Inference Under Random Limit Bootstrap Measures

Cavaliere, Giuseppe; Georgiev, Iliyan

doi:10.3982/ecta16557

Cited by 21 publications

(31 citation statements)

References 38 publications

(43 reference statements)

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“…That is, the limiting distribution can be written in terms of an Ornstein-Uhlenbeck process with a random drift, distributed as τ ∞ , that is, as a Dickey-Fuller distribution. Similar arguments are applied in Cavaliere et al (2015), see also the next section, and in terms of random bootstrap measures in Cavaliere and Georgiev (2019) and Boswijk et al (2019).…”

Section: Unrestricted (Iid) Bootstrapmentioning

confidence: 81%

A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

Cavaliere

Rahbek

2020

Econom. Theory

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In this article, we discuss the bootstrap as a tool for statistical inference in econometric time series models. Importantly, in the context of testing, properties of the bootstrap under the null (size) as well as under the alternative (power) are discussed. Although properties under the alternative are crucial to ensure the consistency of bootstrap-based tests, it is often the case in the literature that only validity under the null is discussed. We provide new results on bootstrap inference for the class of double-autoregressive (DAR) models. In addition, we review key examples from the bootstrap time series literature in order to emphasize the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics, while also providing an up-to-date review of existing approaches. DAR models are particularly interesting for bootstrap inference: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, even second-order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing nonstationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of the bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid. That is, it is able to replicate, under the null hypothesis, the correct limiting distribution. Importantly, we also show that the behavior of this bootstrap under the alternative hypothesis may be more involved because of possible lack of finite second-order moments of the bootstrap innovations. This feature makes for some parameter configurations, the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this possible drawback can be fixed by using a novel bootstrap in this framework. For this “hybrid bootstrap,” the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from unrestricted residuals. We show that the hybrid bootstrap mimics the correct asymptotic null distribution, irrespective of the null being true or false. Monte Carlo simulations illustrate the behavior of both the restricted and the hybrid bootstrap, and we find that both perform very well even for small sample sizes.

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Section: Unrestricted (Iid) Bootstrapmentioning

confidence: 81%

A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

Cavaliere

Rahbek

2020

Econom. Theory

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“…. This is well-known for the bootstrap in time series models, where if the residuals are not centered, their O p (T −1/2 ) sample mean will induce randomness in the limit distribution of the bootstrap statistics (Cavaliere et al, 2015;Cavaliere and Georgiev, 2020).…”

Section: Non-parametric Fib and Ribmentioning

confidence: 91%

Bootstrap Inference for Hawkes and General Point Processes

Cavaliere¹,

Lu²,

Rahbek³

et al. 2021

Preprint

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Inference and testing in general point process models such as the Hawkes model is predominantly based on asymptotic approximations for likelihoodbased estimators and tests, as originally developed in Ogata (1978). As an alternative, and to improve finite sample performance, this paper considers bootstrap-based inference for interval estimation and testing. Specifically, for a wide class of point process models we consider a novel bootstrap scheme labeled 'fixed intensity bootstrap' (FIB), where the conditional intensity is kept fixed across bootstrap repetitions. The FIB, which is very simple to implement and fast in practice, naturally extends previous ideas from the bootstrap literature on time series in discrete time, where the so-called 'fixed design' and 'fixed volatility' bootstrap schemes have shown to be particularly useful and effective. We compare the FIB with the classic recursive bootstrap, which is here labeled 'recursive intensity bootstrap' (RIB). In RIB algorithms, the intensity is stochastic in the bootstrap world and implementation of the bootstrap is more involved, due to its sequential structure. For both bootstrap schemes, no asymptotic theory is available; we therefore provide here a new bootstrap (asymptotic) theory, which allows to assess bootstrap validity. We also introduce novel 'nonparametric' FIB and RIB schemes, which are based on resampling time-changed transformations of the original waiting times. We show effectiveness of the different bootstrap schemes in finite samples through a set of detailed Monte Carlo experiments. As far as we are aware, this is the first detailed Monte Carlo study of bootstrap implementations for Hawkes-type processes. Finally, in order to illustrate, we provide applications of the bootstrap to both financial data and social media data.

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“…This allows us to establish that, although the presence of a random limiting distribution for the bootstrap statistic makes the bootstrap unable to estimate the unconditional distribution of the statistic of interest, the bootstrap can still deliver hypothesis tests with the desired size. In particular, we do this by establishing that the high-level conditions for bootstrap validity in Cavaliere and Georgiev (2020) can be shown to hold for a large class of models with stochastic volatility, including the aforementioned near-integrated GARCH model and the non-stationary stochastic volatility model. We do so by showing new weak convergence results conditional on volatility paths.…”

Section: Introductionmentioning

confidence: 95%

“…Specifically, in this paper we analyze the wild bootstrap under (non-stationary) stochastic volatility by adopting a new approach to assess bootstrap validity under random limit bootstrap measures. Thus, rather than focusing on the usual weak convergence in probability of the bootstrap conditional distribution, we make use of the concept of weak convergence in distribution (see Cavaliere and Georgiev, 2020, for a general introduction) to develop novel conditions for validity of the wild bootstrap, conditional on the volatility process. This allows us to establish that, although the presence of a random limiting distribution for the bootstrap statistic makes the bootstrap unable to estimate the unconditional distribution of the statistic of interest, the bootstrap can still deliver hypothesis tests with the desired size.…”

Section: Introductionmentioning

confidence: 99%

Bootstrapping Non-Stationary Stochastic Volatility

Boswijk¹,

Cavaliere²,

Rahbek³

et al. 2021

Preprint

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In this paper we investigate to what extent the bootstrap can be applied to conditional mean models, such as regression or time series models, when the volatility of the innovations is random and possibly non-stationary. In fact, the volatility of many economic and financial time series displays persistent changes and possible non-stationarity. However, the theory of the bootstrap for such models has focused on deterministic changes of the unconditional variance and little is known about the performance and the validity of the bootstrap when the volatility is driven by a non-stationary stochastic process. This includes near-integrated exogenous volatility processes as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit; a further important example is the case where volatility exhibits infrequent jumps. This paper fills this gap in the literature by developing conditions for bootstrap validity in time series and regression models with nonstationary, stochastic volatility. We show that in such cases the distribution of bootstrap statistics (conditional on the data) is random in the limit. Consequently, the conventional approaches to proofs of bootstrap consistency, based on the notion of weak convergence in probability of the bootstrap statistic, fail to deliver the required validity results. Instead, we use the concept of 'weak convergence in distribution' to develop and establish novel conditions for validity of the wild bootstrap, conditional on the volatility process. We apply our results to several testing problems in the presence of non-stationary stochastic volatility, including testing in a location model, testing for structural change using CUSUM-type functionals, and testing for a unit root in autoregressive models. Importantly, we work under sufficient conditions for bootstrap validity that include the absence of statistical leverage effects, i.e., correlation between the error process and its future conditional variance. The results of the paper are illustrated using Monte Carlo simulations, which indicate that a wild bootstrap approach leads to size control even in small samples.

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Inference Under Random Limit Bootstrap Measures

Cited by 21 publications

References 38 publications

A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

Bootstrap Inference for Hawkes and General Point Processes

Bootstrapping Non-Stationary Stochastic Volatility

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