Inference in VARs with Conditional Heteroskedasticity of Unknown Form

We derive a framework for asymptotically valid inference in stable vector autoregressive (VAR) models with conditional heteroskedasticity of unknown form. We prove a joint central limit theorem for the VAR slope parameter and innovation covariance parameter estimators and address bootstrap inference as well. Our results are important for correct inference on VAR statistics that depend both on the VAR slope and the variance parameters as e.g. in structural impulse response functions (IRFs). We also show that wild and pairwise bootstrap schemes fail in the presence of conditional heteroskedasticity if inference on (functions) of the unconditional variance parameters is of interest because they do not correctly replicate the relevant fourth moments' structure of the error terms. In contrast, the residual-based moving block bootstrap results in asymptotically valid inference. We illustrate the practical implications of our theoretical results by providing simulation evidence on the finite sample properties of different inference methods for IRFs. Our results point out that estimation uncertainty may increase dramatically in the presence of conditional heteroskedasticity. Moreover, most inference methods are likely to understate the true estimation uncertainty substantially in finite samples

Bootstrapping Sample Quantiles of Discrete Data

Sample quantiles are consistent estimators for the true quantile and satisfy central limit theorems (CLTs) if the underlying distribution is continuous. If the distribution is discrete, the situation is much more delicate. In this case, sample quantiles are known to be not even consistent in general for the population quantiles. In a motivating example, we show that Efron’s bootstrap does not consistently mimic the distribution of sample quantiles even in the discrete independent and identically distributed (i.i.d.) data case. To overcome this bootstrap inconsistency, we provide two different and complementing strategies. In the first part of this paper, we prove that m-out-of-n-type bootstraps do consistently mimic the distribution of sample quantiles in the discrete data case. As the corresponding bootstrap confidence intervals tend to be conservative due to the discreteness of the true distribution, we propose randomization techniques to construct bootstrap confidence sets of asymptotically correct size. In the second part, we consider a continuous modification of the cumulative distribution function and make use of mid-quantiles studied in Ma, Genton and Parzen (2011). Contrary to ordinary quantiles and due to continuity, mid-quantiles lose their discrete nature and can be estimated consistently. Moreover, Ma, Genton and Parzen (2011) proved (non-)central limit theorems for i.i.d. data, which we generalize to the time series case. However, as the mid-quantile function fails to be differentiable, classical i.i.d. or block bootstrap methods do not lead to completely satisfactory results and m-out-of-n variants are required here as well. The finite sample performances of both approaches are illustrated in a simulation study by comparing coverage rates of bootstrap confidence intervals

Proxy SVARs : asymptotic theory, bootstrap inference, and the effects of income tax changes in the United States

Proxy structural vector autoregressions (SVARs)identify structural shocks in vector autoregressions (VARs) with external proxy variables that are correlated with the structural shocks of interest but uncorrelated with other structural shocks. We provide asymptotic theory for proxy SVARs when the VAR innovations and proxy variables are jointly a-mixing. We also prove the asymptotic validity of a residual-based moving block bootstrap (MBB) for inference on statistics that depend jointly on estimators for the VAR coeffcients and for covariances of the VAR innovations and proxy variables. These statistics include structural impulse response functions (IRFs). Conversely, wild bootstraps are invalid, even when innovations and proxy variables are either independent and identically distributed or martingale difference sequences, and simulations show that their coverage rates for IRFs can be badly mis-sized. Using the MBB to re-estimate confidence intervals for the IRFs in Mertens and Ravn (2013), we show that inferences cannot be made about the effects of tax changes on output, labor, or investment

Prototypes as Explanation for Time Series Anomaly Detection

Detecting abnormal patterns that deviate from a certain regular repeating pattern in time series is essential in many big data applications. However, the lack of labels, the dynamic nature of time series data, and unforeseeable abnormal behaviors make the detection process challenging. Despite the success of recent deep anomaly detection approaches, the mystical mechanisms in such black-box models have become a new challenge in safety-critical applications. The lack of model transparency and prediction reliability hinders further breakthroughs in such domains. This paper proposes ProtoAD, using prototypes as the example-based explanation for the state of regular patterns during anomaly detection. Without significant impact on the detection performance, prototypes shed light on the deep black-box models and provide intuitive understanding for domain experts and stakeholders. We extend the widely used prototype learning in classification problems into anomaly detection. By visualizing both the latent space and input space prototypes, we intuitively demonstrate how regular data are modeled and why specific patterns are considered abnormal

Goodness of fit testing based on graph functionals for homogenous Erd\"os Renyi graphs

The Erd\"os Renyi graph is a popular choice to model network data as it is parsimoniously parametrized, straightforward to interprete and easy to estimate. However, it has limited suitability in practice, since it often fails to capture crucial characteristics of real-world networks. To check the adequacy of this model, we propose a novel class of goodness-of-fit tests for homogeneous Erd\"os Renyi models against heterogeneous alternatives that allow for nonconstant edge probabilities. We allow for asymptotically dense and sparse networks. The tests are based on graph functionals that cover a broad class of network statistics for which we derive limiting distributions in a unified manner. The resulting class of asymptotic tests includes several existing tests as special cases. Further, we propose a parametric bootstrap and prove its consistency, which allows for performance improvements particularly for small network sizes and avoids the often tedious variance estimation for asymptotic tests. Moreover, we analyse the sensitivity of different goodness-of-fit test statistics that rely on popular choices of subgraphs. We evaluate the proposed class of tests and illustrate our theoretical findings by extensive simulations

