Autoregressive Moving Average Errors Research Articles

Parameter prediction based on Improved Process neural network and ARMA error compensation in Evaporation Process

The traditional model of evaporation process parameters have continuity and cumulative characteristics of the prediction error larger issues, based on the basis of the process proposed an adaptive particle swarm neural network forecasting method parameters established on the autoregressive moving average (ARMA) error correction procedure compensated prediction model to predict the results of the neural network to improve prediction accuracy. Taking a alumina plant evaporation process to analyze production data validation, and compared with the traditional model, the new model prediction accuracy greatly improved, can be used to predict the dynamic process of evaporation of sodium aluminate solution components.

Oracally Efficient Estimation and Consistent Model Selection for Auto-Regressive Moving Average Time Series with Trend

Summary Most time series that are encountered in practice contain non-zero trend, yet textbook approaches to time series analysis are typically focused on zero-mean stationary auto-regressive moving average (ARMA) processes. Trend is often estimated by ad hoc methods and subtracted from time series, and the residuals are used as the true ARMA noise for data analysis and inference, including parameter estimation, lag selection and prediction. We propose a theoretically justified two-step method to analyse time series consisting of a smooth trend function and ARMA error term, which is computationally efficient and easy for practitioners to implement. The trend is estimated by B-spline regression, and the maximum likelihood estimator based on residuals is shown to be oracally efficient in the sense that it is asymptotically as efficient as if the true trend function were known and then removed to obtain the ARMA errors. In addition, consistency of the Bayesian information criterion for model selection is established for the detrended residual sequence. Finite sample performance of the procedure is illustrated by simulation studies and real data analysis.

Testing for uncorrelated errors in ARMA models: non‐standard Andrews‐Ploberger tests

A problem of interest in economic and finance applications is testing whether ARMA (Autoregressive moving average) errors are uncorrelated under weak assumptions, namely assumptions where the errors are neither iid nor a martingale difference. In this paper, non-standard versions of the tests of serial correlation introduced by Andrews and Ploberger (1996, hereafter AP) are proposed for diagnostic checking of ARMA errors. The original AP tests are designed for the case where the observed time series is generated by ARMA(1,1) models under the alternative and use asymptotic critical values computed by AP. The non-standard testing procedure uses AP statistics calculated from residuals and critical values based on asymptotic distribution theory derived under weak assumptions. The motivation for modifying the original AP tests is that they have attractive properties for the case for which they were originally designed: They are consistent against all non-white noise alternatives and have good all-round power against non-seasonal alternatives compared to several widely used tests in the literature, including those of Box and Pierce (1970, hereafter BP) and Ljung and Box (1978, hereafter LB) tests. A further advantage of the AP tests is that there is no need to specify a cutoff lag-length as is necessary for the BP and LB tests. We compare the non-standard AP tests with the non-standard BP and LB tests proposed by Francq et al. (2005), the tests of Hong and Lee (2007), and the tests using standardized residuals proposed by Chen (2008). In Monte Carlo experiments using ARMA models with GARCH (Generalised autoregressive conditional heteroskedasticity), EGARCH (Exponential GARCH) and non-MDS (Martingale difference sequence) innovations, the non-standard AP tests generally have better power than the other tests we consider. This suggests that the power advantage of the original AP tests extends to the more general framework considered in this paper.

Bayesian Uncertainty Analysis of the Distributed Hydrological Model HYDROTEL

AbstractIn this study, a Bayesian, inference-based, Markov chain Monte Carlo (MCMC) method coupled with an autoregressive moving average (ARMA) error model framework was used to assess the uncertainty of the process-based, continuous, distributed hydrological model HYDROTEL when simulating daily streamflows. The uncertainty analysis was performed, as a case study, in two distinct watersheds (Montmorency, Quebec, Canada, and Sassandra, Ivory Coast, West Africa). The MCMC uncertainty analysis showed to be effective, primarily with respect to the fulfillment of the statistical assumptions of the error model. The results of the uncertainty analyses demonstrated that almost 95% of the observed daily outlet flows were bracketed by the 95% prediction uncertainty bands. This indicates that the parameter uncertainty associated with the ARMA error model could reach the prediction uncertainty. It was possible to mimic the prediction uncertainty using only the most sensitive model parameters for the Montmorency and S...

Role of Exposure in Analysis of Road Accidents: A Belgian Case Study

Exposure is a key variable in traffic safety research. In the literature, it is noted as the first and primary determinant of traffic safety. In many cases, however, no valid exposure measure is available. In Belgium, monthly traffic counts for 12 years are available. This offers the opportunity to investigate the added value of exposure in models, next to legal, economic, and climatologic variables. Multiple regression with autoregressive moving average (ARMA) errors is used to quantify the impact of these factors on aggregated traffic safety. For each dependent variable, a model with and without exposure is constructed. The models show that exposure is significantly related to the number of accidents with persons killed and seriously injured and to the corresponding victims, but not to the lightly injured outcomes. Moreover, the addition or deletion of exposure does not influence the effects of the remaining variables in the model. The effects of exposure clearly depend on the type of measure used and on the time horizon considered. The framework of a regression model with ARMA errors allows for missing variables being accounted for by the error term. Even without a variable such as exposure, valid models can be constructed.

