Eldorado Collection:
http://hdl.handle.net/2003/82
2024-06-18T19:54:17ZEssays in time series econometrics
http://hdl.handle.net/2003/42002
Title: Essays in time series econometrics
Authors: Reichold, Karsten
Abstract: This cumulative dissertation consists of three self-contained papers all contributing to the cointegrating regression literature. The first chapter is devoted to classical linear cointegrating regressions, i.e., regressions that contain integrated processes as regressors. It combines traditional and self-normalized Wald-type test statistics with a vector autoregressive sieve bootstrap to reduce size distortions of hypothesis tests on the cointegrating vector. The second chapter focuses on panels of cointegrating polynomial regressions, i.e., panels of regressions that include an integrated process and its powers as regressors. It derives the asymptotic properties of a group-mean fully modified OLS estimator and hypothesis tests based upon it in a fixed cross-section and large time series dimension. The third chapter is devoted to testing for a cointegrating relationship between a fixed number of integrated processes. In particular, it derives asymptotic theory for an existing nonparametric variance ratio unit root test (originally proposed to test for an unit root in an observed univariate time series) when applied to regression residuals.2023-01-01T00:00:00ZEssays on cointegration analysis in the state space framework
http://hdl.handle.net/2003/40515
Title: Essays on cointegration analysis in the state space framework
Authors: Matuschek, Lukas
Abstract: Cointegration analysis is by now a standard tool in multivariate time series analysis with application ranging from economics to climate science. It was formalized by Soren Johansen and Katarina Juselius and their co-authors for VAR processes. This dissertation, consisting of three chapters corresponding to three articles written in collaboration with my co-authors Professor Dietmar Bauer, Patrick de Matos Ribeiro and Professor Martin Wagner, extends the cointegration theory to VARMA processes using a representation by state space systems.
Chapter 1 focuses on theoretical results regarding the sets of transfer functions corresponding to VARMA systems with similar cointegrating properties, summarized in the so-called state space unit root structure. We develop and discuss different parameterizations for vector autoregressive moving average processes with arbitrary unit roots and (co)integration orders and discuss their topological properties. The general results are exemplified in detail for the empirically most relevant cases, the (multiple frequency or seasonal) I(1) and the I(2) case.
In Chapter 2 we show that the Johansen framework for testing hypotheses on the cointegrating ranks and spaces for MFI(1) processes can be extended to the class of VARMA processes and introduce a state space error correction representation. The estimated cointegrating vectors are asymptotically mixed Gaussian and pseudo likelihood ratio tests for the cointegrating ranks have the same distributions under the null hypothesis in the VARMA case as in the VAR case. In a simulation study our tests outperform the tests by Johansen and Schaumburg in small samples.
In Chapter 3 we develop estimation and inference techniques for I(2) cointegrated VARMA processes cast in state space format. We show consistency and derive the asymptotic distributions of estimators maximizing the Gaussian pseudo likelihood function. Furthermore, we discuss hypothesis tests for the state space unit root structure, leading to the well-known limiting distributions for VAR I(2) processes. Again, a small simulation study shows favorable results for small samples, with our test leading to better performance in determining these integer parameters.2021-01-01T00:00:00ZLimit theorems for locally stationary processes
http://hdl.handle.net/2003/40226
Title: Limit theorems for locally stationary processes
Authors: Kawka, Rafael
Abstract: We present limit theorems for locally stationary processes that have a one sided time-varying moving average representation. In particular, we prove a central limit theorem (CLT), a weak and a strong law of large numbers (WLLN, SLLN) and a law of the iterated logarithm (LIL) under mild assumptions using a time-varying Beveridge–Nelson decomposition.2020-10-01T00:00:00ZPseudo maximum likelihood estimation of cointegrated multiple frequency I(1) VARMA processes using the state space framework
http://hdl.handle.net/2003/40060
Title: Pseudo maximum likelihood estimation of cointegrated multiple frequency I(1) VARMA processes using the state space framework
Authors: Ribeiro, Patrick de Matos
Abstract: Since the seminal contribution of Clive W.J. Granger that introduced the concept of cointegration, the modeling of multivariate (economic) time series with models and methods that allow for unit roots and cointegration has become standard econometric practice with applications ranging from macroeconomics through finance to climate science. With some early exceptions most authors focus on the VAR framework, most notably Johaansen who developed vector error correction models for the empirically most relevant cases, the I(1) and the I(2) case. Limiting cointegration analysis to VAR processes may be too restrictive. For several reasons discussed in this theses it may be advantageous to use the more general VARMA framework. However, cointegration analysis in theVARMA framework is complicated, in particular in the case of higher integration orders or multiple unit roots. One possibility to overcome the difficulties for the cointegration analysis of VARMA processes is the usage of the state space framework. This dissertation provides important tools for cointegration analysis in the state space framework, namely a continuous parameterization and a pseudo maximum likelihood estimator for the multiple frequency I(1) case.
