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Summary This paper develops a unified finite-time theory for the ordinary least squares estimation of possibly unstable and even slightly explosive vector autoregressive models under linear restrictions, with the applicable region $\rho(A)\leqslant 1+c/n$, where $\rho(A)$ is the spectral radius of the transition matrix $A$ in the var(1) representation, $n$ is the time horizon and $c>0$ is a universal constant. The linear restriction framework encompasses various existing models such as banded/network vector autoregressive models. We show that the restrictions reduce the error bounds via not only the reduced dimensionality, but also a scale factor resembling the asymptotic covariance matrix of the estimator in the fixed-dimensional set-up: as long as the model is correctly specified, this scale factor is decreasing in the number of restrictions. It is revealed that the phase transition from slow to fast error rate regimes is determined by the smallest singular value of $A$, a measure of the least excitable mode of the system. The minimax lower bounds are derived across different regimes. The developed non-asymptotic theory not only bridges the theoretical gap between stable and unstable regimes, but precisely characterizes the effect of restrictions and its interplay with model parameters. Simulations support our theoretical results.

Summary We consider the weighted bootstrap approximation to the distribution of a class of M-estimators for the parameters of the generalized autoregressive conditional heteroscedastic model. We prove that the bootstrap distribution, given the data, is a consistent estimate in probability of the distribution of the M-estimator, which is asymptotically normal. We propose an algorithm for the computation of M-estimates which at the same time is useful for computing bootstrap replicates from the given data. Our simulation study indicates superior coverage rates for various weighted bootstrap schemes compared with the rates based on the normal approximation and existing bootstrap methods for the generalized autoregressive conditional heteroscedastic model, such as percentile $t$-subsampling schemes. Since some familiar bootstrap schemes are special cases of the weighted bootstrap, this paper thus provides a unified theory and algorithm for bootstrapping in generalized autoregressive conditional heteroscedastic models.

Summary Causal inference in multivariate time series is challenging because the sampling rate may not be as fast as the time scale of the causal interactions, so the observed series is a subsampled version of the desired series. Furthermore, series may be observed at different sampling rates, yielding mixed-frequency series. To determine instantaneous and lagged effects between series at the causal scale, we take a model-based approach that relies on structural vector autoregressive models. We present a unifying framework for parameter identifiability and estimation under subsampling and mixed frequencies when the noise, or shocks, is non-Gaussian. By studying the structural case, we develop identifiability and estimation methods for the causal structure of lagged and instantaneous effects at the desired time scale. We further derive an exact expectation-maximization algorithm for inference in both subsampled and mixed-frequency settings. We validate our approach in simulated scenarios and on a climate and an econometric dataset.

SUMMARYThis paper discusses edge correction for a large class of conditional and intrinsic autoregressions on two-dimensional finite regular arrays. The proposed method includes a novel reparameterization, retains the simple neighbourhood structure, ensures the nonnegative definiteness of the precision matrix, and enables scalable matrix-free statistical computation. The edge correction provides new insight into how higher-order differencing enters into the precision matrix of a conditional autoregression.

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 consider a class of vector autoregressive models with banded coefficient matrices. The setting represents a type of sparse structure for high-dimensional time series, though the implied autocovariance matrices are not banded. The structure is also practically meaningful when the order of component time series is arranged appropriately. The convergence rates for the estimated banded autoregressive coefficient matrices are established. We also propose a Bayesian information criterion for determining the width of the bands in the coefficient matrices, which is proved to be consistent. By exploring some approximate banded structure for the autocovariance functions of banded vector autoregressive processes, consistent estimators for the auto-covariance matrices are constructed.

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.