Role of Exposure in Analysis of Road Accidents

Exposure is a key variable in traffic safety research. In the literature, it is noted as the first and primary determinant of traffic safety. In many cases, however, no valid exposure measure is available. In Belgium, monthly traffic counts for 12 years are available. This offers the opportunity to investigate the added value of exposure in models, next to legal, economic, and climatologic variables. Multiple regression with autoregressive moving average (ARMA) errors is used to quantify the impact of these factors on aggregated traffic safety. For each dependent variable, a model with and without exposure is constructed. The models show that exposure is significantly related to the number of accidents with persons killed and seriously injured and to the corresponding victims, but not to the lightly injured outcomes. Moreover, the addition or deletion of exposure does not influence the effects of the remaining variables in the model. The effects of exposure clearly depend on the type of measure used and on the time horizon considered. The framework of a regression model with ARMA errors allows for missing variables being accounted for by the error term. Even without a variable such as exposure, valid models can be constructed.

M‐Estimation for regressions with integrated regressors and arma errors

Abstract. General M‐estimation is developed for regression models with integrated regressors and autoregressive moving average (ARMA) errors, in which the ARMA parameters are jointly estimated with the regression parameters. The large sample distribution of the M‐estimator is derived. Allowing the regressors to be dependent on the error terms, a parametric ‘fully modified’ (FM) M‐estimator is proposed. In cases of ARMA errors, a Monte‐Carlo experiment reveals superiority of the parametric estimators over the semiparametric FM M‐estimator of Phillips Econometric Theory 11 (1995, p 912) in terms of empirical mean squared error.

Understanding Time-Series Regression Estimators

A large number of methods have been developed for estimating time-series regression parameters. Students and practitioners have a difficult time understanding what these various methods are, let alone picking the most appropriate one for their application. This article explains how these methods are related. A chronology for the development of the various methods is presented, followed by a logical characterization of the methods. An examination of current computational techniques and computing power leads to the conclusion that exact maximum likelihood estimators should be used in almost all cases where regression models have autoregressive, moving average, or mixed autoregressive-moving average error structures.

Linear Trend with Fractionally Integrated Errors

We consider the estimation of linear trend for a time series in the presence of additive long‐memory noise with memory parameter d∈[0, 1.5). Although no parametric model is assumed for the noise, our assumptions include as special cases the random walk with drift as well as linear trend with stationary invertible autoregressive moving‐average errors. Moreover, our assumptions include a wide variety of trend‐stationary and difference‐stationary situations. We consider three different trend estimators: the ordinary least squares estimator based on the original series, the sample mean of the first differences and a class of weighted (tapered) means of the first differences. We present expressions for the asymptotic variances of these estimators in the form of one‐dimensional integrals. We also establish the asymptotic normality of the tapered means for d∈[0, 1.5) −{0.5} and of the ordinary least squares estimator for d∈ (0.5, 1.5). We point out connections with existing theory and present applications of the methodology.

Relationship Between Missing Data Likelihoods and Complete Data Restricted Likelihoods for Regression Time Series Models: An Application to Total Ozone Data

SUMMARY The relationship is shown between the likelihood of autoregressive moving average (ARMA) models, or the restricted likelihood of a regression model with ARMA errors with possibly non-consecutive data, and the restricted likelihood when the missing values are filled in with Os and regression terms are added to account for the missing values. The latter method is also useful to model additive outliers in a time series setting. This relationship is then used to fit the models with standard computer packages and the results are applied to the analysis of total ozone time series data that involve missing values.

Examples of Regression with Serially Correlated Errors

Three data sets are analysed to illustrate methods of modelling regression errors which are serially correlated. An autoregressive-moving average error process is used in fitting a regression equation to the energy demands of a mechanical model of a suckler cow. Drug-induced currents in ion- channels are represented by a realisation of a stochastic compartment system. First-order linear stochastic difference equations are used to model milk yield of cows. It is concluded that error models should be used with caution. situations it is assumed that the function is deficient, and it is changed. But there are cases where the assumption of independent errors is not wholly plausible. For example, some sources of error will persist over several observations when repeated measurements are made on a single experimental unit. Systematic departures may then be modelled either by another regression function, or by correlated errors. The modelling objective determines the choice: for a simple summary it may be preferable for the regression function to explain all systematic variability, whereas a correlated stochastic component may be of more assistance in understanding the data generating mechanism. A succinct summary of data is often achieved by using the regression function to describe the long-term trends and the correlations the short-term fluctua- tions. In the presence of correlated errors, ordinary least squares regression parameter estimators may be inefficient and the conventional estimators of the variances of these estimators are usually biased. The simplest way round these problems is to discard the biased standard errors; the argument being that least-squares estimation is often not very inefficient, and is intuitively appealing because of its simplicity. This approach is most useful when no estimate of precision is required, for example when data are available from independent units and within-unit variability is of little importance. (See, for example, Rowell & Walters, 1976.) Alternatively, if it can be assumed that the errors arose from a particular stochastic model, any parameters can be estimated jointly with the regression ones by maximising the likelihood. Empirical and mechanistic approaches to modelling errors will be considered in the following two sections. In essence, the mechanistic approach requires knowledge of the processes by which the data were generated, whereas the empirical method is purely data-based (Thornley, 1976, pp. 4-6).

