This thesis presents the results of research into the use of factor models for stationary economic time series. Two basic scenarios are considered. The first is a situation where a large number of observations are available on a relatively small number variables, and a dynamic factor model is specified. It is shown that a dynamic factor model may be derived as a representation of a VARMA model of reduced spectral rank observed subject to measurement error. In some cases the resulting factor model corresponds to a minimal state-space representation of the VARMA plus noise model. Identification is discussed and proved for a fairly general class of dynamic factor model, and a frequency domain estimation procedure is proposed which has the advantage of generalising easily to models with rich dynamic structures. The second scenario is one where both the number of variables and the number of observations jointly diverge to infinity. The principal components estimator is considered in this case, and consistency is proved under assumptions which allow for much more error cross-correlation than the previously published theorems. Ancillary results include finite sample/variables bounds linking population principal components to population factors, and consistency results for principal components in a dual limit framework under a `gap' condition on the eigenvalues. A new factor model, named the Grouped Variable Approximate Factor Model, is introduced. This factor model allows for arbitrarily strong correlation between some of the errors, provided that the variables corresponding to the strongly correlated errors may be arranged into groups. An approximate instrumental variables estimator is proposed for the model and consistency is proved.

Financial time-series forecasting has long been discussed because of its vitality for making informed investment decisions. This kind of problem, however, is intrinsically challenging due to the data dynamics in nature. Most of the research works in the past focus on artificial neural network (ANN)-based approaches. It has been pointed out that such approaches suffer from explanatory power and generalized prediction ability though. The objective of this research is thus to propose a hybrid approach for stock price forecasting. Independent component analysis (ICA) is employed to reveal the latent structure of the observed time-series and remove noise and redundancy in the structure. It further assists clustering analysis. Support vector regression (SVR) models are then applied to enhance the generalization ability with separate models built based on the time-series data of companies in each individual cluster. Two experiments are conducted accordingly. The results show that SVR has robust accuracy performance. More importantly, SVR models with ICA-based clustered data perform better than the single SVR model with all data involved. Our proposed approach does enhance the generalization ability of the forecasting models, which justifies the feasibility of its applications.

published_or_final_version / Statistics / Master / Master of Social Sciences

Ο σκοπός αυτής της μεταπτυχιακής εργασίας είναι διπλός και συγκεκριμένα αφορά στη μελέτη του απλού / γενικευμένου πολλαπλού μοντέλου παλινδρόμησης όταν σε αυτό παραβιάζεται μια από τις συνθήκες των Gauss-Markov και πιο συγκεκριμένα όταν, Cov{ε_i,ε_j }≠0, ∀ i≠j και στην ανάλυση χρονοσειρών. Αρχικά, γίνεται συνοπτική αναφορά στο απλό και στο πολλαπλό γραμμικό μοντέλο παλινδρόμησης, στις ιδιότητες καθώς και στις εκτιμήσεις των συντελεστών παλινδρόμησης. Περιγράφονται οι ιδιότητες των τυχαίων όρων όπως μέση τιμή, διασπορά, συντελεστές συσχέτισης κ.α., εφόσον υπάρχει παραβίαση της ιδιότητας της συνδιασποράς αυτών. Τέλος, περιγράφεται ο έλεγχος για αυτοσυσχέτιση των τυχαίων όρων των Durbin-Watson καθώς και μια ποικιλία διορθωτικών μέτρων με σκοπό την εξάλειψή της. Στο δεύτερο μέρος, αρχικά αναφέρονται βασικές έννοιες της θεωρίας των χρονοσειρών. Στη συνέχεια, γίνεται ανάλυση διαφόρων στάσιμων χρονοσειρών και συγκεκριμένα, ξεκινώντας από το λευκό θόρυβο, παρουσιάζονται οι χρονοσειρές κινητού μέσου (ΜΑ), οι αυτοπαλινδρομικές χρονοσειρές (ΑR), οι χρονοσειρές ARMA, καθώς και η γενική περίπτωση μη στάσιμων χρονοσειρών, των ΑRΙΜΑ χρονοσειρών και παρατίθενται συνοπτικά τα πρώτα στάδια ανάλυσης μιας χρονοσειράς για κάθε μια από τις περιπτώσεις αυτές. Η εργασία αυτή βασίστηκε σε δύο σημαντικά βιβλία διακεκριμένων επιστημόνων, του κ. Γεώργιου Κ. Χρήστου, Εισαγωγή στην Οικονομετρία και στο βιβλίο των John Neter, Michael H. Kutner, Christofer J. Nachtsheim και William Wasserman, Applied Linear Regression Models. / The purpose of this thesis is twofold, namely concerns the study of the simple / generalized multiple regression model when this violated one of the conditions of Gauss-Markov specifically when, Cov {e_i, e_j} ≠ 0, ∀ i ≠ j and time series analysis. Initially, there is a brief reference to the simple and multiple linear regression model, the properties and estimates of regression coefficients. Describe the properties of random terms such as mean, variance, correlation coefficients, etc., if there is a breach of the status of their covariance. Finally, described the test for autocorrelation of random terms of the Durbin-Watson and a variety of corrective measures to eliminate it. In the second part, first mentioned basic concepts of the theory of time series. Then, various stationary time series analyzes and specifically, starting from the white noise, the time series moving average presented (MA), the aftopalindromikes time series (AR) time series ARMA, and the general case of non-stationary time series of ARIMA time series and briefly presents the first analysis steps in a time series for each of these cases. This work was based on two important books of distinguished scientists, Mr. George K. Christou, Introduction to Econometrics, and in the book of John Neter, Michael H. Kutner, Christofer J. Nachtsheim and William Wasserman, Applied Linear Regression Models.