Antedependence models, also known as transition models, have proven to be useful for longitudinal data exhibiting serial correlation, especially when the variances and/or same-lag correlations are time-varying. Statistical inference procedures associated with normal antedependence models are well-developed and have many nice properties, but they are not appropriate for longitudinal data that exhibit considerable skewness. We propose two direct extensions of normal antedependence models to skew-normal antedependence models. The first is obtained by imposing antedependence on a multivariate skew-normal distribution, and the second is a sequential autoregressive model with skew-normal innovations. For both models, necessary and sufficient conditions for [Formula: see text]th-order antedependence are established, and likelihood-based estimation and testing procedures for models satisfying those conditions are developed. The procedures are applied to simulated data and to real data from a study of cattle growth.

Self-exciting threshold autoregressive time series models have been used extensively, and the conditional mean obtained from these models can be used to predict the future value of a random variable. In this paper we consider quantile forecasts of a time series based on the quantile self-exciting threshold autoregressive time series models proposed by Cai and Stander (2008) and present a new forecasting method for them. Simulation studies and application to real time series show that the method works very well. Copyright 2010, Oxford University Press.

This paper proposes a simple new model for stationary time series of integer counts. Previous work has focused on thinning methods and classical time series autoregressive moving-average difference equations; in contrast, our methods use a renewal process to generate a correlated sequence of Bernoulli trials. By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric or any other discrete marginal distribution can be readily constructed. The model class proposed is parsimonious, non-Markov and readily generates series with either short- or long-memory autocovariances. The model can be fitted with linear prediction techniques for stationary series. As an example, a stationary series with binomial marginal distributions is fitted to the number of rainy days in 210 consecutive weeks at Key West, Florida. Copyright 2009, Oxford University Press.

We introduce the class of Student t-mixture autoregressive models, which is promising for financial time series modelling. The model is able to capture serial correlations, time-varying means and volatilities, and the shape of the conditional distributions can be time varied from short-tailed to long-tailed, or from unimodal to multimodal. The use of t-distributed errors in each component of the model allows conditional leptokurtic distributions that account for the commonly observed excess unconditional kurtosis in financial data. Methods of parameter estimation and model selection are given. Finally, the proposed modelling procedure is illustrated through a real example. Copyright 2009, Oxford University Press.

We investigate the nonstationary double ar(1) model, where ω > 0, α > 0, the η t are independent standard normal random variables and Elog |φ + η t √α| ⩾ 0. We show that the maximum likelihood estimator of (φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling ([11]) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of φ for any φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical ar(1) model, corresponding to α = 0. Copyright 2008, Oxford University Press.

SUMMARY Typical alternative hypotheses in the analysis of residuals of a standard regression model are considered, and for each one a Bayesian diagnostic based on a symmetric form of the Kullback-Leibler divergence is determined. The results include an explicit expression for the diagnostic when the alternative hypothesis is that the errors are generated by an unknown distribution function with a Dirichlet process prior. This expression is immediately interpretable, exactly computable and endowed with important asymptotic connections. A linear approximation of the diagnostic reveals close links with the class of Lagrange multiplier test statistics. When the alternative hypothesis is that the errors are generated by an autoregressive process the linear approximation is proportional to the Box-Pierce statistic or to the Ljung-Box statistic, according to the characteristics of the prior, if the observations have zero mean; it depends on the Durbin-Watson statistic if the errors are first-order autoregressive, and it is related to the Cliff-Ord statistic if they are generated by a first-order spatial autoregression. The sensitivity to the prior of the diagnostic and of its linear approximation is also discussed.

The recent paper by Peng & Yao (2003) gave an interesting extension of least absolute deviation estimation to generalised autoregressive conditional heteroscedasticity, GARCH, time series models. The asymptotic distributions of absolute residual autocorrelations and squared residual autocorrelations from the GARCH model estimated by the least absolute deviation method are derived in this paper. These results lead to two useful diagnostic tools which can be used to check whether or not a GARCH model fitted by using the least absolute deviation method is adequate. Some simulation experiments give further support to the asymptotic theory and a real data example is also reported.

