Beta seasonal autoregressive moving average models

In this paper, we introduce the class of beta seasonal autoregressive moving average (βSARMA) models for modelling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha AV and Cribari-Neto F. Beta autoregressive moving average models. Test. 2009;18(3):529–545] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed.     

Bootstrap-based evaluation of markov-switching time series models

This paper explores the possibility of evaluating the adequacy of Markov-switching time series models by comparing selected functionals (such as the spectral density function and moving empirical moments) obtained from the data with those of the fitted model using a bootstrap algorithm. The proposed model checking procedure is easy to implement and flexible enough to be adapted to a wide variety of models with parameters subject to Markov regime-switching. Examples with real and artificial data illustrate the potential of the methodology.     

Markov-switching generalized additive models

We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.     

Time Series Properties of the Class of Generalized First-Order Autoregressive Processes with Moving Average Errors

A new class of time series models known as Generalized Autoregressive of order one with first-order moving average errors has been introduced in order to reveal some hidden features of certain time series data. The variance and autocovariance of the process is derived in order to study the behaviour of the process. It is shown that in special cases these new results reduce to the standard ARMA results. Estimation of parameters based on the Whittle procedure is discussed. We illustrate the use of this class of model by using two examples.     

On Markov-switching periodic ARMA models

In this work, we propose a generalization of the classical Markov-switching ARMA models to the periodic time-varying case. Specifically, we propose a Markov-switching periodic ARMA (MS-PARMA) model. In addition of capturing regime switching often encountered during the study of many economic time series, this new model also captures the periodicity feature in the autocorrelation structure. We first provide some probabilistic properties of this class of models, namely the strict periodic stationarity and the existence of higher-order moments. We thus propose a procedure for computing the autocovariance function where we show that the autocovariances of the MS-PARMA model satisfy a system of equations similar to the PARMA Yule–Walker equations. We propose also an easily implemented algorithm which can be used to obtain parameter estimates for the MS-PARMA model. Finally, a simulation study of the performance of the proposed estimation method is provided.     

Transformed symmetric generalized autoregressive moving average models

Cordeiro and Andrade [Transformed generalized linear models. J Stat Plan Inference. 2009;139:2970–2987] incorporated the idea of transforming the response variable to the generalized autoregressive moving average (GARMA) model, introduced by Benjamin et al. [Generalized autoregressive moving average models. J Am Stat Assoc. 2003;98:214–223], thus developing the transformed generalized autoregressive moving average (TGARMA) model. The goal of this article is to develop the TGARMA model for symmetric continuous conditional distributions with a possible nonlinear structure for the mean that enables the fitting of a wide range of models to several time series data types. We derive an iterative process for estimating the parameters of the new model by maximum likelihood and obtain a simple formula to estimate the parameter that defines the transformation of the response variable. Furthermore, we determine the moments of the original dependent variable which generalize previous published results. We illustrate the theory by means of real data sets and evaluate the results developed through simulation studies.     

Bayesian analysis of contaminated quarter plane moving average models

This article deals with Bayesian analysis of quarter plane moving average (MA) models observed on a rectangular part of a lattice. We present some properties concerning the autocorrelation function of MA models. These properties relate correlation parameters with the original model parameters providing much more understandable interpretation of results concerning the model. Simulation experiment is developed to explore the sensitivity of the posterior distribution when the process is contaminated with innovation and additive contamination. We show by simulation that the correlation structure of the model is seriously affected when the process contains additive contamination. We then propose a more general class of MA models which automatically deals with the contamination phenomenon [contaminated MA (CMA) model]. Also, we establish theoretical properties of the correlation function analogous with those in the previous model. Finally, we consider two applications of the CMA model. The results obtained in numerical examples show the goodness of the CMA model under contaminated data.     

Inference on Filtered and Smoothed Probabilities in Markov-Switching Autoregressive Models

ABSTRACT

We derive a statistical theory that provides useful asymptotic approximations to the distributions of the single inferences of filtered and smoothed probabilities, derived from time series characterized by Markov-switching dynamics. We show that the uncertainty in these probabilities diminishes when the states are separated, the variance of the shocks is low, and the time series or the regimes are persistent. As empirical illustrations of our approach, we analyze the U.S. GDP growth rates and the U.S. real interest rates. For both models, we illustrate the usefulness of the confidence intervals when identifying the business cycle phases and the interest rate regimes.     

Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Generalized normal ARMA model applied to the areas of economy,hydrology, and public policy

The main purpose of this article is the presentation of a new class of time series models which is the merge output of the generalized normal distribution with ideas from the GARMA model. Symmetrically, tails that may be lighter or heavier than the Gaussian distribution, and Gaussian and Laplace distributions as special cases, are the main advantages of the use of generalized normal distribution. The proposed model is called generalized normal autoregressive moving average (GN-ARMA). We exemplify the application of the proposed model adjusting it to the three time series, which are from the areas of economy, hydrology, and public policy.     

Markov-switching BILINEAR − GARCH models: Structure and estimation

Markov-switching (MS) models are becoming increasingly popular as efficient tools of modeling various phenomena in different disciplines, in particular for non Gaussian time series. In this articlept", we propose a broad class of Markov-switching BILINEAR–GARCH processes (MS ? BLGARCH hereafter) obtained by adding to a MS ? GARCH model one or more interaction components between the observed series and its volatility process. This parameterization offers remarkably rich dynamics and complex behavior for modeling and forecasting financial time-series data which exhibit structural changes. In these models, the parameters of conditional variance are allowed to vary according to some latent time-homogeneous Markov chain with finite state space or “regimes.” The main aim of this new model is to capture asymmetric and hence purported to be able to capture leverage effect characterized by the negativity of the correlation between returns shocks and subsequent shocks in volatility patterns in different regimes. So, first, some basic structural properties of this new model including sufficient conditions ensuring the existence of stationary, causal, ergodic solutions, and moments properties are given. Second, since the second-order structure provides a useful information to identify an appropriate time-series model, we derive the expression of the covariance function of for MS ? BLGARCH and for its powers. As a consequence, we find that the second (resp. higher)-order structure is similar to some linear processes, and hence MS ? BLGARCH (resp. its powers) admit an ARMA representation. This finding allows us for parameter estimation via GMM procedure proved by a Monte Carlo study and applied to foreign exchange rate of the Algerian Dinar against the single European currency.     

Random Level-Shift Time Series Models,ARIMA Approximations,and Level-Shift Detection

The main purpose of this article is to assess the performance of autoregressive integrated moving average (ARIMA) models when occasional level shifts occur in the time series under study. A random level-shift time series model that allows the level of the process to change occasionally is introduced. Between two consecutive changes, the process behaves like the usual autoregressive moving average (ARMA) process. In practice, a series generated from a random level-shift ARMA (RLARMA) model may be misspecified as an ARIMA process. The efficiency of this ARIMA approximation with respect to estimation of current level and forecasting is investigated. The results of examining a special case of an RLARMA model indicate that the ARIMA approximations are inadequate for estimating the current level, but they are robust for forecasting future observations except when there is a very low frequency of level shifts or when the series are highly negatively correlated. A level-shift detection procedure is presented to handle the low-frequency level-shift phenomena, and its usefulness in building models for forecasting is demonstrated.     

Fitting Time Series Models by Minimizing Multistep-ahead Errors: a Frequency Domain Approach

This paper brings together two topics in the estimation of time series forecasting models: the use of the multistep-ahead error sum of squares as a criterion to be minimized and frequency domain methods for carrying out this minimization. The methods are developed for the wide class of time series models having a spectrum which is linear in unknown coefficients. This includes the IMA(1, 1) model for which the common exponentially weigh-ted moving average predictor is optimal, besides more general structural models for series exhibiting trends and seasonality. The method is extended to include the Box–Jenkins `air line' model. The value of the multistep criterion is that it provides protection against using an incorrectly specified model. The value of frequency domain estimation is that the iteratively reweighted least squares scheme for fitting generalized linear models is readily extended to construct the parameter estimates and their standard errors. It also yields insight into the loss of efficiency when the model is correct and the robustness of the criterion against an incorrect model. A simple example is used to illustrate the method, and a real example demonstrates the extension to seasonal models. The discussion considers a diagnostic test statistic for indicating an incorrect model.     

