In this thesis, a Bayesian approach is adopted to handle parameter estimation and model uncertainty in autoregressive moving average (ARMA) time series models and dynamic linear models (DLM). Bayesian model uncertainty is handled in a parametric fashion through the use of posterior model probabilities computed via Markov chain Monte Carlo (MCMC) simulation techniques. Attention is focused on reversible jump Markov chain Monte Carlo (RJMCMC) samplers, which can move between models of different dimensions, to address the problem of model order uncertainty and strategies for proposing efficient sampling schemes in autoregressive moving average time series models and dynamic linear models are developed. The general problem of assessing convergence of the sampler in a dimension-changing context is addressed by computing estimates of the probabilities of moving to higher and lower dimensional spaces. Graphical and numerical techniques are used to compare different updating schemes. The methodology is illustrated by applying it to both simulated and real data sets and the results for the Bayesian model selection and parameter estimation procedures are compared with the classical model selection criteria and maximum likelihood estimation.

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Στο πρώτο κεφάλαιο εισάγουμε τις βασικές έννοιες της διπλωματικής εργασίας. Αναφέρουμε τους ορισμούς και τον σκοπό της ανάλυσης χρονολογικών σειρών. Επίσης εισάγονται ορισμένα βασικά χαρακτηριστικά των χρονολογικών σειρών όπως η έννοια της στασιμότητας και της συνάρτησης αυτοσυσχέτισης και αναφέρουμε τις τρεις βασικές κατηγορίες στοχαστικών υποδειγμάτων χρονολογικών σειρών που αφορούν στις στάσιμες στοχαστικές διαδικασίες, οι οποίες θα αναλυθούν στα επόμενα κεφάλαια. Στο δεύτερο κεφάλαιο αναλύουμε τα αυτοπαλίνδρομα υποδείγματα, πρώτης, δεύτερης και γενικά p τάξης. Αναφέρονται παραδείγματα. Στο τρίτο κεφάλαιο αναλύουμε τα υποδείγματα κινητού μέσου πρώτης και γενικά q τάξης καθώς και μεικτά υποδείγματα πρώτης και γενικά (p,q) τάξης. Αναφέρονται παραδείγματα. Στο τέταρτο κεφάλαιο αναλύουμε χρονολογικές σειρές που δεν έχουν τα χαρακτηριστικά στάσιμων στοχαστικών διαδικασιών. Επίσης αναλύουμε την μεθοδολογία Box-Jenkins, η οποία είναι μία μέθοδος εξεύρεσης ενός στατιστικού υποδείγματος (ARIMA). Τέλος εφαρμόζεται η παραπάνω μέθοδος σε ένα παράδειγμα με τη χρήση του SPSS. / At the first chapter we introduce the basic concepts. We present the main definitions and the objectives of the time series analysis. Furthermore, we introduce some basic characteristics of the time series such as the concepts of “stationary process” and “autocorrelation”. Finally we mention three basic categories of time series models that concern stationary stochastic processes. Following in the second chapter we analyze the autoregressive models of first, second and generally “p” order. We present various relative examples. At the third chapter we analyze the moving average models of first and generally “q” order. Additionally, we analyze the mixed models of first and generally (p,q) order. Various relative examples are presented. Finally, at the forth chapter we analyze time series that don’t have the characteristics of stationary stochastic proceedings. Also we analyze the method Box-Jenkins. Furthermore, the later method is studied using the statistic software package SPSS.

Forecasting is an important analysis task and there is a need of integrating time series models and estimation methods in database systems. The main issue is the computationally expensive maintenance of model parameters when new data is inserted. In this paper, we examine how an important class of time series models, the AutoRegressive Integrated Moving Average (ARIMA) models, can be maintained with respect to inserts. Therefore, we propose a novel approach, on-demand estimation, for the efficient maintenance of maximum likelihood estimates from numerically implemented estimators. We present an extensive experimental evaluation on both real and synthetic data, which shows that our approach yields a substantial speedup while sacrificing only a limited amount of predictive accuracy.

The main objective of the thesis was to investigate the feasibility of accurately and precisely fore- casting the number of both national and provincial bene ciaries of social security grants in South Africa, using simple autoregressive integrated moving average (ARIMA) models. The series of the monthly number of bene ciaries of the old age, child support, foster care and disability grants from April 2004 to March 2010 were used to achieve the objectives of the thesis. The conclusions from analysing the series were that: (1) ARIMA models for forecasting are province and grant-type spe- ci c; (2) for some grants, national forecasts obtained by aggregating provincial ARIMA forecasts are more accurate and precise than those obtained by ARIMA modelling national series; and (3) for some grants, forecasts obtained by modelling the latest half of the series were more accurate and precise than those obtained from modelling the full series. / Mathematical Sciences / M.Sc. (Statistics)

The main objective of the thesis was to investigate the feasibility of accurately and precisely fore- casting the number of both national and provincial bene ciaries of social security grants in South Africa, using simple autoregressive integrated moving average (ARIMA) models. The series of the monthly number of bene ciaries of the old age, child support, foster care and disability grants from April 2004 to March 2010 were used to achieve the objectives of the thesis. The conclusions from analysing the series were that: (1) ARIMA models for forecasting are province and grant-type spe- ci c; (2) for some grants, national forecasts obtained by aggregating provincial ARIMA forecasts are more accurate and precise than those obtained by ARIMA modelling national series; and (3) for some grants, forecasts obtained by modelling the latest half of the series were more accurate and precise than those obtained from modelling the full series. / Mathematical Sciences / M.Sc. (Statistics)