Chapter 1 discusses the parameterization of state space processes of arbitrary integration orders. Since the state space representation of a stochastic process is not unique, a canonical form is necessary which selects one unique state space representation. Since this canonical form places restrictions on the system matrices, not all entries of the matrices are free parameters. Some entries are restricted to be zero or depend on other entries. The parametrization is based on the canonical form of Bauer and Wagner (2012), which is particuarly well suited for cointegration analysis. Since there is no continuous parameterization for all state space systems of a given system order, we partition the set of all systems into subsets on which a continuous parameterization is possible. For this we use a multi-index which is chosen in such a way that properties like the unit roots, integration orders and dimensions of the cointegrating spaces remain constant in each subset. In addition to deriving a continuous parametrization, which is almost everywhere continuously invertible, we find a generic subset which is open and dense in the set of all integrated processes with a state space representation of a given system order. Additionally, we discuss the topological structure of the subsets, defining a partial ordering of the multi-indices.
Finally, we discuss the implementation of hypotheses on the cointegrating ranks and spaces in the parametrization for the empirically most relevant cases, the multiple frequency I(1) and the I(2) case. We show that all hypotheses commonly tested for VAR processes in these cases can be implemented in the state space framework. This potentially allows for the derivation of pseudo likelihood ratio tests for these hypotheses.
Chapter 2 examines pseudo maximum likelihood estimation for multiple frequency I(1) processes. We derive the likelihood function for MFI(1) processes and show that the pseudo maximum likelihood estimator is consistent under relatively mild conditions. Additionally, we show that setting the starting values of the state process to zero does not affect the asymptotic properties of the pseudo maximum likelihood estimator. For the case of a correctly chosen multi-index we additionally derive the asymptotic distribution of the pseudo maximum likelihood estimator, providing the ground work for pseudo likelihood ratio tests. \\
Finally, Chapter 3 consists of an useful tutorial for the analysis of economic time series using the state space framework. Using the analysis of King, Plosser, Stock and Watson (1991) as an illustrative example, we demonstrate that all economically relevant questions examined by these authors can also be analyzed using the state space framework. The analysis of King, Plosser, Stock and Watson (1991) is based on quarterly US economic data from 1949 to 1988. We compare the methods developed for the state space framework, namely the pseudo maximum likelihood estimator from Chapter 2 and the tests based upon it to the methods used by King, Plosser, Stock and Watson (1991), i.e., the DOLS estimator of and the tests for the cointegrating rank of Stock and Watson (1988) and to the vector error correction model for I(1) processes by Johansen (1995). The results obtained with the three different approaches differ, which indicates that the results of empirical applications to time series of dimension six or more of sample sizes below two or three hundred should be interpreted with care.
