A generalized least squares estimation method for invertible vector moving average models (original) (raw)

Vector Autoregresive Moving Average Identification for Macroeconomic Modeling: Algorithms and Theory

2009

This paper develops a new methodology for identifying the structure of VARMA time series models. The analysis proceeds by examining the echelon canonical form and presents a fully automatic data driven approach to model specification using a new technique to determine the Kronecker invariants. A novel feature of the inferential procedures developed here is that they work in terms of a canonical scalar ARMAX representation in which the exogenous regressors are given by predetermined contemporaneous and lagged values of other variables in the VARMA system. This feature facilitates the construction of algorithms which, from the perspective of macroeconomic modeling, are efficacious in that they do not use AR approximations at any stage. Algorithms that are applicable to both asymptotically stationary and unit-root, partially nonstationary (cointegrated) time series models are presented. A sequence of lemmas and theorems show that the algorithms are based on calculations that yield stro...

Closed-form results for vector moving average models with a univariate estimation approach

Econometrics and Statistics, 2019

The estimation of a vector moving average (VMA) process represents a challenging task since the likelihood estimator is extremely slow to converge, even for smalldimensional systems. An alternative estimation method is provided, based on computing several aggregations of the variables of the system and applying likelihood estimators to the resulting univariate processes; the VMA parameters are then recovered using linear algebra tools. This avoids the complexity of maximizing the multivariate likelihood directly. Closed-form results are presented and used to compute the parameters of the process as a function of its autocovariances, using linear algebra tools. Then, an autocovariance estimation method based on the estimation of univariate models only is introduced. It is proved that the resulting estimator is consistent and asymptotically normal. A Monte Carlo simulation shows the good performance of this estimator in small samples.

On least-squares estimation of the residual variance in the first-order moving average model

Computational Statistics & Data Analysis, 1999

In the ÿrst-order moving average model we analyze the behavior of the estimator of the variance of the random residual coming from the method of least squares. This procedure is incorporated into some widely used computer programs. We show through simulations that the asymptotic formulas for the bias and variance of the maximum likelihood estimator, can be used as approximations for the least-squares estimator, at least when the model parameter is far from the region of non-invertibility. Asymptotic results are developed using the "long autoregression" idea, and this leads to a closed-form expression for the least-squares estimator. In turn this is compared with the maximum likelihood estimator under normality, both in its exact and in an approximated version, which is obtained by approximating the matrix in the exponent of the Gaussian likelihood function. This comparison is illustrated by some numerical examples. The dependency of the results about biases on the values of the model parameter is emphasized.

Exact Maximum Likelihood Estimation of Stationary Vector ARMA Models

Journal of the American Statistical Association, 1995

The problems of evaluating and maximizing the exact likelihood function of vector ARMA models are considered separately. A new and efficient procedure far evaluating the exact likelihood function is presented. This method puts together a set of useful features which can only be found separately in currently available algoritluns. A procedure for maximizing the exact likeliliood function, which takes full advantage of the properties offered by the evaluation algorithm, is also considered, Combining these two procedures, a * This paper summarizes the author's doctoral

A Unified Approach to Arma Model Identification and Preliminary Estimation

Journal of Time Series Analysis, 1984

This paper reviews several different methods for identifying the orders of autoregressive-moving average models for time series data. The case is made that these have a common basis, and that a unified approach may be found in the analysis of a matrix G, defined to be the covariance matrix of forecast values. The estimation of this matrix is considered, emphasis being placed on the use of high order autoregression to approximate the predictor coefficients. Statistical procedures are proposed for analysing G, and identifying the model orders. A simulation example and three sets of real data are used to illustrate the procedure, which appears to be a very useful tool for order identification and preliminary model estimation.

Partially adaptive estimation of ARMA time series models

International Journal of Forecasting, 1989

This paper presents a new partially adaptive estimator of ARMA models which includes least absolute deviation (LAD or L~), least squares, Lp, and 'optimal' Lp as special or limiting cases. This estimator is based upon a generalized t(GT) distribution for the residuals. The GT distribution has a mixture interpretation based on an innovative outlier framework, and results in an estimator with a bounded @function for finite 'degrees of freedom.' Joint estimation of distributional parameters and ARMA parameters allows the estimation technique to 'adjust' or 'adapt' to the data type and provides a natural way to test for normality of residuals. Monte Carlo simulations compare the performance of these estimators across different distributional assumptions including a 'contaminated' normal. The partially adaptive estimators perform well across diverse data types including the innovative outlier and 'contaminated normal' models. The distributions of corresponding Q statistics and likelihood ratio statistics are considered.

Estimation of parameters using an updated vector autoregressive model

International Journal of Contemporary Mathematical Sciences

Many statistical models, be it deterministic or stochastic, usually contain a number of parameters that make up the model(s). Ordinarily, the maximum likelihood estimation (MLE) and least squares estimation (LSE) methods are the most applied methods of estimation. However, in the two approaches, the main focus is on the estimation of the parameters since parameter estimation is a key step that cannot be avoided as far as modelling or model building is concerned. In this paper, parameter estimation in the updated vector autoregressive model is shown. We consider estimation of the parameters by use of the dual estimation approach, precisely using joint estimation which can estimate both the state and the parameters, applied to some VAR models in one dimension and in two dimension. From the results, it is observed that there is convergence of the parameters to the true parameter values as time evolves.

A New Robust Estimation Method for ARMA Models

IEEE Transactions on Signal Processing, 2000

This paper presents a new robust method to estimate the parameters of ARMA models. This method makes use of the autocorrelations estimates based on the ratio of medians together with a robust filter cleaner able to reject a large fraction of outliers, and a Gaussian maximum likelihood estimation which handles missing values. The main advantages of the procedure are its easiness, robustness and fast execution. Its effectiveness is demonstrated on an example of the forecasting of the French daily electricity consumptions.

Change Point Estimation of the Stationary State in Auto Regressive Moving Average Models, Using Maximum Likelihood Estimation and Singular Value Decomposition-based Filtering

International journal of engineering. Transactions B: Applications, 2019

In this paper, for the first time, the subject of change point estimation has been utilized in the stationary state of auto regressive moving average (ARMA) (1, 1). In the monitoring phase, in case the features of the question pursue a time series, i.e., ARMA(1,1), on the basis of the maximum likelihood technique, an approach will be developed for the estimation of the stationary state's change point. To estimate unidentified parameters following the change point, the dynamic linear model's filtering was utilized on the basis of the singular decomposition of values. The proposed model has wide applications in several fields such as finance, stock exchange marks and rapid production. The results of simulation showed the suggested estimator's effectiveness. In addition, a real example on stock exchange market is offered to delineate the application.