Seasonally persistent models were first introduced by Andel (1986) and Gray et al. (1989) to extend autoregressive moving-average and fractionally differenced models and to encompass long-memory quasi-periodic behaviour. These models are, for certain ranges of parameters, stationary, and we prove here that the behaviour of the periodogram and other tapered estimators cannot be simply extended from the work of Kunsch (1986) and Hurvich & Beltrao (1993) on long memory induced by a pole at the origin. We demonstrate that potentially large both positive and negative bias can be found from the same value of the long-memory parameter, and that the new distribution can be easily written down in the case of Gaussian processes. We also consider using both the cosine taper and the sine taper. The extended least squares estimator is also considered in this context.

SUMMARY We consider the estimation of parameters of a multiple regression model with non stationary errors. We assume the nonstationary errors satisfy a time-dependent auto regressive process and describe a method for estimating the parameters of the regressors and the time-dependent autoregressive parameters. The parameters are rescaled as in nonparametric regression to obtain the asymptotic sampling properties of the estimators. The method is illustrated with an example taken from global temperature anomalies.

This paper studies properties of ordinary and generalised least squares estimators in a simple linear regression with stationary autocorrelated errors. Explicit expressions for the variances of the regression parameter estimators are derived for some common time series autocorrelation structures, including a first-order autoregression and general moving averages. Applications of the results include confidence intervals and an example where the variance of the trend slope estimator does not increase with increasing autocorrelation.

We propose a test for multimodality with dependent data by resampling from a suitably constructed transition probability kernel, which includes Silverman's test with independent data as a special case. We extend some theoretical properties of Silverman's test with independent and identically distributed data to weakly dependent data, and also discuss the robustness of Silverman's test against departure from independence.

Hall & Yao (2003) showed that, for ARCH/GARCH, i.e. autoregressive conditional heteroscedastic/ generalised autoregressive conditional heteroscedastic, models with heavy-tailed errors, the conventional maximum quasilikelihood estimator suffers from complex limit distributions and slow convergence rates. In this paper three types of absolute deviations estimator have been examined, and the one based on logarithmic transformation turns out to be particularly appealing. We have shown that this estimator is asymptotically normal and unbiased. Furthermore it enjoys the standard convergence rate of n 1/2 regardless of whether the errors are heavy-tailed or not. Simulation lends further support to our theoretical results.

SUMMARY We consider the testing and estimation of thresholds in heteroscedastic threshold autoregressive models with an unknown number of thresholds. A test statistic based on empirical wavelet coefficients is proposed. The asymptotic distribution of the test statistic is established and consistent estimators of the thresholds and the number of thresholds are given. A Monte Carlo study and a real example are used to assess the performance of our method.

SUMMARY We present a general class of nonlinear regression and time series models that we call generalised structured models. The class is a natural generalisation of generalised additive models, and it includes generalised interaction models, structured volatility models, visual GARCH, generalised autoregressive conditional heteroscedasticity, models and varying coefficient models. We discuss estimation principles including smoothing splines and a generalisation of the projection approach of Mammen et al. (1999). We finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models.

SUMMARY There is a well-known Bayesian interpretation for function estimation by spline smoothing using a limit of proper normal priors. The limiting prior and the conditional and intrinsic autoregressive priors popular for spatial modelling have a common form, which we call partially informative normal. We derive necessary and sufficient conditions for the propriety of the posterior for this class of partially informative normal priors with noninformative priors on the variance components, a condition crucial for successful implementation of the Gibbs sampler. The results apply for fully Bayesian smoothing

SUMMARY We introduce a family of multivariate binary distributions with certain conditional linear property. This family is particularly useful for efficient and easy simulation of correlated binary variables with a given marginal mean vector and correlation matrix.

Structural vector autoregressions allow contemporaneous series dependence and assume errors with no contemporaneous correlation. Models of this form, that also have a recursive structure, can be described by a directed acyclic graph.An important tool for identification of these models is the conditional independence graph constructed from the contemporaneous and lagged values of the process. We determine the large-sample properties of statistics used to test for the presence of links in this graph. A simple example illustrates how these results may be applied.

