JEL classification: C15 C23Keywords: Block bootstrap Panel unit root test Cross-sectional dependence a b s t r a c t In this paper we consider the issue of unit root testing in cross-sectionally dependent panels. We consider panels that may be characterized by various forms of cross-sectional dependence including (but not exclusive to) the popular common factor framework. We consider block bootstrap versions of the groupmean (Im et al., 2003) and the pooled (Levin et al., 2002) unit root coefficient DF tests for panel data, originally proposed for a setting of no cross-sectional dependence beyond a common time effect. The tests, suited for testing for unit roots in the observed data, can be easily implemented as no specification or estimation of the dependence structure is required. Asymptotic properties of the tests are derived for T going to infinity and N finite. Asymptotic validity of the bootstrap tests is established in very general settings, including the presence of common factors and cointegration across units. Properties under the alternative hypothesis are also considered. In a Monte Carlo simulation, the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests. The power properties of the bootstrap tests appear to be similar to those of the asymptotic tests.

In this paper we study and compare the properties of several bootstrap unit root tests recently proposed in the literature. The tests are Dickey-Fuller or Augmented DF-tests, either based on residuals from an autoregression and the use of the block bootstrap or on first differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences versus residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests.We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among them using response surfaces for their size and power in a simulation study.This study leads to the following conclusions: (i) augmented DF-tests are always preferred to standard DF-tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties but residual-based tests appear more powerful.

propose decision rules based on a four-way union of rejections of quasi-differenced (QD) and ordinary least squares (OLS) detrended tests, both with and without a linear trend, to deal with the first two problems. In this paper we first discuss, again under homoskedasticity, how these union tests may be validly bootstrapped using the sieve bootstrap principle combined with either the independent and identically distributed (i.i.d.) or wild bootstrap resampling schemes. This serves to highlight the complications that arise when attempting to bootstrap the union tests. We then demonstrate that in the presence of nonstationary volatility the union test statistics have limit distributions that depend on the form of the volatility process, making tests based on the standard asymptotic critical values or, indeed, the i.i.d. bootstrap principle invalid. We show that wild bootstrap union tests are, however, asymptotically valid in the presence of nonstationary volatility. The wild bootstrap union tests therefore allow for a joint treatment of all three of the aforementioned issues in practice.

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Reproduction permitted only if source is stated.ISBN 978-3-95729-217-9 (Printversion) Non-technical summary Research QuestionWhen dealing with time series sampled at various frequencies it has become common practice to directly incorporate high-frequency information into the econom(etr)ic model at hand. These specifications were first restricted to the single regression case; with the development of the (stacked) mixed-frequency vector autoregressive (MF-VAR) system (Ghysels, 2015) it is now possible to treat all series similarly and investigate causal effects between them. However, if the difference in frequencies between the series involved is large (as, e.g., in a month/working day scenario), estimation accuracy of the system coefficients is exacerbated, implying the detection of causal effects to be potentially inaccurate. To overcome this issue various parameter reduction techniques are introduced and analyzed. These methods are then evaluated in terms of their ability to detect causality patterns between the series under consideration in the resulting restricted model. ContributionTwo parameter reduction techniques are discussed in detail: three reduced rank regression (RRR) model variants and a Bayesian MF-VAR. Using a Monte Carlo experiment both approaches are compared in terms of their Granger causality testing behavior with an unrestricted VAR, a (time-aggregated) low-frequency VAR and the max-test (Ghysels, Motegi, and Hill, 2015a). To further enhance their finite sample properties we develop and evaluate (whenever possible) two bootstrap variants of these tests. Finally, the methods are applied to U.S. data by investigating channels of causality between the monthly growth rate of the industrial production index (IPI) and daily bipower variation (BV) of the S&P500 index. ResultsWe find that, depending on the direction of causality under consideration, a different set of tests results in the best Granger non-causality testing behavior. For the direction from the high-to the low-frequency series, standard testing within the Bayesian MF-VAR, the max-test and the restricted bootstrap version of the Wald test in two RRR model versions performs best. For the reverse direction, the unrestricted bootstrap variants of the Bonferroni-corrected Wald tests within the unrestricted VAR and the RRR models dominate. As far as our application is concerned, Granger causality from BV to IPIgrowth is clearly supported by the data; evidence for causality in the reverse direction, however, only comes from a subset of tests. Maastricht UniversityAbstract We analyze Granger causality testing in a mixed-frequency VAR, where the difference in sampling frequencies of the variables is large. Given a realistic sample size, the number of high-frequency observations per low-frequency period leads to parameter proliferation problems in case we attempt to estimate the model unrestrictedly. We propose several tests based on reduced rank restrictions, and implement bootstrap versions to account for the uncertainty when estimating fac...

