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The conditional approach of Hill's estimator depends on a threshold choice. This may give different results in a statistical test when different thresholds are used. Motivated by the uniformly most...

This paper considers the mixture autoregressive panel (MARP) model. This model can capture the burst and multi-modal phenomenon in some panel data sets. It also enlarges the stationarity region of the tra- ditional AR model. An estimation method based on the EM algorithm is proposed and the assumption required of the model is quite low. To illus- trate the method, we fitted the MARP model to the gray-sided voles data. Another MARP model with less restriction is also proposed.

We consider the Autoregressive Conditional Marked Duration (ACMD) model and apply it to 16 stocks traded in Hong Kong Stock Ex- change (SEHK). By examining the orderings of appropriate sets of model parameters, market microstructure phenomena can be explained. To sub- stantiate these conclusions, likelihood ratio test is used for testing the sig- nificance of the parameter orderings of the ACMD model. While some of our results resolve a few controversial market microstructure hypotheses and echo some of the existing empirical evidence, we discover some interesting market microstructure phenomena that may be characteristic to SEHK.

The proposition of tail risk as a new asset pricing factor has gained traction in recent years. Recent work by Almeida, Ardison, Garcia, and Vicente (Nonparametric tail risk, stock returns, and the...

We distinguish the evaluation methods for two main kinds of investment strategies, namely, passive and active portfolio management. Passive portfolio management aims at tracking an underlying index as close as possible with the most important measure being the tracking error. To claim the tracking error not exceeding a certain threshold, we apply the concept of noninferiority test as opposed to the common malpractice of one‐sided test, which tries to accept the null hypothesis when there is insufficient evidence to reject it. In contrast, the normal one‐sided test should be adopted in active portfolio management, which requires another crucial statistic, the information ratio, of an active portfolio to exceed the underlying benchmark in a risk adjusted sense. The asymptotic variances of the tracking error and the difference between two information ratios are derived, which allow proper evaluation and comparison of strategies within the passive and active portfolio management frameworks.

This article investigates a portmanteau test statistic for checking model adequacy of smooth transition autoregressive (STAR) models. The asymptotic distribution of residual autocorrelations and the least‐squares estimators are also derived. Hence, the correct asymptotic standard errors for residual autocorrelations are also obtained facilitating model diagnostic checking. Through the graphical display of the simulation results concerning the size and power, for commonly used nominal sizes (), the portmanteau test appears to be more advantageous than the Lagrange multiplier tests in checking serial independence for the errors of STAR models.

The estimation of portfolio value-at-risk (VaR) requires a good estimate of the covariance matrix. As it is well known that a sample covariance matrix based on some historical rolling window is noisy and is a poor estimate for the high-dimensional population covariance matrix, to estimate the conditional portfolio VaR we develop a framework using the dynamic conditional covariance model, within which various de-noising tools are employed for the estimation of the unconditional covariance target. Various de-noising treatments in our study include shrinkage methods, random matrix theory methods and regularization methods. We validate the model empirically by using various coverage tests and loss function measures and discover that the choice of de-noising treatments for the covariance target plays a critical role in measuring the accuracy of the dynamic portfolio VaR estimates. In our large-scale empirical evaluation of de-noising tools, the regularization methods seem to produce the poorest VaR estimates under various coverage tests and loss function measures, while the shrinkage methods and the random matrix theory methods produce comparable results.

Pairs trading can be regarded as conditional mean reversion strategies. The conditions are usually imposed in two stages: Identification of pairs’ relationship and the opening (and closing) mechanism sequentially as a ‘pass or fail’ test. Nevertheless, as cointegration relationship is often not a ‘yes or no’ question but a ‘strong or weak’ one, dichotomizing the relationship through screening may not be optimal. This research presents a new single-stage approach to pairs trading based on a single ‘power statistic’. Its superiority in attaining better risk-to-reward ratios is demonstrated empirically in a large scale backtest study. 1

Realized moments of higher order computed from intraday returns are introduced in recent years. The literature indicates that realized skewness is an important factor in explaining future asset returns. However, the literature mainly focuses on the whole market and on the monthly or weekly scale. In this paper, we conduct an extensive empirical analysis to investigate the forecasting abilities of realized skewness and realized kurtosis towards individual stock's future return and variance in the daily scale. It is found that realized kurtosis possesses significant forecasting power for the stock's future variance. In the meanwhile, realized skewness is lack of explanatory power for the future daily return for individual stocks with a short horizon, in contrast with the existing literature.