SUMMARY We have implemented a Lagrange multiplier test for the alternative hypothesis of a nonlinear continuous-time autoregressive model with the instantaneous mean having multiple degrees of nonlinearity. This test is an extension of a Lagrange multiplier test proposed by Tsai & Chan (2000), with the alternative model analogous to the model used in Tsay's (1986) discrete-time work. The performance of the test in the finite-sample case is compared with several existing tests for nonlinearity including Keenan's (1985) test, Petruccelli & Davies' (1986) test, Tsay's (1986, 1989) tests and Tsai & Chan's (2000) test. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregressive models, bilinear models and the nonlinear continuous-time autoregressive models for which the Lagrange multiplier test is designed. In general, the test is more powerful than all the other tests. The test is further illustrated with the annual sunspot data and the lynx data.

SUMMARY We generalise the mixture autoregressive, MAR, model to the logistic mixture autoregressive with exogenous variables, LMARX, model for the modelling of nonlinear time series. The models consist of a mixture of two Gaussian transfer function models with the mixing proportions changing over time. The model can also be considered as a generalisation of the self-exciting threshold autoregressive, SETAR, model and the open-loop threshold autoregressive, TARSO, model. The advantages of the LMARX model over other nonlinear time series models include a wider range of shape-changing predictive distributions, the ability to handle cycles and conditional heteroscedasticity in the time series and better point prediction. Estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The models are applied to two real datasets and compared with other competing models.

We have implemented a Lagrange multiplier test specifically for the alternative of a nonlinear continuous-time autoregressive model with the instantaneous mean having one degree of nonlinearity. The test is then extended to testing for the alternative of general nonlinear continuous-time autoregressive models with multiple degrees of nonlinearity. The performance of the test in the finite-sample case is compared with several existing tests for nonlinearity including Keenan's (1985) test. Petruccelli & Davies' (1986) test and Tsay's (1986, 1989) tests. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregressive models, bilinear models and the nonlinear continuous-time autoregressive models which the Lagrange multiplier test is designed to detect. The Lagrange multiplier test outperforms the other tests in detecting the model for which it is designed. Compared with the other tests, the test has excellent power in detecting bilinear models, but seems less powerful in detecting self-exciting threshold autoregressive nonlinearity. The test is further illustrated with the Hong Kong beach water quality data.

SUMMARY For regularly-spaced observations with the Gaussian autocorrelation function, the finite and infinite autoregressive and moving-average representations can be obtained theoretically. This allows exact Gaussian maximum likelihood to be performed very accurately, and gives a simple exact simulation method.

Asymptotic simultaneous confidence bands for all ω in [0, 0.5] are obtained for each scalar function in a spectral density matrix, as well as for the logarithm of the autospectra, the squared coherence and the phase spectra. The bands are illustrated using a simulation study and surface elevation data.

This paper addresses both theoretical and methodological issues related to the prediction of long-memory models with incomplete data. Estimates and forecasts are calculated by means of state space models and the influence of data gaps on the performance of short and long run predictions is investigated. These techniques are illustrated with a statistical analysis of the minimum water levels of the Nile river, a time series exhibiting strong dependency.

SUMMARY Many physical/biological processes involve variability over both space and time. As a result of difficulties caused by large datasets and the modelling of space, time and spatiotemporal interactions, traditional space-time methods are limited. In this paper, we present an approach to space-time prediction that achieves dimension reduction and uses a statistical model that is temporally dynamic and spatially descriptive. That is, it exploits the unidirectional flow of time, in an autoregressive framework, and is spatially 'descriptive' in that the autoregressive process is spatially coloured. With the inclusion of a measurement equation, this formulation naturally leads to the development of a spatio-temporal Kalman filter that achieves dimension reduction in the analysis of large spatio-temporal datasets. Unlike other recent space-time Kalman filters, our model also allows a nondynamic spatial component. The method is applied to a dataset of near-surface winds over the topical Pacific ocean. Spatial predictions with this dataset are improved by considering the additional non-dynamic spatial process. The improvement becomes more pronounced as the signal-to-noise ratio decreases.

