This thesis concerns the asymptotic behaviour of the sequence of 1-step recursive Chow statistics and various tests derived therefrom. The 1-step statistics are produced as diagnostic output in standard econometrics software, and are expected to reflect model misspecification. Such misspecification testing is important in validating the assumptions of a model and so ensuring that subsequent inference is correct. Original contributions to the theory of misspecification testing include (i) a result on the pointwise convergence of the 1-step statistics; (ii) a result on the extreme-value convergence of the maximum of the statistics; and (iii) a result on the weak convergence of an empirical process formed by the statistics. In Chapter 2, we describe the almost sure pointwise convergence of the 1-step statistic for a broad class of time series models and processes, including unit root and explosive processes. We develop an asymptotic equivalence result, and use this to establish the asymptotic distribution of the maximum of a sequence of 1-step statistics with normal errors. This allows joint consideration of the sequence of 1-step tests via its maximum: the sup-Chow test. In Chapter 3, we use simulation to investigate the power properties of this test and compare it with benchmark tests of structural stability. We find that the sup-Chow test may have advantages when the nature of instability is unknown. In Chapter 4, we consider how the test may be adapted to situations in which the errors cannot be assumed normal. We evaluate several promising approaches, but also note a trade-off between robustness and power. In Chapter 5 we analyse an empirical process formed from the 1-step statistics, and prove a weak convergence result. Under the assumption of normal errors, the limiting distribution reduces to that of a Brownian bridge. The asymptotic approximation appears to works well even in small samples.

We are interested in the stochastic properties, individual and joint, of mis- specification testing when the data are generated by an autoregressive process. Good mis-specification tests are invariant to the dynamic properties of the pro- cess summarized by its characteristic roots, and to irrelevant misspecifications. Invariance in parameter space obviates inference prior to mis-specification test- ing. This is important as the latter is used to validate the former. Mutual independence of the tests allows calibration of the overall significance level. Es- tablishing such results requires work on individual tests and on their stochastic interactions. In Chapter 2, we derive the asymptotic distribution of two types of CUSUM of squares test, one implemented with standardized one-step-ahead OLS pre- diction errors and another implemented with OLS residuals. The latter is found to be valid in all but singular explosive cases, but the former only in purely non-explosive, or regular explosive cases with all roots in the explosive region of the parameter space; in Chapter 3, we show that a nuisance term arises in the mixed case. In Chapter 4, we derive numerically a finite sam- ple correction to render the tests implementable into software, and Chapter 1 contains two examples of applications. In Chapter 5, we consider inference on the parameters associated with the stationary part of the process, together with tests for a unit root, lag length, variance constancy, and normality of the regression innovations. In character- izing the joint distribution of these tests, we rely on asymptotic theory, and show independence in the limit. A simulation experiment suggests that finite sample correlations between some of the tests are statistically significant but small. Asymptotically, then, control of the overall significance level of the test procedure is feasible, and there is no reason to discount inference for the use of these mis-specification tests in model selection.

This thesis analyses, derives and evaluates specification tests of Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH) regression models, both univariate and multivariate. Of particular interest, in the first half of the thesis, is the derivation of robust test procedures designed to assess the Constant Conditional Correlation (CCC) assumption often employed in multivariate GARCH (MGARCH) models. New asymptotically valid conditional moment tests are proposed which are simple to construct, easily implementable following the full or partial Quasi Maximum Likelihood (QML) estimation and which are robust to non-normality. In doing so, a non-normality robust version of the Tse's (2000) LM test is provided. In addition, a new and easily programmable expressions of the expected Hessian matrix associated with the QMLE is obtained. The finite sample performances of these tests are investigated in an extensive Monte Carlo study, programmed in GAUSS.In the second half of the thesis, attention is devoted to nonparametric testing of GARCH regression models. First simultaneous consistent nonparametric tests of the conditional mean and conditional variance structure of univariate GARCH models are considered. The approach is developed from the Integrated Generalized Spectral (IGS) and Projected Integrated Conditional Moment (PICM) procedures proposed recently by Escanciano (2008 and 2009, respectively) for time series models. Extending Escanciano (2008), a new and simple wild bootstrap procedure is proposed to implement these tests. A Monte Carlo study compares the performance of these nonparametric tests and four parametric tests of nonlinearity and/or asymmetry under a wide range of alternatives. Although the proposed bootstrap scheme does not strictly satisfy the asymptotic requirements, the simulation results demonstrate its ability to control the size extremely well and therefore the power comparison seems justified. Furthermore, this suggests there may exist weaker conditions under which the tests are implementable. The simulation exercise also presents the new evidence of the effect of conditional mean misspecification on various parametric tests of conditional variance. The testing procedures are also illustrated with the help of the S&P 500 data. Finally the PICM and IGS approaches are extended to the MGARCH case. The procedure is illustrated with the help of a bivariate CCC-GARCH model, but can be generalized to other MGARCH specifications. Simulation exercise shows that these tests have satisfactory size and are robust to non-normality. The marginal mean and variance tests have excellent power; however the covariance marginal tests lack power for some alternatives.

