# Manual Bootstrapping Stationary ARMA-GARCH Models

Bootstrap methods are applied to improve accuracies of finite sample distributions of estimators or test statistics over those based on large sample theories. For trend-volatility models, Franke et al. Shimizu demonstrated the cases where bootstrap does not work for heteroscedastic time series models. Diverse dependent structures are imposed on the observation by both trend and volatility for data from trend-volatility models.

Therefore, block bootstrapping may be a better choice because the bootstrap sampling should capture the dependence structure of the original data in order for the bootstrap sample to have the same dependence structure as the observation, instead of i. For example, volatility clustering features prevalent in log-returns of financial assets such as stock prices and foreign exchange rates and in macroeconomic variables are reserved in block bootstrap samples but not in i.

Kreiss and Paparoditis and references provide a review of the bootstrap methods for dependent data. We apply the stationary bootstrap method in this work as a bootstrap applicability for the ARCH model. The stationary bootstrap proposed by Politis and Romano is characterized by resampling blocks of geometrically distributed random length. As recent stationary bootstrap works, we mention Swensen , Paparoditis and Politis , Parker et al.

In this paper, the nonparametric ARCH regression model is considered and studied for asymptotic validity by means of the stationary bootstrap. We prove asymptotic validity of the stationary bootstrap estimator for the unknown ARCH function by showing that it has the same limiting distribution as the Nadaraya-Watson estimator. Our result is an extension of Shimizu , who showed the weak consistency of the Nadaraya-Watson estimator for the ARCH part as well as proposed the residual and wild bootstrap estimators in the nonparametric AR-nonparametric ARCH model.

This paper is organized as follows. In Section 2, the model, the assumptions and the existing theories are stated. In Section 3, the main result is presented along with description of the stationary bootstrap procedure.

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The conclusion is given in Section 4. Proof is reported in Appendix. The ARCH 1 error in 2. The kernel smoothing estimator is denoted by m? A8 below. A1 : The distribution of the i. From 2. For the AR part we need the following assumptions on the kernel smoothing estimator m?

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Theorem 1. Franke et al. A8 above. Theorem 2.

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Shimizu, Assume A1? For notational simplicity, we use u t for? To do this, we generate random variables I 1 ,. Note that. In addition to A1? Uh-oh, it looks like your Internet Explorer is out of date. For a better shopping experience, please upgrade now. Javascript is not enabled in your browser. Enabling JavaScript in your browser will allow you to experience all the features of our site.

## Table of Contents: Bootstrapping Stationary ARMA-GARCH Models

Learn how to enable JavaScript on your browser. See All Customer Reviews. Shop Books. Add to Wishlist. USD Our result is an extension of Shimizu , who showed the weak consistency of the Nadaraya-Watson estimator for the ARCH part as well as proposed the residual and wild bootstrap estimators in the nonparametric AR-nonparametric ARCH model. This paper is organized as follows. In Section 2, the model, the assumptions and the existing theories are stated.

In Section 3, the main result is presented along with description of the stationary bootstrap procedure. The conclusion is given in Section 4. Proof is reported in Appendix. The ARCH 1 error in 2. The kernel smoothing estimator is denoted by m? A8 below.

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A1 : The distribution of the i. From 2. For the AR part we need the following assumptions on the kernel smoothing estimator m? Theorem 1. Franke et al. A8 above.

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Theorem 2. Shimizu, Assume A1? For notational simplicity, we use u t for? To do this, we generate random variables I 1 ,. Note that. In addition to A1? A8 , two additional conditions are needed for the consistency of the stationary bootstrap. Large sample validity of the proposed bootstrap procedure is proved in a rigorous manner. The asymptotic validity will enable us to construct statistical inference methods such as confidence intervals and hypothesis testing. It would be worth to compare the bootstrap methods with other non-bootstrap methods in a Monte-Carlo experiment.

This issue may be a topic for future studies. Title Author Keyword Volume Vol. CrossRef 0. Dong Wan Shin a , and Eunju Hwang 1,b. E-mail: ehwang gachon. Keywords : stationary bootstrap, ARCH, nonparametric regression, consistency. Other Sections Abstract 1. Introduction 2.