In this paper the subsampling method of Carlstein (1986) is used to estimate the risk of prediction for time series data. We extend Carlstein's result by proving strong consistency of the subsampling estimator and we propose a procedure for selecting a time series model empirically from a set of possibly nonnested and misspecified models by using estimated risk of prediction as a selection criterion. When this procedure is applied to the selection of the order of an autoregressive model, it is shown to be a consistent order selector if an appropriate subsample size is chosen. We propose a practical model selection procedure with a common subsample size chosen by Hall & Jing's (1996) procedure.

We propose a method of modelling panel time series data with both inter- and intra-individual correlation, and of fitting an autoregressive model to such data. Estimators are obtained by a conditional likelihood argument. If there are few observations in each series, the estimators can be dramatically improved by Burg-type estimators taking edge effects into account. The consequences of ignoring the intercorrelation term are analysed. Partial lack of consistency is demonstrated in this situation. Moreover, a break-even point is found for the strength of the intercorrelation, beyond which a conventional estimator, ignoring correlation, will become increasingly inferior. Asymptotic normality of estimators is established, and our results are illustrated on a real data example, where it is seen that choosing the right type of estimator is crucial.

SUMMARY Stable autoregressive models with exogenous covariates can be used to mimic nonstationary, stabilising growth profiles in longitudinal data. If different individuals have very different trajectories, the autoregressive parameters can be interpreted as random variables varying from one individual to the next. If all random effects are assumed Gaussian, the model will be quite easy to estimate although the marginal distributions of the observations will no longer be Gaussian. However, this kind of model can only be used when the observations are equally spaced.

The modelling of extremes of a time series has progressed from the assumption of independent observations to more realistic forms of temporal dependence. In this paper, we focus on Markov chains, deriving a class of models for their joint tail which allows the degree of clustering of extremes to decrease at high levels, overcoming a key Limitation in current methodologies. Theoretical aspects of the model are examined and a simulation algorithm is developed through which the stochastic properties of summaries of the extremal txhaviour of the chain are evaluated. The approach is illustrated through a simulation study of extremal events of Gaussian autoregressive processes and an application to temperature data.

SUMMARY The question of model choice for the class of stationary and nonstationary, fractional and nonfractional autoregressive processes is considered. This class is defined by the property that the dth difference, for -2 < d < oo, is a stationary autoregressive process of order p0 < 00. A version of the Akaike information criterion, AIC, for determining an appropriate autoregressive order when d and the autoregressive parameters are estimated simultaneously by a maximum likelihood procedure (Beran, 1995) is derived and shown to be of the same general form as for a stationary autoregressive process, but with d treated as an additional estimated parameter. Moreover, as in the stationary case, this criterion is shown not to provide a consistent estimator of p0. The corresponding versions of the BIC of Schwarz (1978) and the HIC of Hannan & Quinn (1979) are shown to yield consistent estimators of po. The results provide a unified treatment of fractional and nonfractional, stationary and integrated nonstationary autoregressive models.

The continuous threshold autoregressive model is a sub-class of the threshold autoregressive model subject to the requirement that the piece-wise linear autoregressive function be continuous everywhere. In contrast with the discontinuous case, it is shown that, under suitable regularity conditions, the conditional least squares estimator of the parameters including the threshold parameter is root-n consistent and asymptotically normally distributed. The theory is illustrated by a simulation study and is applied to the quarterly U.S. unemployment rates.

SUMMARY We examine simple estimators for general ARMA models and a corresponding identification method. Both estimation and identification are based on a matrix formed from the coefficients of an autoregressive approximation to the process of interest. We show that a zero determinant of this matrix is necessary and sufficient for the existence of a common factor in autoregressive and moving average lag polynomials, and therefore for redundant parameters in the model. Simulation results suggest a close match between the empirical finite-sample distribution of the test statistic for model order reduction and its asymptotic distribution.

