Estimation of a multiple-threshold AR(p) model

The present paper deals with the multiple-threshold p-order autoregressive model which has been introduced by Tong and Lim [H. Tong, K.S. Lim, Threshold autoregression, limit cycles and cyclical data, J. R. Stat. Soc. Ser. B 42 (1980) 245–292] in nonlinear system modelling. Under some conditions on the coefficients of the model which ensure the stationarity, the existence of moments and the strong mixing property of this process and under other mild assumptions, we establish the asymptotic properties (consistency and asymptotic normality) of the minimum Hellinger distance estimates of the autoregressive coefficients of the model.     

Seemingly unrelated regression on the autoregressive (Ar(p)) singular equation system

This paper studies the estimation of seemingly unrelated regressions (SUR) of singular equation systems with an autoregressive error process (AR(p)) for each equation.Parameter estimates of the autoregressive singular equation system are not generally invariant to the equation deleted. Under the model specification restriction on the autoregressive parameters, the invariance property is preserved, and this paper shows that a single equation generalized least squares (GLS) estimation for a general autoregressive error process is equivalent to the SURGLS estimation of the AR(p) singular equation system.     

The Asymptotic Behavior of INAR (p) Models

An integer-valued autoregressive model with random time delay under random environment is presented. The geometric ergodicity of the iterative sequence determined by this new model is discussed. Moreover, sufficient conditions for stationarity and β-mixing property with exponential decay for the INAR model with random time delay under random environment are developed.     

Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

Bayesian model selection in linear mixed effects models with autoregressive(p) errors using mixture priors

In this article, we apply the Bayesian approach to the linear mixed effect models with autoregressive(p) random errors under mixture priors obtained with the Markov chain Monte Carlo (MCMC) method. The mixture structure of a point mass and continuous distribution can help to select the variables in fixed and random effects models from the posterior sample generated using the MCMC method. Bayesian prediction of future observations is also one of the major concerns. To get the best model, we consider the commonly used highest posterior probability model and the median posterior probability model. As a result, both criteria tend to be needed to choose the best model from the entire simulation study. In terms of predictive accuracy, a real example confirms that the proposed method provides accurate results.     

On the efficiency of regression analysis with AR(p) errors

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.     

Objective priors for causal AR(p) with partial autocorrelations

We consider the problem of deriving formal objective priors for the causal/stationary autoregressive model of order p. We compare the frequentist behaviour of the most common default priors, namely the uniform (over the stationarity region) prior, the Jeffreys’ prior and the reference prior.     

A (Semi)Parametric Functional Coefficient Logarithmic Autoregressive Conditional Duration Model

In this article, we propose a class of logarithmic autoregressive conditional duration (ACD)-type models that accommodates overdispersion, intermittent dynamics, multiple regimes, and asymmetries in financial durations. In particular, our functional coefficient logarithmic autoregressive conditional duration (FC-LACD) model relies on a smooth-transition autoregressive specification. The motivation lies on the fact that the latter yields a universal approximation if one lets the number of regimes grows without bound. After establishing sufficient conditions for strict stationarity, we address model identifiability as well as the asymptotic properties of the quasi-maximum likelihood (QML) estimator for the FC-LACD model with a fixed number of regimes. In addition, we also discuss how to consistently estimate a semiparametric variant of the FC-LACD model that takes the number of regimes to infinity. An empirical illustration indicates that our functional coefficient model is flexible enough to model IBM price durations.     

An optimal k of kth MA-ARIMA models under AR(p) models

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated autoregressive AR(p) model. We run a simulation using the three above examining method under specific conditions. The main finding is that the optimal k = 4 and then k = 3. Especially, the fourth WMA ARIMA model, fourth EWMA ARIMA model, and third EWMA ARIMA model are the best forecasting models among others, respectively. For all the six real data reveal the similar results of simulation study.     

Improved model selection criteria for SETAR time series models

The purpose of this paper is threefold. First, we obtain the asymptotic properties of the modified model selection criteria proposed by Hurvich et al. (1990. Improved estimators of Kullback-Leibler information for autoregressive model selection in small samples. Biometrika 77, 709–719) for autoregressive models. Second, we provide some highlights on the better performance of this modified criteria. Third, we extend the modification introduced by these authors to model selection criteria commonly used in the class of self-exciting threshold autoregressive (SETAR) time series models. We show the improvements of the modified criteria in their finite sample performance. In particular, for small and medium sample size the frequency of selecting the true model improves for the consistent criteria and the root mean square error (RMSE) of prediction improves for the efficient criteria. These results are illustrated via simulation with SETAR models in which we assume that the threshold and the parameters are unknown.     

