Multivariate Autoregressive Moving-average Models Research Articles

Equity Returns and Volatilities Before and After the 2007-08 Financial Crisis

Abstract This paper investigates the linkages among equity markets of four European countries (Germany, France, Italy, UK) and the USA in terms of market returns and transmission of volatilities. We use daily exchange traded funds (ETF) data from January 2002 to March 2016 and utilize both a Multivariate Autoregressive Moving Average model (MARMA) and a Generalized Autoregressive Conditional Heteroskedasticity model (GARCH). We divided the data into three separate periods: before the 2007-08 financial crisis, during the crisis and after the crisis. The results show the existence of significant co-movement of returns in all three selected periods although some important differences before and after the financial crisis are noted. Findings also include marked increases in integration of the markets and thus diminishing diversification opportunities for investors. Volatilities appear to react strongly to market movements and their shocks fade away slowly in all five countries during the crisis period. There is also strong evidence of volatility spillovers particularly during and after the crisis periods.

Effects of cerebral ischemia on human neurovascular coupling, CO2 reactivity, and dynamic cerebral autoregulation.

Cerebral blood flow (CBF) regulation can be impaired in acute ischemic stroke but the combined effects of dynamic cerebral autoregulation (CA), CO2 cerebrovascular reactivity (CVR), and neurovascular coupling (NVC), obtained from simultaneous measurements, have not been described. CBF velocity in the middle cerebral artery (MCA) (CBFv, transcranial Doppler), blood pressure (BP, Finometer), and end-tidal Pco2 (PetCO2 , infrared capnography) were recorded during a 1-min passive movement of the arm in 27 healthy controls [mean age (SD) 61.4 (6.0) yr] and 27 acute stroke patients [age 63 (11.7) yr]. A multivariate autoregressive-moving average model was used to separate the contributions of BP, arterial Pco2 (PaCO2 ), and the neural activation to the CBFv responses. CBFv step responses for the BP, CO2, and stimulus inputs were also obtained. The contribution of the stimulus to the CBFv response was highly significant for the difference between the affected side [area under the curve (AUC) 104.5 (4.5)%] and controls [AUC 106.9 (4.3)%; P = 0.008]. CBFv step responses to CO2 [affected hemisphere 0.39 (0.7), unaffected 0.55 (0.8), controls 1.39 (0.9)%/mmHg; P = 0.01, affected vs. controls; P = 0.025, unaffected vs. controls] and motor stimulus inputs [affected hemisphere 0.20 (0.1), unaffected 0.22 (0.2), controls 0.37 (0.2) arbitrary units; P = 0.009, affected vs. controls; P = 0.02, unaffected vs. controls] were reduced in the stroke group compared with controls. The CBFv step responses to the BP input at baseline and during the paradigm were not different between groups (P = 0.07), but PetCO2 was lower in the stroke group (P < 0.05). These results provide new insights into the interaction of CA, CVR, and NVC in both health and disease states.

Influence of CO2 on neurovascular coupling: interaction with dynamic cerebral autoregulation and cerebrovascular reactivity.

PaCO2 affects cerebral blood flow (CBF) and its regulatory mechanisms, but the interaction between neurovascular coupling (NVC), cerebral autoregulation (CA), and cerebrovascular reactivity to CO2 (CVR), in response to hypercapnia, is not known. Recordings of cerebral blood flow velocity (CBFv), blood pressure (BP), heart rate, and end‐tidal CO2 (EtCO2) were performed in 18 subjects during normocapnia and 5% CO2 inhalation while performing a passive motor paradigm. Together with BP and EtCO2, a gate signal to represent the effect of stimulation was used as input to a multivariate autoregressive‐moving average model to calculate their separate effects on CBFv. Hypercapnia led to a depression of dynamic CA at rest and during stimulation in both hemispheres (P <0.02) as well as impairment of the NVC response, particularly in the ipsilateral hemisphere (P <0.01). Neither hypercapnia nor the passive motor stimulation influenced CVR. Dynamic CA was not influenced by the motor paradigm during normocapnia. The CBFv step responses to each individual input (BP, EtCO2, stimulation) allowed identification of the influences of hypercapnia and neuromotor stimulation on CA, CVR, and NVC, which have not been previously described, and also confirmed the depressing effects of hypercapnia on CA and NVC. The stability of CVR during these maneuvers and the lack of influence of stimulation on dynamic CA are novel findings which deserve further investigation. Dynamic multivariate modeling can identify the complex interplay between different CBF regulatory mechanisms and should be recommended for studies involving similar interactions, such as the effects of exercise or posture on cerebral hemodynamics.