The research is based on financial time series modelling with special application / Thesis (M.Sc.) - University of KwaZulu-Natal, Pietermaritzburg, 2009.

This thesis presents the results of research into the use of factor models for stationary economic time series. Two basic scenarios are considered. The first is a situation where a large number of observations are available on a relatively small number variables, and a dynamic factor model is specified. It is shown that a dynamic factor model may be derived as a representation of a VARMA model of reduced spectral rank observed subject to measurement error. In some cases the resulting factor model corresponds to a minimal state-space representation of the VARMA plus noise model. Identification is discussed and proved for a fairly general class of dynamic factor model, and a frequency domain estimation procedure is proposed which has the advantage of generalising easily to models with rich dynamic structures. The second scenario is one where both the number of variables and the number of observations jointly diverge to infinity. The principal components estimator is considered in this case, and consistency is proved under assumptions which allow for much more error cross-correlation than the previously published theorems. Ancillary results include finite sample/variables bounds linking population principal components to population factors, and consistency results for principal components in a dual limit framework under a `gap' condition on the eigenvalues. A new factor model, named the Grouped Variable Approximate Factor Model, is introduced. This factor model allows for arbitrarily strong correlation between some of the errors, provided that the variables corresponding to the strongly correlated errors may be arranged into groups. An approximate instrumental variables estimator is proposed for the model and consistency is proved.

[Truncated abstract] Time series of natural phenomena usually show irregular fluctuations. Often we want to know the underlying system and to predict future phenomena. An effective way of tackling this task is by time series modelling. Originally, linear time series models were used. As it became apparent that nonlinear systems abound in nature, modelling techniques that take into account nonlinearity in time series were developed. A particularly convenient and general class of nonlinear models is the pseudolinear models, which are linear combinations of nonlinear functions. These models can be obtained by starting with a large dictionary of basis functions which one hopes will be able to describe any likely nonlinearity, selecting a small subset of it, and taking a linear combination of these to form the model. The major component of this thesis concerns how to build good models for nonlinear time series. In building such models, there are three important problems, broadly speaking. The first is how to select basis functions which reflect the peculiarities of the time series as much as possible. The second is how to fix the model size so that the models can reflect the underlying system of the data and the influences of noise included in the data are removed as much as possible. The third is how to provide good estimates for the parameters in the basis functions, considering that they may have significant bias when the noise included in the time series is significant relative to the nonlinearity. Although these problems are mentioned separately, they are strongly interconnected

[pt] O presente trabalho trata da melhoria da estatí­stica-teste Multiplicador de Lagrange com distribuição qui-quadrado até ordem n (-1) , baseando-se na expansão de Harris (1985) e na melhoria obtida para os testes Escore, fornecida por Cordeiro e Ferrari (1991 e 1994), Apresentamos uma abordagem totalmente ambientada aos modelos estruturais de séries temporais, utilizando-se tais testes na detecção de ciclos. O trabalho apresenta também uma série de simulações comparando as performances destes testes aperfeiçoados com os tradicionalmente utilizados. / [en] The presente work discusses the improvement of the statistics-test Lagrange Multipliers with chi-squared distribution to order n (-1) , basing itself in Harris´ (1995) expansion and in the improvement for the score tests, furnished by Cordeiro and Ferrari (1991 and 1994). We present a totally adapted aproach to time series structural models, utilizing these tests in the cycles detection. The work aldo presents a serie simulations comparing the perfomances of these improved tests with the ones traditionally utilized.