ARIMA modelling of birth data

In this article the fitting of ARIMA models to time series relating to the births at Edendale Hospital in Natal, South Africa, over a 16-year period is discussed. The model (011)X(011)12 provides andexcellent fit to the monthly totals of mothers delivered but serious discrepancies between estimates of the moving average parameters obtained by the method of unconditional least squares using various statistical computer packages were observed. These can be ascribed to the fact that certain of the packages use the method of backcasting to calculate the unconditional sum-of-squares function in the estimation procedure and this approach breaks down for values of the moving average parameters close to the invertibility boundary. A simulation study to compare the different methods of estimation for the model (011) X (011)12 and to assess the seriousness of discrepancies in the ULS estimates is described. ARIMA models which provided the best fit to the series of monthly totals of caesarean sections performed, breechs births and instrumental deliveries are non-seasonal and reflect a dependence of present on past observations. In contrast the series involving stillbirths and neonatal deaths are white noise indicating that perinatal deaths are random events.     

Evidence for hedge fund predictability from a multivariate Student's t full-factor GARCH model

Extending previous work on hedge fund return predictability, this paper introduces the idea of modelling the conditional distribution of hedge fund returns using Student's t full-factor multivariate GARCH models. This class of models takes into account the stylized facts of hedge fund return series, that is, heteroskedasticity, fat tails and deviations from normality. For the proposed class of multivariate predictive regression models, we derive analytic expressions for the score and the Hessian matrix, which can be used within classical and Bayesian inferential procedures to estimate the model parameters, as well as to compare different predictive regression models. We propose a Bayesian approach to model comparison which provides posterior probabilities for various predictive models that can be used for model averaging. Our empirical application indicates that accounting for fat tails and time-varying covariances/correlations provides a more appropriate modelling approach of the underlying dynamics of financial series and improves our ability to predict hedge fund returns.     

Time series models with asymmetric Laplace innovations

We propose autoregressive moving average (ARMA) and generalized autoregressive conditional heteroscedastic (GARCH) models driven by asymmetric Laplace (AL) noise. The AL distribution plays, in the geometric-stable class, the analogous role played by the normal in the alpha-stable class, and has shown promise in the modelling of certain types of financial and engineering data. In the case of an ARMA model we derive the marginal distribution of the process, as well as its bivariate distribution when separated by a finite number of lags. The calculation of exact confidence bands for minimum mean-squared error linear predictors is shown to be straightforward. Conditional maximum likelihood-based inference is advocated, and corresponding asymptotic results are discussed. The models are particularly suited for processes that are skewed, peaked, and leptokurtic, but which appear to have some higher order moments. A case study of a fund of real estate returns reveals that AL noise models tend to deliver a superior fit with substantially less parameters than normal noise counterparts, and provide both a competitive fit and a greater degree of numerical stability with respect to other skewed distributions.     

Markov-Switching and Stochastic Volatility Diffusion Models of Short-Term Interest Rates

This article empirically compares the Markov-switching and stochastic volatility diffusion models of the short rate. The evidence supports the Markov-switching diffusion model. Estimates of the elasticity of volatility parameter for single-regime models unanimously indicate an explosive volatility process, whereas the Markov-switching models estimates are reasonable. Itis found that either Markov switching or stochastic volatility, but not both, is needed to adequately fit the data. A robust conclusion is that volatility depends on the level of the short rate. Finally, the Markov-switching model is the best for forecasting. A technical contribution of this article is a presentation of quasi-maximum likelihood estimation techniques for the Markov-switching stochastic-volatility model.     

Conditional Heteroscedasticity in Qualitative Response Models of Time Series: A Gibbs-Sampling Approach to the Bank Prime Rate

Previous time series applications of qualitative response models have ignored features of the data, such as conditional heteroscedasticity, that are routinely addressed in time series econometrics of financial data. This article addresses this issue by adding Markov-switching heteroscedasticity to a dynamic ordered probit model of discrete changes in the bank prime lending rate and estimating via the Gibbs sampler. The dynamic ordered probit model of Eichengreen, Watson, and Grossman allows for serial autocorrelation in probit analysis of a time series, and this article demonstrates the relative simplicity of estimating a dynamic ordered probit using the Gibbs sampler instead of the Eichengreen et al. maximum likelihood procedure. In addition, the extension to regime-switching parameters and conditional heteroscedasticity is easy to implement under Gibbs sampling. The article compares tests of goodness of fit between dynamic ordered probit models of the prime rate that have constant variance and conditional heteroscedasticity.     

M‐estimation for general ARMA Processes with Infinite Variance

Abstract. General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non‐Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M‐estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non‐Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996) , we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M‐estimator. We also consider bootstrapping the M‐estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M‐estimation and bootstrap procedures. An empirical example of financial time series is also provided.