Rongmao Zhang

Group LASSO for Structural Break Time Series

Consider a structural break autoregressive (SBAR) process where j = 1, …, m + 1, {t1, …, tm} are change-points, 1 = t0 < t1 < ⋅⋅⋅ < tm + 1 = n + 1, σ( · ) is a measurable function on , and {ϵt} are white noise with unit variance. In practice, the number of change-points m is usually assumed to be known and small, because a large m would involve a huge amount of computational burden for parameters estimation. By reformulating the problem in a variable selection context, the group least absolute shrinkage and selection operator (LASSO) is proposed to estimate an SBAR model when m is unknown. It is shown that both m and the locations of the change-points {t1, …, tm} can be consistently estimated from the data, and the computation can be efficiently performed. An improved practical version that incorporates group LASSO and the stepwise regression variable selection technique are discussed. Simulation studies are conducted to assess the finite sample performance. Supplementary materials for this article are available online.

On a Threshold Double Autoregressive Model

This article first proposes a score-based test for a double autoregressive model against a threshold double autoregressive (AR) model. It is an asymptotically distribution-free test and is easy to implement in practice. The article further studies the quasi-maximum likelihood estimation of a threshold double autoregressive model. It is shown that the estimated threshold is n-consistent and converges weakly to a functional of a two-sided compound Poisson process and the remaining parameters are asymptotically normal. Our results include the asymptotic theory of the estimator for threshold AR models with autoregressive conditional heteroscedastic (ARCH) errors and threshold ARCH models as special cases, each of which is also new in literature. Two portmanteau-type statistics are also derived for checking the adequacy of fitted model when either the error is nonnormal or the threshold is unknown. Simulation studies are conducted to assess the performance of the score-based test and the estimator in finite samples. The results are illustrated with an application to the weekly closing prices of Hang Seng Index. This article also includes the weak convergence of a score-marked empirical process on the space under an α-mixing assumption, which is independent of interest.

Tests for covariance matrix with fixed or divergent dimension

Testing covariance structure is of importance in many areas of statistical analysis, such as microarray analysis and signal processing. Conventional tests for finite-dimensional covariance cannot be applied to high-dimensional data in general, and tests for high-dimensional covariance in the literature usually depend on some special structure of the matrix. In this paper, we propose some empirical likelihood ratio tests for testing whether a covariance matrix equals a given one or has a banded structure. The asymptotic distributions of the new tests are independent of the dimension.

TAIL INDEX OF AN AR(1) MODEL WITH ARCH(1) ERRORS

Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interval for the tail index, one needs to estimate the complicated asymptotic variance of the tail index estimator, however. In this paper the asymptotic normality of the tail index estimator is first derived, and a profile empirical likelihood method to construct a confidence interval for the tail index is then proposed. A simulation study shows that the proposed empirical likelihood method works better than the bootstrap method in terms of coverage accuracy, especially when the process is nearly nonstationary.

ASYMPTOTIC INFERENCE FOR AR MODELS WITH HEAVY-TAILED G-GARCH NOISES

It is well known that the least squares estimator (LSE) of an AR( p ) model with i.i.d. (independent and identically distributed) noises is n 1/ α L ( n )-consistent when the tail index α of the noise is within (0,2) and is n 1/2 -consistent when α ≥ 2, where L ( n ) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR( p ) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α e [2,4], it is shown that the LSE is n 1–2/ α -consistent if α e (2,4), log n -consistent if α = 2, and n 1/2 / log n -consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

Maximum likelihood estimation for nearly non-stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Yt = ρnYt-1 ɛt with α‐stable noise {ɛt} is studied in this paper. Herein ρn = 1 − γ/n, γ ≥ 0 is a constant, Yo is a fixed random variable and et is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 1 and Eɛ1 = 0, the limit distribution of the MLE of ρn and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and Eɛ1 ≠ 0, the limit distribution of the MLE of ρn and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.

Non‐Stationary Autoregressive Processes with Infinite Variance

Consider an AR(p) process Yt=β1Yt-1+…+βpYt-p+ɛt, where {ɛt} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0Yt} is said to be a non‐stationary AR(p) process if at least one of its characteristic roots lies on the unit circle. The limit distribution of the least squares estimator (LSE) of β=(β1,…,βp)T for {Yt} with infinite variance innovation {ɛt} is established in this paper. In particular, by virtue of the result of Kurtz and Protter (1991) of stochastic integrals, it is shown that the limit distribution of the LSE is a functional of integrated stable process. Simulations for the estimator of β and its limit distribution are also given.

Jackknife-blockwise empirical likelihood methods under dependence

Empirical likelihood for general estimating equations is a method for testing hypothesis or constructing confidence regions on parameters of interest. If the number of parameters of interest is smaller than that of estimating equations, a profile empirical likelihood has to be employed. In case of dependent data, a profile blockwise empirical likelihood method can be used. However, if too many nuisance parameters are involved, a computational difficulty in optimizing the profile empirical likelihood arises. Recently, Li et al. (2011) [9] proposed a jackknife empirical likelihood method to reduce the computation in the profile empirical likelihood methods for independent data. In this paper, we propose a jackknife-blockwise empirical likelihood method to overcome the computational burden in the profile blockwise empirical likelihood method for weakly dependent data.

Limit Theory for a General Class of GARCH Models with Just Barely Infinite Variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ɛ = ση and σ = f (σ, σ,…, σ, ɛ, ɛ,…, ɛ), when {ɛ} is a process with just barely infinite variance, that is, {ɛ} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.

Marked empirical processes for non-stationary time series

Consider a first-order autoregressive process