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This paper proposes a three-step method for detecting multiple structural breaks for piecewise stationary vector autoregressive processes. The number of structural breaks can be large and unknown with the locations of the breaks being different among different components. The proposed method is established via a link between a structural break problem and a high-dimensional regression problem. By means of this connection, a group orthogonal greedy algorithm, originated from the high-dimensional variable selection context, is developed for efficiently screening out potential break-points in the first step. A high-dimensional information criterion is proposed for consistent structural breaks estimation in the second step. In the third step, the information criterion further determines the specific components in which structural breaks occur. Monte Carlo experiments are conducted to demonstrate the finite sample performance, and applications to stock data are provided to illustrate the proposed method.

The tail of the logarithmic degree distribution of networks decays linearly with respect to the logarithmic degree is known as the power law and is ubiquitous in daily lives. A commonly used technique in modeling the power law is preferential attachment (PA), which sequentially joins each new node to the existing nodes according to the conditional probability law proportional to a linear function of their degrees. Although effective, it is tricky to apply PA to real networks because the number of nodes and that of edges have to satisfy a linear constraint. This paper enables real application of PA by making each new node as an isolated node that attaches to other nodes according to PA scheme in some later epochs. This simple and novel strategy provides an additional degree of freedom to relax the aforementioned constraint to the observed data and uses the PA scheme to compute the implied proportion of the unobserved zero-degree nodes. By using martingale convergence theory, the degree distribution of the proposed model is shown to follow the power law and its asymptotic variance is proved to be the solution of a Sylvester matrix equation, a class of equations frequently found in the control theory (see Hansen and Sargent (2008, 2014)). These results give a strongly consistent estimator for the power-law parameter and its asymptotic normality. Note that this statistical inference procedure is non-iterative and is particularly applicable for big networks such as the World Wide Web presented in Section 6 . Moreover, the proposed model offers a theoretically coherent framework that can be used to study other network features, such as clustering and connectedness, as given in Cheung (2016).

This paper studies the maximum likelihood estimation of nonlinear state space models. Particle Markov chain Monte Carlo method is introduced to implement the Monte Carlo expectation maximization algorithm for more accurate and robust estimation. Under this framework, an automated sample size selection criterion is constructed via renewal theory. This criterion would increase the sample size when the relative likelihood indicates that the parameters are close to each other. The proposed methodology is applied to the stochastic volatility model and another nonlinear state space model for illustration, where the results show better estimation performance.

Customized personal rate offering is of growing importance in the insurance industry. To achieve this, an important step is to identify subgroups of insureds from the corresponding heterogeneous claim frequency data. In this paper, a penalized Poisson regression approach for subgroup analysis in claim frequency data is proposed. Subjects are assumed to follow a zero-inflated Poisson regression model with group-specific intercepts, which capture group characteristics of claim frequency. A penalized likelihood function is derived and optimized to identify the group-specific intercepts and effects of individual covariates. To handle the challenges arising from the optimization of the penalized likelihood function, an alternating direction method of multipliers algorithm is developed and its convergence is established. Simulation studies and real applications are provided for illustrations.

This paper proposes a new portmanteau-type statistic by combining several lags of the sample autocorrelations to test for the presence of a unit-root of an autoregressive model. The proposed method is nonparametric in nature, which is model free and easy to implement. It avoids modeling the fitted residuals and does not require estimation of nuisance parameters, as commonly done in the augmented Dickey–Fuller or Phillips–Perron procedure. Asymptotic properties of the test are established under general stationary conditions on the noises. Finite sample studies are also reported to illustrate the superior power of the proposed method. Applications to test for cointegration are also given.

An effective model for time-varying quantiles of a time series is of considerable practical importance across various disciplines. In particular, in financial risk management, computation of Value-at-risk (VaR), one of the most popular risk measures, involves knowledge of quantiles of portfolio returns. This paper examines the random walk behavior of VaRs constructed under two most common approaches, viz. historical simulation and the parametric approach using GARCH models. We find that sequences of historical VaRs appear to follow a unit root model, which can be an artifact under some settings, whereas its counterpart constructed via the parametric approach does not follow a random walk model by default.

This paper develops a novel approach for estimating a threshold autoregressive (TAR) model with multiple-regimes and establishes its large sample properties. By reframing the problem in a regression variable selection context, a least absolute shrinkage and selection operator (LASSO) procedure is proposed to estimate a TAR model with an unknown number of thresholds, where the computation can be performed efficiently. It is further shown that the number and the location of the thresholds can be consistently estimated. A near optimal convergence rate of the threshold parameters is also established. Simulation studies are conducted to assess the performance in finite samples. The results are illustrated with an application to the quarterly US real GNP data over the period 1947–2009.