Additionally, we test the robustness of the vector error correction model and the state space framework by repeating the analysis on an extended data set with quarterly US economic data from 1949 to 2018 and on the subset with data from 1989 to 2018. The results of both approaches differ for the three data sets. This may be a hint that there are structural breaks in the economic time series.2020-01-01T00:00:00ZEssays on cointegrating polynomial regressions with applications to the EKC
http://hdl.handle.net/2003/36054
Title: Essays on cointegrating polynomial regressions with applications to the EKC
Authors: Grabarczyk, Peter
Abstract: We consider cointegrating polynomial regression (CPR) relationships as a special case of nonlinear cointegrating relationships, which steadily gain interest recently in many different areas of applied research, e. g., empirical macroeconomics, empirical finance or environmental economics. CPRs include deterministic variables as well as polynomially transformed integrated variables as explanatory variables and stationary errors. The stochastic regressors are allowed to be endogenous and the errors are allowed to be serially correlated. In this case, the limiting distribution of the OLS estimator is contaminated by second order bias terms. Therefore, various modified OLS estimators are considered that have a zero mean Gaussian mixture limiting distribution and allow for valid asymptotic chi-squared inference. The main motivation for developing estimation and testing techniques for CPRs is the analysis of the environmental Kuznets curve (EKC) hypothesis, which postulates an inverted U-shaped relationship between measures of economic activity, typically proxied by GDP per capita, and pollution. The logarithm of GDP per capita is often found to be integrated of order one. The hypothesized inverted U-shape requires the inclusion of log GDP per capita and its square, or even higher order powers, as explanatory variables. Polynomially transformed integrated processes are not integrated themselves, which is neglected in the large part of the EKC literature. Instead, powers of integrated processes are considered to be integrated and standard unit root and cointegration techniques are applied. We show that the standard fully modified OLS estimator has the same asymptotic distribution in the CPR setting as its tailor-made CPR extension. For cointegration testing, however, the use of standard techniques in CPRs has an effect also asymptotically, since invalid critical values are carried out. A simulation study underlines the theoretical findings in the sense that the difference between standard cointegration methods and their CPR extensions are small in terms of bias and root mean squared error. However, tests based upon the latter outperform tests based upon the former in terms of lower over-rejections under the null and larger (size-corrected) power for hypothesis testing as well as cointegration testing. We also consider an extension of the recently developed integrated modified OLS (IM-OLS) estimator for CPRs, which is shown to have a zero mean Gaussian mixture limiting distribution that forms the basis for asymptotic standard inference. Furthermore, we provide fixed-b asymptotic theory to capture the impact of kernel and bandwidth choices on the sampling distributions of the estimators. A simulation study shows that IM-OLS based tests can lead to substantially smaller size distortions for hypothesis testing under the null at the cost of some minor loss in size-corrected power compared to other modified OLS based tests. We also apply the developed methods to analyze the EKC hypothesis based on a data set containing carbon dioxide emissions and gross domestic product for 19 early industrialized countries over the period 1870-2013. By means of an IM-OLS residual based cointegration test, we find evidence for the existence of a quadratic EKC relationship for six countries and in one additional country for a cubic EKC relationship. Finally, we analyze the EKC hypothesis for carbon dioxide emissions in a multi-country system of equations approach for six early industrialized countries. A seemingly unrelated cointegrating polynomial regressions approach allows to take into account cross-sectional dependence and parameter heterogeneity. Wald-type tests are carried out to test for poolability, i.e. equality of parameters, for subsets of coefficients over potentially different subsets of cross-sections. Subsequent estimation in a group-wise pooled setting reduces the number of estimated parameters about one third in the empirical application, whereas the estimation results are similar to those obtained in unrestricted individual CPRs. We also show that estimation in a panel-type approach including cross-sectional parameter homogeneity, as typically pursued in the empirical EKC literature, is rejected by poolability testing and performs severely worse.2017-01-01T00:00:00ZIn search of Q: results on identification in structural vector autoregressive models
http://hdl.handle.net/2003/36044
Title: In search of Q: results on identification in structural vector autoregressive models
Authors: Uhrin, Gábor B.
Abstract: The thesis consists of three independent research papers in the area of identification of structural vector autoregressive (SVAR) models.
