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The Bartlett correction is a desirable feature of the likelihood inference, which yields the confidence region for parameters with improved coverage probability. This study examines the Bartlett correction for the frequency domain empirical likelihood (FDEL), based on the Whittle likelihood of linear time series models. Nordman and Lahiri (Ann Stat 34:3019–3050, 2006) showed that the FDEL does not have an ordinary Chi-squared limit when the innovation is non-Gaussian with unknown variance, which restricts the use of the FDEL inference in time series. We show that, by profiling the innovation variance out of the Whittle likelihood function, the FDEL is Chi-squared-distributed and Bartlett correctable. In particular, the order of the coverage error of the confidence region can be reduced from $$O(n^{-1})$$ to $$O(n^{-2})$$ .

ENTHIS LINK GOES TO A ENGLISH SECTIONFRTHIS LINK GOES TO A FRENCH SECTION Bartlett correction constitutes one of the attractive features of empirical likelihood because it enables the construction of confidence regions for parameters with improved coverage probabilities. We study the Bartlett correction of spatial frequency domain empirical likelihood (SFDEL) based on general spectral estimating functions for regularly spaced spatial data. This general formulation can be applied to testing and estimation problems in spatial analysis, for example testing covariance isotropy, testing covariance separability as well as estimating the parameters of spatial covariance models. We show that the SFDEL is Bartlett correctable. In particular, the improvement in coverage accuracies of the Bartlett‐corrected confidence regions depends on the underlying spatial structures. The Canadian Journal of Statistics 47: 455–472; 2019 © 2019 Statistical Society of Canada

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.

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.

Analysis of high-dimensional time series data is of increasing interest among different fields. This article studies high-dimensional time series from a dimension reduction perspective using factor modelling. Statistical inference is conducted using eigen-analysis of a certain non-negative definite matrix related to autocovariance matrices of the time series, which is applicable to fixed or increasing dimension. When the dimension goes to infinity, the rate of convergence and limiting distributions of estimated factors are established. Using the limiting distributions of estimated factors, a high-dimensional final prediction error criterion is proposed to select the number of factors. Asymptotic properties of the criterion are illustrated by simulation studies and real applications.

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log-likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short-memory time series with possibly non-Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett-corrected confidence regions can be reduced from O(n−1) to O(n−2).

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.

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 av...

Testing for causality is of critical importance for many econometric applications. For bivariate AR(1) processes, the limit distributions of causality tests based on least squares estimation depend on the presence of nonstationary processes. When nonstationary processes are present, the limit distributions of such tests are usually very complicated, and the full-sample bootstrap method becomes inconsistent as pointed out in Choi (2005, Statistics and Probability Letters 75, 39–48). In this paper, a profile empirical likelihood method is proposed to test for causality. The proposed test statistic is robust against the presence of nonstationary processes in the sense that one does not have to determine the existence of nonstationary processes a priori. Simulation studies confirm that the proposed test statistic works well.

Consider an AR(p) process , where {ɛt} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0

An empirical likelihood–based confidence interval is proposed for interval estimations of the autoregressive coefficient of a first-order autoregressive model via weighted score equations. Although the proposed weighted estimate is less efficient than the usual least squares estimate, its asymptotic limit is always normal without assuming stationarity of the process. Unlike the bootstrap method or the least squares procedure, the proposed empirical likelihood–based confidence interval is applicable regardless of whether the underlying autoregressive process is stationary, unit root, near-integrated, or even explosive, thereby providing a unified approach for interval estimation of an AR(1) model to encompass all situations. Finite-sample simulation studies confirm the effectiveness of the proposed method.

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near-unit root autoregressive process Yt = ?nYt-1 + ?t with a-stable noise {?t} is studied in this paper. Herein ?n = 1 - ?/n, ? = 0 is a constant, Y0 is a fixed random variable and et is an a-stable random variable with characteristic function f(t,?) for some parameter ?. It is shown that when 0 1 and E?1 = 0, the limit distribution of the MLE of ?n and ? are mixtures of a stable process and Gaussian processes. On the other hand, when a > 1 and E?1 ? 0, the limit distribution of the MLE of ?n and ? 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 a is less than 1.

It is known that the tail index of a GARCH model is determined by a moment equation, which involves the underlying unknown parameters of the model. A tail index estimator can therefore be constructed by solving the sample moment equation with the unknown parameters being replaced by its quasi-maximum likelihood estimates (QMLE). To construct a confidence interval for the tail index, one needs to estimate the non-trivial asymptotic variance of the QMLE. In this paper, an empirical likelihood method is proposed for interval estimation of the tail index. One advantage of the proposed method is that interval estimation can still be achieved without having to estimate the complicated asymptotic variance. A simulation study confirms the advantage of the proposed method.

