Three essays in financial economics

This dissertation consists of three essays. In the first essay, we extend the maximum likelihood estimation results of Ait-Sahalia (1998) for time-homogeneous diffusions to the time-inhomogeneous setup.{09}We derive a closed-form approximation of the likelihood function for discretely sampled time-inhomogeneous diffusions and prove that it converges to the true likelihood function and yields consistent parameter estimates. Monte Carlo simulations for several financial models reveal that our method largely outperforms other widely used numerical procedures in approximating the likelihood function, and produces parameter estimates which are very close to the true maximum likelihood estimators and superior to estimates obtained from the Euler approximation. A version of this essay has appeared in the Journal of Econometrics jointly with Alexei Egorov and Haitao Li.; In second essay, we characterize the dynamics of short-term interate rate using a Markov regime switching model. A formal statistical test for the number of regimes in such a model is developed and applied to the U.S. short-term interest rate data.{09}The test rejects the null hypothesis of a single regime model in favor of a two regime model, where in one regime the interest rate exhibits strong mean-reversion and high volatility and in another it behaves like a random walk with low volatility. We show that the sensitivity of volatility to the level of interest rate in the two regime model is much lower than 1.5, and a two regime analog of the classical CIR model fits the data well. These findings lends support nonlinear drift in the recent nonparametric literature are shown to be consistent with our two regime model.; In third essay, we assess the relevance of survival bias for empirical research in finance that uses historical data. As we show, the model of market survival in Brown, Goetzmann and Ross (1995) fails to generate significant survival bias in the equity premium. Using a simple alternative model, we demonstrate that in order to generate even a modest 2% of survival bias in stock returns, the probability for the market to survive for 100 years has to be as low as 1%. Given that no theory in the existing literature predicts that survival bias in the U.S. equity premium should be significant, we believe that the current concerns for survival bias are probably without grounds. An extented version of this paper has been published in the Journal of Finance, jointly with Haitao Li.