Processus de Lévy et leurs applications en finance / No English summary available.

The idea of time series forecasting techniques is that the past has certain information about future. So, the question of how the information is encoded in the past can be interpreted and later used to extrapolate events of future constitute the crux of time series analysis and forecasting. Several methods such as qualitative techniques (e.g., Delphi method), causal techniques (e.g., least squares regression), quantitative techniques (e.g., smoothing method, time series models) have been developed in the past in which the concept lies in establishing a model either theoretically or mathematically from past observations and estimate future from it. Of all the models, time series methods such as autoregressive moving average (ARMA) process have gained popularity because of their simplicity in implementation and accuracy in obtaining forecasts. But, these models were formulated based on certain properties that a time series is assumed to possess. Classical decomposition techniques were developed to supplement the requirements of time series models. These methods try to define a time series in terms of simple patterns called trend, cyclical and seasonal patterns along with noise. So, the idea of decomposing a time series into component patterns, later modeling each component using forecasting processes and finally combining the component forecasts to obtain actual time series predictions yielded superior performance over standard forecasting techniques. All these methods involve basic principle of moving average computation. But, the developed classical decomposition methods are disadvantageous in terms of containing fixed number of components for any time series, data independent decompositions. During moving average computation, edges of time series might not get modeled properly which affects long range forecasting. So, these issues are to be addressed by more efficient and advanced decomposition techniques such as Wavelets and Empirical Mode Decomposition (EMD). Wavelets and EMD are some of the most innovative concepts considered in time series analysis and are focused on processing nonlinear and nonstationary time series. Hence, this research has been undertaken to ascertain the predictability of nonstationary time series using wavelet and Empirical Mode Decomposition (EMD) based ARMA models. The development of wavelets has been made based on concepts of Fourier analysis and Window Fourier Transform. In accordance with this, initially, the necessity of involving the advent of wavelets has been presented. This is followed by the discussion regarding the advantages that are provided by wavelets. Primarily, the wavelets were defined in the sense of continuous time series. Later, in order to match the real world requirements, wavelets analysis has been defined in discrete scenario which is called as Discrete Wavelet Transform (DWT). The current thesis utilized DWT for performing time series decomposition. The detailed discussion regarding the theory behind time series decomposition is presented in the thesis. This is followed by description regarding mathematical viewpoint of time series decomposition using DWT, which involves decomposition algorithm. EMD also comes under same class as wavelets in the consequence of time series decomposition. EMD is developed out of the fact that most of the time series in nature contain multiple frequencies leading to existence of different scales simultaneously. This method, when compared to standard Fourier analysis and wavelet algorithms, has greater scope of adaptation in processing various nonstationary time series. The method involves decomposing any complicated time series into a very small number of finite empirical modes (IMFs-Intrinsic Mode Functions), where each mode contains information of the original time series. The algorithm of time series decomposition using EMD is presented post conceptual elucidation in the current thesis. Later, the proposed time series forecasting algorithm that couples EMD and ARMA model is presented that even considers the number of time steps ahead of which forecasting needs to be performed. In order to test the methodologies of wavelet and EMD based algorithms for prediction of time series with non stationarity, series of streamflow data from USA and rainfall data from India are used in the study. Four non-stationary streamflow sites (USGS data resources) of monthly total volumes and two non-stationary gridded rainfall sites (IMD) of monthly total rainfall are considered for the study. The predictability by the proposed algorithm is checked in two scenarios, first being six months ahead forecast and the second being twelve months ahead forecast. Normalized Root Mean Square Error (NRMSE) and Nash Sutcliffe Efficiency Index (Ef) are considered to evaluate the performance of the proposed techniques. Based on the performance measures, the results indicate that wavelet based analyses generate good variations in the case of six months ahead forecast maintaining harmony with the observed values at most of the sites. Although the methods are observed to capture the minima of the time series effectively both in the case of six and twelve months ahead predictions, better forecasts are obtained with wavelet based method over EMD based method in the case of twelve months ahead predictions. It is therefore inferred that wavelet based method has better prediction capabilities over EMD based method despite some of the limitations of time series methods and the manner in which decomposition takes place. Finally, the study concludes that the wavelet based time series algorithm could be used to model events such as droughts with reasonable accuracy. Also, some modifications that could be made in the model have been suggested which can extend the scope of applicability to other areas in the field of hydrology.