Essays in long memory and stock market volatility

This document consists of three essays in econometrics, one theoretical and two applied.; In the first essay, I study the asymptotics of the nonstationary fractional integrated time series, the long memory time series with {dollar}dge{lcub}1over2{rcub}{dollar}. There is considerable empirical evidence showing long memory of this magnitude in a lot of nominal economics time series including inflation rate and stock market volatility, a study of the large sample property is therefore needed and useful. Also, I found the asymptotics of nonstationary fractional integrated time series useful in the study of the large sample theory of the KPSS test.; In the second essay, I conduct an empirical investigation of the long memory in stock market volatility with a large collection of NYSE stocks. I first confirm the existence of long memory in stock market volatility in all of the individual stocks. I then investigate the issue of estimating the long memory magnitude in the stock market volatility. In general I find that GPH (Geweke and Porter-Hudak (1983)) and even FXTQ (Fox-Taqqu frequency domain QMLE) with finite ARFIMA model tend to underestimate the magnitude of long memory. In order to get a more accurate estimation of the long memory coefficient, I propose to decompose the original series into the long memory part and the short memory part according to the idea of Inclan and Tiao (1994) and Liu (1995) and estimate the long memory magnitude from the long memory part. The empirical result of this paper can be summarized as (1) Long memory in stock market volatility has a typical magnitude around or above 0.40 and the magnitude of long memory seems clustered according to industry. (2) Long memory can be successfully filtered out if we take proper care of the regime switching in the stock market volatility.; Inspired by the idea that regime switching could give rise to persistence observationally equivalent to a unit root, in the third essay, I propose that persistence of the form long memory might be given rise by regime switching. This idea is adapted to the stochastic volatility model, as a result, the Regime Switching Stochastic Volatility (RSSV) model proposed in this essay will generate long memory in the volatility of stock prices. The simulation-based Gallant-Tauchen EMM framework is employed to evaluate the empirical relevance of this model. In the case of stock prices of several individual companies, I find significant persistence of the regime and I cannot reject either that the observed dynamics of stock prices is actually coming from the RSSV model or that the observed long memory is related to the underlying regime switching.