Essays on optimal tests

In the following chapters optimal tests in the Markov Switching framework are proposed and discussed.; The first chapter proposes an optimal test for the null hypothesis of parameter constancy against the alternative where the parameters vary according to an unobserved Markov chain. Our test only requires estimating the model under the null hypothesis, which has great computational advantage over competing tests. Local alternatives are discussed and asymptotic optimality is derived. Monte Carlo studies show that the proposal test has a good finite sample performance. In an application to US stock prices and dividends, our test successfully detects the asymmetry in stock markets.; Stationary data under the null hypothesis are assumed in the first chapter. In the second chapter, we look at nonstationary data. We test the null hypothesis of unit root against the alternative of stochastic unit root with Markov Switching. The test statistic is of different form. Order of the local alternatives and asymptotic distribution of the test statistic are also different due to the nonstationary property of the data. Small sample performance is investigated. When applying the test to the well known US macroeconomic datasets, unit root hypotheses are rejected in favor of Markov Switching alternatives.; Empirically, a routine finding is that the estimated sum of GARCH parameters is close to unity, suggesting that the conditional volatility of most stock return data are likely to follow an integrated GARCH (IGARCH) process. However, such an extremely high persistence in the unconditional variance may be overstated because of neglected structural breaks or parameter changes in GARCH models. As a result it is important to distinguish between these two processes, one being a globally stationary regime-switching process and the other being a nonstationary IGARCH process. The third chapter addresses these issues by applying our general testing procedure. Monte Carlo studies show that our tests have a good finite sample performance. In an application to stock return data, we find strong evidence in favor of MS-GARCH models. This result also supports a view that neglecting the parameter changes would result in the spurious finding of IGARCH.