Abstract: This dissertation considers two issues related to the long-run variance estimation. These issues are concerned with small sample properties of estimators and tests which require a robust long-run variance estimator. In Chapter 2, I consider the size property of a t-test for a simple location model using an autoregressive (AR) spectral variance estimator when observations are generated by a linear process. A valid third-order Edgeworth expansion of such a test is established to find an error in rejecting the null hypothesis in finite samples. I show that under mild conditions, the size property can be significantly improved by using an AR spectral method in lieu of a wide class of kernel variance estimators. Chapter 3 studies generalized method of moments (GMM) and generalized empirical likelihood (CEL) for an over-identified model. When the kernel variance estimator is used to construct an optimal weight matrix for those estimators, I clarify that performances of estimators depend on choosing a smoothing parameter. In contrast to the common guideline minimizing estimation error of the long-run variance, that is, a weight matrix, I use performances of an estimator itself as criteria. In terms of the higher-order bias and mean squared error (MSE), optimal smoothing parameters for GMM and GEL are characterized. Implementing an optimal smoothing parameter for the higher-order MSE is a ISO discussed.