaussian switching regime model to the new class of models. When the distributions of the outcome errors are asymmetric, we show that an additional correction term accounting for skewness in the outcome error distribution (besides the analogue of the well known inverse mill’s ratio) needs to be included in the second step regression. We use the two-step estimators of parameters in the model to construct simple estimators of average treatment effects and establish their asymptotic properties. Simulation results confirm the importance of accounting for skewness in the outcome errors in estimating both model parameters and the average treatment effect and the treatment effect for the treated.     

Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models

This paper deals with a nonlinear errors-in-variables model where the distributions of the unobserved predictor variables and of the measurement errors are nonparametric. Using the instrumental variable approach, we propose method of moments estimators for the unknown parameters and simulation-based estimators to overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals. Both estimators are consistent and asymptotically normally distributed under fairly general regularity conditions. Moreover, root-n consistent semiparametric estimators and a rank condition for model identifiability are derived using the combined methods of the nonparametric technique and Fourier deconvolution.     

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.     

Partial rank estimation of duration models with general forms of censoring

In this paper we propose estimators for the regression coefficients in censored duration models which are distribution free, impose no parametric specification on the baseline hazard function, and can accommodate general forms of censoring. The estimators are shown to have desirable asymptotic properties and Monte Carlo simulations demonstrate good finite sample performance. Among the data features the new estimators can accommodate are covariate-dependent censoring, double censoring, and fixed (individual or group specific) effects. We also examine the behavior of the estimator in an empirical illustration.     

Asymptotic properties of Monte Carlo estimators of diffusion processes

This paper studies the limit distributions of Monte Carlo estimators of diffusion processes. We examine two types of estimators based on the Euler scheme, one applied to the original processes, the other to a Doss transformation of the processes. We show that the transformation increases the speed of convergence of the Euler scheme. We also study estimators of conditional expectations of diffusions. After characterizing expected approximation errors, we construct second-order bias-corrected estimators. We also derive new convergence results for the Mihlstein scheme. Illustrations of the results are provided in the context of simulation-based estimation of diffusion processes.     

Estimation of partial differential equations with applications in finance

Linear parabolic partial differential equations (PDE’s) and diffusion models are closely linked through the celebrated Feynman–Kac representation of solutions to PDE’s. In asset pricing theory, this leads to the representation of derivative prices as solutions to PDE’s. Very often implied derivative prices are calculated given preliminary estimates of the diffusion model for the underlying variable. We demonstrate that the implied derivative prices are consistent and derive their asymptotic distribution under general conditions. We apply this result to three leading cases of preliminary estimators: Nonparametric, semiparametric and fully parametric ones. In all three cases, the asymptotic distribution of the solution is derived. We demonstrate the use of these results in obtaining confidence bands and standard errors for implied prices of bonds, options and other derivatives. Our general results also are of interest for the estimation of diffusion models using either historical data of the underlying process or option prices; these issues are also discussed.     

Riesz estimators

We consider properties of estimators that can be written as vector lattice (Riesz space) operations. Using techniques widely used in economic theory and functional analysis, we study the approximation properties of these estimators paying special attention to additive models. We also provide two algorithms RIESZVAR(i-ii) for the consistent parametric estimation of continuous multivariate piecewise linear functions.     

Quadratic Engel Curves with Measurement Error: Evidence from a Budget Survey*

This paper examines the importance of accounting for measurement error in total expenditure in the estimation of Engel curves, based on the 1994 Ethiopian Urban Household Survey. Using Lewbel's [Review of Economics and Statistics (1996 ), Vol. 78, pp. 718–725] estimator for demand models with correlated measurement errors in the dependent and independent variables, we find robust evidence of a quadratic relationship between food share and total expenditure in the capital city, and significant biases in various estimators that do not correct for correlated measurement errors.     