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.

Fan and Yao (1998) proposed an efficient method to estimate the conditional variance of heteroskedastic regression models. Chen, Cheng, and Peng (2009) applied variance reduction techniques to the estimator of Fan and Yao (1998) and proposed a new estimator for conditional variance to account for the skewness of financial data. In this paper, we apply empirical likelihood methods to construct confidence intervals for the conditional variance based on the estimator of Fan and Yao (1998) and the reduced variance modification of Chen et al. (2009). Simulation studies and data analysis demonstrate the advantage of the empirical likelihood method over the normal approximation method.

A desirable feature of the empirical likelihood (EL) method is its Bartlett correctability. Previous studies have only demonstrated that the Bartlett correctability of EL for independent cases. This paper considers the Bartlett correctability of EL in time series models. The validity of the formal Edgeworth expansion for the EL ratio statistic in the short-memory case is established and through meticulous calculations, a closed form expansion for the statistic is deduced. The order of the coverage error of the EL confidence region for time series is obtained based on such an Edgeworth expansion of the EL ratio statistic. It is further demonstrated that the coverage error can be reduced by an order of magnitude after using a Bartlett correction. Finally, a simulation study is presented to illustrate the Bartlett correctability of EL in the short-memory case.

A robust local linear regression smoothing estimator for a nonparametric regression model with heavy-tailed dependent errors is considered in this paper. Under certain regularity conditions, the weak consistency and asymptotic distribution of the proposed estimators are obtained. If the errors are short-range dependent, then the limiting distribution of the estimator is normal. If the data are long-range dependent, then the limiting distribution of the estimator is a stable distribution.

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.

This paper studies the optimal expected gain/loss of a portfolio at a given risk level when the initial investment is zero and the number of stocks [Formula: see text] grows with the sample size [Formula: see text]. A new estimator of the optimal expected gain/loss of such a portfolio is proposed after examining the behavior of the sample mean vector and the sample covariance matrix based on conditional expectations. It is found that the effect of the sample mean vector is additive and the effect of the sample covariance matrix is multiplicative, both of which over-predict the optimal expected gain/loss. By virtue of a shrinkage method, a new estimate is proposed when the sample covariance matrix is not invertible. The superiority of the proposed estimator is demonstrated by matrix inequalities and simulation studies.

A nonparametric test statistic based on the distance between the joint and marginal densities is developed to test for the serial dependence for a given sequence of time series data. The key idea lies in observing that, under the null hypothesis of independence, the joint density of the observations is equal to the product of their individual marginals. Histograms are used in constructing such a statistic which is nonparametric and consistent. It possesses high power in capturing subtle or diffuse dependence structure. A bilinear time series model is used to illustrate its performance with the classical correlation approach.

This paper reviews some of the developments of the unit root and near unit root time series. It gives an overview of this important topic and describes the impact of some of the recent progress on subsequent research.

This paper develops an empirical likelihood approach for regular generalized autoregressive conditional heteroskedasticity (GARCH) models and GARCH models with unit roots. For regular GARCH models, it is shown that the log empirical likelihood ratio statistic asymptotically follows a χ2 distribution. For GARCH models with unit roots, two versions of the empirical likelihood methods, the least squares score and the maximum likelihood score functions, are considered. For both cases, the limiting distributions of the log empirical likelihood ratio statistics are established. These two statistics can be used to test for unit roots under the GARCH framework. Finite-sample performances are assessed through simulations for GARCH models with unit roots.This research was supported in part by Hong Kong Research grants Council Grants CUHK4043/02P and HKUST6273/03H. The authors thank two referees and the Co-Editor, Bruce Hansen, for insightful and helpful comments about the relationship between QMLE and MELE, which led to substantial improvement of the presentation. Computational assistance from Jerry Wong and Chun-Yip Yau is also gratefully acknowledged.

Since the seminal works by Granger and Joyeux (1980) and Hosking (1981), estimations of long-memory time series models have been receiving considerable attention and a number of parameter estimation procedures have been proposed. This paper gives an overview of this plethora of methodologies with special focus on likelihood-based techniques. Broadly speaking, likelihood-based techniques can be classified into the following categories: the exact maximum likelihood (ML) estimation (Sowell, 1992; Dahlhaus, 1989), ML estimates based on autoregressive approximations (Granger & Joyeux, 1980; Li & McLeod, 1986), Whittle estimates (Fox & Taqqu, 1986; Giraitis & Surgailis, 1990), Whittle estimates with autoregressive truncation (Beran, 1994a), approximate estimates based on the Durbin–Levinson algorithm (Haslett & Raftery, 1989), state-space-based maximum likelihood estimates for ARFIMA models (Chan & Palma, 1998), and estimation of stochastic volatility models (Ghysels, Harvey, & Renault, 1996; Breidt, Crato, & de Lima, 1998; Chan & Petris, 2000) among others. Given the diversified applications of these techniques in different areas, this review aims at providing a succinct survey of these methodologies as well as an overview of important related problems such as the ML estimation with missing data (Palma & Chan, 1997), influence of subsets of observations on estimates and the estimation of seasonal long-memory models (Palma & Chan, 2005). Performances and asymptotic properties of these techniques are compared and examined. Inter-connections and finite sample performances among these procedures are studied. Finally, applications to financial time series of these methodologies are discussed.