房地产价格区域间联动与泡沫的空间扩散 ——基于2000－2005年中国35个大中城市面板数据的实证检验

 利用中国35个大中城市2000－2005年间的面板数据，本文首先对房地产价格中的泡沫成分进行测度；然后构建泡沫自回归模型，并对其残差进行CSD检验，结果显示不同城市间房地产泡沫的演化过程相互影响。与前人研究不同，以房地产泡沫为研究对象，不仅可以确认中国不同城市间房地产价格存在联动性，而且可以证明消费者的适应性预期是其中重要的传导机制之一。     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

Comparison of predictions by kriging and spatial autoregressive models

In geostatistics, the prediction of unknown quantities at given locations is commonly made by the kriging technique. In addition to the kriging technique for modeling regular lattice spatial data, the spatial autoregressive models can also be used. In this article, the spatial autoregressive model and the kriging technique are introduced. We extend prediction method proposed by Basu and Reinsel for SAR(2,1) model. Then, using a simulation study and real data, we compare prediction accuracy of the spatial autoregressive models with that of the kriging prediction. The results of simulation study show that predictions made by the autoregressive models are good competitor for the kriging method.     

Resampling for Order Estimation of Autoregressive Models with Missing Data

In this article, we consider the order estimation of autoregressive models with incomplete data using the expectation–maximization (EM) algorithm-based information criteria. The criteria take the form of a penalization of the conditional expectation of the log-likelihood. The evaluation of the penalization term generally involves numerical differentiation and matrix inversion. We introduce a simplification of the penalization term for autoregressive model selection and we propose a penalty factor based on a resampling procedure in the criteria formula. The simulation results show the improvements yielded by the proposed method when compared with the classical information criteria for model selection with incomplete data.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

A study on geographically weighted spatial autoregression models with spatial autoregressive disturbances

Abstract

Spatial heterogeneity and correlation are both considered in the geographical weighted spatial autoregressive model. At present, this kind of model has aroused the attention of some scholars. For the estimation of the model, the existing research is based on the assumption that the error terms are independent and identically distributed. In this article we use a computationally simple procedure for estimating the model with spatially autoregressive disturbance terms, both the estimates of constant coefficients and variable coefficients are obtained. Finally, we give the large sample properties of the estimators under some ordinary conditions. In addition, application study of the estimation methods involved will be further explored in a separate study.     

Bootstrapping p Values and Power in the First-Order Autoregression: A Monte Carlo Investigation

The small-sample behavior of the bootstrap is investigated as a method for estimating p values and power in the stationary first-order autoregressive model. Monte Carlo methods are used to examine the bootstrap and Student-t approximations to the true distribution of the test statistic frequently used for testing hypotheses on the underlying slope parameter. In contrast to Student's t, the results suggest that the bootstrap can accurately estimate p values and power in this model in sample sizes as small as 5–10.     

Bayesian inference and forecasting in the stationary bilinear model

A stationary bilinear (SB) model can be used to describe processes with a time-varying degree of persistence that depends on past shocks. This study develops methods for Bayesian inference, model comparison, and forecasting in the SB model. Using monthly U.K. inflation data, we find that the SB model outperforms the random walk, first-order autoregressive AR(1), and autoregressive moving average ARMA(1,1) models in terms of root mean squared forecast errors. In addition, the SB model is superior to these three models in terms of predictive likelihood for the majority of forecast observations.     

On the estimation problem of periodic autoregressive time series: symmetric and asymmetric innovations

Periodic autoregressive (PAR) models with symmetric innovations are widely used on time series analysis, whereas its asymmetric counterpart inference remains a challenge, because of a number of problems related to the existing computational methods. In this paper, we use an interesting relationship between periodic autoregressive and vector autoregressive (VAR) models to study maximum likelihood and Bayesian approaches to the inference of a PAR model with normal and skew-normal innovations, where different kinds of estimation methods for the unknown parameters are examined. Several technical difficulties which are usually complicated to handle are reported. Results are compared with the existing classical solutions and the practical implementations of the proposed algorithms are illustrated via comprehensive simulation studies. The methods developed in the study are applied and illustrate a real-time series. The Bayes factor is also used to compare the multivariate normal model versus the multivariate skew-normal model.     

A Bayesian method of distinguishing unit root from stationary processes based on panel data models with cross-sectional dependence

In this paper we develop a Bayesian approach to detecting unit roots in autoregressive panel data models. Our method is based on the comparison of stationary autoregressive models with and without individual deterministic trends, to their counterpart models with a unit autoregressive root. This is done under cross-sectional dependence among the error terms of the panel units. Simulation experiments are conducted with the aim to assess the performance of the suggested inferential procedure, as well as to investigate if the Bayesian model comparison approach can distinguish unit root models from stationary autoregressive models under cross-sectional dependence. The approach is applied to real exchange rate series for a panel of the G7 countries and to a panel of US nominal interest rates data.