Nonparametric estimation of conditional distribution functions with longitudinal data and time-varying parametric models

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies. We propose in this paper an estimation approach based on time-varying parametric models. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model, but the parameters are smooth functions of time. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Asymptotic properties, including the asymptotic biases, variances and mean squared errors, are derived for the local polynomial smoothed estimators. Applicability of our two-step estimation method is demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through simulation study.     

Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models

Two classes of semiparametric diffusion models are considered, where either the drift or the diffusion term is parameterized, while the other term is left unspecified. We propose a pseudo-maximum likelihood estimator (PMLE) of the parametric component that maximizes the likelihood with a preliminary estimator of the unspecified term plugged in. It is demonstrated how models and estimators can be used in a two-step specification testing strategy of semiparametric and fully parametric models, and shown that approximate/simulated versions of the PMLE inherit the properties of the actual but infeasible estimator. A simulation study investigates the finite sample performance of the PMLE.     

A family of empirical likelihood functions and estimators for the binary response model

The minimum discrimination information principle is used to identify an appropriate parametric family of probability distributions and the corresponding maximum likelihood estimators for binary response models. Estimators in the family subsume the conventional logit model and form the basis for a set of parametric estimation alternatives with the usual asymptotic properties. Sampling experiments are used to assess finite sample performance.     

Matching and semi-parametric IV estimation, a distance-based measure of migration, and the wages of young men

Our paper estimates the effect of US internal migration on wage growth for young men between their first and second job. Our analysis of migration extends previous research by: (i) exploiting the distance-based measures of migration in the National Longitudinal Surveys of Youth 1979 (NLSY79); (ii) allowing the effect of migration to differ by schooling level and (iii) using propensity score matching to estimate the average treatment effect on the treated (ATET) for movers and (iv) using local average treatment effect (LATE) estimators with covariates to estimate the average treatment effect (ATE) and ATET for compliers.We believe the Conditional Independence Assumption (CIA) is reasonable for our matching estimators since the NLSY79 provides a relatively rich array of variables on which to match. Our matching methods are based on local linear, local cubic, and local linear ridge regressions. Local linear and local ridge regression matching produce relatively similar point estimates and standard errors, while local cubic regression matching badly over-fits the data and provides very noisy estimates.We use the bootstrap to calculate standard errors. Since the validity of the bootstrap has not been investigated for the matching estimators we use, and has been shown to be invalid for nearest neighbor matching estimators, we conduct a Monte Carlo study on the appropriateness of using the bootstrap to calculate standard errors for local linear regression matching. The data generating processes in our Monte Carlo study are relatively rich and calibrated to match our empirical models or to test the sensitivity of our results to the choice of parameter values. The estimated standard errors from the bootstrap are very close to those from the Monte Carlo experiments, which lends support to our using the bootstrap to calculate standard errors in our setting.From the matching estimators we find a significant positive effect of migration on the wage growth of college graduates, and a marginally significant negative effect for high school dropouts. We do not find any significant effects for other educational groups or for the overall sample. Our results are generally robust to changes in the model specification and changes in our distance-based measure of migration. We find that better data matters; if we use a measure of migration based on moving across county lines, we overstate the number of moves, while if we use a measure based on moving across state lines, we understate the number of moves. Further, using either the county or state measures leads to much less precise estimates.We also consider semi-parametric LATE estimators with covariates (Frölich 2007), using two sets of instrumental variables. We precisely estimate the proportion of compliers in our data, but because we have a small number of compliers, we cannot obtain precise LATE estimates.     

