The first chapter considers the asymptotic validity of bootstrap methods in a linear trend model with a change in slope at an unknown time. Perron and Zhu (2005) analyzed the consistency, rate of convergence, and limiting distributions of the parameter estimates in this model. I provide theoretical results for the asymptotic validity of bootstrap methods related to forming confidence intervals for the break date. I consider two bootstrap schemes, the residual (for white noise errors) and the sieve bootstrap (for correlated errors). Simulation experiments confirm that confidence intervals obtained using bootstrap methods perform well in terms of exact coverage rate. The second chapter extends Perron and Zhu's (2005) analysis to cover more general fractionally integrated errors with memory parameter d in the interval (-0.5,1.5). My theoretical results uncover some interesting features. For example, with a concurrent level shift allowed, the rate of convergence of the estimate of the break date is the same for all values of d in the interval (-0.5,0.5), a feature linked to the contamination induced by allowing a level shift. In all other cases, the rate of convergence is decreasing as d increases. I also provide results about the spurious break issue. The third chapter considers constructing confidence intervals for the break date in linear regressions. I compare the performance of various procedures in terms of the exact coverage rates and lengths: Bai's (1997) based on the asymptotic distribution with shrinking shifts, Elliott and Müller's (EM) (2007) based on inverting a test locally invariant to the magnitude of the change, Eo and Morley's (2013) based on inverting a likelihood ratio test, and various bootstrap procedures. In terms of coverage rates, EM's approach is the best but with a high cost in terms of length. With serially correlated errors and a change in intercept or in the coefficient of a regressor with a high signal-to-noise ratio, or when a lagged dependent variable is present, the length approaches the whole sample as the magnitude of the change increases. This drawback is not present for the other methods. Theoretical results are provided to explain the drawbacks of EM's method.

Duas ou mais séries não estacionárias são cointegradas se existir uma relação de equilíbrio de longo prazo entre elas. Nas últimas décadas, o interesse na literatura sobre o tema cointegração aumentou de maneira expressiva. Os modelos tradicionais supõem que o vetor de cointegração não varia ao longo do tempo. Entretanto, existem evidências na literatura de que esta suposição pode ser considerada muito restritiva. Utilizando o conceito de ondaletas, propomos um modelo de correção de erros vetorial em que é permitido ao vetor de cointegração variar ao longo do tempo. Diferente de trabalhos similares, é permitido ao vetor de cointegração variar suave ou abruptamente, dependendo da família de ondaletas considerada. Experimentos de Monte Carlo foram utilizados para estudar os quantis e o poder do teste de razão de verossimilhanças entre as hipóteses de cointegração usual e a de cointegração variando com o tempo. Os experimentos sugerem que o teste possui poder contra alternativas que variam ao longo do tempo. Foi demonstrada a capacidade do modelo em lidar satisfatoriamente com séries cointegradas simuladas, que apresentavam mudança de regime para o vetor de cointegração. O modelo foi empregado ainda para testar a validade da hipótese de paridade de poder de compra entre Estados Unidos e doze países da Organização para Cooperação e Desenvolvimento Econômico (OECD): Canadá, Japão e mais dez países europeus. Assim como em trabalhos similares, foram verificadas evidências de cointegração variando com o tempo entre os países. Foram utilizados valores-p bootstrap para verificar a significância da estatística do teste. / Two or more non-stationary time series are cointegrated if there is a long-run equilibrium relationship between them. In recent decades, interest in the literature on the subject of cointegration increased expressively. Traditional models that address this issue assume that the cointegration vector does not vary over time. However, there is evidence in the literature that this assumption can be considered very restrictive. Using the concept of wavelets, we propose a vector error correction model in which is allowed to the cointegration vector vary over time. Unlike similar works, the cointegration vector is allowed to vary smoothly or abruptly, depending on the considered family of wavelets. Monte Carlo experiments were used to study the quantiles and the power of the likelihood ratio test of the hypotheses of usual cointegration versus the time-varying cointegration. The experiments suggest that the test has power against alternatives that vary over time. It was demonstrated the ability of the model to deal satisfactorily with simulated cointegrated series, which presented regime change for the cointegration vector. The model was also used to test the validity of the Purchasing Power Parity hypothesis between United States and twelve countries of the Organization for Economic Cooperation and Development (OECD): Canada, Japan and ten other European countries. As in similar works, evidence of time-varying cointegration was verified among countries. Bootstrap p-values were used to verify the significance of the likelihood ratio of the test.