By allowing deviations from equilibrium to follow a fractionally integrated process, the notion of fractional cointegration analysis encompasses a wide range of mean-reverting behaviors. For fractional cointegrations, asymptotic theories have been extensively studied, and numerous empirical studies have been conducted in finance and economics. But as far as testing for fractional cointegration is concerned, most of the testing procedures have restrictions on the integration orders of observed time series or integrating error and some tests involve determination of bandwidth. In this paper, a general fractional cointegration model with the observed series and the cointegrating error being fractional processes is considered, and a residual-based testing procedure for fractional cointegration is proposed. Under some regularity conditions, the test statistic has an asymptotic standard normal distribution under the null hypothesis of no fractional cointegration and diverges under the alternatives. This test procedure is easy to implement and works well in finite samples, as reported in a Monte Carlo experiment.

Let X"t=@?"j"="0^~c"j@e"t"-"j be a moving average process with GARCH (1, 1) innovations {@e"t}. In this paper, the asymptotic behavior of the quadratic form Q"n=@?"j"="1^n@?"s"="1^nb(t-s)X"tX"s is derived when the innovation {@e"t} is a long-memory and heavy-tailed process with tail index @a, where {b(i)} is a sequence of constants. In particular, it is shown that when 1 =4, Q"n has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations.

A unified asymptotic theory for nearly unstable higher order autoregressive processes and their least squares estimates is established. A novel version of Jordan’s canonical decomposition with perturbations together with a suitable plug-in principle is proposed to develop the underlying theories. Assumptions are stated in terms of the domain of attraction of partial Fourier transforms. The machinery is applied to recapture some of the classical results with the driving noise being martingale differences. Further, we show how to extend the results to higher order fractional ARIMA models in nearly unstable settings, thereby offering a comprehensive theory to analyse nearly unstable time series.

Credit migrations constitute the building blocks of modern risk management. A firm-specific structural model of credit migration that incorporates the firm's capital structure and the risk perception of rating agencies is proposed. The proposed model employs the notion of distance-to-default, which quantifies default probability. The properties of Brownian excursions play an essential role in the analysis. The proposed model not only allows the derivation of closed-form credit transition probability, but also provides plausible explanations for certain empirical evidence, such as the default probability overlaps in ratings and the slow-to-respond feature of rating agencies. The proposed model is calibrated through simulations and applied to empirical data, which show rating agencies' risk perceptions to be significant. The calibrated model allows calculation of the firm-specific transition probabilities of rated companies.

Consider the nonparametric heteroscedastic regression model Y = m ( X ) + σ ( X ) ɛ , where m ( · ) is an unknown conditional mean function and σ ( · ) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ ( X ) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.

Bandwidth selection has been an important topic in nonparametric density estimation. In this paper an effective method for local bandwidth selection is proposed. For local bandwidth selection, due to data sparsity and other reasons, extremely small bandwidths are sometimes selected, which lead to severe undersmoothing. To circumvent this difficulty, the main idea behind the proposed method is to choose the largest bandwidth that still achieves the optimal rate. When coupled with practical bias reduction techniques, the bandwidth selected from this method can be applied simultaneously to conduct both local point and interval estimation. Simulation studies demonstrate the effectiveness of the proposed approach, which compares favorably with other existing approaches.

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.

A method of assessing an economist's subjective prior for a unit root model is given, and applied. The methods extend previous work by allowing a family of piecewise conjugate prior distribution that permit different opinions when ϱ 1. This larger family is still closed under sampling, so it retains the simplicity that is the principal advantage of conjugate analysis.

An autoregressive time series is said to be unstable if all of its characteristics roots lie on or outside the unit circle. This paper aims at developing the asymptotic inference for the least squares estimates of unstable autoregressive time series with drifts. Our framework allows for both constant and periodic drifts. The presence of a nonzero drift in the series affects the asymptotic behavior in a surprising and interesting way. For models with constant nonzero drifts, conventional asymptotic normal theory is attained only for unit root time series but not in general. For models with periodic drifts, degenerate nonnormal asymptotics are resulted for unit root series. However for models with roots lying on the unit circle which are not equal to unity, nonnormal asymptotics are always attained for both constant and periodic drifts unless there are confounding effects between the periodicities of the drifts and the locations of the roots. Our approach is based on a componentwise argument used in Chan and Wei (1988). The theories developed here extend much earlier work and help to clarify the intrinsic critical phenomena in the field of unstable time series.