This paper develops an order selection criterion for a continuous autoregressive (CAR) time series. Based on the quadratic variation consideration of a CAR(p) process, a new order selection criterion, the quadratic variation criterion (QVC) is proposed. It is shown that this new order selection criterion is consistent and provides an effective means to estimate the order of a CAR(p) model. Simulation studies suggest that the proposed method is efficient and outperforms other order selection criteria. The QVC is applied to select the order of the cumulative excess return process and the effect of the risk premium of a GARCH-M model when changing variance is taken into account.

In this article, accumulated sudden infant death syndrome (SIDS) data, from 1974–1978 and 1979–1984 for the counties of North Carolina, are analyzed. After a spatial exploratory data analysis, Markov random-field models are fit to the data. The (spatial) trend is meant to capture the large-scale variation in the data, and the variance and spatial dependence are meant to capture the small-scale variation. The trend could be a function of other explanatory variables or could simply be modeled as a function of spatial location. Both models are fit and compared. The results give an excellent illustration of a phenomenon already well-known in time series, that autocorrelation in data can be due to an undiscovered explanatory variable. Indeed, for 1974–1978 we confirm a dependence of SIDS rate on proportion of nonwhite babies born, along with insignificant spatial correlation. Without this regressor variable, however, the spatial correlation is significant. In 1979–1984, perhaps due to reporting bias o...

A time series is said to be nearly nonstationary if some of its characteristic roots are close to the unit circle. For a seasonal time series, such a notion of near-nonstationarity is studied in a double-array setting. This approach not only furnishes a natural transition between stationarity and nonstationarity, but also unifies the corresponding asymptotic theories in a seasonal-time-series context. The general theory is expressed in terms of functionals of independent diffusion processes. The asymptotic results have applications to estimation and testing in a nearly nonstationary situation and serve as a useful alternative to the common practice of seasonal adjustment. Une serie chronologique est dite presque non stationnaire si certaines des racines de son polynǒme caracteristique sont proches du cercle unite. Une telle notion de presque non stationnarite est considered pour une serie chronologique saisonniere. Cette approche permet non seulement une transition naturelle entre stationnarite et non stationnarite, mais aussi unifie les theories asymptotiques correspondantes. La theorie generale est exprimee en termes de fonctionnelles de processus de diffusion independants. Les resultats asymptotiques trouvent des applications a ľestimation et aux tests dans une situation de presque non stationnarite. Ils fournissent aussi une alternative a ľajustement saisonnier habituellement utlise.

Consider a near-integrated time series driven by a heavy-tailed and long-memory noise e t = ∑ j = 0 ∞ c j η t − j , where { η j } is a sequence of i . i . d random variables belonging to the domain of attraction of a stable law with index α . The limit distribution of the quantile estimate and the semi-parametric estimate of the autoregressive parameters with long- and short-range dependent innovations are established in this paper. Under certain regularity conditions, it is shown that when the noise is short-memory, the quantile estimate converges weakly to a mixture of a Gaussian process and a stable Ornstein–Uhlenbeck (O–U) process while the semi-parametric estimate converges weakly to a normal distribution. But when the noise is long-memory, the limit distribution of the quantile estimate becomes substantially different. Depending on the range of the stable index α , the limit distribution is shown to be either a functional of a fractional stable O–U process or a mixture of a stable process and a stable O–U process. These results indicate that although the quantile estimate tends to be more efficient for infinite variance time series, extreme caution should be exercised in the long-memory situation.

An autoregressive process $y_n = \beta_1y_{n-1} + \cdots + \beta_py_{n-p} + \varepsilon_n$ is said to be unstable if the characteristic polynomial $\phi(z) = 1 - \beta_1z - \cdots - \beta_pz^p$ has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of $(\beta_1, \cdots, \beta_p)$ is derived and characterized as a functional of stochastic integrals under a $2 + \delta$ moment assumption on $\varepsilon_n$. Up to the present, distributional results were available only with substantial restrictions on the possible roots which did not suggest the form of the distribution for the general case. To establish the limiting distribution, a result concerning the weak convergence of a sequence of random variables to a stochastic integral, which is of independent interest, is also developed.