Duas ou mais séries não estacionárias são cointegradas se existir uma relação de equilíbrio de longo prazo entre elas. Nas últimas décadas, o interesse na literatura sobre o tema cointegração aumentou de maneira expressiva. Os modelos tradicionais supõem que o vetor de cointegração não varia ao longo do tempo. Entretanto, existem evidências na literatura de que esta suposição pode ser considerada muito restritiva. Utilizando o conceito de ondaletas, propomos um modelo de correção de erros vetorial em que é permitido ao vetor de cointegração variar ao longo do tempo. Diferente de trabalhos similares, é permitido ao vetor de cointegração variar suave ou abruptamente, dependendo da família de ondaletas considerada. Experimentos de Monte Carlo foram utilizados para estudar os quantis e o poder do teste de razão de verossimilhanças entre as hipóteses de cointegração usual e a de cointegração variando com o tempo. Os experimentos sugerem que o teste possui poder contra alternativas que variam ao longo do tempo. Foi demonstrada a capacidade do modelo em lidar satisfatoriamente com séries cointegradas simuladas, que apresentavam mudança de regime para o vetor de cointegração. O modelo foi empregado ainda para testar a validade da hipótese de paridade de poder de compra entre Estados Unidos e doze países da Organização para Cooperação e Desenvolvimento Econômico (OECD): Canadá, Japão e mais dez países europeus. Assim como em trabalhos similares, foram verificadas evidências de cointegração variando com o tempo entre os países. Foram utilizados valores-p bootstrap para verificar a significância da estatística do teste. / Two or more non-stationary time series are cointegrated if there is a long-run equilibrium relationship between them. In recent decades, interest in the literature on the subject of cointegration increased expressively. Traditional models that address this issue assume that the cointegration vector does not vary over time. However, there is evidence in the literature that this assumption can be considered very restrictive. Using the concept of wavelets, we propose a vector error correction model in which is allowed to the cointegration vector vary over time. Unlike similar works, the cointegration vector is allowed to vary smoothly or abruptly, depending on the considered family of wavelets. Monte Carlo experiments were used to study the quantiles and the power of the likelihood ratio test of the hypotheses of usual cointegration versus the time-varying cointegration. The experiments suggest that the test has power against alternatives that vary over time. It was demonstrated the ability of the model to deal satisfactorily with simulated cointegrated series, which presented regime change for the cointegration vector. The model was also used to test the validity of the Purchasing Power Parity hypothesis between United States and twelve countries of the Organization for Economic Cooperation and Development (OECD): Canada, Japan and ten other European countries. As in similar works, evidence of time-varying cointegration was verified among countries. Bootstrap p-values were used to verify the significance of the likelihood ratio of the test.

Ce mémoire propose une adaptation lisse de méthodes bootstrap par pseudo-population aux fins d'estimation de la variance et de formation d'intervalles de confiance pour des quantiles de population finie. Dans le cas de données i.i.d., Hall et al. (1989) ont montré que l'ordre de convergence de l'erreur relative de l’estimateur bootstrap de la variance d’un quantile échantillonnal connaît un gain lorsque l'on rééchantillonne à partir d’une estimation lisse de la fonction de répartition plutôt que de la fonction de répartition expérimentale. Dans cet ouvrage, nous étendons le principe du bootstrap lisse au contexte de population finie en le mettant en œuvre au sein des méthodes bootstrap par pseudo-population. Étant donné un noyau et un paramètre de lissage, cela consiste à lisser la pseudo-population dont sont issus les échantillons bootstrap selon le plan de sondage initial. Deux plans sont abordés, soit l'échantillonnage aléatoire simple sans remise et l'échantillonnage de Poisson. Comme l'utilisation des algorithmes proposés nécessite la spécification du paramètre de lissage, nous décrivons une méthode de sélection par injection et des méthodes de sélection par la minimisation d'estimés bootstrap de critères d'ajustement sur une grille de valeurs du paramètre de lissage. Nous présentons des résultats d'une étude par simulation permettant de montrer empiriquement l'efficacité de l'approche lisse par rapport à l'approche standard pour ce qui est de l'estimation de la variance d'un estimateur de quantile et des résultats plus mitigés en ce qui concerne les intervalles de confiance. / This thesis introduces smoothed pseudo-population bootstrap methods for the purposes of variance estimation and the construction of confidence intervals for finite population quantiles. In an i.i.d. context, Hall et al. (1989) have shown that resampling from a smoothed estimate of the distribution function instead of the usual empirical distribution function can improve the convergence rate of the bootstrap variance estimator of a sample quantile. We extend the smoothed bootstrap to the survey sampling framework by implementing it in pseudo-population bootstrap methods. Given a kernel function and a bandwidth, it consists of smoothing the pseudo-population from which bootstrap samples are drawn using the original sampling design. Two designs are discussed, namely simple random sampling and Poisson sampling. The implementation of the proposed algorithms requires the specification of the bandwidth. To do so, we develop a plug-in selection method along with grid search selection methods based on bootstrap estimates of two performance metrics. We present the results of a simulation study which provide empirical evidence that the smoothed approach is more efficient than the standard approach for estimating the variance of a quantile estimator together with mixed results regarding confidence intervals.

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