Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. We do not assume any speci¯c data generating process, and employ the sieve bootstrap based on the approximated¯nite-order vector autoregressions for the regression errors and the¯rst di®erences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the¯nite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not e±cient. For the e±cient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen (1991) and Stock and Watson (1993) relying on the regression augmented with the leads and lags of di®erenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do inferences that are asymptotically valid. A Monte Carlo study is conducted to investigate the¯nite sample performances of the proposed bootstrap methods.

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“…by the bootstrap invariance principle and Kurtz and Protter (1991). Moreover, it can be deduced analogously as in Lemma A4 that…”

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confidence: 61%

Bootstrapping cointegrating regressions

Chang

Park

Song

2006

Journal of Econometrics

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“…Chan and Wei [4, Lemma 2.4] derive the distribution of 490 BRUCE E. HANSEN yet published for martingale arrays are contained in Kurtz and Protter [10]. These authors consider a variety of limit theorems when V, is a semimartingale.…”

Section: Introductionmentioning

confidence: 99%

“…The third example assumes that the second differences of U, are strong mixing, which has applications in the theory of cointegration among I(2) variables [13]. The final example assumes that Un is a vector process with a root local to unity, which has applications in the theory of near-integration [3,17,20] and continuous time approximations [ [10]. Their results encompass a broad range of processes, including semimartingales with weaker moment conditions.…”

Section: Introductionmentioning

confidence: 99%

Convergence to Stochastic Integrals for Dependent Heterogeneous Processes

1992

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This paper provides conditions to establish the weak convergence of stochastic integrals. The theorems are proved under the assumption that the innovations are strong mixing with uniformly bounded 2+ moments. Several applications of the results are given, relevant for the theories of estimation with I(1) processes, I(2) processes, processes with nonstationary variances, nearintegrated processes, and continuous time approximations.

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“…The fluid limit is relatively straightforward once the convergence of the limit established. The key to establish the diffusion limits comes from a nice result of Kurtz and Protter [44], which is a generalization of Wong and Zakai's [64] earlier work concerning passing the convergence relation between the càdlàg processes (X n , Y n ) to (X, Y ) in the Skorohod topology to the convergence relation between X n dY n to X dY . With the fluid and diffusion limit for Z(t), it is possible to calculate various quantities of interest for the LOB, including (i) the probability that an order is filled before the price increases, (ii) the expected price to pay if the order is not executed, (iii) the expected cost to place N 0 orders at the bid and N − N 0 at the ask, and (iv) the optimal strategy for placing orders.…”

Section: Continuous-time Model With Reduced Formmentioning

confidence: 99%

Optimal Placement in a Limit Order Book

Guo¹

2013

Theory Driven by Influential Applications

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Please scroll down for article-it is on subsequent pagesINFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org Abstract Algorithmic trading refers to the automatic and rapid trading of large quantities with orders specified and implemented by an algorithm. Roughly speaking, algorithmic trading is based on two different time scales: the daily or weekly scale and a smaller (10-to 100-second) time scale. These two time scales essentially reflect the two steps by which the traders slice and place orders. The first step is to optimally slice big orders into smaller ones on a daily basis with the goal to minimize the price impact and/or to maximize the expected utility; the second step is to optimally place the orders within seconds. The former is the well-known optimal execution problem, and the latter is the much less studied optimal placement problem. This paper reviews several simple models and approaches for the optimal placement problem. Several most relevant statistical issues are presented, together with a brief discussion on the key differences between the system of limit order book and the multiclass queues with reneging.

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Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

Cited by 448 publications

References 30 publications

Bootstrapping cointegrating regressions

Bootstrapping cointegrating regressions

Convergence to Stochastic Integrals for Dependent Heterogeneous Processes

Optimal Placement in a Limit Order Book

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