Active, passive, and motor imagery paradigms: component analysis to assess neurovascular coupling

The association between neural activity and cerebral blood flow (CBF) has been used to assess neurovascular coupling (NVC) in health and diseases states, but little attention has been given to the contribution of simultaneous changes in peripheral covariates. We used an innovative approach to assess the contributions of arterial blood pressure (BP), PaCO2, and the stimulus itself to changes in CBF velocities (CBFv) during active (MA), passive (MP), and motor imagery (MI) paradigms. Continuous recordings of CBFv, beat-to-beat BP, heart rate, and breath-by-breath end-tidal CO2 (EtCO2) were performed in 17 right-handed subjects before, during, and after motor-cognitive paradigms performed with the right arm. A multivariate autoregressive-moving average model was used to calculate the separate contributions of BP, EtCO2, and the neural activation stimulus (represented by a metronome on-off signal) to the CBFv response during paradigms. Differences were found in the bilateral CBFv responses to MI compared with MA and MP, due to the contributions of stimulation (P < 0.05). BP was the dominant contributor to the initial peaked CBFv response in all paradigms with no significant differences between paradigms, while the contribution of the stimulus explained the plateau phase and extended duration of the CBFv responses. Separating the neural activation contribution from the influences of other covariates, it was possible to detect differences between three paradigms often used to assess disease-related NVC. Apparently similar CBFv responses to different motor-cognitive paradigms can be misleading due to the contributions from peripheral covariates and could lead to inaccurate assessment of NVC, particularly during MI.

Contribution of arterial blood pressure and PaCO2 to the cerebrovascular responses to motor stimulation

Motor stimulation induces a neurovascular response that can be detected by continuous measurement of cerebral blood flow (CBF). Simultaneous changes in arterial blood pressure (ABP) and Pa(CO(2)) have been reported, but their influence on the CBF response has not been quantified. Continuous bilateral recordings of CBF velocity (CBFV), ABP, and end-tidal CO(2) (ET(CO(2))) were obtained in 10 healthy middle-aged subjects at rest and during 60 s of repetitive, metronome-controlled (1 Hz) elbow flexion. A multivariate autoregressive-moving average model was adopted to quantify the relationship between beat-to-beat changes in ABP, breath-by-breath ET(CO(2)), and the motor stimulus, represented by the metronome on-off signal (inputs), and the CBFV response to stimulation (output). All three inputs contributed to explain CBFV variance following stimulation. For the ipsi- and contralateral hemispheres, ABP explained 20.3 ± 17.3% (P = 0.0007) and 19.5 ± 17.2% (P = 0.01) of CBFV variance, respectively. Corresponding values for ET(CO(2)) and metronome signals were 22.0 ± 24.2% (P = 0.008), 24.0 ± 24.1% (P = 0.037), 32.7 ± 22.5% (P = 0.0015), and 43.2 ± 25.1% (P = 0.013), respectively. Synchronized population averages suggest that the initial sudden change in CBFV was largely due to ABP, while the influence of ET(CO(2)) was more erratic. The component due to elbow flexion showed a well-defined pattern, with rise time slower than the main CBFV change but reaching a stable plateau after 15 s of stimulation. Identifying and removing the influences of ABP and Pa(CO(2)) to motor-induced changes in CBF should lead to more robust estimates of neurovascular coupling and better understanding of its physiological covariates.

