Testing for threshold moving average with conditional heteroscedasticity

The recent paper by Ling and Tong (2005) considered a quasi-likelihood ratio test for the threshold in moving average models with i.i.d. errors. This article generalizes their results to the case with GARCH errors, and a new quasi-likelihood ratio test is derived. The generalization is not direct since the techniques developed for TMA models heavily depend on the property of p-dependence that is no longer satisfied by the time series models with conditional heteroscedasticity. The new test statistic is shown to converge weakly to a functional of a centered Gaussian process under the null hypothesis of no threshold, and it is also proved that the test has nontrivial asymptotic power under local alternatives. Monte Carlo experiments demonstrate the necessity of our test when a moving average time series has a time varying conditional variance. As further support, two data examples are reported.published_or_final_versio

Least absolute deviation estimation for unit root processes with garch errors

This paper considers a local least absolute deviation estimation for unit root processes with generalized autoregressive conditional heteroskedastic (GARCH) errors and derives its asymptotic properties under only finite second-order moment for both errors and innovations. When the innovations are symmetrically distributed, the asymptotic distribution of the estimated unit root is shown to be a functional of a bivariate Brownian motion, and then two unit root tests are derived. The simulation results demonstrate that the tests outperform those based on the Gaussian quasi maximum likelihood estimators with heavy-tailed innovations and those based on the simple least absolute deviation estimators. © 2009 Copyright Cambridge University Press 2009.published_or_final_versio

A new hyperbolic GARCH model

There are two commonly used hyperbolic GARCH processes, the FIGARCH and HYGARCH processes, in modeling the long-range dependence in volatility. However, the FIGARCH process always has infinite variance, and the HYGARCH model has a more complicated form. This paper builds a simple bridge between a common GARCH model and an integrated GARCH model, and hence a new hyperbolic GARCH model along the lines of FIGARCH models. The new model remedies the drawback of FIGARCH processes by allowing the existence of finite variance as in HYGARCH models, while it has a form nearly as simple as the FIGARCH model. Two inference tools, including the Gaussian QMLE and a portmanteau test for the adequacy of the fitted model, are derived, and an easily implemented test for hyperbolic memory is also constructed. Their finite sample performances are evaluated by simulation experiments, and an empirical example gives further support to our new model. © 2015 Elsevier B.V.postprin

On the least squares estimation of threshold autoregressive moving-average models

In the financial market, the volatility of financial assets plays a key role in the problem of measuring market risk in many investment decisions. Insights into economic forces that may contribute to or amplify volatility are thus important. The financial market is characterized by regime switching between phases of low volatility and phases of high volatility. Nonlinearity and long memory are two salient features of volatility. To jointly capture the features of long memory and nonlinearity, a new threshold time series model with hyperbolic generalized autoregressive conditional heteroscedasticity is considered in this article. A goodness of fit test is derived to check the adequacy of the fitted model. Simulation and empirical results provide further support to the proposed model.postprin

A threshold approach for peaks-over-threshold modeling using maximum product of spacings

We propose a threshold model extending the generalized Pareto distribution for exceedances over a threshold. The threshold is solely determined within the model and is shown to be super-consistent under the maximum product of spacings estimation method. We apply the model to some insurance data and demonstrate the merit of having a full parametric model for the entire data set.published_or_final_versio

Time Varying Spatio-temporal Covariance Models

In this paper, we introduce valid parametric covariance models for univariate and multivariate spatio-temporal random fields. In contrast to the traditional models, we allow the model parameters to vary over time. Since variables in applications usually exhibit seasonality or changes in dependency structures, the allowance of varying parameters would be beneficial in terms of improving model flexibility. Conditions in constructing valid covariance models and discussions on practical implementation will be provided. As an application, a set of air pollution data observed from a monitoring network will be modeled. It is found that the time varying model performs better in prediction compared with the traditional models. © 2015 Elsevier Ltd.postprin

A note on the estimation of extreme value distributions using maximum product of spacings

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Testing for double threshold autoregressive conditional heteroscedastic model

The testing problem for the hypothesis of linearity against the double threshold autoregressive conditional heteroscedastic model is addressed. The, problem is nonstandard as the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We will show that the asymptotic null distribution of the Lagrange-multiplier test statistic is a functional of a zero-mean Gaussian process. In some cases, we give the upper percentage points of the test statistic. The performance of the test statistic is illustrated by extensive simulation experiments and an example.published_or_final_versio

A bootstrapped spectral test for adequacy in weak ARMA models

This paper proposes a Cramér-von Mises (CM) test statistic to check the adequacy of weak ARMA models. Without posing a martingale difference assumption on the error terms, the asymptotic null distribution of the CM test is obtained. Moreover, this CM test is consistent, and has nontrivial power against the local alternative of order n-1/2. Due to the unknown dependence of error terms and the estimation effects, a new block-wise random weighting method is constructed to bootstrap the critical values of the test statistic. The new method is easy to implement and its validity is justified. The theory is illustrated by a small simulation study and an application to S&P 500 stock index. © 2015 Elsevier B.V.postprin

A New Pearson-Type QMLE for Conditionally Heteroscedastic Models

This article proposes a novel Pearson-type quasi-maximum likelihood estimator (QMLE) of GARCH(p, q) models. Unlike the existing Gaussian QMLE, Laplacian QMLE, generalized non-Gaussian QMLE, or LAD estimator, our Pearsonian QMLE (PQMLE) captures not just the heavy-tailed but also the skewed innovations. Under strict stationarity and some weak moment conditions, the strong consistency and asymptotic normality of the PQMLE are obtained. With no further efforts, the PQMLE can be applied to other conditionally heteroscedastic models. A simulation study is carried out to assess the performance of the PQMLE. Two applications to four major stock indexes and two exchange rates further highlight the importance of our new method. Heavy-tailed and skewed innovations are often observed together in practice, and the PQMLE now gives us a systematic way to capture these two coexisting features. © 2015 American Statistical Association.postprin