Este trabalho investiga a aplicação de critérios de teste para o teste de especificações do aspecto comportamental de Sistemas Reativos, descritos em Estelle e em Statecharts. A utilização de Sistemas Reativos em várias atividades humanas, requer uma maior qualidade tanto do produto como do processo de desenvolvimento, pois falhas nesses sistemas podem ocasionar riscos para vidas humanas e perdas econômicas. Os critérios de teste propostos nesta tese visam a fornecer uma medida de cobertura dos testes, permitindo que a qualidade da atividade de teste possa ser mensurada e avaliada. Esta tese apresenta contribuições para as três atividades fundamentais no contexto de teste de software, que são: definição de critérios de teste, desenvolvimento de estudos teóricos/empíricos e desenvolvimento de ferramentas. Com relação à definição de critérios de teste, é proposta a aplicação do Teste de Mutação para Estelle e a aplicação de critérios de Fluxo de Controle para Estelle e Statecharts. Para o Teste de Mutação, são identificados os tipos de erros em especificações Estelle, definidos os operadores de mutação, estratégias de teste incrementais e critérios de mutação alternativa que visam a diminuir o custo de aplicação desse critério. Para os critérios de Fluxo de Controle, foram definidas duas famílias de critérios: FCCS - Família de Critérios de Cobertura para Statecharts e FCCE - Família de Critérios de Cobertura para Estelle. Estudos teóricos são realizados visando a analisar a complexidade do Teste de Mutação para Estelle e a relação de inclusão dos critérios FCCS e FCCE. Estudos empíricos são realizados visando a comparar os critérios de teste definidos e a analisar a sua aplicação durante a simulação de especificações Estelle e Statecharts. Com relação ao desenvolvimento de ferramentas, a família de ferramentas Proteum, que apóia a aplicação do teste de Mutação, e os ambientes para simulação de especificações Estelle (EDT) e Statecharts (StaSim) fornecem uma base essencial para o desenvolvimento das ferramentas. São apresentadas algumas ponderações que devem ser consideradas para a definição de ferramentas de apoio à aplicação dos critérios propostos. / Reactive Systems are applied to several human activities and as failures in these systems may cause human or economical losses, it is required the use of high-quality software development processes that would lead to the production of high-quality products. This thesis investigates criteria for testing of Reactive Systems\' behavior specifications, specified either in Estelle or in Statecharts. These criteria systematize the testing activity and provide mechanisms for the software tests quality assessment. This thesis presents contributions to the three fundamental activities in the context of software testing, which are: definition of testing criteria, theoretical studies and tool development. In relation to the definition of testing criteria, it is proposed the use of Mutation Testing for Estelle specifications and the use of Control Flow Testing for Estelle and Statecharts specifications. For Mutation Testing, the errors types in Estelle specifications are identified; mutation operators are defined and incremental testing strategies are established. In this context, it is explored the alternative mutation criteria, which aim at reducing the cost of application of the Mutation Testing. For Control Flow Testing, two families of criteria are defined: SCCF - Statechart Coverage Criteria Family and ECCF - Estelle Coverage Criteria Family. Theoretical studies are accomplished to analyze the complexity of the Mutation Testing to Estelle and the inclusion relation for the FCCS and FCCE criteria. Case studies are conducted to evaluate the testing criteria defined in this thesis. The application of these criteria during the simulation of Estelle and Statecharts specifications is analyzed. The Proteum family tools, that support the application of Mutation Testing, and the simulation environments to Estelle (EDT) and Statecharts (StatSim) supply an essential base for tools development. Considerations about the definition of supporting tools to the application of the proposed criteria are realized.

