Most of the literature on Value at Risk concentrates on the unconditional nonparametric or parametric approach to VaR estimation and much less on the direct modeling of conditional quantiles. This thesis focuses on the direct conditional VaR modeling, using the flexible quantile regression and hence imposing no restrictions on the return distribution. We apply semiparamet- ric Conditional Autoregressive Value at Risk (CAViaR) models that allow time-variation of the conditional distribution of returns and also different time-variation for different quantiles on four stock price indices: Czech PX, Hungarian BUX, German DAX and U.S. S&P 500. The objective is to inves- tigate how the introduction of dynamics impacts VaR accuracy. The main contribution lies firstly in the primary application of this approach on Cen- tral European stock market and secondly in the fact that we investigate the impact on VaR accuracy during the pre-crisis period and also the period covering the global financial crisis. Our results show that CAViaR models perform very well in describing the evolution of the quantiles, both in abso- lute terms and relative to the benchmark parametric models. Not only do they provide generally a better fit, they are also able to produce accurate forecasts. CAViaR models may be therefore used as a...

This research develops and applies nonparametric estimation tools in two sectors of interest of financial econometrics: risk management and market microstructure. In the first part we address the problem of estimating conditional quantiles in financial and economic time series. Research in this field received great impulse since quantile based risk measures such as Value at Risk (VaR) have become essential tools to assess the riskiness of trading activities. The great amounts of data available in financial time series allows building nonparametric estimators that are not subject to the risk of specification error of parametric models. A wavelet based estimator is developed. With this approach, minimum regularity conditions of the underlying process are required. Moreover the specific choice of the wavelets in this work leads to the constructions of shape preserving estimators of probability functions. In other words, estimates of probability functions, both densities and cumulative distribution functions, are probability functions themselves. This method is compared with competing methods through simulations and applications to real data. In the second part we carry out a nonparametric analysis of financial durations, that is of the waiting times between particular financial events, such as trades, quote updates, volume accumulation, that happen in financial markets. These data display very peculiar stylized facts one has to take into account when attempting to model them. We make use of an existing algorithm to describe nonparametrically the dynamics of the process in terms of its lagged realizations and of a latent variable, its conditional mean. The estimation devices needed to effectively apply the algorithm to our dataset are presented in this part of the work.

Quantile regression, a method used to estimate conditional quantiles of a set of data ( X, Y ), was popularized by Koenker and Bassett (1978). For a particular quantile q , the q th quantile estimate of Y given X = x can be found using an asymmetrically-weighted, absolute-loss criteria. This form of regression is considered to be robust, in that it is less affected by outliers in the data set than least-squares regression. However, like standard L 1 regression, this form of quantile regression can still be affected by multiple outliers. In this thesis, we propose a method for improving robustness in quantile regression through an application of Scott's L 2 Estimation (2001). Theoretic and asymptotic results are presented and used to estimate properties of our method. Along with simple linear regression, semiparametric extensions are examined. To verify our method and its extensions, simulated results are considered. Real data sets are also considered, including estimating the effect of various factors on the conditional quantiles of child birth weight, using semiparametric quantile regression to analyze the relationship between age and personal income, and assessing the value distributions of Major League Baseball players.

Correctly specified models to forecast returns of indices are important for in- vestors to minimize risk on financial markets. This thesis focuses on conditional Value at Risk modeling, employing flexible quantile regression framework and hence avoiding the assumption on the return distribution. We apply semi- parametric linear quantile regression (LQR) models with realized variance and also models with positive and negative semivariance which allows for direct modelling of the quantiles. Four European stock price indices are taken into account: Czech PX, Hungarian BUX, German DAX and London FTSE 100. The objective is to investigate how the use of realized variance influence the VaR accuracy and the correlation between the Central & Eastern and Western European indices. The main contribution is application of the LQR models for modelling of conditional quantiles and comparison of the correlation between European indices with use of the realized measures. Our results show that linear quantile regression models on one-step-ahead forecast provide better fit and more accurate modelling than classical VaR model with assumption of nor- mally distributed returns. Therefore LQR models with realized variance can be used as accurate tool for investors. Moreover we show that diversification benefits are...

This thesis investigates forecasting performance of Quantile Regression Neural Networks in forecasting multiperiod quantiles of realized volatility and quantiles of returns. It relies on model-free measures of realized variance and its components (realized variance, median realized variance, integrated variance, jump variation and positive and negative semivariances). The data used are S&P 500 futures and WTI Crude Oil futures contracts. Resulting models of returns and volatility have good absolute performance and relative performance in comparison to the linear quantile regression models. In the case of in- sample the models estimated by Quantile Regression Neural Networks provide better estimates than linear quantile regression models and in the case of out-of-sample they are equally good.