Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the _L2$L_2$ distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.     

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.     

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.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

The effects of dynamic feedbacks on LS and MM estimator accuracy in panel data models

The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined.     

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

In this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.     

Binary choice models with discrete regressors: Identification and misspecification

This paper explores the inferential question in semiparametric binary response models when the continuous support condition is not satisfied and all regressors have discrete support. I focus mainly on the models under the conditional median restriction, as in Manski (1985). I find sharp bounds on the components of the parameter of interest and outline several applications. The formulas for bounds obtained using a recursive procedure help analyze cases where one regressor’s support becomes increasingly dense. Furthermore, I investigate asymptotic properties of estimators of the identification set. I describe a relation between the maximum score estimation and support vector machines and propose several approaches to address the problem of empty identification sets when the model is misspecified.     

Semiparametric robust estimation of truncated and censored regression models

Many estimation methods of truncated and censored regression models such as the maximum likelihood and symmetrically censored least squares (SCLS) are sensitive to outliers and data contamination as we document. Therefore, we propose a semiparametric general trimmed estimator (GTE) of truncated and censored regression, which is highly robust but relatively imprecise. To improve its performance, we also propose data-adaptive and one-step trimmed estimators. We derive the robust and asymptotic properties of all proposed estimators and show that the one-step estimators (e.g., one-step SCLS) are as robust as GTE and are asymptotically equivalent to the original estimator (e.g., SCLS). The finite-sample properties of existing and proposed estimators are studied by means of Monte Carlo simulations.     

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.     

Regression models with mixed sampling frequencies

We study regression models that involve data sampled at different frequencies. We derive the asymptotic properties of the NLS estimators of such regression models and compare them with the LS estimators of a traditional model that involves aggregating or equally weighting data to estimate a model at the same sampling frequency. In addition we propose new tests to examine the null hypothesis of equal weights in aggregating time series in a regression model. We explore the above theoretical aspects and verify them via an extensive Monte Carlo simulation study and an empirical application.     

Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

Simplified Implementation of the Heckman Estimator of the Dynamic Probit Model and a Comparison with Alternative Estimators*

This paper presents a convenient shortcut method for implementing the Heckman estimator of the dynamic random effects probit model and other dynamic nonlinear panel data models using standard software. It then compares the estimators proposed by Heckman, Orme and Wooldridge, based on three alternative approximations, first in an empirical model for the probability of unemployment and then in a set of simulation experiments. The results indicate that none of the three estimators dominates the other two in all cases. In most cases, all three estimators display satisfactory performance, except when the number of time periods is very small.     

Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

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.     

An integrated maximum score estimator for a generalized censored quantile regression model

Quantile regression techniques have been widely used in empirical economics. In this paper, we consider the estimation of a generalized quantile regression model when data are subject to fixed or random censoring. Through a discretization technique, we transform the censored regression model into a sequence of binary choice models and further propose an integrated smoothed maximum score estimator by combining individual binary choice models, following the insights of Horowitz (1992) and Manski (1985). Unlike the estimators of Horowitz (1992) and Manski (1985), our estimators converge at the usual parametric rate through an integration process. In the case of fixed censoring, our approach overcomes a major drawback of existing approaches associated with the curse-of-dimensionality problem. Our approach for the fixed censored case can be extended readily to the case with random censoring for which other existing approaches are no longer applicable. Both of our estimators are consistent and asymptotically normal. A simulation study demonstrates that our estimators perform well in finite samples.     

A theory of robust long-run variance estimation

Long-run variance estimation can typically be viewed as the problem of estimating the scale of a limiting continuous time Gaussian process on the unit interval. A natural benchmark model is given by a sample that consists of equally spaced observations of this limiting process. The paper analyzes the asymptotic robustness of long-run variance estimators to contaminations of this benchmark model. It is shown that any equivariant long-run variance estimator that is consistent in the benchmark model is highly fragile: there always exists a sequence of contaminated models with the same limiting behavior as the benchmark model for which the estimator converges in probability to an arbitrary positive value. A class of robust inconsistent long-run variance estimators is derived that optimally trades off asymptotic variance in the benchmark model against the largest asymptotic bias in a specific set of contaminated models.     

Information criteria for impulse response function matching estimation of DSGE models

We propose new information criteria for impulse response function matching estimators (IRFMEs). These estimators yield sampling distributions of the structural parameters of dynamic stochastic general equilibrium (DSGE) models by minimizing the distance between sample and theoretical impulse responses. First, we propose an information criterion to select only the responses that produce consistent estimates of the true but unknown structural parameters: the Valid Impulse Response Selection Criterion (VIRSC). The criterion is especially useful for mis-specified models. Second, we propose a criterion to select the impulse responses that are most informative about DSGE model parameters: the Relevant Impulse Response Selection Criterion (RIRSC). These criteria can be used in combination to select the subset of valid impulse response functions with minimal dimension that yields asymptotically efficient estimators. The criteria are general enough to apply to impulse responses estimated by VARs, local projections, and simulation methods. We show that the use of our criteria significantly affects estimates and inference about key parameters of two well-known new Keynesian DSGE models. Monte Carlo evidence indicates that the criteria yield gains in terms of finite sample bias as well as offering tests statistics whose behavior is better approximated by the first order asymptotic theory. Thus, our criteria improve existing methods used to implement IRFMEs.     

Indirect estimation of large conditionally heteroskedastic factor models,with an application to the Dow 30 stocks

We derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a Kalman-filter approximation with inequality constraints on the auxiliary model parameters. We also propose alternative indirect estimators for large-scale models, and explain how to apply our procedures to many other dynamic latent variable models. We analyse the small sample behaviour of our indirect estimators and several likelihood-based procedures through an extensive Monte Carlo experiment with empirically realistic designs. Finally, we apply our procedures to weekly returns on the Dow 30 stocks.     

Detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility

In this paper, we propose two estimators, an integral estimator and a discretized estimator, for the wavelet coefficient of regression functions in nonparametric regression models with heteroscedastic variance. These estimators can be used to test the jumps of the regression function. The model allows for lagged-dependent variables and other mixing regressors. The asymptotic distributions of the statistics are established, and the asymptotic critical values are analytically obtained from the asymptotic distribution. We also use the test to determine consistent estimators for the locations of change points. The jump sizes and locations of change points can be consistently estimated using wavelet coefficients, and the convergency rates of these estimators are derived. We perform some Monte Carlo simulations to check the powers and sizes of the test statistics. Finally, we give practical examples in finance and economics to detect changes in stock returns and short-term interest rates using the empirical wavelet method.     

Estimation with overidentifying inequality moment conditions

This paper derives limit distributions of empirical likelihood estimators for models in which inequality moment conditions provide overidentifying information. We show that the use of this information leads to a reduction of the asymptotic mean-squared estimation error and propose asymptotically uniformly valid tests and confidence sets for the parameters of interest. While inequality moment conditions arise in many important economic models, we use a dynamic macroeconomic model as a data generating process and illustrate our methods with instrumental variable estimators of monetary policy rules. The results obtained in this paper extend to conventional GMM estimators.