In this paper we conduct an out-of-sample forecasting exercise for monthly Swedish air traveler volumes. The models considered are multiplicative seasonal ARIMA, Neural network autoregression, Exponential smoothing, the Prophet model and a Random Walk as a benchmark model. We divide the out-of-sample data into three different evaluation periods: Pre-COVID-19, during COVID-19 and Post-COVID-19 for which we calculate the MAE, MAPE and RMSE for each model in each of these evaluation periods. The results show that for the Pre-COVID-19 period all models produce accurate forecasts, in comparison to the Random Walk model. For the period during COVID-19, no model outperforms the Random Walk, with only Exponential smoothing performing as well as the Random Walk. For the period Post-COVID-19, the best performing models are Random Walk, SARIMA and Exponential smoothing, with all aforementioned models having similar performance.

Anomaly detection on time series forecasts can be used by many industries in especially forewarning systems that can predict anomalies before they happen. Infor (Sweden) AB is software company that provides Enterprise Resource Planning cloud solutions. Infor is interested in predicting anomalies in their data and that is the motivation for this thesis work. The general idea is firstly to forecast the time series and then secondly detect and classify anomalies on the forecast. The first part is time series forecasting and the second part is anomaly detection and classification done on the forecasted values. In this thesis work, the time series forecasting to predict anomalous behaviour is done using two strategies namely the recursive strategy and the direct strategy. The recursive strategy includes two methods; AutoRegressive Integrated Moving Average and Neural Network AutoRegression. The direct strategy is done with ForecastML-eXtreme Gradient Boosting. Then the three methods are compared concerning performance of forecasting. The anomaly detection and classification is done by setting a decision rule based on a threshold. In this thesis work, since the true anomaly thresholds were not previously known, an arbitrary initial anomaly threshold is set by using a combination of statistical methods for outlier detection and then human judgement by the company commissioners. These statistical methods include Seasonal and Trend decomposition using Loess + InterQuartile Range, Twitter + InterQuartile Range and Twitter + GESD (Generalized Extreme Studentized Deviate). After defining what an anomaly threshold is in the usage context of Infor (Sweden) AB, then a decision rule is set and used to classify anomalies in time series forecasts. The results from comparing the classifications of the forecasts from the three time series forecasting methods are unfortunate and no recommendation is made concerning what model or algorithm to be used by Infor (Sweden) AB. However, the thesis work concludes by recommending other methods that can be tried in future research.

In dieser Arbeit untersuchen wir einige Möglichkeiten, financial und soziale Netzwerke zu analysieren, ein Thema, das in letzter Zeit in der ökonometrischen Literatur große Beachtung gefunden hat. Kapitel 2 untersucht den Risiko-Spillover-Effekt über das in White et al. (2015) eingeführte multivariate bedingtes autoregressives Value-at-Risk-Modell. Wir sind an der Anwendung auf nicht stationäre Zeitreihen interessiert und entwickeln einen sequentiellen statistischen Test, welcher das größte verfügbare Homogenitätsintervall auswählt. Unser Ansatz basiert auf der Changepoint-Teststatistik und wir verwenden einen neuartigen Multiplier Bootstrap Ansatz zur Bewertung der kritischen Werte. In Kapitel 3 konzentrieren wir uns auf soziale Netzwerke. Wir modellieren Interaktionen zwischen Benutzern durch ein Vektor-Autoregressivmodell, das Zhu et al. (2017) folgt. Um für die hohe Dimensionalität kontrollieren, betrachten wir ein Netzwerk, das einerseits von Influencers und Andererseits von Communities gesteuert wird, was uns hilft, den autoregressiven Operator selbst dann abzuschätzen, wenn die Anzahl der aktiven Parameter kleiner als die Stichprobegröße ist. Kapitel 4 befasst sich mit technischen Tools für die Schätzung des Kovarianzmatrix und Kreuzkovarianzmatrix. Wir entwickeln eine neue Version von der Hanson-Wright- Ungleichung für einen Zufallsvektor mit subgaußschen Komponenten. Ausgehend von unseren Ergebnissen zeigen wir eine Version der dimensionslosen Bernstein-Ungleichung, die für Zufallsmatrizen mit einer subexponentiellen Spektralnorm gilt. Wir wenden diese Ungleichung auf das Problem der Schätzung der Kovarianzmatrix mit fehlenden Beobachtungen an und beweisen eine verbesserte Version des früheren Ergebnisses von (Lounici 2014). / In this work we explore some ways of studying financial and social networks, a topic that has recently received tremendous amount of attention in the Econometric literature. Chapter 2 studies risk spillover effect via Multivariate Conditional Autoregressive Value at Risk model introduced in White et al. (2015). We are particularly interested in application to non-stationary time series and develop a sequential test procedure that chooses the largest available interval of homogeneity. Our approach is based on change point test statistics and we use a novel Multiplier Bootstrap approach for the evaluation of critical values. In Chapter 3 we aim at social networks. We model interactions between users through a vector autoregressive model, following Zhu et al. (2017). To cope with high dimensionality we consider a network that is driven by influencers on one side, and communities on the other, which helps us to estimate the autoregressive operator even when the number of active parameters is smaller than the sample size. Chapter 4 is devoted to technical tools related to covariance cross-covariance estimation. We derive uniform versions of the Hanson-Wright inequality for a random vector with independent subgaussian components. The core technique is based on the entropy method combined with truncations of both gradients of functions of interest and of the coordinates itself. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an improved high probability version of the recent result of Lounici (2014).