Vector Autoregressive Process Research Articles

Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.

A test for a regime shift

AbstractThis note describes a formal statistical approach to testing for a regime shift based on population time series for a system of interacting species or groups. Under this approach, a regime shift is said to occur if the system switches abruptly from varying around one locally stable steady state to varying around another. This allows the problem to be formulated as testing for a changepoint in a vector autoregressive process. The test is illustrated using data from the North Sea covering the period 1963–97.

Granger's representation theorem: A closed‐form expression for I(1) processes

Summary The Granger representation theorem states that a cointegrated vector autoregressive process can be decomposed into four components: a random walk, a stationary process, a deterministic part, and a term that depends on the initial values. In this paper, we present a new proof of the theorem. This proof enables us to derive closed-form expressions of all terms of the representation and allows a unified treatment of models with different deterministic specifications. The applicability of our results is illustrated by examples. For example, the closed-form expressions are useful for impulse response analyses and facilitate the analysis of cointegration models with structural changes.

Time series modeling of endocardial border motion in ultrasonic images comparing support vector machines, multilayer perceptrons and linear estimation techniques

This paper presents a novel study of endocardial boundary motion tracking from sequences of echocardiogram images using neural network and linear estimation techniques. Contrary to the majority of previous studies theoretically analyzing endocardial motion physical parameters, a time series modeling approach is herein adopted. Such a modeling approach is demonstrated, by extensive experimentation, to be very efficient in terms of endocardial border motion tracking performance. The tracking performance of the different modeling techniques involved is evaluated quantitatively by defining suitable error measures as well as qualitatively. A thorough experimental investigation shows the importance of the highly correlated nature of endocardial contour motion within a cardiac cycle. Moreover, it shows that its short term dynamics can be almost equally well captured by support vector machines (SVM) for non-linear regression, multilayer perceptrons (MLP) and two matrix-parameters vector autoregressive (VAR) process models. Longer term dynamics, however, can be described more effectively using SVM for non-linear regression rather than MLPs. Additionally, the latter is shown to describe more effectively longer term dynamics than the VAR modeling approach. Such results are important for modeling the endocardial motion process, aiming at introducing improved adaptive focusing of ultrasonic scanners in order to enhance the quality of heart ultrasonic images.

Asymptotic inference results for multivariate long‐memory processes

Summary In this paper, we extend the well-known Sims, Stock and Watson (SSW) (Sims et al. 1990; Econometrica 56, 113–44), analysis on estimation and testing in vector autoregressive process (VARs) with integer unit roots and deterministic components to a more general set-up where non-stationary fractionally integrated (NFI) processes are considered. In particular, we focus on partial VAR models where the conditioning variables are NFI since this is the only finite-lag VAR model compatible with such processes. We show how SSW's conclusions remain valid. This means that whenever a block of coefficients in the partial VAR can be written as coefficients on zero-mean I(0) regressors in models including a constant term, they will have a joint asymptotic normal distribution. Monte Carlo simulations and an empirical application of our theoretical results are also provided.

Testing for the Cointegrating Rank of a VAR Process with Level Shift at Unknown Time

A systems cointegration rank test is proposed that is applicable for vector autoregressive (VAR) processes with a structural shift at unknown time. The structural shift is modeled as a simple shift in the level of the process. It is proposed to estimate the break date first on the basis of a full unrestricted VAR model. Two alternative estimators are considered and their asymptotic properties are derived. In the next step the deterministic part of the process including the shift size is estimated and the series are adjusted by subtracting the estimated deterministic part. A Johansen type test for the cointegrating rank is applied to the adjusted series. The test statistic is shown to have a well-known asymptotic null distribution that does not depend on the break date. The performance of the procedure in small samples is investigated by simulations.

Feedback, causality and distance between arma models

The purpose of this paper is to analyze in bivariate vector autoregression the relationship between feedback in stochastic systems, Granger causality and a measure of dissimilarity between ARMA models. In particular, we consider a bivariate vector autoregressive processes of order p (a bivariate VAR( p) process) and we prove if the distance between the univariate ARMA models implied by the VAR representation is greater than a certain number that is a function of p, then Granger causality must exist in at least one direction in the variables.

Estimating the risk premium of swap spreads. Two econometric GARCH-based techniques

Two ‘reduced-form’ GARCH-M models are used to estimate the German swap spreads from a risk premium point of view. The first model makes use of a parametric GARCH in mean model that has been extended to the case of a vector autoregressive process. The second is a semiparametric model where the conditional variance is formalized as a GARCH process while conditional mean is an arbitrary function of it. It is shown that the monotonic relation implied by both GARCH in mean models between the delta swap spreads and its conditional variance holds for all maturities considered. Not surprisingly, the semiparametric model leads to a better explanation of the swap spreads dynamic than the parametric specification.

