This dissertation consists of three chapters on econometrics. The first chapter, Are all firms inefficient? , is related to the stochastic frontier model of Aigner et al. (1977). In the usual stochastic frontier model, all firms are inefficient, because inefficiency is nonnegative and the probability that inefficiency is exactly zero equals zero. We modify this model by adding a parameter p which equals the probability that a firm is fully efficient. This model has also been considered by Kumbhakar et al. (2013). We extend their paper in several ways. We discuss some identification issues that arise if all firms are inefficient or no firms are inefficient. We show that the likelihood has a stationary point at parameters that indicate no inefficiency and that this point is a local maximum if the OLS residuals are positively skewed. We consider the case that a logit or probit model determines the probability of full efficiency in terms of some observable variables. Finally, we consider problems involved in testing the hypothesis that p = 0. We provide some simulations and an empirical example. The simulation results suggest that the proposed model appears to be useful when (i) it is reasonable to suppose that some firms are fully efficient, and (ii) the inefficiency levels of the inefficient firms are not small relative to statistical noise. The focus of the second and third chapters lies on asymptotic theory for test statistics in time series that are robust to heteroskedasticity and autocorrelation (HAC) especially under the fixed-b asymptotic framework proposed by Kiefer and Vogelsang (2005). In the second chapter, Serial Correlation Robust Inference with Missing Data, we investigate the properties of HAC robust test statistics when there is missing data. We characterize the time series with missing observations as amplitude modulated series following Parzen (1963). For estimation and inference this amounts to plugging in zeros for missing observations. We also investigate an alternative approach where the missing observations are simply ignored. There are three main theoretical findings. First, when the missing process is random and satisfies strong mixing conditions, HAC robust t and Wald statistics computed from the amplitude modulated series follow the usual fixed-b limits as in Kiefer and Vogelsang (2005). Second, when the missing process is non-random, the fixedb limits depend on the locations of missing observations but are otherwise pivotal. Third, when missing observations are ignored we obtain the surprising result that fixed-b limits of the robust t and Wald statistics have the standard fixed-b limits whether the missing process is random or non-random. We discuss methods for obtaining fixed-b critical values with a focus on bootstrap methods. We find that the naive i.i.d. bootstrap is the most effective and practical way to obtain fixed-b critical values when data is missing especially when the bootstrap conditions on the locations of the missing data. In the third chapter, Inference in time series models using smoothed clustered standard errors, we propose a long run variance estimator for conducting inference in time series regression models that combines the traditional nonparametric kernel approach with a cluster approach. The basic idea is to divide the time periods into non-overlapping clusters and construct the long run variance estimator by first aggregating within clusters and then kernel smoothing across clusters. We derive asymptotic results holding the number of clusters fixed and also treating the clusters as increasing with the sample size. We find that the "fixed number of clusters" asymptotic approximation works well whether the number of clusters (G) is small or large. Also, we find that the naive i.i.d. bootstrap mimics the fixed number of clusters critical values regardless of G. Finite sample simulations suggest that clustering before kernel smoothing can reduce over-rejections caused by strong serial correlation without a great cost in terms of power.
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