The thesis is dedicated to time series analysis for functional data and contains three original parts. In the first part, we derive statistical tests for the presence of a periodic component in a time series of functions. We consider both the traditional setting in which the periodic functional signal is contaminated by functional white noise, and a more general setting of a contaminating process which is weakly dependent. Several forms of the periodic component are considered. Our tests are motivated by the likelihood principle and fall into two broad categories, which we term multivariate and fully functional. Overall, for the functional series that motivate this research, the fully functional tests exhibit a superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data. In the second part, we consider vector autoregressive processes (VARs) with innovations having a singular covariance matrix (in short singular VARs). These objects appear naturally in the context of dynamic factor models. The Yule-Walker estimator of such a VAR is problematic, because the solution of the corresponding equation system tends to be numerically rather unstable. For example, if we overestimate the order of the VAR, then the singularity of the innovations renders the Yule-Walker equation system singular as well. Moreover, even with correctly selected order, the Yule-Walker system tends be close to singular in finite sample. We show that this has a severe impact on predictions. While the asymptotic rate of the mean square prediction error (MSPE) can be just like in the regular (non-singular) case, the finite sample behavior is suffering. This effect turns out to be particularly dramatic in context of dynamic factor models, where we do not directly observe the so-called common components which we aim to predict. Then, when the data are sampled with some additional error, the MSPE often gets severely inflated. We explain the reason for this phenomenon and show how to overcome the problem. Our numerical results underline that it is very important to adapt prediction algorithms accordingly. In the third part, we set up theoretical foundations and a practical method to forecast multiple functional time series (FTS). In order to do so, we generalize the static factor model to the case where cross-section units are FTS. We first derive a representation result. We show that if the first r eigenvalues of the covariance operator of the cross-section of n FTS are unbounded as n diverges and if the (r+1)th eigenvalue is bounded, then we can represent the each FTS as a sum of a common component driven by r factors and an idiosyncratic component. We suggest a method of estimation and prediction of such a model. We assess the performances of the method through a simulation study. Finally, we show that by applying our method to a cross-section of volatility curves of the stocks of S&P100, we have a better prediction accuracy than by limiting the analysis to individual FTS. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished

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Táto diplomová práca sa zaoberá novým prístupom k zhlukovaniu časových rád na základe dynamického faktorového modelu. Dynamický faktorový model je technika redukujúca dimenziu a rozširuje klasickú faktorovú analýzu o požiadavku autokorelačnej štruktúry latentných faktorov. Parametre modelu sa odhadujú pomocou EM algoritmu za použitia Kalmanovho filtra a vyhladzovača a taktiež sú aplikované nevyhnutné podmienky na model, aby sa stal identifikovateľným. Po tom, ako je v práci predstavený teoretický koncept prístupu, dynamický faktorový model je aplikovaný na skutočné pozorované časové rady a práca skúma jeho správanie a vlastnosti na jednomesačných meteorologických dátach požiarneho indexu (Fire Weather Index) na 108 požiarnych staniciach umiestnených v Britskej Kolumbii. Postup výpočtu modelu odhadne záťažovú maticu (loadings matrix) spolu so zodpovedajúcim malým počtom latentných faktorov a kovariačnou maticou modelovaných časových rád. Diplomová práca aplikuje k-means zhlukovanie na výslednú záťažovú maticu a ponúka rozdelenie meteorologických staníc do zhlukov založené na redukovanej dimenzionalite pôvodných dát. Vďaka odhadnutým priemerom zhlukov a odhadnutým latentným faktorom je možné získať aj priemerné trendy každého zhluku. Následne sú dosiahnuté výsledky porovnané s výsledkami získanými na dátach z rovnakých staníc avšak iného mesiaca, aby sa stanovila stabilita zhlukovania. Práca sa taktiež zaoberá efektom varimax rotácie záťažovej matice. Diplomová práca naviac navrhuje metódu detekovania odľahlých časových rád založenú na odhadnutej kovariačnej matici modelu a rozoberá dôsledky odľahlých hodnôt na odhanutý model.