Submitted by Paulo Gustavo Grahl (pgrahl@fgvmail.br) on 2013-10-18T05:32:44Z No. of bitstreams: 1 DoutoradoPG_final.pdf: 23501670 bytes, checksum: 55b15051b9acc69ac74e639efe776fae (MD5) / Approved for entry into archive by ÁUREA CORRÊA DA FONSECA CORRÊA DA FONSECA (aurea.fonseca@fgv.br) on 2013-10-28T18:22:53Z (GMT) No. of bitstreams: 1 DoutoradoPG_final.pdf: 23501670 bytes, checksum: 55b15051b9acc69ac74e639efe776fae (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2013-10-29T18:24:15Z (GMT) No. of bitstreams: 1 DoutoradoPG_final.pdf: 23501670 bytes, checksum: 55b15051b9acc69ac74e639efe776fae (MD5) / Made available in DSpace on 2013-10-29T18:25:35Z (GMT). No. of bitstreams: 1 DoutoradoPG_final.pdf: 23501670 bytes, checksum: 55b15051b9acc69ac74e639efe776fae (MD5) Previous issue date: 2012-12-22 / Esta dissertação concentra-se nos processos estocásticos espaciais definidos em um reticulado, os chamados modelos do tipo Cliff & Ord. Minha contribuição nesta tese consiste em utilizar aproximações de Edgeworth e saddlepoint para investigar as propriedades em amostras finitas do teste para detectar a presença de dependência espacial em modelos SAR (autoregressivo espacial), e propor uma nova classe de modelos econométricos espaciais na qual os parâmetros que afetam a estrutura da média são distintos dos parâmetros presentes na estrutura da variância do processo. Isto permite uma interpretação mais clara dos parâmetros do modelo, além de generalizar uma proposta de taxonomia feita por Anselin (2003). Eu proponho um estimador para os parâmetros do modelo e derivo a distribuição assintótica do estimador. O modelo sugerido na dissertação fornece uma interpretação interessante ao modelo SARAR, bastante comum na literatura. A investigação das propriedades em amostras finitas dos testes expande com relação a literatura permitindo que a matriz de vizinhança do processo espacial seja uma função não-linear do parâmetro de dependência espacial. A utilização de aproximações ao invés de simulações (mais comum na literatura), permite uma maneira fácil de comparar as propriedades dos testes com diferentes matrizes de vizinhança e corrigir o tamanho ao comparar a potência dos testes. Eu obtenho teste invariante ótimo que é também localmente uniformemente mais potente (LUMPI). Construo o envelope de potência para o teste LUMPI e mostro que ele é virtualmente UMP, pois a potência do teste está muito próxima ao envelope (considerando as estruturas espaciais definidas na dissertação). Eu sugiro um procedimento prático para construir um teste que tem boa potência em uma gama de situações onde talvez o teste LUMPI não tenha boas propriedades. Eu concluo que a potência do teste aumenta com o tamanho da amostra e com o parâmetro de dependência espacial (o que está de acordo com a literatura). Entretanto, disputo a visão consensual que a potência do teste diminui a medida que a matriz de vizinhança fica mais densa. Isto reflete um erro de medida comum na literatura, pois a distância estatística entre a hipótese nula e a alternativa varia muito com a estrutura da matriz. Fazendo a correção, concluo que a potência do teste aumenta com a distância da alternativa à nula, como esperado. / This dissertation focus on spatial stochastic process on a lattice (Cliff & Ord--type of models). My contribution consists of using Edgeworth and saddlepoint series to investigate small sample size and power properties of tests for detecting spatial dependence in spatial autoregressive (SAR) stochastic processes, and proposing a new class of spatial econometric models where the spatial dependence parameters that enter the mean structure are different from the ones in the covariance structure. This allows a clearer interpretation of models' parameters and generalizes the set of local and global models suggested by Anselin (2003) as an alternative to the traditional Cliff & Ord models. I propose an estimation procedure for the model's parameters and derive the asymptotic distribution of the parameters' estimators. The suggested model provides some insights on the structure of the commonly used mixed regressive, spatial autoregressive model with spatial autoregressive disturbances (SARAR). The study of the small sample properties of tests to detect spatial dependence expands on the existing literature by allowing the neighborhood structure to be a nonlinear function of the spatial dependence parameter. The use of series approximations instead of the often used Monte Carlo simulation allows a simple way to compare test properties across different neighborhood structures and to correct for size when comparing power. I obtain the power envelope for testing the presence of spatial dependence in the SAR process using the optimal invariant test statistic, which is also locally uniformly most powerful invariant (LUMPI). I have found that the LUMPI test is virtually UMP since its power is very close to the power envelope. I suggest a practical procedure to build a test that, while not UMP, retain good power properties in a wider range for the spatial parameter when compared to the LUMPI test. I find that power increases with sample size and with the spatial dependence parameter -- which agrees with the literature. However, I call into question the consensus view that power decreases as the spatial weight matrix becomes more densely connected. This finding in the literature reflects an error of measure because the hypothesis being compared are at very different statistical distance from the null. After adjusting for this, the power is larger for alternative hypothesis further away from the null -- as one would expect.