The first paper studies the government spending shocks on output. New sign restrictions on the impulse responses are introduced to the fiscal policy literature. Upon imposing the new sign restrictions, we find contractionary output reactions to positive government spending shocks. The results are robust to several alternative model and sign restriction specifications.
The second paper investigates the implications of including more (forward-looking) information in a classical monetary policy SVAR. We augmenting the classical specification with federal funds futures and factors extracted from a large macroeconomic dataset. It is established that information-augmentation does not necessarily yield shocks that are correlated more strongly with benchmark monetary policy shocks. The empirical conclusions based on the baseline and augmented specifications are similar: monetary policy shocks contribute only marginally to the evolution of (real) macroeconomic variables.
The third paper‘s empirical motivation is to quantify the effects of monetary policy shocks on asset prices. We set up a monetary SVAR including the S&P500 series, and we use set identifying sign and zero restrictions. It is established that the majority of monetary policy shocks estimated from admissible structural models correlate only weakly with benchmark monetary policy shocks. Structural models with medium correlations, imply impulse responses that vary widely in their shapes and magnitudes. Concentrating, however, only on the 100 models with the highest correlations uncovers negative, but near-zero asset price responses to positive monetary policy shocks, coupled with mildly positive output responses.2017-01-01T00:00:00ZOptimal estimation of nonlinear functions of distribution parameters
http://hdl.handle.net/2003/26535
Title: Optimal estimation of nonlinear functions of distribution parameters
Authors: Moya, Ana2009-12-01T13:56:50ZModellierung von Kapitalmarktrenditen mittels asymmetrischer GARCH-Modelle
http://hdl.handle.net/2003/2784
Title: Modellierung von Kapitalmarktrenditen mittels asymmetrischer GARCH-Modelle
Authors: Schoffer, Olaf
Abstract: In der vorliegenden Arbeit wird die Modellierung von Kapitalmarktrenditen mittels asymmetrischer GARCH-Prozesse betrachtet. Insbesondere werden die Eigenschaften des A-PARCH-Modells sowie seiner Erweiterungen untersucht. Dieses Modell ermöglicht nicht allein die Modellierung von Volatilitätsschwankungen und Hochgipfligkeit wie das GARCH-Modell. Es ist ebenso geeignet, das Charakteristikum der Asymmetrie nachzubilden. Ferner können bedingte Volatilitäten als nichtganzzahlige Potenz vom Absolutbetrag der Beobachtungen beschrieben werden. Langes Gedächtnis in den bedingten Volatilitäten kann durch einen A-PARCH-Prozeß jedoch nicht modelliert werden, da ihre Autokorrelationsfunktion exponentiell und nicht hyperbolisch abfällt.Zunächst wird ein Vergleich von Modellanpassungen für Renditen von Aktienkursdaten durchgeführt. Dabei weist der A-PARCH-Prozeß mit bedingter t-Verteilung gegenüber anderen symmetrischen und asymmetrischen GARCH-Prozessen die kleinsten Werte für das Informationskriterium SBC (Schwarz-Bayes-Informationskriterium) sowie für das Maß für die Vorhersagegüte MSPE (mittlerer quadratischer Prognosefehler) auf. Er wird somit bei der Modellierung von Strukturen dieser Kapitalmarktdaten bevorzugt.Weiter wird die Hebelwirkungshypothese als mögliche Ursache für Asymmetrie in Kapitalmarktdaten mittels symmetrischer und asymmetrischer GARCH-Modelle untersucht. Unter anderem wird dabei für Modellanpassungen an Renditen von Aktien- und Wechselkursdaten sowie Edelmetallpreisen und Zinssätzen der Wert für SBC betrachtet. Die jeweilige Bevorzugung von symmetrischen bzw. asymmetrischen Modellen für die Anpassung an die beschriebenen Kapitalmarktrenditen stützt die Plausibilität der Hebelwirkungshypothese. Obwohl es auch weiterhin keine endgültige Aussage über die Gültigkeit dieser Hypothese gibt, sollte ihr bei der