Inflation, unemployment and government popularity—Dynamic models for the United States, Britain and West Germany

There has been numerous attempts to model the relationship between unemployment, inflation, other economic variables and government popularity in a variety of industrial countries. However, there is conflicting evidence about the magnitude and significance of effects both between different countries, and within the same country at different points of time. The purpose of this article is to examine the existing literature, to provide a critique of the theoretical and statistical validity of many existing studies, and to specify and estimate a dynamic model of the relationship between inflation, unemployment and government popularity in three countries over the post-war period. This model is a multivariate transfer function with an autoregressive-moving average error structure which has been developed in its general form by Box and Jenkins. The results demonstrate significant relationships between inflation, unemployment and government popularity, but relationships which are relatively weak and unstable over time.

Effectiveness of Seat Belt Legislation on the Queensland Road Toll—An Australian Case Study in Intervention Analysis

Abstract The effect of seat belt legislation introduced in the state of Queensland was examined by using two approaches. The first approach consisted of modeling time series of road deaths before intervention (the legislation) as a Box-Jenkins univariate time series model. As we expected, this model proved to be inadequate in explaining the postintervention period, and the model was modified to incorporate the expected form of the intervention effects. Results showed that the legislation produced significant reduction in the road toll, but this analytical approach was limited in describing long-run effects. The second approach, involving causal model and incorporating a proxy explanatory variable and autoregressive-moving average error, was fruitful in overcoming the limitation of the previous approach. The long-run legislative effect was quantified, at a specific level of the explanatory variable, to be a 46 percent reduction in deaths.

Maximum Likelihood Estimation of Stochastic Linear Difference Equations with Autoregressive Moving Average Errors

Abstract : A method is proposed for the estimation of a general class of scalar linear time series models. The model takes the form of a stochastic difference equation for the dependent variable with exogenous variable inputs, and the disturbances are autocorrelated through an autoregressive moving average process. In the present paper an asymptotically efficient yet computationally simple estimation procedure (in the time domain) is derived for this model. The resulting estimator is shown to be asymptotically equivalent to the maximum likelihood estimator and to possess a limiting multivariate normal distribution. (Author)

Maximum likelihood estimation of regression models with autoregressive-moving average disturbances

SUMMARY The regression model with autoregressive-moving average disturbances may be cast in a form suitable for the application of Kalman filtering techniques. This enables the generalized least squares estimator to be calculated without evaluating and inverting the covariance matrix of the disturbances. The problem of forecasting future values of the dependent variable is also effectively solved when the Kalman filter technique is applied. Furthermore, the properties of the residuals produced by the filter suggest that they may be useful for diagnostic checking of the model. The Kalman filter algorithm also forins the basis of a method for the exact maximum likelihood estimation of the model. This may well have computational, as well as theoretical, advantages over other methods. In a general formulation of the linear regression model, the disturbances may be assumed to be generated by an autoregressive-moving average process. Most work, however, has been restricted to the estimation of models with purely autoregressive disturbances. Models with autoregressive-moving average, or even pure moving average, errors have been much less popular, the main reason for this being computational difficulties. At first sight, it is necessary to evaluate the covariance matrix of the disturbances, and then to invert it. Since the matrix is n x n, where n is the sample size, this can be time consuming, although, as Akaike (1973) and Galbraith & Galbraith (1974) have shown, the special form of the covariance matrix means that it can be inverted relatively efficiently. Maximum likelihood estimation of the regression model with autoregressive-moving average errors has been considered by Pierce (1971). The method of estimation he adopts is basically an extension of the familiar 'conditional sum of squares' approach employed by Box & Jenkins (1970) in their treatment of autoregressive-moving average time series models. This method avoids any large-scale matrix inversions, but it does lead to an estimation problem which is nonlinear in the vector of regression coefficients as well as in the parameters of the autoregressive-moving average process. Furthermore, the estimators obtained in this way are only approximations to the full maximum likelihood estimators. A number of authors, for example Newbold (1974) and Osborn (1976), have recently stressed the desirability of computing estimators of autoregressive-moving average parameters using the exact likelihood function, and this view is reiterated in the context of regression models

Least squares estimation in the regression model with autoregressive-moving average errors

SUMMARY To treat the problem of correlated errors in regression, a model in which the errors follow a stationary autoregressive-moving average time series is suggested. Simultaneous least squares estimation of the regression and the time series parameters is discussed, and it is shown that asymptotically the estimates obtained in this manner possess normal distributions, whether or not the errors themselves are normally distributed. The estimates of the regression parameters are uncorrelated with those of the time series parameters; the former are distributed as if they had arisen from a certain transformed model with uncorrelated errors, while the latter have the same covariance matrix as those from a stationary series with no deterministic component. The estimate of variance is also asymptotically normal. A Monte Carlo sampling study indicates that these results can serve as a useful approximation for samples of moderate size.