Cette thèse est dédiée à une classe particulière de modèles à facteurs dynamiques non linéaires, les modèles à facteurs dynamiques à changement de régime markovien (MS-DFM). Par la combinaison des caractéristiques du modèle à facteur dynamique et celui du modèle à changement de régimes markoviens(i.e. la capacité d’agréger des quantités massives d’information et de suivre des processus fluctuants), ce cadre s’est révélé très utile et convenable pour plusieurs applications, dont le plus important est l’analyse des cycles économiques.La connaissance de l’état actuel des cycles économiques est crucial afin de surveiller la santé économique et d’évaluer les résultats des politiques économiques. Néanmoins, ce n’est pas une tâche facile à réaliser car, d’une part, il n’y a pas d’ensemble de données et de méthodes communément reconnus pour identifier les points de retournement, d’autre part, car les institutions officielles annoncent un nouveau point de retournement, dans les pays où une telle pratique existe, avec un délai structurel de plusieurs mois.Le MS-DFM est en mesure de résoudre ces problèmes en fournissant des estimations de l’état actuel de l’économie de manière rapide, transparente et reproductible sur la base de la composante commune des indicateurs macroéconomiques caractérisant le secteur réel.Cette thèse contribue à la vaste littérature sur l’identification des points de retournement du cycle économique dans trois direction. Dans le Chapitre 3, on compare les deux techniques d’estimation de MS-DFM, les méthodes en une étape et en deux étapes, et on les applique aux données françaises pour obtenir la chronologie des points de retournement du cycle économique. Dans Chapitre 4, sur la base des simulations de Monte Carlo, on étudie la convergence des estimateurs de la technique retenue - la méthode d’estimation en deux étapes, et on analyse leur comportement en échantillon fini. Dans le Chapitre 5, on propose une extension de MS-DFM - le MS-DFM à l’influence dynamique (DI-MS-DFM)- qui permet d’évaluer la contribution du secteur financier à la dynamique du cycle économique et vice versa, tout en tenant compte du fait que l’interaction entre eux puisse être dynamique. / This thesis is dedicated to the study of a particular class of non-linear Dynamic Factor Models, the Dynamic Factor Models with Markov Switching (MS-DFM). Combining the features of the Dynamic Factor model and the Markov Switching model, i.e. the ability to aggregate massive amounts of information and to track recurring processes, this framework has proved to be a very useful and convenient instrument in many applications, the most important of them being the analysis of business cycles.In order to monitor the health of an economy and to evaluate policy results, the knowledge of the currentstate of the business cycle is essential. However, it is not easy to determine since there is no commonly accepted dataset and method to identify turning points, and the official institutions announce a newturning point, in countries where such practice exists, with a structural delay of several months. The MS-DFM is able to resolve these issues by providing estimates of the current state of the economy in a timely, transparent and replicable manner on the basis of the common component of macroeconomic indicators characterizing the real sector. The thesis contributes to the vast literature in this area in three directions. In Chapter 3, I compare the two popular estimation techniques of the MS-DFM, the one-step and the two-step methods, and apply them to the French data to obtain the business cycle turning point chronology. In Chapter 4, on the basis of Monte Carlo simulations, I study the consistency of the estimators of the preferred technique -the two-step estimation method, and analyze their behavior in small samples. In Chapter 5, I extend the MS-DFM and suggest the Dynamical Influence MS-DFM, which allows to evaluate the contribution of the financial sector to the dynamics of the business cycle and vice versa, taking into consideration that the interaction between them can be dynamic.

