Learning mixture models via component-wise parameter smoothing

The task of obtaining an optimal set of parameters to fit a mixture model has many applications in science and engineering domains and is a computationally challenging problem. A novel algorithm using a convolution based smoothing approach to construct a hierarchy (or family) of smoothed log-likelihood surfaces is proposed. This approach smooths the likelihood function and applies the EM algorithm to obtain a promising solution on the smoothed surface. Using the most promising solutions as initial guesses, the EM algorithm is applied again on the original likelihood. Though the results are demonstrated using only two levels, the method can potentially be applied to any number of levels in the hierarchy. A theoretical insight demonstrates that the smoothing approach indeed reduces the overall gradient of a modified version of the likelihood surface. This optimization procedure effectively eliminates extensive searching in non-promising regions of the parameter space. Results on some benchmark datasets demonstrate significant improvements of the proposed algorithm compared to other approaches. Empirical results on the reduction in the number of local maxima and improvements in the initialization procedures are provided.

Information Asymmetries: Three Essays in Market Microstructure

This dissertation aims at revisiting existing empirical market microstructure models for the measurement of information asymmetries on financial markets and to develop alternative estimation approaches that (1) reduce existing bias, (2) reduce data requirement, and (3) increase the applicability of these models. The first chapter of this thesis reconsiderates trade indicator models and is a joint work with Erik Theissen. More precisely, it reconsiderates the models of Madhavan Richardson, and Roomans (1997) and Huang and Stoll (1997). Trade indicator models divide the spread into an adverse selection component and remaining components. As a byproduct an estimate of the spread becomes available. It is a stylized fact that trade indicator models (e.g. Madhavan, Richardson, and Roomans (1997) and Huang and Stoll (1997)) underestimate the bid-ask spread. We argue that this negative bias is due to an endogeneity problem, which is caused by a negative correlation between the arrival of public information and trade direction. In our sample (the component stocks of the DAX 30 index) we find that the average correlation between these variables is -0.193. We develop modified estimators and show that they yield essentially unbiased spread estimates. The second and third chapters of this thesis build an entity and consider another way to measure information asymmetries on financial markets. In the second chapter, a joint work with Joachim Grammig, and Erik Theissen, we propose a methodology to estimate the probability of informed trading (PIN) that only requires data on the daily number of transactions (but not on the number of buyer-initiated and seller-initiated trades). Because maximum likelihood estimation of the model is problematic we propose a Bayesian estimation approach. We perform extensive simulations to evaluate the performance of our estimator. Our methodology increases the applicability of PIN estimation to situations in which the data necessary for trade classification is unavailable, or in which trade classification is inaccurate. The third chapter investigates information asymmetries in U.S. corporate bond markets using transaction data from the Trade Reporting and Compliance Engine (TRACE) for constituents of the S&P 500 in the first half-year of 2011. As a measurement of information asymmetry I employ the probability of informed trading (PIN) proposed by Easley, Kiefer, O’Hara, and Paperman (1996). In a cross-sectional regression of 4,155 fixed-income securities on bond characteristics, market variables, and stock statistics, I find that nearly 50% of the variation in PINs is explained. All estimated coefficients conform to expectations. While a comparison of PINs in bond and corresponding equity markets confirms prior findings of lower PINs on more active stock markets, it indicates the reverse for fixed-income securities: Less-frequently traded bonds exhibit lower PINs. These findings accord with there being lower transaction costs on less active bond markets as found by Goldstein, Hotchkiss, and Sirri (2007). However, as news probabilities for bonds from the same issuer and bonds and corresponding stocks differ significantly, I question the appropriateness of traditional models for measuring information asymmetries. The probability of informed trading might not be a suitable measure for highly fragmented markets such as the U.S. corporate bond market