Essays on Weak Instrument Regression

This dissertation looks at the problem with weak instrument regression. Chapter 1 derives the optimal rotational invariant similar tests for weak instrument regression. The model is based on the existing theories on optimal test for weak instrument regression with some novelty. I use the Bayesian framework and assign equal weights on the strengths of the instrument variables. So relatively speak small values receive more weights and thus the test perform better for extremely weak instrument. Chapter 2 considers the problem of consistent estimation for weak instrument regression. In particular, I derive the necessary and sufficient condition for the existence of consistent estimator when the each individual instrument is weak but the number of instruments, satisfying certain conditions, goes to infinity. Chapter 3 is focused on the estimator of intraday evolution of realized volatility. We use multiple filtration so that the influence of microstructure noise is mitigated to a large extent and the resulting estimator for intraday realized volatilities is shown to be consistent