<p> My dissertation consists of three essays which contribute new theoretical results to large panel data models with cross-sectional dependence. These essays try to answer or partially answer some prominent questions such as how to detect the presence of cross-sectional dependence and how to capture the latent structure of cross-sectional dependence and estimate parameters efficiently by removing its effects.</p><p> Chapter 2 introduces a nonparametric test for cross-sectional contemporaneous dependence in large dimensional panel data models based on the squared distance between the pair-wise joint density and the product of the marginals. The test can be applied to either raw observable data or residuals from local polynomial time series regressions for each individual to estimate the joint and marginal probability density functions of the error terms. In either case, we establish the asymptotic normality of our test statistic under the null hypothesis by permitting both the cross section dimension <i> n</i> and the time series dimension <i>T</i> to pass to infinity simultaneously and relying upon the Hoeffding decomposition of a two-fold <i> U</i>-statistic. We also establish the consistency of our test. A small set of Monte Carlo simulations is conducted to evaluate the finite sample performance of our test and compare it with that of Pesaran (2004) and Chen, Gao, and Li (2009).</p><p> Chapter 3 analyzes nonparametric dynamic panel data models with interactive fixed effects, where the predetermined regressors enter the models nonparametrically and the common factors enter the models linearly but with individual specific factor loadings. We consider the issues of estimation and specification testing when both the cross-sectional dimension <i>N</i> and the time dimension <i> T</i> are large. We propose sieve estimation for the nonparametric function by extending Bai's (2009) principal component analysis (PCA) to our nonparametric framework. Following Moon and Weidner's (2010, 2012) asymptotic expansion of the Gaussian quasilog-likelihood function, we derive the convergence rate for the sieve estimator and establish its asymptotic normality. The sources of asymptotic biases are discussed and a consistent bias-corrected estimator is provided. We also propose a consistent specification test for the linearity of the nonparametric functional form by comparing the linear and sieve estimators. We establish the asymptotic distributions of the test statistic under both the null hypothesis and a sequence of Pitman local alternatives.</p><p> To improve the finite sample performance of the test, we also propose a bootstrap procedure to obtain the bootstrap <i>p</i>-values and justify its validity. Monte Carlo simulations are conducted to investigate the finite sample performance of our estimator and test. We apply our model to an economic growth data set to study the relationship between capital accumulation and real GDP growth rate.</p><p> Chapter 4 proposes a nonparametric test for common trends in semiparametric panel data models with fixed effects based on a measure of nonparametric goodness-of-fit (<i>R</i>²). We first estimate the model under the null hypothesis of common trends by the method of profile least squares, and obtain the augmented residual which consistently estimates the sum of the fixed effect and the disturbance under the null.</p><p> Then we run a local linear regression of the augmented residuals on a time trend and calculate the nonparametric <i>R</i>² for each cross section unit. The proposed test statistic is obtained by averaging all cross sectional nonparametric <i>R</i>²'s, which is close to 0 under the null and deviates from 0 under the alternative. We show that after appropriate standardization the test statistic is asymptotically normally distributed under both the null hypothesis and a sequence of Pitman local alternatives. We prove test consistency and propose a bootstrap procedure to obtain <i>p</i>-values. Monte Carlo simulations indicate that the test performs well infinite samples. Empirical applications are conducted exploring the commonality of spatial trends in UK climate change data and idiosyncratic trends in OECD real GDP growth data. Both applications reveal the fragility of the widely adopted common trends assumption.</p>