Modellauswahl, d.h. symmetrischer gegenüber asymmetrischem Ansatz, Rechnung getragen werden. Asymmetrische Modelle sollten demnach nur für Daten verwendet werden, für die das Zugrundeliegen einer Hebelwirkung sinnvoll ist.Um langes Gedächtnis als Eigenschaft von Kapitalmarktrenditen nachbilden zu können, gibt es verschiedene Erweiterungen des ursprünglichen GARCH-Ansatzes. In der vorliegenden Arbeit wird das FI-A-PARCH-Modell sowie sein Spezialfall FIGARCH betrachtet. Diese ergeben sich aus A-PARCH- bzw. GARCH-Modellen durch geeignetes Hinzufügen des fraktionalen Differenzenoperators (1 - B)d. Zur praktischen Umsetzung dieser Modelle ist jedoch eine Abschätzung dieses Operators mittels binomischer Reihe notwendig. Über die in dieser Abschätzung zu verwendende Anzahl von Beobachtungen gibt es jedoch bisher kaum konkrete Aussagen. Daher werden Parameterschätzungen in FI-A-PARCH-Modellen in Abhängigkeit von dieser Beobachtungszahl betrachtet. Es kann festgestellt werden, daß sich die Werte der Schätzungen für die vorliegenden Modellanpassungen etwa ab einem Wert von 300 Beobachtungen stabilisieren. Basierend auf diesem Ergebnis wird empfohlen, für ähnliche Probleme eine Anzahl von ca. 300 Beobachtungen für die Abschätzung von (1 - B)d zu verwenden.Obwohl der FI-A-PARCH-Prozeß zur Modellierung von langem Gedächtnis entwickelt wurde, besitzen seine bedingten Volatilitäten nicht die Eigenschaft langen Gedächtnisses in dem Sinne, daß die Autokorrelationsfunktion hyperbolisch fällt. Da die zweiten Momente analog zu denen des FIGARCH-Prozesses unendlich sind, existiert die betrachtete Autokorrelationsfunktion nicht. In einem weiteren Vergleich von Modellanpassungen nach MSPE wird der FI-A-PARCH-Prozeß mit bedingter t-Verteilung zwar nicht in jedem Fall bevorzugt, er erzielt jedoch jeweils die kleinsten Werte für SBC und kann somit als sinnvolle Ergänzung der zuvor vorgestellten Modelle angesehen werden.Ein Modell, welches die Nachbildung von langem Gedächtnis in den bedingten Volatilitäten ermöglicht, ist das Hyperbolische GARCH-Modell. Jedoch kann damit keine Asymmetrie beschrieben werden. Analog zur Herleitung des HYGARCH-Modells wird daher in der vorliegenden Arbeit das A-PARCH-Modell zum Hyperbolischen A-PARCH-Modell erweitert. Eigenschaften dieses Modells können unter anderem mittels Volterra-Reihenentwicklung von Asymmetrischen Power-GARCH-Modellen hergeleitet werden. Die Existenz der zweiten Momente wird gezeigt sowie die dazu notwendigen Bedingungen hergeleitet. Weiterhin kann das Vorliegen von langem Gedächtnis in der Transformation {(|yt| - yt)d} des Hyperbolischen A-PARCH-Prozesses {yt} nachgewiesen werden. Für den Prozeß {|yt|d}, welcher die bedingten Volatilitäten repräsentiert, kann daraus zunächst jedoch keine Aussage über die Autokorrelationsstruktur abgeleitet werden. Das HY-A-PARCH-Modell ermöglicht also die Beschreibung der Charakteristika Volatilitätsschwankungen, Hochgipfligkeit und Asymmetrie sowie das Vorliegen von langem Gedächtnis zumindest für eine Transformation der bedingten Volatilitäten. Der Nachweis von langem Gedächtnis in der Reihe {|yt|d} von HY-A-PARCH-Prozessen bleibt somit zunächst der künftigen Forschung vorbehalten. Als Ansatz zur Lösung dieses Problems wird die Methode der Appell-Polynome vorgeschlagen.; The available thesis contemplates the modeling of returns of financial markets using asymmetric GARCH processes. In particular the characteristics of the A-PARCH model as well as its extensions are examined. This model not only allows for modeling volatility clustering and leptocurtosis like the GARCH model. It is suitable to reproduce the characteristic of asymmetry as well. Furthermore, conditional volatilities can be described as none-integer-numbered power by the absolute value of the observations. However, long memory in the conditional volatilities cannot be modeled by an A-PARCH process since their autocorrelation function decays exponentially but not hyperbolically.First a comparison of model fits for returns of asset