Identification of Generalized Dynamic Factor Models from mixed-frequency data

Modeling of high dimensional time series by linear time series models such as vector autoregressive models is often marred by the so-called “curse of dimensionality”. In order to overcome this problem generalized linear dynamic factor models (GDFM’s) maybe used. In high-dimensional time series the single univariate time series are often sampled at different frequencies. This is the so-called mixed-frequency situation. We consider identifiability of the underlying high-frequency GDFM (i.e. the GDFM generating the data at the highest sampling frequency occurring) in the case of mixed frequency data and we shortly describe two estimation procedures in this situation based on the EM algorithm.Brian Anderson is supported by Data-61, CSIRO and by the Australian Research Council's Discovery Project DP-160104500

Weak and strong cross section dependence and estimation of large panels

This paper introduces the concepts of time-specific weak and strong cross section dependence. A double- indexed process is said to be cross sectionally weakly dependent at a given point in time, t, if its weighted average along the cross section dimension (N) converges to its expectation in quadratic mean, as N is increased without bounds for all weights that satisfy certain ‘granularity’ conditions. Relationship with the notions of weak and strong common factors is investigated and an application to the estimation of panel data models with an infinite number of weak factors and a finite number of strong factors is also considered. The paper concludes with a set of Monte Carlo experiments where the small sample properties of estimators based on principal components and CCE estimators are investigated and compared under various assumptions on the nature of the unobserved common effects. JEL Classification: C10, C31, C33Panels, Strong and Weak Cross Section Dependence, Weak and Strong Factors

Weak and Strong Cross Section Dependence and Estimation of Large Panels

This paper introduces the concepts of time-specific weak and strong cross section dependence. A double-indexed process is said to be cross sectionally weakly dependent at a given point in time, t, if its weighted average along the cross section dimension (N) converges to its expectation in quadratic mean, as N is increased without bounds for all weights that satisfy certain ‘granularity’ conditions. Relationship with the notions of weak and strong common factors is investigated and an application to the estimation of panel data models with an infinite number of weak factors and a finite number of strong factors is also considered. The paper concludes with a set of Monte Carlo experiments where the small sample properties of estimators based on principal components and CCE estimators are investigated and compared under various assumptions on the nature of the unobserved common effects.panels, strong and weak cross section dependence, weak and strong factors

Generalized Linear Dynamic Factor Models: An Approach via Singular Autoregressions

We consider generalized linear dynamic factor models. These models have been developed recently and they are used for high dimensional time series in order to overcome the "curse of dimensionality". We present a structure theory with emphasis on the zeroless case, which is generic in the setting considered. Accordingly the latent variables are modeled as a possibly singular autoregressive process and (generalized) Yule-Walker equations are used for parameter estimation. The Yule-Walker equations do not necessarily have a unique solution in the singular case, and the resulting complexities are examined with a view to find a stable and coprime system

Generalized Linear Dynamic Factor Models - An Approach via Singular Autoregressions

We consider generalized linear dynamic factor models. These models have been developed recently and they are used for high dimensional time series in order to overcome the "curse of dimensionality". We present a structure theory with emphasis on the zeroless case, which is generic in the setting considered. Accordingly the latent variables are modeled as a possibly singular autoregressive process and (generalized) Yule-Walker equations are used for parameter estimation. The Yule-Walker equations do not necessarily have a unique solution in the singular case, and the resulting complexities are examined with a view to find a stable and coprime system

Comparing three estimation methods in the context of generalized dynamic factor models