Comparative Forecasting Models Accuracy of Short-term Natural Rubber Prices

The study describes a number of short-term expost forecasts of econometric models and univariate model of autoregressive-integrated-moving average (ARIMA) model of Natural Rubber (NR) prices SMR20 (Standard Malaysia Rubber Grade 20) in the Malaysia market. The econometric models include single equation model using Vector Error Correction Method (VECM) and multivariate autoregressive-moving average (MARMA) model. The models were developed to determine the inter-relationships between NR production, consumption and prices, to forecast the price of NR (SMR20) and to determine which model is more efficient in their price forecasting accuracy. The models based on data from period January 1990-December 2008. Comparative forecasting models accuracy between single equation model, MARMA model and univariate model of ARIMA, were made in terms of their estimation accuracy based on RMSE, MAE, RMPE and (U-Theil) criteria. The results revealed that the values of the RMSE, MAE, RMPE and U of MARMA model were comparatively smaller than the values generated by single equation model and ARIMA model. These statistics suggested that MARMA model is more accurate and efficient measured in terms of its statistical criteria than single equation model and ARIMA model in forecasting the NR price of SMR20 in the next 6 months or so.

Application of the multi-model partitioning theory for simultaneous order and parameter estimation of multivariate ARMA models

In this paper, a study on how to perform simultaneous order and parameter estimation of multivariate (MV) ARMA (autoregressive moving average) models under the presence of noise is addressed. The proposed method, which is computationally efficient, is an extension of a previously presented method for MV AR models and is based on the well established and widely applied multi-model partitioning theory. A series of computer simulations indicate that the method is infallible in selecting the correct model order in very few steps. The simultaneous estimation of the multivariate ARMA parameters is also another benefit of the proposed method. The results are compared with two other established order selection criteria namely Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC). Finally, it is shown that the method is also successful in tracking model order changes, in real time.

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving‐average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one‐step‐ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.

ESTIMATION OF THE MULTI‐VARIATE AUTOREGRESSIVE MOVING AVERAGE HAVING PARAMETER RESTRICTIONS AND AN APPLICATION TO ROTATIONAL SAMPLING

. The vector autoregressive moving average model with nonlinear parametric restrictions is considered. A simple and easy-to-compute Newton-Raphson estimator is proposed that approximates the restricted maximum likelihood estimator which takes full advantage of the information contained in the restrictions. In the case when there are no parametric restrictions, our Newton-Raphson estimator is equivalent to the estimator proposed by Reinsel et al. (Maximum likelihood estimators in the multivariate autoregressive moving-average model from a generalized least squares view point. J. Time Ser. Anal. 13 (1992), 133–45). The Newton-Raphson estimation procedure also extends to the vector ARMAX model. Application of our Newton-Raphson estimation method in rotational sampling problems is discussed. Simulation results are presented for two different restricted models to illustrate the estimation procedure and compare its performance with that of two alternative procedures that ignore the parametric restrictions.

ESTIMATION OF THE MULTIVARIATE AUTOREGRESSIVE MOVING AVERAGE HAVING PARAMETER RESTRICTIONS AND AN APPLICATION TO ROTATIONAL SAMPLING

Abstract. The vector autoregressive moving average model with nonlinear parametric restrictions is considered. A simple and easy‐to‐compute Newton‐Raphson estimator is proposed that approximates the restricted maximum likelihood estimator which takes full advantage of the information contained in the restrictions. In the case when there are no parametric restrictions, our Newton‐Raphson estimator is equivalent to the estimator proposed by Reinsel et al. (Maximum likelihood estimators in the multivariate autoregressive moving‐average model from a generalized least squares view point. J. Time Ser. Anal. 13 (1992), 133–45). The Newton‐Raphson estimation procedure also extends to the vector ARMAX model. Application of our Newton‐Raphson estimation method in rotational sampling problems is discussed. Simulation results are presented for two different restricted models to illustrate the estimation procedure and compare its performance with that of two alternative procedures that ignore the parametric restrictions.

Recursive Parameter Estimation for ARMA Simulations

The parameter estimation algorithms for the autoregressive moving average (ARMA) simulation of multivariate random processes is considered. The recursive procedure for estimating the parameters of multivariate AR models is extended to the parameter estimation of multivariate ARMA models of same order for both the AR and MA components, which are usually used in the ARMA simulation of stationary multivariate random processes. The recursive parameter estimation procedure for ARMA models relies on the knowledge of the input-output cross correlation of the model, which is obtained with a procedure from the two-stage least-squares method, in that an AR model of high order is used to estimate the cross correlation. A numerical example shows that the recursive procedure for multivariate ARMA models leads to good ARMA representation of random processes, which is characterized by a specified target spectrum.