SUMMARY The usual conditions for a Bartlett correction are not fulfilled when testing for a unit root in a Gaussian autoregressive model. However, by expanding the moments of the likelihood ratio statistic it can be shown that the Bartlett correction improves the asymptotic distribution approximation.

SUMMARY Ljung's (1989) method for estimating missing values and evaluating the corresponding likelihood function in scalar time series is extended to the vector case. The series is assumed to be generated by a possibly partially nonstationary and noninvertible vector autoregressive-moving average process. No particular pattern of missing data is assumed. Future and past values are special cases of missing data that can be estimated in the same way. The method does not use Kalman filter iterations and hence avoids initialisation problems. It does not require the series to be differenced and thus avoids complications caused by over-differencing. The estimators of the missing data are provided by the normal equations of an appropriate regression problem. These equations are adapted to cope with temporally aggregated data; the procedure parallels a matrix treatment of contour conditions in the analysis of variance. Autoregressive processes are considered in some detail.

For an arbitrary covariance matrix and any number of treatments and periods, necessary and sufficient conditions for the optimality of a repeated measurements, i.e. crossover or changeover, design were given in the, as yet, unpublished report by H. B. Kushner 'Optimal repeated measurements designs: the linear optimality equations' (Kushner, report). Here we specialise the results to the two-treatment case. We give linear equations whose solutions are the proportions of subjects allocated to each treatment sequence in the most general optimal design. The treatment effects information matrix of an optimal design is also presented. The results are applied to several situations. For arbitrary covariance matrices, we show that designs consisting of at most two treatment sequences and their duals must necessarily be dual balanced to be optimal. For autoregressive covariance matrices, we confirm that designs of Matthews (1987) for three and four periods are optimal and we derive new optimal designs for five and six periods. Some optimality results of Laska & Meisner (1985) are extended. A simple formula for the efficiency of a dual balanced nonoptimal design is provided and used to compute the exact efficiencies of some of Kunert's (1991) designs.

A constructive result on time series decomposition is presented and illustrated. Developed through dynamic linear models, the decomposition is useful in analysis of an observed time series through inference about underlying, latent component series that may have physical interpretations. Particular special cases include state space autoregressive component models, in which the decomposition is useful for isolating latent, quasi-cyclical components, in particular. Brief summaries of analyses of some geological records related to climatic change illustrate the result.

Tsay (1987) developed the conditional heteroscedastic autoregressive moving-average model, which includes the conditional heteroscedastic autoregressive and random coefficient autoregressive models as special cases. This paper establishes the multivariate conditional heteroscedastic autoregressive moving-average model, and considers its theoretical properties and applications. Maximum likelihood estimation of the model is discussed in detail. A representation of the information matrix is obtained using the star product. This enhances estimation and statistical inferences procedures. Some simulation results and an application to the volatility of the Standard & Poor's 500 and Sydney's All Ordinaries indices are also considered.

SUMMARY Model selection for time series data based upon joint multiperiod forecasts is investigated. Some popular selection criteria are applied to a suitable multivariate regression model to create new selection criteria. Monte Carlo experiments show that the proposed selection procedure performs more efficiently than the corresponding regular procedure, particularly when the data are generated from a high order autoregressive model. Analysis of some real data supports the applicability of the proposed procedure, especially when the degree of differencing required is uncertain.

SUMMARY This paper is concerned with the null hypothesis that errors in a regression equation for time series data follow a random walk. We examine the power properties of most powerful invariant tests for the unit root null hypotheses against exact stationary and nonstationary first order autoregressive models. The analysis shows the importance of a constant term and a linear trend variable in certain cases. The implications of the results for models estimated using seasonal data are briefly discussed.

SUMMARY We propose bootstrap tests for unit roots in first-order autoregressive models and we establish their asymptotic validity both for independent and for autoregressive errors; in this case, the bootstrap methodology directly approaches the asymptotic distribution, making unnecessary the usual corrections due to dependence of innovations. We also present a Monte Carlo power study comparing these tests with existing alternative methods. For small samples, the power of the bootstrap test outperforms that of previous proposals.