Verallgemeinerte lineare dynamische Faktormodelle (GDFM's) dienen der Analyse hoch-dimensionaler Zeitreihen, wo die Anzahl der einzelnen Zeitreihen relativ groß im Verhältnis zur Stichprobengröße ist. Sie wurden in Forni et al. (2000), Forni und Lippi (2001) sowie Stock und Watson (2002a) eingeführt, und verallgemeinern lineare dynamische Faktormodelle mit strikt idiosynkratischen Fehlern (Geweke (1977)) bzw. kombinieren diese mit verallgemeinerten statischen Faktormodellen (Chamberlain (1983), Chamberlain und Rothschild (1983)). Erfolgreiche Anwendungsgebiete dieser Modelle sind die Prognose sowie die Strukturanalyse von länderübergreifenden disaggregierten Finanz- und makroökonomischen Daten. GDFM's erlauben zwar eine schwache Abhängigkeit in den Fehlern, sind dadurch jedoch nur noch asymptotisch identifizierbar. Diese Diplomarbeit betrachtet drei Schätzmethoden für den nicht beobachtbaren statischen Faktorprozess, die auf vereinfachten identifizierbaren Modellen beruhen. Das Ziel dieser Arbeit ist es in einer Simulationsstudie die Rolle der Missspezifikation bei Anwendung der Schätzer im verallgemeinerten Kontext herauszuarbeiten, sowie herauszufinden, unter welchen Umständen eine Schätzmethode die andere dominiert. Der principal component estimator (PC-Schätzer) entsteht aus der Spektralzerlegung der Kovarianzmatrix der Beobachtungen. Er schätzt die Faktoren durch die Berechnung der ersten Hauptkomponenten des Beobachtungsprozesses. Der two-stage estimator (TS-Schätzer) berechnet jene Linearkombination aller Beobachtungen, die den quadratischen Abstand zu den statischen Faktoren komponentenweise minimieren. Er entspricht also der komponentenweise orthogonalen Projekten der statischen Faktoren auf den linearen Raum, der von den Komponenten aller Beobachtungen aufgespannt wird. Berechnet wird dieser durch die Aufstellung eines Zustandsraum-Modells und die Anwendung der Kalman Filter Rekursionen. Die Parameter werden durch die PC-Schätzer sowie die Lösung der Yule-Walker Gleichungen bestimmt. Der quasi-maximum likelihood estimator (QML-Schätzer) entspricht ebenfalls der orthogonalen Projektion der statischen Faktoren auf den Raum aller Beobachtungen, jedoch unter anderen geschätzten Parametern für das Zustandsraum-Modell. Es wird gezeigt, dass im Falle von white-noise Fehlern mit diagonaler Kovarianzmatrix, der zu erwartende relative Vorteil der TS- und QML-Schätzer gegenüber dem PC-Schätzer eintritt. Kommen die Daten aus einem verallgemeinerten dynamischen Faktormodell, welches durch Missspezifikation des Fehlerspektrums gekennzeichnet ist, so geht dieser Vorteil zurück bzw. verloren, und bei hoher lokaler Abhängigkeit zwischen den idiosynkratischen Komponenten kann er sich sogar in einen Nachteil verwandeln. Ein langes Gedächtnis der statischen Faktoren hat ebenso Einfluss auf die absolute wie die relative Schätzgenauigkeit, wie das Ausmaß des Rausch-Anteils am latenten Signal.This thesis considers three different estimation methods for the unobserved static factors z(t) in the context of generalized dynamic factor models. These models assume that a high dimensional multivariate time-series x(t) is driven by a very low dimensional factor process z(t). This unobserved factors partly explain the observations x(t) through the equation x(t) = Lz(t) + u(t). The models are generalized, because they allow for weak cross-sectional dependence among the idiosyncratic components u(i,t) . A drawback of this general assumption is that the common (Lz(t)) and idiosyncratic component u(t) are only asymptotically (for the number of different time-series going to infinity) identified. Therefore all three estimation methods for the static factors are based on restricted models. Despite these restrictions they consistently estimate the linear space of static factors under generalized assumptions. However, no further analytical properties for all three methods are available. Therefore a simulation study was conducted, in order to understand their behavior and compare their performances for finite dimensional panels. Also a data set from the US macroeconomy was analyzed in order to get authentic parameters for the simulation study. The aim of this simulation study was to study the performance of the estimators for different models and parameters. The results suggest that all three estimators suffer from neglecting the generalized dependence structure of the idiosyncratic component. If the idiosyncratic process is white noise, a ranking of the estimators seems to be possible. However, high local dependencies could weaken or even erase relative advantages that could be observed in the white noise case. This effect seems to be stronger for the cases where the dimension of the dynamic factors is smaller than the dimension of the static factors. Different noise-to-signal ratios and especially a long-memory static factor process z(t) clearly influence absolute and relative performance of the estimators