Estimação robusta em modelos lineares de equações simultâneas

Os Modelos de Equações Simultâneas (SEM) são modelos estatísticos com muita tradição em estudos de Econometria, uma vez que permitem representar e estudar uma vasta gama de processos económicos. Os estimadores mais usados em SEM resultam da aplicação do Método dos Mínimos Quadrados ou do Método da Máxima Verosimilhança, os quais não são robustos. Em Maronna e Yohai (1997), os autores propõem formas de “robustificar” esses estimadores. Um outro método de estimação com interesse nestes modelos é o Método dos Momentos Generalizado (GMM), o qual também conduz a estimadores não robustos. Estimadores que sofrem de falta de robustez são muito inconvenientes uma vez que podem conduzir a resultados enganadores quando são violadas as hipóteses subjacentes ao modelo assumido. Os estimadores robustos são de grande valor, em particular quando os modelos em estudo são complexos, como é o caso dos SEM. O principal objectivo desta investigação foi o de procurar tais estimadores tendo-se construído um estimador robusto a que se deu o nome de GMMOGK. Trata-se de uma versão robusta do estimador GMM. Para avaliar o desempenho do novo estimador foi feito um adequado estudo de simulação e foi também feita a aplicação do estimador a um conjunto de dados reais. O estimador robusto tem um bom desempenho nos modelos heterocedásticos considerados e, nessas condições, comporta-se melhor do que os estimadores não robustos usados no estudo. Contudo, quando a análise é feita em cada equação separadamente, a especificidade de cada equação individual e a estrutura de dependência do sistema são dois aspectos que influenciam o desempenho do estimador, tal como acontece com os estimadores usuais. Para enquadrar a investigação, o texto inclui uma revisão de aspectos essenciais dos SEM, o seu papel em Econometria, os principais métodos de estimação, com particular ênfase no GMM, e uma curta introdução à estimação robusta.

Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Characterization of RSOA modulators in radio over fiber links

In this work physical and behavioral models for a bulk Reflective Semiconductor Optical Amplifier (RSOA) modulator in Radio over Fiber (RoF) links are proposed. The transmission performance of the RSOA modulator is predicted under broadband signal drive. At first, the simplified physical model for the RSOA modulator in RoF links is proposed, which is based on the rate equation and traveling-wave equations with several assumptions. The model is implemented with the Symbolically Defined Devices (SDD) in Advanced Design System (ADS) and validated with experimental results. Detailed analysis regarding optical gain, harmonic and intermodulation distortions, and transmission performance is performed. The distribution of the carrier and Amplified Spontaneous Emission (ASE) is also demonstrated. Behavioral modeling of the RSOA modulator is to enable us to investigate the nonlinear distortion of the RSOA modulator from another perspective in system level. The Amplitude-to-Amplitude Conversion (AM-AM) and Amplitude-to-Phase Conversion (AM-PM) distortions of the RSOA modulator are demonstrated based on an Artificial Neural Network (ANN) and a generalized polynomial model. Another behavioral model based on Xparameters was obtained from the physical model. Compensation of the nonlinearity of the RSOA modulator is carried out based on a memory polynomial model. The nonlinear distortion of the RSOA modulator is reduced successfully. The improvement of the 3rd order intermodulation distortion is up to 17 dB. The Error Vector Magnitude (EVM) is improved from 6.1% to 2.0%. In the last part of this work, the performance of Fibre Optic Networks for Distributed and Extendible Heterogeneous Radio Architectures and Service Provisioning (FUTON) systems, which is the four-channel virtual Multiple Input Multiple Output (MIMO), is predicted by using the developed physical model. Based on Subcarrier Multiplexing (SCM) techniques, four-channel signals with 100 MHz bandwidth per channel are generated and used to drive the RSOA modulator. The transmission performance of the RSOA modulator under the broadband multi channels is depicted with the figure of merit, EVM under di erent adrature Amplitude Modulation (QAM) level of 64 and 254 for various number of Orthogonal Frequency Division Multiplexing (OFDM) subcarriers of 64, 512, 1024 and 2048.