Abstract

The subject of the dissertation concerns the problem of testing the conditional distribution in the GARCH time series models. The concept is based on the Neyman's idea to embed the hypothetical density into a k-parametric exponential family of densities. Thus, for the GARCH(1,1) case the score statistic and the data driven test statistic were derived. The GARCH(1,1) model parameters are the nuisance parameters in the testing problem. Firstly, the simple hypothesis was considered, i.e. when the hypothetical density was fully specified on the real line. Simulation study showed that the data driven goodness of tit test is sensitive against wide range of alternatives to the null distribution and competes well or even outperforms two other tests proposed in recent literature. In the sequel, the theoretical results were extended to the case of composite hypothesis, i.e. when the hypothetical density is represented by a parametric family of distributions. Main limit theorem establishing the asymptotic distribution of the test statistic was drawn and proved. The multi-stage proof is composed of several auxiliary lemmas. Most of technical difficulties in the proof is associated with a number of remainder terms resulting from replacing the nuisance parameters with their square root consistent estimators. With certain regularity assumtions imposed on the hypothetical density, the asymptotic distribution of the test statistic was proved to be the same after plugging the estimators of the nuisance parameters. The main theorem encompasses the previous case of the simple hypothesis, too. Therefore the problem of adaptive testing the conditional distribution in the GARCH(1,1) time series has been fully solved. Theoretical results were followed by numerical simulations of the test power for the case of GED hypothetical family. The simulation study confirmed high sensitivity of the test, which makes it a useful tool for practical purposes.