Summary The estimation of time series models with heavy-tailed innovations has been widely discussed, but corresponding goodness-of-fit tests have attracted less attention, primarily because the autocorrelation function commonly used in constructing goodness-of-fit tests necessarily imposes certain moment conditions on the innovations. As a bounded random variable has finite moments of all orders, we address the problem by first transforming the residuals with a bounded function. More specifically, we consider the sample autocorrelation function of the transformed absolute residuals of a fitted generalized autoregressive conditional heteroscedastic model. With the corresponding residual empirical distribution function naturally employed as the transformation, a robust goodness-of-fit test is then constructed. The asymptotic distributions of the test statistic under the null hypothesis and local alternatives are derived, and Monte Carlo experiments are conducted to examine finite-sample properties. The proposed test is shown to be more powerful than existing tests when the innovations are heavy-tailed.

We extend a current result in the literature by introducing a parametric family of matrix-valued cross-covariance functions, which would help describe multivariate space–time dependence structures for any number of variables. All the direct and cross-covariance functions belong to the Matern class. The smoothness, space–time separability and scale parameters are allowed to be different for each variable.

It is well known that in finance variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high-frequency returns as the ex-post realization of the covariance matrix of low-frequency returns. For the analysis of dynamics of RCOV matrices, we propose the generalized conditional autoregressive Wishart (GCAW) model. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The GCAW is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments, and stationarity are discussed. Empirical examples including sequences of daily RCOV matrices from the New York Stock Exchange ill...

This article proposes a mixture double autoregressive model by introducing the flexibility of mixture models to the double autoregressive model, a novel conditional heteroscedastic model recently proposed in the literature. To make it more flexible, the mixing proportions are further assumed to be time varying, and probabilistic properties including strict stationarity and higher order moments are derived. Inference tools including the maximum likelihood estimation, an expectation–maximization (EM) algorithm for searching the estimator and an information criterion for model selection are carefully studied for the logistic mixture double autoregressive model, which has two components and is encountered more frequently in practice. Monte Carlo experiments give further support to the new models, and the analysis of an empirical example is also reported.

We propose a new Conditional BEKK matrix-F (CBF) model for the time-varying realized covariance (RCOV) matrices. This CBF model is capable of capturing heavy-tailed RCOV, which is an important stylized fact but could not be handled adequately by the Wishart-based models. To further mimic the long memory feature of the RCOV, a special CBF model with the conditional heterogeneous autoregressive (HAR) structure is introduced. Moreover, we give a systematical study on the probabilistic properties and statistical inferences of the CBF model, including exploring its stationarity, establishing the asymptotics of its maximum likelihood estimator, and giving some new inner-product-based tests for its model checking. In order to handle a large dimensional RCOV matrix, we construct two reduced CBF models -- the variance-target CBF model (for moderate but fixed dimensional RCOV matrix) and the factor CBF model (for high dimensional RCOV matrix). For both reduced models, the asymptotic theory of the estimated parameters is derived. The importance of our entire methodology is illustrated by simulation results and two real examples.

Dispersion regression is often used to predict the expected deviance in a generalised linear model. Using the individual deviance residual as the response variable in that model is considered the standard approach in dispersion modelling. In this paper, we investigate an alternative approach by fitting the dispersion model on the individual Pearson residual responses, which is more straightforward than and has superior interpretability to the deviance approach because no transformation on the observed and expected responses via the likelihood function is required. However, the mean and dispersion parameters are non-orthogonal if the model parameter estimates are obtained by maximising the pseudo-likelihood function. Consequently, the mean and dispersion regression parameters must be estimated simultaneously, and the estimation algorithm is multidimensional and hence more complex. As the asymptotic behaviour of both the deviance and Pearson residuals suggests that they should converge, we expect Pearson residual dispersion models to perform in the same way as or even better than deviance residual models.

