ARMA Form Research Articles

Generalized ARMA models with martingale difference errors

The analysis of non-Gaussian time series has been studied extensively and has many applications. Many successful models can be viewed as special cases or variations of the generalized autoregressive moving average (GARMA) models of Benjamin et al. (2003), where a link function similar to that used in generalized linear models is introduced and the conditional mean, under the link function, assumes an ARMA structure. Under such a model, the ‘transformed’ time series, under the same link function, assumes an ARMA form as well. Unfortunately, unless the link function is an identity function, the error sequence defined in the transformed ARMA model is usually not a martingale difference sequence. In this paper we extend the GARMA model in such a way that the resulting ARMA model in the transformed space has a martingale difference sequence as its error sequence. The benefit of such an extension are four-folds. It has easily verifiable conditions for stationarity and ergodicity; its Gaussian pseudo-likelihood estimator is consistent; standard time series model building tools are ready to use; and its MLE’s asymptotic distribution can be established. We also proposes two new classes of non-Gaussian time series models under the new framework. The performance of the proposed models is demonstrated with simulated and real examples.

The Lognormal Autoregressive Conditional Duration (LNACD) Model and a Comparison with an Alternative ACD Models

Engle and Russell (1998) introduce the autoregressive conditional duration (ACD) model to model the dynamics of financial duration. It is recognized that the ACD model can be specified in ARMA form. We show that as long as the innovations of the ACD model follows a lognormal distribution, the equivalent ARMA model will be Gaussian distributed. Motivated by this fact, we develop a lognormal autoregressive conditional duration (LNACD) model. The LNACD model permits a humped-shaped hazard function with one free shape parameter, which has a computational advantage compared to the existing ACD specification in the literature. We compare the performance of the LNACD model with alternative specification of ACD model. The empirical results show that the LNACD model is always superior to Exponential and Weibull ACD models and its performance is similar to the Burr and Generalized Gamma ACD models.

Maturity Dispersion, Stock Auto-Correlation, and Management Strategy in Exploited Populations

Fishery management policies need to be based on historical summaries of stock status which are well correlated with the size of the group of individuals who will be affected by any harvest. This paper is motivated by the problem of managing stocks of Atlantic salmon, which can be accurately monitored during the riverine stages of their life-history, but which spend a lengthy period at sea before returning to spawn. We begin by formulating a minimal stochastic model of stock-recruitment driven population dynamics, which linearises to a standard ARMA form. We investigate the relation between maturity dispersion and the auto-covariance of stock fluctuations driven by process noise in the recruitment process and/or random variability in survival from recruitment to spawning. We demonstrate that significant reductions in fluctuation intensity and/or increases in long-run average yield can be achieved by controlling harvesting in response to the value of a historical summary focussed on lags at which the uncontrolled population dynamics produce strong correlations. We apply our minimal model to two well-characterised Atlantic salmon populations, and find poor agreement between predicted and observed stock fluctuation ACF. Re-examination of the ancilliary data available for one of our two exemplary systems leads us to propose an extended model which also linearises to ARMA form, and which predicts a fluctuation ACF more closely in agreement with that observed, and could thus form a satisfactory vehicle for policy discussion.

MIXREG: a computer program for mixed-effects regression analysis with autocorrelated errors

MIXREG is a program that provides estimates for a mixed-effects regression model (MRM) for normally-distributed response data including autocorrelated errors. This model can be used for analysis of unbalanced longitudinal data, where individuals may be measured at a different number of timepoints, or even at different timepoints. Autocorrelated errors of a general form or following an AR(1), MA(1), or ARMA(1,1) form are allowable. This model can also be used for analysis of clustered data, where the mixed-effects model assumes data within clusters are dependent. The degree of dependency is estimated jointly with estimates of the usual model parameters, thus adjusting for clustering. MIXREG uses maximum marginal likelihood estimation, utilizing both the EM algorithm and a Fisher-scoring solution. For the scoring solution, the covariance matrix of the random effects is expressed in its Gaussian decomposition, and the diagonal matrix reparameterized using the exponential transformation. Estimation of the individual random effects is accomplished using an empirical Bayes approach. Examples illustrating usage and features of MIXREG are provided.

Transforming the error-components model for estimation with general ARMA disturbances

Baltagi and Li (1991) give a transformation which may be applied to certain autocorrelated disturbances in an error-components model to yield spherical disturbances. They derive the transformation for AR(1) and AR(2) cases. We show that the results of Galbraith and Zinde-Walsh (1992) on the general ARMA form of the transformation matrix can be applied to allow estimation of the error-components model with disturbances which follow any ARMA ( p, q) process. We also show that, for models with a cross-sectional dimension which is large relative to the time dimension, it is possible to use this method to perform a general autocorrelation transformation without specifying a particular ARMA process or estimating ARMA parameters.

Exact ARMA lattice predictors from autocorrelation functions

The paper derives an optimal linear L/sup 2/-predictor of ARMA-type in the lattice form of arbitrarily fixed dimensions for a process whose autocorrelation function is known. The algorithm preserves exact optimality at each step, as opposed to asymptotic convergence of more usual algorithms, at the expense of hereditary computation. Only the discrete-time case is examined. It is shown how the unnormalized (respectively normalized) lattice form may be reduced to only 4n-2 parameters (respectively 2n+1) for a nth-order projection on the past. The normalization algorithm for the forward and backward residuals uses only scalar square root computations. Some examples that show the accuracy of this technique compared with those using the classical ARMA form for the predictor, are given. >

System order and structure indices of linear systems in polynomial form

Linear time-invariant, finite-dimensional discrete time systems are very often specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). In this paper it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker observability indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Treating dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, the notions of McMillan degree and observability indices can be generalized to non-controllable as well as to non-causal systems.

On the Order and Structural Indices of Linear Systems Represented in Polynomial Form

Linear time-invariant, finite dimensional discrete time systems very often are specified in a polynomial form representation in either a forward or backward shift operator (sometimes called MFD and ARMA form). Considering dynamical systems in terms of their behaviour, i.e. the set of admissible signal trajectories, it is shown that there exists a unifying theory for the determination of McMillan degree and Kronecker indices of systems represented by polynomial matrices in the two shift operators, considering the MFD and ARMA forms as special cases. Moreover the notions of McMillan degree and Kronecker indices can be generalized to noncontrollable and noncausal systems.

Effects of a single outlier on arma identification

Fox (1972), Box and Tiao (1975), and Abraham and Box (1979) have proposed methods for detecting outliers in time series whose ARMA form is known (or identified). We show that the existence of a single aberrant observation, innovation, or intervention causes an ARMA model to be misidentified using unadjusted autocorrelation (acf) and partial autocorrelation estimates. The magnitude, location, type of outlier, and in some cases the ARMA's parameters, affect the identification outcome. We use variance inflation, signal-to-noise ratios, and acf critical values to determine an ARMA model's susceptibility to misidentifi-cation. Numerical and simulation examples suggest how to iteratively use the outlier detection methods in practice.

The teachers/practitioners corner the effects of indexing ARMA series using the consumer price index

AbstractThe authors demonstrate that indexing a time series with an ARMA representation using the Consumer Price Index does not materially alter the ARMA form of the model. They further demonstrate that the forecasting error of the indexed series and of the product of the forecasts of the index and the time series are, for practical purpose, the same. Simulation results are reported for five model classes.