A flexible two-piece normal dynamic linear model

We construct a flexible dynamic linear model for the analysis and prediction of multivariate time series, assuming a two-piece normal initial distribution for the state vector. We derive a novel Kalman filter for this model, obtaining a two components mixture as predictive and filtering distributions. In order to estimate the covariance of the error sequences, we develop a Gibbs-sampling algorithm to perform Bayesian inference. The proposed approach is validated and compared with a Gaussian dynamic linear model in simulations and on a real data set

Inventory planning with dynamic demand. A state of art review

Proper inventory planning should incorporate factors changing over time, since static factors are not robust to this apparent variability. In models of inventories is necessary to recognize the great demand uncertainty. This paper reviews the state of the art of the most significant developments related to inventory models, especially those who consider dynamic demands in time. In addition, demand forecasting models and some techniques for optimizing inventories are analyzed, considering costs and service levels, among others. In the literature review, gaps have been identified related to the treatment of multivariate inventories as well as the use of Bayesian statistics for the purpose of optimization and the development of demand forecasts

A Journey Into State-Space Models

Questa tesi riguarda la modellizzazione di serie storiche generate da processi latenti, utilizzando modelli "state-space". Vengono proposti nuovi modelli e metodologie per poi applicarli ad una variet\ue0 di casi tipici presenti in finanza ed economia. La tesi \ue8 suddivisa in sei capitoli. Il primo capitolo presenta le motivazioni della ricerca, i suoi obiettivi e la presentazione dei contenuti. Il secondo capitolo approfondisce il concetto di modelli "state-space", riporta e discute le procedure di filtraggio pi\uf9 comuni, e chiarisce alcune definizioni, propriet\ue0 e concetti matematici che verranno usati nei capitoli successivi. Nel Capitolo 3 viene proposto un nuovo modello "state-space" per tener conto delle asimmetrie ("skewness") nelle osservazioni, nel quale l'assunzione di normalit\ue0 non \ue8 pi\uf9 necessaria. La distribuzione normale viene, infatti, sostituita con la distribuzione "close skew-normal" che \ue8 pi\uf9 flessibile ed include la distribuzione normale. Imponendo una struttura auto-regressiva all'equazione di stato e un errore di misura distribuito secondo una "close skew-normal", si costruisce una versione "skewed" del noto filtro di Kalman. Quindi, nel Capitolo 4 si considera la metodologia di filtraggio robusta proposta da Calvet, Czellar and Ronchetti (2015, "Robust Filtering", Journal of the American Statistical Association) con una distribuzione t di Student per ottenere previsioni accurate che tengono conto di valori anomali e di errori di specificazione, sia per i modelli "finite state-space" sia "infinite state-space". Il Capitolo 5 presenta i fondamenti per la costruzione di modelli a volatilit\ue0 stocastica con errori "close skew-normal" nelle osservazioni. Infine, il Capitolo 6 riassume il contributo della tesi e discute possibili future estensioni della ricerca.This thesis is concerned with the modelling of time series driven by unobservable processes using state space models. New models and methodologies are proposed and applied on a variety of real life examples arising from finance and economics. The dissertation is comprised of six chapters. The first chapter motivates the thesis, provides the objectives and discusses the outline of the dissertation contents. In the second chapter, we define the concept of state space modelling, review some popular filtering procedures and recall some important definitions, properties and mathematical concepts that will be used in the subsequent chapters. In Chapter three, we propose a new state-space model that accounts for asymmetry, relaxing the assumption of normality and exploiting the close skew-normal distribution which is more flexible and extends the Gaussian distribution. By allowing a stationary autoregressive structure in the state equation, and a close skew-normal distributed measurement error, we also construct a skewed version of the well known Kalman filter. Then in Chapter four, we adapt the robust filtering methodology of Calvet, Czellar and Ronchetti (2015, "Robust Filtering", Journal of the American Statistical Association) to build a robust filter with Student-t observation density that provides accurate state inference accounting for outliers and misspecification; this for both finite and infinite state-space models. In the fifth chapter, we provide the foundations for the construction of stochastic volatility models with close skew-normal errors in the observation equation. The summary of the thesis, future works and possible extensions appear in Chapter six