Spatio-temporal processes involving more than one variable emerge in various fields. Any serious attempt of statistical inference and prediction for multivariate data require knowledge about the dependency structures within and across variables. In this work, we provide general conditions leading to positive semi-definiteness of the overall matrix-valued covariance functions. Both the marginal and cross-covariance functions belong to a generally non-separable Matern class spatio-temporal covariance functions, but with possibly different scale, smoothness and space–time separability parameters. The main focus of this work is on bivariate spatio-temporal random fields. As an illustration, the model is fitted on a set of bivariate air pollution data.

Asymmetric power GARCH models have been widely used to study the higher order moments of financial returns, while their quantile estimation has been rarely investigated. This paper introduces a simple monotonic transformation on its conditional quantile function to make the quantile regression tractable. The asymptotic normality of the resulting quantile estimators is established under either stationarity or non-stationarity. Moreover, based on the estimation procedure, new tests for strict stationarity and asymmetry are also constructed. This is the first try of the quantile estimation for non-stationary ARCH-type models in the literature. The usefulness of the proposed methodology is illustrated by simulation results and real data analysis.

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.

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.

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.

This paper proposes a Cramer-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 by using the Hillbert space approach. 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.

This paper extends the classical two-regime threshold autoregressive model by introducing hysteresis to its regime-switching structure, which leads to a new model: the hysteretic autoregressive model. The proposed model enjoys the piecewise linear structure of a threshold model but has a more flexible regime switching mechanism. A sufficient condition is given for geometric ergodicity. Conditional least squares estimation is discussed, and the asymptotic distributions of its estimators and information criteria for model selection are derived. Simulation results and an example support the model.

This paper proposes some novel one-sided omnibus tests for independence between two multivariate stationary time series. These new tests apply the Hilbert-Schmidt independence criterion (HSIC) to test the independence between the innovations of both time series. Under regular conditions, the limiting null distributions of our HSIC-based tests are established. Next, our HSIC-based tests are shown to be consistent. Moreover, a residual bootstrap method is used to obtain the critical values for our HSIC-based tests, and its validity is justified. Compared with the existing cross-correlation-based tests for linear dependence, our tests examine the general (including both linear and non-linear) dependence to give investigators more complete information on the causal relationship between two multivariate time series. The merits of our tests are illustrated by some simulation results and a real example.

This paper extends the work of Yuen et al. (2013), who obtained explicit results for the discount-free Gerber–Shiu function for a compound binomial risk model in the presence of delayed claims and a randomized dividend strategy with a zero threshold level. Specifically, we establish a recursion method for computing the Gerber–Shiu expected discounted penalty function, which entails a number of important quantities in ruin theory, within the framework of the compound binomial aggregate claims with delayed by-claims and randomized dividends payable at a non-negative threshold level.

Generalized Threshold Model (GTM) is a non-linear time series model which generalizes the Threshold Autoregressive Model (TAR) to implement the idea of the Generalized Linear Model under the threshold time series framework. However, the dispersion parameter is usually assumed as constant in the context of Generalized Linear Model which does not hold in general. In this paper, the GTM is extended to a Double Generalized Threshold Model (DGTM) where the dispersion parameter, defined as the expected deviance of the individual response about its mean, varies throughout the entire sample. The variation of the dispersion parameter can be predicted by another threshold type generalized linear model, which is interlinked with the threshold model for the mean and can be estimated simultaneously.

The problem of heavy tail in regression models is studied. It is proposed that regression models are estimated by a standard procedure and a statistical check for heavy tail using residuals is conducted as a tool for regression diagnostic. Using the peaks-over-threshold approach, the generalized Pareto distribution quantifies the degree of heavy tail by the extreme value index. The number of excesses is determined by means of an innovative threshold model which partitions the random sample into extreme values and ordinary values. The overall decision on a significant heavy tail is justified by both a statistical test and a quantile–quantile plot. The usefulness of the approach includes justification of goodness of fit of the estimated regression model and quantification of the occurrence of extremal events. The proposed methodology is supplemented by surface ozone level in the city center of Leeds.