The thesis, entitled "Essays on macroeconometrics and short-term forecasting",<p>is composed of three chapters. The first two chapters are on nowcasting,<p>a topic that has received an increasing attention both among practitioners and<p>the academics especially in conjunction and in the aftermath of the 2008-2009<p>economic crisis. At the heart of the two chapters is the idea of exploiting the<p>information from data published at a higher frequency for obtaining early estimates<p>of the macroeconomic variable of interest. The models used to compute<p>the nowcasts are dynamic models conceived for handling in an efficient way<p>the characteristics of the data used in a real-time context, like the fact that due to the different frequencies and the non-synchronicity of the releases<p>the time series have in general missing data at the end of the sample. While<p>the first chapter uses a small model like a VAR for nowcasting Italian GDP,<p>the second one makes use of a dynamic factor model, more suitable to handle<p>medium-large data sets, for providing early estimates of the employment in<p>the euro area. The third chapter develops a topic only marginally touched<p>by the second chapter, i.e. the estimation of dynamic factor models on data characterized by block-structures.<p>The firrst chapter assesses the accuracy of the Italian GDP nowcasts based<p>on a small information set consisting of GDP itself, the industrial production<p>index and the Economic Sentiment Indicator. The task is carried out by using<p>real-time vintages of data in an out-of-sample exercise over rolling windows<p>of data. Beside using real-time data, the real-time setting of the exercise is<p>also guaranteed by updating the nowcasts according to the historical release calendar. The model used to compute the nowcasts is a mixed-frequency Vector<p>Autoregressive (VAR) model, cast in state-space form and estimated by<p>maximum likelihood. The results show that the model can provide quite accurate<p>early estimates of the Italian GDP growth rates not only with respect<p>to a naive benchmark but also with respect to a bridge model based on the<p>same information set and a mixed-frequency VAR with only GDP and the industrial production index.<p>The chapter also analyzes with some attention the role of the Economic Sentiment<p>Indicator, and of soft information in general. The comparison of our<p>mixed-frequency VAR with one with only GDP and the industrial production<p>index clearly shows that using soft information helps obtaining more accurate<p>early estimates. Evidence is also found that the advantage from using soft<p>information goes beyond its timeliness.<p>In the second chapter we focus on nowcasting the quarterly national account<p>employment of the euro area making use of both country-specific and<p>area wide information. The relevance of anticipating Eurostat estimates of<p>employment rests on the fact that, despite it represents an important macroeconomic<p>variable, euro area employment is measured at a relatively low frequency<p>(quarterly) and published with a considerable delay (approximately<p>two months and a half). Obtaining an early estimate of this variable is possible<p>thanks to the fact that several Member States publish employment data and<p>employment-related statistics in advance with respect to the Eurostat release<p>of the euro area employment. Data availability represents, nevertheless, a<p>major limit as country-level time series are in general non homogeneous, have<p>different starting periods and, in some cases, are very short. We construct a<p>data set of monthly and quarterly time series consisting of both aggregate and<p>country-level data on Quarterly National Account employment, employment<p>expectations from business surveys and Labour Force Survey employment and<p>unemployment. In order to perform a real time out-of-sample exercise simulating<p>the (pseudo) real-time availability of the data, we construct an artificial<p>calendar of data releases based on the effective calendar observed during the first quarter of 2012. The model used to compute the nowcasts is a dynamic<p>factor model allowing for mixed-frequency data, missing data at the beginning<p>of the sample and ragged edges typical of non synchronous data releases. Our<p>results show that using country-specific information as soon as it is available<p>allows to obtain reasonably accurate estimates of the employment of the euro<p>area about fifteen days before the end of the quarter.<p>We also look at the nowcasts of employment of the four largest Member<p>States. We find that (with the exception of France) augmenting the dynamic<p>factor model with country-specific factors provides better results than those<p>obtained with the model without country-specific factors.<p>The third chapter of the thesis deals with dynamic factor models on data<p>characterized by local cross-correlation due to the presence of block-structures.<p>The latter is modeled by introducing block-specific factors, i.e. factors that<p>are specific to blocks of time series. We propose an algorithm to estimate the model by (quasi) maximum likelihood and use it to run Monte Carlo<p>simulations to evaluate the effects of modeling or not the block-structure on<p>the estimates of common factors. We find two main results: first, that in finite samples modeling the block-structure, beside being interesting per se, can help<p>reducing the model miss-specification and getting more accurate estimates<p>of the common factors; second, that imposing a wrong block-structure or<p>imposing a block-structure when it is not present does not have negative<p>effects on the estimates of the common factors. These two results allow us<p>to conclude that it is always recommendable to model the block-structure<p>especially if the characteristics of the data suggest that there is one. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished