Vector Autoregressive Process Research Articles

Quantile approximations in auto-regressive portfolio models

This paper develops an analytical approximation for the distribution function of a terminal value of a periodic series of buy-and-hold investments placed over a fixed time horizon for the case when log-returns of assets follow a p -th order vector auto-regressive process. The derivation is based on a first order Taylor conditioned approximation with a suitably chosen conditioning variable. The results indicate a remarkably good fit between the approximating procedure and simulations based on realistic parameters.

Markov-chain approximations of vector autoregressions: Application of general multivariate-normal integration techniques

Discrete Markov chains are helpful for approximating vector autoregressive processes in computational work. We relax G. Tauchen (1986) [Finite state Markov-chain approximations to univariate and vector autoregressions. Economics Letters 20, 177–181] in practice using multivariate-normal integration techniques to allow for arbitrary positive-semidefinite covariance structures. Examples are provided for non-diagonal and singular non-diagonal error covariances.

Subset selection for vector autoregressive processes via adaptive Lasso

Subset selection is a critical component of vector autoregressive (VAR) modeling. This paper proposes simple and hybrid subset selection procedures for VAR models via the adaptive Lasso. By a proper choice of tuning parameters, one can identify the correct subset and obtain the asymptotic normality of the nonzero parameters with probability tending to one. Simulation results show that for small samples, a particular hybrid procedure has the best performance in terms of prediction mean squared errors, estimation errors and subset selection accuracy under various settings. The proposed method is also applied to modeling the IS-LM data for illustration.

Asset-liability management under time-varying investment opportunities

Stochastic linear programming is a suitable numerical approach for solving practical asset-liability management problems. In this paper, we consider a multi-stage setting under time-varying investment opportunities and propose a decomposition of the benefits in dynamic re-allocation and predictability effects. We use a first-order unrestricted vector autoregressive process to model asset returns and state variables and include, in addition to equity returns and dividend-price ratios, Nelson/Siegel parameters to account for the evolution of the yield curve. The objective is to minimize the Conditional Value at Risk of shareholder value, i.e., the difference between the mark-to-market value of (financial) assets and the present value of future liabilities.

The cost of a short-selling constraint – welfare implications for investors under uncertainty

In this article, I analyse investors' welfare losses from being restricted from short-selling. To measure those losses, I use the concept of the proportionate opportunity cost along with various Constant Relative Risk Aversion (CRRA) utility functions. Two sets of asset returns are used with a Vector Autoregressive process for generating joint returns distributions: the original historical asset returns data set and the historical asset returns with extreme values exaggerated. In each case, 1000 alternative sets of assets, including one with a risk-free nominal return, are randomly made available for investment. I show that the optimal portfolio strategy with the short-selling constraint performs almost as well as the unconstrained portfolio strategy for investors with medium and high levels of risk aversion. The results, derived from the original historical asset returns data set, show that investors' welfare losses reach 12.8% of initial wealth when risk aversion is low and reach 13.5% of initial wealth with extreme returns exaggerated. The results in both cases indicate that less risk-averse investors experience greater welfare losses and that the short-selling constraint reduces the cost of sub-optimal diversification.

A liability-relative drawdown approach to pension asset liability management

Defined benefit pension schemes accumulate assets with the ultimate objective of honoring their obligation to the beneficiaries. Liabilities should be at the center of designing investment policies and serve as the ultimate reference point for evaluating and allocating risks and measuring performance. The goal of the investment policy should be to maximize expected excess returns over liabilities subject to an acceptable level of risk that is expressed relative to liabilities. In this article, we argue for the use of a liability-relative drawdown optimization approach to construct investment portfolios. Asset and liability returns are simulated using a vector autoregressive process with state variables. We find that drawdown optimal portfolios provide better downside protection, are better diversified and tend to be less equity centric while providing higher expected returns compared to surplus variance portfolios.

Evidence on a DSGE Business Cycle Model Subject to Neutral and Investment-Specific Technology Shocks Using Bayesian Model Averaging

The empirical support for a real business cycle model with two technology shocks is evaluated using a Bayesian model averaging procedure. This procedure makes use of a finite mixture of many models within the class of vector autoregressive (VAR) processes. The linear VAR model is extended to permit cointegration, a range of deterministic processes, equilibrium restrictions and restrictions on long-run responses to technology shocks. We find support for a number of the features implied by the real business cycle model. For example, restricting long run responses to identify technology shocks has reasonable support and important implications for the short run responses to these shocks. Further, there is evidence that savings and investment ratios form stable relationships, but technology shocks do not account for all stochastic trends in our system. There is uncertainty as to the most appropriate model for our data, with thirteen models receiving similar support, and the model or model set used has signficant implications for the results obtained.

Hyper-spherical and elliptical stochastic cycles

A univariate first-order stochastic cycle can be represented as an element of a bivariate first-order vector autoregressive process, or VAR(1), where the transition matrix is associated with a rotation along a circle in the plane, and the reduced form is ARMA(2,1). This paper generalizes this representation in two directions. According to the first, the cyclical dynamics originate from the motion of a point along an ellipse. The reduced form is also ARMA(2,1), but the model can account for certain types of asymmetries. The second deals with the multivariate case: the cyclical dynamics result from the projection along one of the coordinate axis of a point moving in along an hyper-sphere. This is described by a VAR(1) process whose transition matrix is obtained by a sequence of n-dimensional Givens rotations. The reduced form of an element of the system is shown to be ARMA(n, n − 1). The properties of the resulting models are analysed in the frequency domain, and we show that this generalization can account for a multimodal spectral density. The illustrations show that the proposed generalizations can be fitted successfully to some well-known case studies of the time series literature.

Speed of adjustment in cointegrated systems

This paper discusses summary measures for the speed of adjustment in possibly cointegrated Vector Autoregressive Processes (VAR). In particular we propose long-run half-lives, based on interim and total multipliers. We discuss their relation with Granger-noncausality and other types of half-life, which are shown to convey different information, except in the univariate AR(1) case. We present likelihood-based inference on long-run half-lives, regarded as discrete functions of parameters in the VAR model. It is shown how asymptotic confidence regions can be defined. An empirical illustration concerning speed of adjustment to purchasing-power parity is provided.

An empirical method for assessing the research relevance gap

There has been much debate on the relevance to firms of the academic research produced by business schools. However, what has not received as much attention is how the relevance of the research to businesses should be measured in a systematic and empirical way. We develop a systematic method to test for the relevance of academic research to businesses. Our method models as a vector autoregressive process the interests of the academic and practitioner communities in some new topic, as expressed by the number of articles published in the academic and the practitioner literature on that topic per calendar quarter, and then studies Granger causality between the academic and practitioner interest processes. This method can be used by academics to empirically demonstrate the impact of their intellectual contributions on practitioners and thence on the business world. We employ our approach to two relatively new and important topics, Real Options and Economic Value Added.

Shocks to Shocks: A Theoretical Foundation for the Information Content of Earnings

This paper rigorously develops and extends the Ball-Brown (1968) and Beaver (1968) measures of the information content of earnings from underlying primitives, using the asset pricing model of Vuolteenaho (2002). The standard Ball-Brown and Beaver measures of information content are obtained by assuming that the time series of (log deflated) earnings are AR(1) with drift and expected future discount rates are inter-temporally constant. The Ball-Brown and Beaver measures of information content are further generalized to the case of dynamic discount rates by assuming initially that (log) expected future discount rates have fixed and dynamic return components such that the dynamic return component is AR(1) with drift. These measures of information content are further generalized to two broad cases: (1) the time series of earnings and expected future discount rates are ARMA(p,q) processes; and (2) the time series of earnings and expected future discount rates follow a log-linear Vector Autoregressive (VAR) process. One of the more ubiquitous applications of Ball-Brown analysis - and recently of the Beaver approach - involves decomposing earnings into components (e.g., cash flows and accruals) in order to determine which earnings component has greater information content. Closed form solutions of the information content of earnings components are obtained for the standard Ball-Brown and Beaver measures and for the generalizations of these measures.

Weighted least squares approximate restricted likelihood estimation for vector autoregressive processes

We derive a weighted least squares approximate restricted likelihood estimator for a k-dimensional pth-order autoregressive model with intercept. Exact likelihood optimization of this model is generally infeasible due to the parameter space, which is complicated and high-dimensional, involving pk2 parameters. The weighted least squares estimator has significantly reduced bias and mean squared error than the ordinary least squares estimator for both stationary and nonstationary processes. Furthermore, at the unit root, the limiting distribution of the weighted least squares approximate restricted likelihood estimator is shown to be the zero-intercept Dickey–Fuller distribution, unlike the ordinary least squares with intercept estimator that has a different distribution with significantly higher bias.

Granger Causality Analysis of Human Cell-Cycle Gene Expression Profiles

Granger causality (GC) tests are ideally suited to investigate time series data generated by bivariate vector autoregressive (VAR) processes. Recent studies have applied GC analysis and its extensions for modeling functional relationships and network structure from temporal gene expression profiles. The present study investigates GC analysis of human cell-cycle gene expression profiles that can be modeled as a first-order bivariate VAR. Analytical results presented establish the contribution of the VAR process parameters, including auto-regulatory feedback and noise variance to the mean-squared forecast error, as a critical component in identifying statistically significant GC relationships. These results in turn discourage blind inference of functional relationship between a given pair of genes solely based on the result of the statistical tests for GC. The presence of significant auto-regulatory feedback and discrepancy in noise variance is demonstrated across the cell-cycle gene expression profiles by VAR parameter estimation. It is emphasized that discrepancies in noise variance can be due to artifacts and can lead to spurious existence of functional relationship between a given pair of genes. VAR parameter estimation is encouraged for better of GC interpretation of the results. Published case studies on GC analysis of the same publicly available cell-cycle gene expression data are reinvestigated for transparency.

Evidence on a Real Business Cycle Model with Neutral and Investment-Specific Technology Shocks Using Bayesian Model Averaging

textabstractThe empirical support for a real business cycle model with two technology shocks is evaluated using a Bayesian model averaging procedure. This procedure makes use of a finite mixture of many models within the class of vector autoregressive (VAR) processes. The linear VAR model is extended to permit cointegration, a range of deterministic processes, equilibrium restrictions and restrictions on long-run responses to technology shocks. We find support for a number of the features implied by the real business cycle model. For example, restricting long run responses to identify technology shocks has reasonable support and important implications for the short run responses to these shocks. Further, there is evidence that savings and investment ratios form stable relationships, but technology shocks do not account for all stochastic trends in our system. There is uncertainty as to the most appropriate model for our data, with thirteen models receiving similar support, and the model or model set used has signficant implications for the results obtained.

Vector autoregressive order selection and forecasting via the modified divergence information criterion

This paper examines the problem of order selection in connection to the forecasting performance for vector autoregressive (VAR) processes. For this purpose we present a generalisation of the modified divergence information criterion (MDIC) for VAR models and compare it with traditional information criteria by Monte Carlo methods for different data generating processes for small, medium, and large sample sizes. The VAR modified divergence information criterion (VAR/MDIC) shows remarkable good results by choosing the correct model more frequently than the known traditional information criteria with the smallest mean squared forecast error.