COINTEGRATION ANALYSIS OF TOURISM REVENUES BY THE MEMBER COUNTRIES OF EUROPEAN UNION TO GREECE

The countries of Europe and especially European Union member countries are, traditionally, amongst the most important tourism markets for Greece. The purpose of this article is to investigate the long-run changes in tourism revenues from European Union (EU) member countries to Greece. In order to explain tourism revenues we use a number of leading macroeconomic variables, including the real income of 15 EU member countries, the exchange rate, and some dummies, which examine the crises periods for tourism. The quarterly data that are used for this article cover the period from January 1960 to December 2000. The augmented Dickey-Fuller for unit root is examined in the univariate framework and Johansen's maximum likelihood procedure is used to test the cointegration method and to estimate the number of cointegrated vectors of ``VAR model''-Vector autoregressive processes. Error correction model is estimated to explain tourism revenues from European Union member countries to Greece. The results show that the real income of 15 EU member countries and the exchange rate have a positive effect on tourism revenues of Greece, whereas some political crises have a negative effect on them

STAP for GPS receiver synchronization

A space-time adaptive processing (STAP) algorithm for delay tracking and acquisition of the GPS signature sequence with interference rejection capability is developed. The interference can consist of both broadband and narrowband jammers, and is mitigated in two steps. The narrowband jammers are modelled as vector autoregressive (VAR) processes and rejected by temporal whitening. The spatial nulling is implicitly achieved by estimating a sample covariance matrix and feeding its inverse into the extended Kalman filter (EKF). The EKF estimates of the code delay and the fading channel are used for a t-test for acquisition detection. Computer simulations demonstrate robust performance of the algorithm in severe jamming, and also show that the algorithm outperforms the conventional delay-locked loop (DLL).

Asymptotic properties of some subset vector autoregressive process estimators

We establish consistency and derive asymptotic distributions for estimators of the coefficients of a subset vector autoregressive (SVAR) process. Using a martingale central limit theorem, we first derive the asymptotic distribution of the subset least squares (LS) estimators. Exploiting the similarity of closed form expressions for the LS and Yule–Walker (YW) estimators, we extend the asymptotics to the latter. Using the fact that the subset Yule–Walker and recently proposed Burg estimators satisfy closely related recursive algorithms, we then extend the asymptotic results to the Burg estimators. All estimators are shown to have the same limiting distribution.

Reduced order cross-directional controller design for sheet forming processes

This paper presents a novel control strategy that efficiently addresses the issue of the large number of cross machine direction actuators with which sheet-forming processes are equipped. The approach relies on using the Karhunen-Loeve expansion (KLE) to model the disturbances affecting the properties of interest together with vector autoregressive processes to design a low-order representation of the disturbances. Subsequently, the control space is also reduced by appropriate transformations that rely on the use of KLE. Model predictive control is then implemented on the reduced order model and an optimal solution is obtained in that space, satisfying various types of actuator constraints defined on the full-order system and minimizing the cross-directional variability. The solution from the reduced order model is then projected back to the original space for implementation on the real plant. Simulation examples are included to illustrate the method.

Modeling and generating multivariate time-series input processes using a vector autoregressive technique

We present a model for representing stationary multivariate time-series input processes with marginal distributions from the Johnson translation system and an autocorrelation structure specified through some finite lag. We then describe how to generate data accurately to drive computer simulations. The central idea is to transform a Gaussian vector autoregressive process into the desired multivariate time-series input process that we presume as having a VARTA (Vector-Autoregressive-To-Anything) distribution. We manipulate the autocorrelation structure of the Gaussian vector autoregressive process so that we achieve the desired autocorrelation structure for the simulation input process. We call this the correlation-matching problem and solve it by an algorithm that incorporates a numerical-search procedure and a numerical-integration technique. An illustrative example is included.

Nonparametric estimation of copulas for time series

We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration containing a comparison with the independent, comotonic and Gaussian copulas is given for European and US stock index returns.

Nonparametric Estimation of Copulas for Time Series

We consider a nonparametric method to estimate copulas, i.e. functions linking joint distributions to their univariate margins. We derive the asymptotic properties of kernel estimators of copulas and their derivatives in the context of a multivariate stationary process satisfactory strong mixing conditions. Monte Carlo results are reported for a stationary vector autoregressive process of order one with Gaussian innovations. An empirical illustration is given for European and US stock index returns. Another empirical illustration deals with Danish data on fire insurance losses.

