Diagnostic Tests Based on Quantile Residuals for Nonlinear Time Series Models

This thesis studies quantile residuals and uses different methodologies to develop test statistics that are applicable in evaluating linear and nonlinear time series models based on continuous distributions. Models based on mixtures of distributions are of special interest because it turns out that for those models traditional residuals, often referred to as Pearson's residuals, are not appropriate. As such models have become more and more popular in practice, especially with financial time series data there is a need for reliable diagnostic tools that can be used to evaluate them. The aim of the thesis is to show how such diagnostic tools can be obtained and used in model evaluation. The quantile residuals considered here are defined in such a way that, when the model is correctly specified and its parameters are consistently estimated, they are approximately independent with standard normal distribution. All the tests derived in the thesis are pure significance type tests and are theoretically sound in that they properly take the uncertainty caused by parameter estimation into account. -- In Chapter 2 a general framework based on the likelihood function and smooth functions of univariate quantile residuals is derived that can be used to obtain misspecification tests for various purposes. Three easy-to-use tests aimed at detecting non-normality, autocorrelation, and conditional heteroscedasticity in quantile residuals are formulated. It also turns out that these tests can be interpreted as Lagrange Multiplier or score tests so that they are asymptotically optimal against local alternatives. Chapter 3 extends the concept of quantile residuals to multivariate models. The framework of Chapter 2 is generalized and tests aimed at detecting non-normality, serial correlation, and conditional heteroscedasticity in multivariate quantile residuals are derived based on it. Score test interpretations are obtained for the serial correlation and conditional heteroscedasticity tests and in a rather restricted special case for the normality test. In Chapter 4 the tests are constructed using the empirical distribution function of quantile residuals. So-called Khmaladze s martingale transformation is applied in order to eliminate the uncertainty caused by parameter estimation. Various test statistics are considered so that critical bounds for histogram type plots as well as Quantile-Quantile and Probability-Probability type plots of quantile residuals are obtained. Chapters 2, 3, and 4 contain simulations and empirical examples which illustrate the finite sample size and power properties of the derived tests and also how the tests and related graphical tools based on residuals are applied in practice.

Fertility in Namibia : Changes in fertility levels in North-Central Namibia 1960-2001, including an assessment of the impact of HIV

The aim of this study was to estimate the development of fertility in North-Central Namibia, former Ovamboland, from 1960 to 2001. Special attention was given to the onset of fertility decline and to the impact of the HIV epidemic on fertility. An additional aim was to introduce parish registers as a source of data for fertility research in Africa. Data used consisted of parish registers from Evangelical Lutheran congregations, the 1991 and 2001 Population and Housing Censuses, the 1992 and 2000 Namibia Demographic and Health Surveys, and the HIV sentinel surveillances of 1992-2004. Both period and cohort fertility were analysed. The P/F ratio method was used when analysing census data. The impact of HIV infection on fertility was estimated indirectly by comparing the fertility histories of women who died at an age of less than 50 years with the fertility of other women. The impact of the HIV epidemic on fertility was assessed both among infected women and in the general population. Fertility in the study population began to decline in 1980. The decline was rapid during the 1980s, levelled off in the early 1990s at the end of war of independence and then continued to decline until the end of the study period. According to parish registers, total fertility was 6.4 in the 1960s and 6.5 in the 1970s, and declined to 5.1 in the 1980s and 4.2 in the 1990s. Adjustment of these total fertility rates to correspond to levels of fertility based on data from the 1991 and 2001 censuses resulted in total fertility declining from 7.6 in 1960-79 to 6.0 in 1980-89, and to 4.9 in 1990-99. The decline was associated with increased age at first marriage, declining marital fertility and increasing premarital fertility. Fertility among adolescents increased, whereas the fertility of women in all other age groups declined. During the 1980s, the war of independence contributed to declining fertility through spousal separation and delayed marriages. Contraception has been employed in the study region since the 1980s, but in the early 1990s, use of contraceptives was still so limited that fertility was higher in North-Central Namibia than in other regions of the country. In the 1990s, fertility decline was largely a result of the increased prevalence of contraception. HIV prevalence among pregnant women increased from 4% in 1992 to 25% in 2001. In 2001, total fertility among HIV-infected women (3.7) was lower than that among other women (4.8), resulting in total fertility of 4.4 among the general population in 2001. The HIV epidemic explained more than a quarter of the decline in total fertility at population level during most of the 1990s. The HIV epidemic also reduced the number of children born by reducing the number of potential mothers. In the future, HIV will have an extensive influence on both the size and age structure of the Namibian population. Although HIV influences demographic development through both fertility and mortality, the effect through changes in fertility will be smaller than the effect through mortality. In the study region, as in some other regions of southern Africa, a new type of demographic transition is under way, one in which population growth stagnates or even reverses because of the combined effects of declining fertility and increasing mortality, both of which are consequences of the HIV pandemic.