We propose an approach to investigate the unit root properties of individual units in a time series panel or large multivariate time series, based on testing user‐defined increasing proportions of hypothesized I(0) units sequentially. Asymptotically valid critical values are obtained using the block bootstrap. This sequential approach has an advantage over multiple testing approaches as it can exploit the (cross‐sectional) dimension of the system, which the multiple testing approaches cannot do effectively. A simulation study and an empirical application are conducted to analyse the relative performance of the approach in comparison with multiple testing approaches. These demonstrate the usefulness of our method, in particular in systems with a relatively small time dimension.

In this paper we propose an autoregressive wild bootstrap method to construct confidence bands around a smooth deterministic trend. The bootstrap method is easy to implement and does not require any adjustments in the presence of missing data, which makes it particularly suitable for climatological applications. We establish the asymptotic validity of the bootstrap method for both pointwise and simultaneous confidence bands under general conditions, allowing for general patterns of missing data, serial dependence and heteroskedasticity. The finite sample properties of the method are studied in a simulation study. We use the method to study the evolution of trends in daily measurements of atmospheric ethane obtained from a weather station in the Swiss Alps, where the method can easily deal with the many missing observations due to adverse weather conditions. JEL classifications: C14, C22.bootstrap samples with a smaller value for . Additionally, it provides a convenient framework for studying the theoretical properties of our method. Specifically, we need → ∞ as n → ∞, such that γ → 1. This is analogous to the block (and dependent wild) bootstrap, where the block size must increase to capture more dependence when the sample size increases. Assumption 7 postulates the formal conditions that needs to satisfy, which imply that γ → 1, but not too fast.Assumption 7. The bootstrap parameter = (n) satisfies → ∞ and / √ nh → 0 as n → ∞.Note that we propose to use a different bandwidthh in Step 1 of the algorithm. This is a common feature in the literature on bootstrap method for nonparametric regression, where by either selecting a larger (oversmoothing) or smaller bandwidth (undersmoothing) than used for the estimator, one can account for the asymptotic bias that is present in the local polynomial estimation, see Hall and Horowitz (2013, Section 1.4) for an extensive literature review. While undersmoothing, such as used in the related paper by Neumann and Polzehl (1998), aims at making the bias asymptotically negligible, oversmoothing aims at producing a consistent estimator of the (non-negligible) bias.Both have advantages and disadvantages, see the extensive discussion in Hall and Horowitz (2013).We follow Bühlmann (1998), also see Härdle and Marron (1991), and consider a solution based on oversmoothing, which we find to work well in practice. After presenting our theoretical results in Section 4, Remark 8 provides an intuition of why oversmoothing allows to consistently estimate the asymptotic bias. 1 We now state the formal conditions thath must satisfy in Assumption 8; one is that h/h → 0 as n → ∞, which ensures the oversmoothing. Assumption 8. The oversmoothing bandwidthh =h(n) satisfies max h , h/h, nh 5h4 → 0 and max h 4 , 1/nh → 0 as n → ∞. Remark 4. There are a number of ways to choose the AWB parameter γ in practice. Using the relation γ = θ 1/ , one can fix θ and choose as a deterministic function of the sample size. Smeekes and Urbain (2014) found in their simulation study that θ = 0.01 paired with = 1.75n ...

We develop an LM test for Granger causality in high-dimensional (HD) vector autoregressive (VAR) models based on penalized least squares estimations. To obtain a test retaining the appropriate size after the variable selection done by the lasso, we propose a post-double-selection procedure to partial out effects of nuisance variables and establish its uniform asymptotic validity. We conduct an extensive set of Monte-Carlo simulations that show our tests perform well under different data generating processes, even without sparsity. We apply our testing procedure to find networks of volatility spillovers and we find evidence that causal relationships become clearer in HD compared to standard low-dimensional VARs.