This dissertation deals with issues of forecasting in financial markets. The first part of my dissertation is motivated by the observation that most parametric volatility models follow Engle's (1982) original idea of modelling the volatility of asset returns as a function of only past information. However, current returns are potentially quite informative for forecasting, yet are excluded from these models. The first and second chapters of this dissertation try to address this question from both a theoretical and an empirical perspective. The second part of this dissertation deals with the important issue of forecast evaluation and selection in unstable environments, where it is known that the existing methodology can generate spurious and potentially misleading results. In my third chapter, I develop a new methodology for forecast evaluation and selection in such an environment.

In the first chapter, $Real-time\; GARCH$, I propose a new parametric volatility model, which retains the simple structure of GARCH models, but models the volatility process as a mixture of past and current information as in the spirit of Stochastic Volatility (SV) models. This provides therefore a link between GARCH and SV models. I show that with this new model I am able to obtain better volatility forecasts than the standard GARCH-type models; improve the empirical fit of the data, especially in the tails of the distribution; and make the model faster in its adjustment to the new unconditional level of volatility. Further, the new model offers a much needed framework for specification testing as it nests the standard GARCH models. This chapter has been published in the $Journal\; of\; Financial\; Econometrics$ (Smetanina E., 2017, Real-time GARCH, $Journal\; of\; Financial\; Econometrics$, 15(4), 561-601.)

In chapter 2, $Asymptotic\; Inference\; for\; Real-time\; GARCH(1,1)\; model$, I investigate the asymptotic properties of the Gaussian Quasi-Maximum-Likelihood estimator (QMLE) for the Real-time GARCH(1,1) model, developed in the first chapter of this dissertation. I establish the ergodicity and $\beta$-mixing properties of the joint process for squared returns and the volatility process. I also prove strong consistency and asymptotic normality for the parameter vector at the usual $T$ rate. Finally, I demonstrate how the developed theory can be viewed as a generalisation of the QMLE theory for the standard GARCH(1,1) model.

In chapter 3, $Forecast\; Evaluation\; Tests\; in\; Unstable\; Environments$, I develop a new methodology for forecast evaluation and selection in the situations where the relative performance between models changes over time in an unknown fashion. Out-of-sample tests are widely used for evaluating models’ forecasts in economics and finance. Underlying these tests is often the assumption of constant relative performance between competing models, however this is invalid for many practical applications. In a world of changing relative performance, previous methodologies give rise to spurious and potentially misleading results, an example of which is the well-known ``splitting point problem''. I propose a new two-step methodology designed specifically for forecast evaluation in a world of changing relative performance. In the first step I estimate the time-varying mean and variance of the series for forecast loss differences, and in the second step I use these estimates to construct new rankings for models in a changing world. I show that the new tests have high power against a variety of fixed and local alternatives.