MODELING DYNAMIC CORRELATIONS AND SPILLOVER EFFECTS OF COUNTRY RISK: EVIDENCE FROM RUSSIA AND KAZAKHSTAN

Oil economies in the Former Soviet Union (FSU) region, with geographical proximity to each other, are usually impacted by some common risk factors, which make their country risks closely correlated. This paper focuses on correlation between country risks and investigates the spillovers of country risk returns (CRR). Taking Russia and Kazakhstan for example, firstly, this paper identifies the structural breaks in CRR series, using iterated cumulative sums of squares (ICSS) algorithm. Secondly, on the assumption that there may be similarity in structural breaks of CRR series of the two countries, Vector Autoregression (VAR) process and Granger causality test are used to identify whether there are mean spillovers of CRR series. Finally, the volatility spillovers are captured by using multivariate conditional volatility models in the framework of the BEKK models. Empirical results show that (1) there are significant unidirectional mean spillovers from Russia to Kazakhstan; (2) there are asymmetric bidirectional volatility spillovers between Russia and Kazakhstan; and volatility spillover effects from Russia to Kazakhstan are stronger.

Estimating causal dependencies in networks of nonlinear stochastic dynamical systems

The inference of causal interaction structures in multivariate systems enables a deeper understanding of the investigated network. Analyzing nonlinear systems using partial directed coherence requires high model orders of the underlying vector-autoregressive process. We present a method to overcome the drawbacks caused by the high model orders. We calculate the corresponding statistics and provide a significance level. The performance is illustrated by means of model systems and in an application to neurological data.

A unified approach for generating cross-correlated and auto-correlated MIMO fading envelope processes

Diversity techniques for various communication and MIMO systems exploit the spatial and temporal diversity attributes to mitigate the ill effects of the fading channels. To evaluate these techniques, a method to generate multiple correlated fading channels is crucial. We propose a unified approach capable of generating correlated flat-fading envelope processes with the desired auto-correlation functions, cross-correlation functions, and probability density functions (pdfs). The proposed approach utilizes the Gaussian vector autoregressive process and the inverse transform sampling techniques. Comparing to the past research focusing on generating fading channels of the same family, the novelty of the proposed approach is its capability to generate fading processes of heterogeneous pdfs. Three examples are demonstrated. In the first example, the autocorrelated Nakagami channel is generated. The second example is designed to generate correlated 2x2 MIMO Rayleigh channels. In the third example, the proposed approach generates three correlated channels having Nakagami, Rician, and Rayleigh pdfs. The settings of the first two examples are adopted from previously published results, which are intended to verify the effectiveness of our proposed approach to tackle previously known problems. The third example demonstrates the novelty of the proposed approach to generate correlated channels of heterogeneous pdfs.

Testing for the cointegrating rank of a vector autoregressive process with uncertain deterministic trend term

Summary When applying Johansen's procedure for determining the cointegrating rank to systems of variables with linear deterministic trends, there are two possible tests to choose from. One test allows for a trend in the cointegration relations and the other one restricts the trend to being orthogonal to the cointegration relations. The first test is known to have reduced power relative to the second one if there is in fact no trend in the cointegration relations, whereas the second one is based on a misspecified model if the linear trend is not orthogonal to the cointegration relations. Hence, the treatment of the linear trend term is crucial for the outcome of the rank determination procedure. We compare three alternative procedures, which are applicable if there is uncertainty regarding the proper trend specification. In the first one a specific cointegrating rank is rejected if one of the two tests rejects, in the second one the trend term is decided upon by a pretest and in the third procedure only tests which allow for an unrestricted trend term are used. We provide theoretical asymptotic and small sample simulation results, which show that the first strategy is preferable in applied work.

Endogenous Completeness of Diffusion Driven Equilibrium Markets

We study the existence of equilibria with endogenously complete markets in a continuous-time, heterogenous agents economy driven by a multidimensional diffusion process. Our main results show that if prices are real analytic as functions of time and the state variables of the model then a sufficient condition for market completeness is that the volatility of dividends be nondegenerate. In contrast to previous research, our formulation does not require that securities pay terminal dividends and thus allows for both finite or infinite horizon economies. We illustrate our results by providing easily applicable conditions for market completeness in two benchmark cases: that where the state variables are given by a vector autoregressive process and that where they are given by a vector of autonomous diffusion processes. We also provide counterexamples which show that real analyticity cannot be dispensed with if one is to deduce dynamic market completeness from the structure of dividends.