Chapter 1 is "Heteroskedasticity and Spatiotemporal Dependence Robust Inference for Linear Panel Models with Fixed Effects." This chapter studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator, which is flexible to nest existing estimators as special cases with certain choices of bandwidths. For distributional approximations, we consider two different types of asymptotics. When the level of smoothing is assumed to increase with the sample size, the proposed estimator is consistent and the associated Wald statistic converges to a chi square distribution. We show that our covariance estimator improves upon existing estimators in terms of robustness and efficiency. When we assume the level of smoothing to be held fixed, the covariance estimator has a random limit and we show by asymptotic expansion that the limiting distribution of the test statistic depends on the bandwidth parameters, the kernel function, and the number of restrictions being tested. As this distribution is nonstandard, we establish the validity of an F-approximation to this distribution, which greatly facilitates the test. For optimal bandwidth selection, we propose a procedure based on the upper bound of asymptotic mean square error criterion. The flexibility of our estimator and proposed bandwidth selection procedure make our estimator adaptive to the dependence structure in data. This adaptiveness automates the selection of covariance estimator. That is, our estimator reduces to the existing estimators which are designed to cope with the particular dependence structures. Simulation results show that the F-approximation and the adaptiveness work reasonably well. Chapter 2 is "Spatial Heteroskedasticity and Autocorrelation Consistent Estimation of Covariance Matrix". This chapter considers spatial heteroskedasticity and autocorrelation consistent (spatial HAC) estimation of covariance matrices of parameter estimators. We generalize the spatial HAC estimators introduced by Kelejian and Prucha (2007) to apply to linear and nonlinear spatial models with moment conditions. We establish its consistency, rate of convergence and asymptotic truncated mean squared error (MSE). Based on the asymptotic truncated MSE criterion, we derive the optimal bandwidth parameter and suggest its data dependent estimation procedure using a parametric plug-in method. The finite sample performances of the spatial HAC estimator are evaluated via Monte Carlo simulation. Chapter 3 is "k-step Bootstrap Bias Correction for Fixed Effects Estimator in Nonlinear Panel Models." Fixed effects estimators in nonlinear panel models with fixed T usually suffer from inconsistency because of the incidental parameters problem first noted by Neyman and Scott (1948). Moreover, even though T grows at the same rate as n, they are asymptotically biased and therefore the associated confidence interval has a large coverage error. This chapter proposes a k-step parametric bootstrap bias corrected estimator. We prove that our estimator is asymptotically normal and is centered at the true parameter if T grows faster than n to a third power. In addition to bias correction, we construct a confidence interval with a double bootstrap procedure, and Monte Carlo experiments confirm that the error in coverage probability of our CI's is smaller than those of the alternatives. We also propose bias correction for average marginal effects

Fixed effects estimators in nonlinear panel models with fixed and short time series length T usually suffer from inconsistency because of the incidental parameters problem first noted by Neyman and Scott (1948). Moreover, even if T grows but at a rate not faster than the cross sectional sample size n, they are asymptotically biased, and therefore the associated confidence intervals have a large coverage error. This paper analyzes the properties of the parametric bootstrap bias corrected maximum likelihood (ML) estimators of nonlinear panel models with fixed effects. We assume that each time series follows a finite order Markov process. We show that the bootstrap bias corrected estimators are asymptotically normal and centered at the true parameter. In particular, we propose using the k-step parametric bootstrap procedure to alleviate the computational cost of implementing the standard bootstrap. We also apply the standard and k-step bootstrap bias correction to average marginal effect estimation and to the double bootstrap for confidence interval construction. Our Monte Carlo simulations show that the k-step bootstrap bias corrected estimator reduces the bias remarkably well in finite samples.

This paper develops robust testing procedures for nonparametric kernel methods in the presence of temporal dependence of unknown forms. Based on the fixed-bandwidth asymptotic variance and the pre-asymptotic variance, we propose a heteroskedasticity and autocorrelation robust (HAR) variance estimator that achieves double robustness --- it is asymptotically valid regardless of whether the temporal dependence is present or not, and whether the kernel smoothing bandwidth is held constant or allowed to decay with the sample size. Using the HAR variance estimator, we construct the studentized test statistic and examine its asymptotic properties under both the fixed-smoothing and increasing-smoothing asymptotics. The fixed-smoothing approximation and the associated convenient t-approximation achieve extra robustness --- it is asymptotically valid regardless of whether the truncation lag parameter governing the covariance weighting grows at the same rate as or a slower rate than the sample size. Finally, we suggest a simulation-based calibration approach to choose smoothing parameters that optimize testing oriented criteria. Simulation shows that the proposed procedures work very well in finite samples.