Asymptotic properties of the efficient estimators for cointegrating regression models with serially dependent errors

In this paper, we analytically investigate three efficient estimators for cointegrating regression models: Phillips and Hansen’s [Phillips, P.C.B., Hansen, B.E., 1990. Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125] fully modified OLS estimator, Park’s [Park, J.Y., 1992. Canonical cointegrating regressions. Econometrica 60, 119–143] canonical cointegrating regression estimator, and Saikkonen’s [Saikkonen, P., 1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] dynamic OLS estimator. We consider the case where the regression errors are moderately serially correlated and the AR coefficient in the regression errors approaches 1 at a rate slower than 1/T$1/T$, where T$T$ represents the sample size. We derive the limiting distributions of the efficient estimators under this system and find that they depend on the approaching rate of the AR coefficient. If the rate is slow enough, efficiency is established for the three estimators; however, if the approaching rate is relatively faster, the estimators will have the same limiting distribution as the OLS estimator. For the intermediate case, the second-order bias of the OLS estimator is partially eliminated by the efficient methods. This result explains why, in finite samples, the effect of the efficient methods diminishes as the serial correlation in the regression errors becomes stronger. We also propose to modify the existing efficient estimators in order to eliminate the second-order bias, which possibly remains in the efficient estimators. Using Monte Carlo simulations, we demonstrate that our modification is effective when the regression errors are moderately serially correlated and the simultaneous correlation is relatively strong.     

A fast resample method for parametric and semiparametric models

We propose a fast resample method for two step nonlinear parametric and semiparametric models, which does not require recomputation of the second stage estimator during each resample iteration. The fast resample method directly exploits the score function representations computed on each bootstrap sample, thereby reducing computational time considerably. This method is used to approximate the limit distribution of parametric and semiparametric estimators, possibly simulation based, that admit an asymptotic linear representation. Monte Carlo experiments demonstrate the desirable performance and vast improvement in the numerical speed of the fast bootstrap method.     

Nonparametric and semiparametric regressions subject to monotonicity constraints: Estimation and forecasting

This paper considers nonparametric and semiparametric regression models subject to monotonicity constraint. We use bagging as an alternative approach to Hall and Huang (2001). Asymptotic properties of our proposed estimators and forecasts are established. Monte Carlo simulation is conducted to show their finite sample performance. An application to predicting equity premium is taken for illustration. We introduce a new forecasting evaluation criterion based on the second order stochastic dominance in the size of forecast errors and compare models over different sizes of forecast errors. Imposing monotonicity constraint can mitigate the chance of making large size forecast errors.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

The Effects of Additive Outliers and Measurement Errors when Testing for Structural Breaks in Variance*

This article discusses the asymptotic and finite‐sample properties of the CUSUM tests for detecting structural breaks in volatility when the data are perturbed with (additive) outliers and/or measurement errors. The special focus is on the parametric and non‐parametric tests in Inclán and Tiao (1994) and Kokoszka and Leipus (2000) . Whereas the asymptotic distribution of the former can be largely affected, the distribution of the latter remains invariant and renders consistent break‐point estimates. In small samples, however, large additive outliers are able to generate sizeable distortions in both tests, which explains some of the contradictory findings in previous literature.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.     

The performance of estimators based on the propensity score

We investigate the finite sample properties of a large number of estimators for the average treatment effect on the treated that are suitable when adjustment for observed covariates is required, like inverse probability weighting, kernel and other variants of matching, as well as different parametric models. The simulation design used is based on real data usually employed for the evaluation of labour market programmes in Germany. We vary several dimensions of the design that are of practical importance, like sample size, the type of the outcome variable, and aspects of the selection process. We find that trimming individual observations with too much weight as well as the choice of tuning parameters are important for all estimators. A conclusion from our simulations is that a particular radius matching estimator combined with regression performs best overall, in particular when robustness to misspecifications of the propensity score and different types of outcome variables is considered an important property.     

Semiparametric and nonparametric estimation of sample selection models under symmetry

This paper considers the semiparametric estimation of binary choice sample selection models under a joint symmetry assumption. Our approaches overcome various drawbacks associated with existing estimators. In particular, our method provides root-n$n$ consistent estimators for both the intercept and slope parameters of the outcome equation in a heteroscedastic framework, without the usual cross equation exclusion restriction or parametric specification for the error distribution and/or the form of heteroscedasticity. Our two-step estimators are shown to be consistent and asymptotically normal. A Monte Carlo simulation study indicates the usefulness of our approaches.