Key Equations in the Tuljapurkar-Lee Model of the Social Security System

We present stochastic forecasts of the Social Security trust fund by modeling key demographic and economic variables as historical time series, and using the fitted models to generate computer simulations of future fund performance. We evaluate several plans for achieving long-term solvency by raising the normal retirement age (NRA), increasing taxes, or investing some portion of the fund in the stock market. Stochastic population trajectories by age and sex are generated using the Lee-Carter and Lee- Tuljapurkar mortality and fertility models. Interest rates, wage growth and equities returns are modeled as vector autoregressive processes. With the exception of mortality, central tendencies are constrained to the Intermediate assumptions of the 2002 Trustees Report. Combining population forecasts with forecasted per-capita tax and benefit profiles by age and sex, we obtain inflows to and outflows from the fund over time, resulting in stochastic fund trajectories and distributions. Under current legislation, we estimate the chance of insolvency by 2038 to be 50%, although the expected fund balance stays positive until 2041. An immediate 2% increase in the payroll tax rate from 12.4% to 14.4% sustains a positive expected fund balance until 2078, with a 50% chance of solvency through 2064. Investing 60% of the fund in the S&P 500 by 2015 keeps the expected fund balance positive until 2060, with a 50% chance of solvency through 2042. An increase in the NRA to age 69 by 2024 keeps the expected fund balance positive until 2047, with a 50% chance of solvency through 2041. A combination of raising the payroll tax to 13.4%, increasing the NRA to 69 by 2024, and investing 25% of the fund in equities by 2015 keeps the expected fund balance positive past 2101 with a 50% chance of solvency through 2077.

Diversity combining for DS/SS systems with time-varying, correlated fading branches

Diversity combining of multiple time varying and correlated fading branches is investigated for direct-sequence spread-spectrum systems. The correlated branches are modeled and estimated jointly as a vector autoregressive (VAR) process. The joint estimation is shown to provide performance gain over separate estimation of the fading branches. The parameter matrices of the VAR model are estimated via the method of expectation maximization (EM) with two algorithms. The first algorithm, using results from Kalman smoothing, provides a closed-form solution to the maximization problem in the iterative EM procedure. However, the iterative EM-Kalman algorithm operates repeatedly on a batch of training data and results in large storage requirements and long processing delays. To overcome these disadvantages, a new algorithm with only forward-time recursions is proposed that approximates the iterative EM solution and efficiently adapts to slowly changing Doppler spreads. As a result, the new algorithm significantly reduces memory and training sequence requirements. Through computer simulations, a near ideal bit-error rate performance is found for both algorithms, and the efficacy of the new adaptive algorithm for channels with changing Doppler spreads is demonstrated.

Comparison of tests for the cointegrating rank of a VAR process with a structural shift

Two different types of tests for the cointegrating rank of vector autoregressive processes with a deterministic shift in the level have been proposed in the literature. The first proposal is based on the likelihood ratio principle using a specific Gaussian model set-up. In the second proposal the time series are adjusted for deterministic terms first and then LR type tests are applied to the adjusted series. The local power of the two types of tests is derived and compared. Moreover, the small sample size and power properties of the tests are explored. It is found that the tests based on adjusted series generally have superior local power and size properties.

On the Recursive Fitting of Subset Autoregressions - Financial and Economic Forecasting (chapter 1)

The paper comprises the preface and chapter 1 of the book titled Financial and Economic Forecasting (Authors: Penm-Penm-Terrell; Publication date: October 2002). The preface provides explanatory remarks at the beginning of the book. It briefly introduces theoretical developments and empirical applications presented in the book. In contrast to conventional methods, the presence of subset structure and zero entries in zero-non-zero (ZNZ) patterned time-series modelling and forecasting has been explicitly explored. In fitting a vector autoregressive process which may include lags up to and including lag K, we may wish to search for the subset vector autoregressive process of size k (where k is the number of lags with non-zero coefficient matrices, k = 1, 2, . . . , K) which has the minimum generalized residual variance. Chapter 1 provides a recursive procedure, which is initialized by evaluating all 'forward' and 'backward' autoregressions in which k = 1. The recursion then allows one to develop successively all subsets of size k = 2, k = 3 up to k = K.

ASYMPTOTIC INFERENCE ON THE MOVING AVERAGE IMPACT MATRIX IN COINTEGRATED I (2) VAR SYSTEMS

This paper provides asymptotic standard errors for the moving average (MA) impact matrix for the second differences of a vector autoregressive (VAR) process integrated of order 2, I(2). Standard errors of the row space of the MA impact matrix are also provided; bases of this row space define the common I(2) trends linear combinations. These standard errors are then used to formulate Wald-type tests. The MA impact matrix is shown to be linked to impact factors that measure the total effect of disequilibrium errors on the growth rate of the system. Most of the relevant limit distributions are Gaussian, and we report artificial regressions that can be used to calculate the estimators of the asymptotic variances. The use of the techniques proposed in the paper is illustrated on UK money data.