Der multiple hybride Bootstrap und Testen auf periodische Stationarität im Spektralbereich

In the first part of this thesis, a new bootstrap procedure for dependent data is proposed and its properties are discussed. Under the assumption of a linear process, the idea of the autoregressive aided periodogram bootstrap (AAPB) of Kreiss and Paparoditis (2003) is reconsidered and in two directions generalized and complemented, respectively. On the one hand, the AAPB is modified in such a way that it is eventually able to generate bootstrap observations in the time domain, which is not possible for the AAPB. On the other hand, multivariate processes of arbitrary dimension are considered. It is shown that the multiple hybrid bootstrap (mHB) that includes the AAPB as a special case, is consistent under quite general assumptions for the sample mean and for kernel spectral density estimates. Moreover, for autocovariances and autocorrelations, different results between the univariate and the multivariate case are discussed. The second part deals with multivariate linear periodically stationary models, which generalize the usual stationary linear models in that effect that their coefficients are no longer assumed to be constant over time, but to behave periodically. These models may be represented as higher-dimensional stationary models and it is shown that their autocovariance structures as well as their spectral densities form upon specific patterns if and only if the underlying processes are actually not just periodically stationary, but also stationary. To test for stationarity, a test statistic based on nonparametric spectral density estimates is constructed that takes advantage of this specific shape. The asymptotic normal distribution of the test statistic is derived and it is shown that the test is asymptotically consistent. Moreover, it is demonstrated how to use the test statistic to test for periodic stationarity with shorter period. Finally, the mHB is used to obtain critical values that are more adequate than those from the CLT.Im ersten Teil wird ein neues Bootstrapverfahren für abhängige Daten vorgeschlagen und dessen Eigenschaften werden diskutiert. Unter der Annahme eines linearen Prozesses, wird die Idee des autoregressive aided periodogram bootstrap (AAPB) von Kreiss und Paparoditis (2003) neu aufgegriffen und in zwei Richtungen verallgemeinert bzw. ergänzt. Zum einen wird das dort untersuchte Bootstrapverfahren so modifiziert, dass es schließlich in der Lage ist, Beobachtungen im Zeitbereich zu erzeugen, was dem AAPB nicht möglich ist. Zum anderen werden multivariate Prozesse beliebiger Dimension betrachtet. Es wird gezeigt, dass der multiple hybride Bootstrap (mHB), der den AAPB als Spezialfall enthält, unter allgemeinen Voraussetzungen für den Mittelwert und Spektraldichteschätzer konsistent ist. Ebenso werden im univariaten und im multivariaten Fall die verschiedenen Resultate für Autokovarianzen und Autokorrelationen diskutiert. Der zweite Teil beschäftigt sich mit multivariaten linearen periodisch stationären Modellen, welche die üblichen stationären linearen Modelle dahingehend verallgemeinern, dass die Modellparameter nicht mehr konstant über die Zeit sind, sondern sich periodisch verhalten. Diese Modelle lassen sich als höherdimensionale stationäre Modelle auffassen und es wird gezeigt, dass deren Autokovarianzstruktur sowie deren Spektraldichten genau dann einem bestimmten Muster folgen, wenn der Prozess tatsächlich nicht nur periodisch stationär, sondern auch stationär ist. Zum Testen auf Stationarität wird eine Teststatistik basierend auf nichtparametrischen Spektraldichteschätzern konstruiert, welche diese Struktur ausnutzt. Die asymptotische Normalität der Teststatistik unter der Hypothese wird hergeleitet und gezeigt, dass der Test konsistent ist. Ebenso wird demonstriert, wie man die Teststatistik benutzen kann, um auf periodische Stationarität kürzerer Periode zu testen. Schließlich wird der mHB benutzt, um geeignetere kritische Werte als mit dem ZGWS zu erhalten

Baxter`s inequality and sieve bootstrap for random fields

The concept of the autoregressive (AR) sieve bootstrap is investigated for the case of spatial processes in Z2. This procedure fits AR models of increasing order to the given data and, via resampling of the residuals, generates bootstrap replicates of the sample. The paper explores the range of validity of this resampling procedure and provides a general check criterion which allows to decide whether the AR sieve bootstrap asymptotically works for a specific statistic of interest or not. The criterion may be applied to a large class of stationary spatial processes. As another major contribution of this paper, a weighted Baxter-inequality for spatial processes is provided. This result yields a rate of convergence for the finite predictor coefficients, i.e. the coefficients of finite-order AR model fits, towards the autoregressive coefficients which are inherent to the underlying process under mild conditions. The developed check criterion is applied to some particularly interesting statistics like sample autocorrelations and standardized sample variograms. A simulation study shows that the procedure performs very well compared to normal approximations as well as block bootstrap methods in finite samples