Este trabalho investiga a aplicação de critérios de teste para o teste de especificações do aspecto comportamental de Sistemas Reativos, descritos em Estelle e em Statecharts. A utilização de Sistemas Reativos em várias atividades humanas, requer uma maior qualidade tanto do produto como do processo de desenvolvimento, pois falhas nesses sistemas podem ocasionar riscos para vidas humanas e perdas econômicas. Os critérios de teste propostos nesta tese visam a fornecer uma medida de cobertura dos testes, permitindo que a qualidade da atividade de teste possa ser mensurada e avaliada. Esta tese apresenta contribuições para as três atividades fundamentais no contexto de teste de software, que são: definição de critérios de teste, desenvolvimento de estudos teóricos/empíricos e desenvolvimento de ferramentas. Com relação à definição de critérios de teste, é proposta a aplicação do Teste de Mutação para Estelle e a aplicação de critérios de Fluxo de Controle para Estelle e Statecharts. Para o Teste de Mutação, são identificados os tipos de erros em especificações Estelle, definidos os operadores de mutação, estratégias de teste incrementais e critérios de mutação alternativa que visam a diminuir o custo de aplicação desse critério. Para os critérios de Fluxo de Controle, foram definidas duas famílias de critérios: FCCS - Família de Critérios de Cobertura para Statecharts e FCCE - Família de Critérios de Cobertura para Estelle. Estudos teóricos são realizados visando a analisar a complexidade do Teste de Mutação para Estelle e a relação de inclusão dos critérios FCCS e FCCE. Estudos empíricos são realizados visando a comparar os critérios de teste definidos e a analisar a sua aplicação durante a simulação de especificações Estelle e Statecharts. Com relação ao desenvolvimento de ferramentas, a família de ferramentas Proteum, que apóia a aplicação do teste de Mutação, e os ambientes para simulação de especificações Estelle (EDT) e Statecharts (StaSim) fornecem uma base essencial para o desenvolvimento das ferramentas. São apresentadas algumas ponderações que devem ser consideradas para a definição de ferramentas de apoio à aplicação dos critérios propostos. / Reactive Systems are applied to several human activities and as failures in these systems may cause human or economical losses, it is required the use of high-quality software development processes that would lead to the production of high-quality products. This thesis investigates criteria for testing of Reactive Systems\' behavior specifications, specified either in Estelle or in Statecharts. These criteria systematize the testing activity and provide mechanisms for the software tests quality assessment. This thesis presents contributions to the three fundamental activities in the context of software testing, which are: definition of testing criteria, theoretical studies and tool development. In relation to the definition of testing criteria, it is proposed the use of Mutation Testing for Estelle specifications and the use of Control Flow Testing for Estelle and Statecharts specifications. For Mutation Testing, the errors types in Estelle specifications are identified; mutation operators are defined and incremental testing strategies are established. In this context, it is explored the alternative mutation criteria, which aim at reducing the cost of application of the Mutation Testing. For Control Flow Testing, two families of criteria are defined: SCCF - Statechart Coverage Criteria Family and ECCF - Estelle Coverage Criteria Family. Theoretical studies are accomplished to analyze the complexity of the Mutation Testing to Estelle and the inclusion relation for the FCCS and FCCE criteria. Case studies are conducted to evaluate the testing criteria defined in this thesis. The application of these criteria during the simulation of Estelle and Statecharts specifications is analyzed. The Proteum family tools, that support the application of Mutation Testing, and the simulation environments to Estelle (EDT) and Statecharts (StatSim) supply an essential base for tools development. Considerations about the definition of supporting tools to the application of the proposed criteria are realized.

Stochastic volatility (SV) models provide a means of tracking and forecasting the variance of financial asset returns. While SV models have a number of theoretical advantages over competing variance modelling procedures they are notoriously difficult to estimate. The distinguishing feature of the SV estimation literature is that those algorithms that provide accurate parameter estimates are conceptually demanding and require a significant amount of computational resources to implement. Furthermore, although a significant number of distinct SV specifications exist, little attention has been paid to how one would choose the appropriate specification for a given data series. Motivated by these facts, a likelihood based joint estimation and specification testing procedure for SV models is introduced that significantly overcomes the operational issues surrounding existing estimators. The estimation and specification testing procedures in this thesis are made possible by the introduction of a discrete nonlinear filtering (DNF) algorithm. This procedure uses the nonlinear filtering set of equations to provide maximum likelihood estimates for the general class of nonlinear latent variable problems which includes the SV model class. The DNF algorithm provides a fast and accurate implementation of the nonlinear filtering equations by treating the continuously valued state-variable as if it were a discrete Markov variable with a large number of states. When the DNF procedure is applied to the standard SV model, very accurate parameter estimates are obtained. Since the accuracy of the DNF is comparable to other procedures, its advantages are seen as ease and speed of implementation and the provision of online filtering (prediction) of variance. Additionally, the DNF procedure is very flexible and can be used for any dynamic latent variable problem with closed form likelihood and transition functions. Likelihood based specification testing for non-nested SV specifications is undertaken by formulating and estimating an encompassing model that nests two competing SV models. Likelihood ratio statistics are then used to make judgements regarding the optimal SV specification. The proposed framework is applied to SV models that incorporate either extreme returns or asymmetries.