MAXIMUM LIKELIHOOD ESTIMATORS IN THE MULTIVARIATE AUTOREGRESSIVE MOVING‐AVERAGE MODEL FROM A GENERALIZED LEAST SQUARES VIEWPOINT

Abstract. Explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function for the multivariate autoregressive moving‐average (ARMA) model. Based on these expressions an explicit description of the Gauss‐Newton iterative procedure to obtain maximum likelihood (ML) estimates of the parameters in the multivariate ARMA model is presented. The resulting computational procedure has the form of a generalized least squares (GLS) estimation involving lagged values of the observed vector series and of the residual series as independent variables. This direct form of the estimator is found to be appealing and useful in understanding and interpreting the ML estimation procedure from a regression point of view, and in comparing the ML procedure with other ‘linear’ estimation procedures that have recently been presented. Simulation results are also presented for a univariate and a multivariate ARMA model to illustrate the ML‐GLS estimation procedure and to compare it with other linear estimation procedures.

Multivariate Periodic ARMA(1,1) Processes

The main objective of this paper is to investigate the properties of multivariate periodic autoregressive moving average (ARMA)(l, 1) processes. Such properties include the covariance structure, the parameter space, and estimation. The parameter space of such periodic processes is derived by aggregation (for instance, monthly models will aggregate to annual models). In particular, the parameter space and estimation are analyzed for contemporaneous ARMA (CARMA) models. It is shown that in general, the aggregation of a multivariate periodic ARMA(1, 1) model leads to a stationary multivariate ARMA(1, 1) model. However, not always a periodic CARMA(1, 1) model aggregates into a stationary CARMA(1, 1) model. Furthermore, it is shown that in estimating the aggregated model parameters, the method based on the periodic model and corresponding parameters, is superior to that based on the aggregated series.

Multisite ARMA(1,1) and Disaggregation Models for Annual Streamflow Generation

Disaggregation and multisite autoregressive moving average (ARMA)(1,1) time‐series models provide simple and efficient frameworks for generation of multisite synthetic streamflow sequences that exhibit long‐term persistence. This paper considers multisite ARMA(1,1) models whose Φ and Θ matrices are diagonal; a Monte Carlo study examined the efficiency of three procedures for estimating individual Φ‐θ values for each site and two estimators of the covariance matrix of the innovations. Also included in the study was a univariate ARMA(1,1) model of the aggregate flows with a simple disaggregation algorithm to generate flows at the individual sites. In the realm of most hydrologic interest, simple diagonal multisite ARMAl(1,1) models performed adequately and it is not necessary to fit the more cumbersome nondiagonal models. The disaggregation procedure coupled with an ARMA(1,1) aggregate flow model did as well as the multivariate diagonal ARMA(1,1) models.

Fixed interval estimation in state space models when some of the data are missing or aggregated

SUMMARY Ansley & Kohn's (1982) geometrical derivation of the Kalman filter fixed-interval smoothing algorithm is extended to the case where some of the observations are missing or aggregated and the state covariance matrix may be singular. Estimates of missing observations are also obtained. The theory is then applied to the interpolation of missing data in multivariate autoregressive-moving average models.

Predictions of multivariate autoregressive-moving average models

SUMMARY A simple formula for multiperiod predictions of multivariate autoregressive-moving average models is derived.The formula is explicitly given as a function of suitably defined parameter matrices and an observation vector, and includes simpler models as its special cases. The asymptotic prediction mean squared error is also derived for predictions with estimated parameters.

The Multivariate Portmanteau Statistic

Abstract Box and Pierce have derived a goodness-of-fit test, the portmanteau test, for univariate autoregressive moving-average (ARMA) time series models. This test is here extended to multivariate ARMA models; the test statistic may be conveniently expressed as a function of the covariances between the residuals of the fitted model. A modified form of the statistic designed to have superior properties in small samples is derived, and the two forms of the statistic are compared via computer simulation.

The exact likelihood function of multivariate autoregressive-moving average models

SUMMARY By making use of the properties of tensor products, this paper describes the derivation of an expression for the exact likelihood function of a stationary process generated by a vector autoregressive-moving average model using concentrated maximum likelihood techniques. Furthermore, in the process of deriving the likelihood function, a closed form expression for the covariance function of the process in terms of the coefficients of the model is derived.