There are four essays on productivity and Panel data econometrics in this dissertation, with the first two essays on empirical research and the last two more focused on theory improvement. The first chapter is study of productivity and efficiency in the Mexican Energy Industry. The second chapter analyzes the productivity and efficiency of U.S. largest banks productivity and efficiency. The third incorporates a Bayesian treatment to two different panel data models. The last chapter introduces a semi-nonparametric method in panel data models. These four chapters have been developed into four working papers. They are Liu et al. (2011), Inanoglu et al. (2012), Liu et al. (2013) and Liu et al. (2014). The first chapter studies the optimizing behavior of Pemex by estimating a cost model of Pemex's production of energy. The estimation using duality between the cost and production function is undertaken, which facilitates the specification. This approach makes it convenient to find the cost shares under different levels of returns to scale. The results indicate the presence of substantial distortions in cost shares. That would be brought back to equilibrium were the Mexican government willing to allow more foreign investment in its energy extraction industry and thus increase the capital use and decrease the labor use. The second chapter utilizes a suite of panel data models in order to examine the extent to which scale economy and efficiencies exist in the largest U.S. banks. The empirical results are assessed based on the consensus among the findings from the various econometric treatments and models. This empirical study is based on a newly developed dataset based on Call Reports from the FDIC for the period 1994-2013. The analyses point to a number of conclusions. First, despite rapid growth over the last 20 years, the largest surviving banks in the U.S. have decreased their level of efficiency as they took on increasing levels of risk (credit, market and liquidity). Second, no measurable scale economies and scope economies are found across our host of models and econometric treatments. In addition to the broad policy implications, this essay also provides an array of econometric techniques, findings from which can be combined to provide a set of robust consensus-based conclusions that can be a valuable analytical tool for supervisors and others involved in the regulatory oversight of financial institutions. The third chapter considers two models for uncovering information about technical change in large heterogeneous panels. The first is a panel data model with nonparametric time effects. The second is a panel data model with common factors whose number is unknown and whose effects are firm-specific. This chapter proposes a Bayesian approach to estimate the two models. Bayesian inference techniques organized around MCMC are applied to implement the models. Monte Carlo experiments are performed to examine the finite-sample performance of this approach and have shown that the method proposed is comparable to the recently proposed estimator of Kneip et al. (2012) (KSS) and dominates a variety of estimators that rely on parametric assumptions. In order to illustrate the new method, the Bayesian approach has been applied to the analysis of efficiency trends in the U.S. largest banks using a dataset based on the Call Report data from FDIC over the period from 1990 to 2009. The fourth chapter introduces a new estimation method under the framework of the stochastic frontier production model. The noise term is assumed to have the traditional normal density but the inefficiency term is spanned by Laguerre Polynomials. This method is a Semi-nonparametric method and follows the spirit of Gallant and Nychka (1987). Finite sample performance of this estimator is shown to dominate the nonparametric estimators via Monte Carlo simulations.