prices is accomplished. In relation to other symmetrical and asymmetrical GARCH processes the A-PARCH process with conditional t-distribution exhibits the smallest values for SBC (Schwarz-Bayes information criterion) as well as for the measure for the forecast quality MSPE (mean square prediction error). Thus, it is preferred by modeling structures of these capital market data.Moreover, the leverage hypothesis is examined as a possible cause for asymmetry in financial market data by symmetrical and asymmetrical GARCH models. Thereby, the value for SBC is inspected for model fits of asset returns and exchange rates as well as for precious metal prices and interest rates. The respective preference of symmetrical or asymmetrical models for the described data supports the plausibility of the leverage hypothesis. This effect should be accounted for model selection, i.e. for symmetrical vs. asymmetrical approach, even though there is still no final statement about the validity of this hypothesis. Therefore, asymmetrical models should be used only for data, for which it is reasonable to imply a leverage effect.In order to be able to reproduce long memory as characteristic of financial market returns, there are several extensions of the original GARCH approach. In the available work the FI-A-PARCH model as well as its special case FIGARCH are considered. These results by suitable adding of the fractional difference operator (1 - B)d from A-PARCH and GARCH models, respectevly. However, an approximation of this operator is necessary for practical implementation of these models using binomial series. But there are hardly concrete statements about the number of observations, which have to be used in this approximation, up to now. Therefore, the parameter estimations in FI-A-PARCH models are examined as a function of this observation number. It can be observed that the values of the estimations for these model fits stabilize from a value of approx. 300 observations. Based on this result, it is recommended to use a number of 300 observations for the estimation of (1 - B)d at similar problems.Although the FI-A-PARCH process was developed to model long term dependence, its conditional volatilities does not possess the characteristic of long memory in the sense that the autocorrelation function decays hyperbolically. Since the second moments are infinite similarly to those of the FIGARCH process, the appropriate autocorrelation function does not exist. In a further comparison of model fits using MSPE the FI-A-PARCH process with conditional t-distribution is not preferred in every case. However, the smallest value for SBC were obtained using this model. Thus it can be regarded as a reasonable addition of the models presented before.A model, which allows for reproduction of long memory in the conditional volatilities, is the Hyperbolic GARCH model. However, it cannot describe asymmetry. Thus, in this work the A-PARCH model is extended to the Hyperbolic A-PARCH model analogous to the derivation of the HYGARCH model. Characteristics of this model can be derived among other things using Volterra series expansion with Asymmetric Power GARCH models. The existence of the second moments is shown as well as the appropriate necessary condition is derived. Further the presence of long memory in the transformation {(|yt| - yt)d} of the hyperbolic A-PARCH-process {yt} is proved. However, for the process {|yt|d}, which represents the conditional volatilities, up to now no conclusion can be derived about the autocorrelation structure. Thus, the HY-A-PARCH model allows for description of the characteristics for volatility clustering, leptocurtosis and asymmetry as well as the presence of long memory at least for a transformation of the conditional volatilities. The proof of long memory in the series {|yt|d} of HY-A-PARCH processes at first remains for future research. The method of Appell polynomials is suggested as approach for solution of this problem.2003-07-24T00:00:00Z